Statistics 222,   Education 351A  Autumn 2020
    
Statistical Methods for Longitudinal Research

Autumn 2020 Remote Asynchronous Instruction


David Rogosa Sequoia 224,   rag{AT}stanford{DOT}edu   
Course web page: http://rogosateaching.com/stat222//

                To see full course materials from Autumn 2018 go here
Course Welcome and Logistics (first day stuff, to be posted in August, call it Week0)
      Lecture slides, week 0 (pdf)             Audio companion, week 0   
For recreation of in-classroom experience, linked below are youtube versions of the music I play
before starting lecture   and    after lecture concludes.      Some may wish to reverse that ordering.


Registrar's information
STATS 222 (Same as EDUC 351A): Statistical Methods for Longitudinal Research   Units: 2
Grading Basis: Letter or Credit/No Credit

Course Description:
 STATS 222: Statistical Methods for Longitudinal Research (EDUC 351A)
Research designs and statistical procedures for time-ordered (repeated-measures) data. 
The analysis of longitudinal panel data is central to empirical research on learning, development, aging, and the effects of interventions. 
Topics include: measurement of change, growth curve models, analysis of durations including survival analysis, 
experimental and non-experimental group comparisons, reciprocal effects, stability. 
See http://rogosateaching.com/stat222/. Prerequisite: intermediate statistical methods
Terms: Aut | Units: 2 | Grading: Letter or Credit/No Credit
Instructors: Rogosa, D. (PI) 


Preliminary Course Outline
    Week 1. Course Overview, Longitudinal Research; Analyses of Individual Histories and Growth Trajectories
    Week 2. Introduction to Data Analysis Methods for assessing Individual Change for Collections of Growth Curves (mixed-effects models)
    Week 3. Analysis of Collections of growth curves: linear, generalized linear and non-linear mixed-effects models
    Week 4. Special case of time-1, time-2 data; Traditional measurement of change for individuals and group comparisons
    Week 5. Assessing Group Growth and Comparing Treatments: Traditional Repeated Measures Analysis of Variance and Linear Mixed-effects Models
    Week 6. Comparing group growth continued: Power calculations, Cohort Designs, Cross-over Designs, Methods for missing data, Observational studies.
    Week 7. Analysis of Durations: Introduction to Survival Analysis and Event History Analysis
    Weeks 8-9. Further topics in analysis of durations: Diagnostics and model modification; Interval censoring, Time-dependence, Recurrent Events, Frailty Models, Behavioral Observations and Series of Events (renewal processes)
    Dead Week. Assorted Special Topics: Assessments of Stability (including Tracking), Reciprocal Effects, (mis)Applications of Structural Equation Models, Longitudinal Network Analysis

Texts and Resources for Course Content
1. Garrett M. Fitzmaurice Nan M. Laird James H. Ware Applied Longitudinal Analysis (Wiley Series in Probability and Statistics; 2nd ed 2011)
  Text Website   second edition website     Text lecture slides   
2. Judith D. Singer and John B. Willett . Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence New York: Oxford University Press, March, 2003.
  Text web page    Text data examples at UCLA IDRE    Powerpoint presentations   good gentle intro to modelling collections of growth curves (and survival analysis) is Willett and Singer (1998)
3. Douglas M. Bates. lme4: Mixed-effects modeling with R  February 17, 2010 Springer (chapters). A merged version of Bates book: lme4: Mixed-effects modeling with R January 11, 2010 has been refound
Manual for R-package lme4    and   mlmRev, Bates-Pinheiro book datasets.    
    Additional Doug Bates materials. Collection of all Doug Bates lme4 talks      Mixed models in R using the lme4 package Part 2: Longitudinal data, modeling interactions Douglas Bates 8th International Amsterdam Conference on Multilevel Analysis 2011-03-16    another version
Original Bates-Pinheiro text (2000).  Mixed-Effects Models in S and S-PLUS (Stanford access). Appendix C has non-linear regression models.
Fitting linear mixed-effects models using lme4, Journal of Statistical Software Douglas Bates Martin Machler Ben Bolker.       Technical topics: Mixed models in R using the lme4 package Part 4: Theory of linear mixed models
4. A handbook of statistical analyses using R (second edition). Brian Everitt, Torsten Hothorn CRC Press, Index of book chapters   Stanford access     Longitudinal chapters: Chap11   Chap12  Chap13. Data sets etc Package 'HSAUR2' August 2014, Title A Handbook of Statistical Analyses Using R (2nd Edition)
   There is now a third edition of HSAUR, but full text not yet available in crcnetbase.com.    CRAN HSAUR3 page  with Vignettes (chapter pieces) and data in reference manual
5. Peter Diggle , Patrick Heagerty, Kung-Yee Liang , Scott Zeger. Analysis of Longitudinal Data 2nd Ed, 2002
   Amazon page     Peter Diggle home page    Book data sets
     A Short Course in Longitudinal Data Analysis Peter J Diggle, Nicola Reeve, Michelle Stanton (School of Health and Medicine, Lancaster University), June 2011     earlier version    associated exercises:  Lab 1  Lab2  Lab3
6. Longitudinal and Panel Data: Analysis and Applications for the Social Sciences by Edward W. Frees (2004). Full book available    and book data and programs (mostly SAS).
7. Growth Curve Analysis and Visualization Using R. Daniel Mirman Chapman and Hall/CRC 2014 Print ISBN: 978-1-4665-8432-7    Stanford Access       Mirman web page (including data links).
8. Longitudinal Data Analysis    Edited by Geert Verbeke , Marie Davidian , Garrett Fitzmaurice , and Geert Molenberghs Chapman and Hall/CRC 2008.   online supplement for LDA book  .
9. Verbeke, G. and Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer Series in Statistics. New-York: Springer.  Extended presentation: Introduction to Longitudinal Data Analysis A shorter exposition: Methods for Analyzing Continuous, Discrete, and Incomplete Longitudinal Data
10. Survival analysis Rupert G. Miller. Available as Stanford Tech Report
11. Event History Analysis with R (Stanford access). Goran Brostrom CRC Press 2012. R-package   eha
12. John D. Kalbfleisch , Ross L. Prentice The Statistical Analysis of Failure Time Data 2nd Ed
  Amazon page    online from Wiley
13. Klein J, Moeschberger M (2003). Survival Analysis, 2nd edition. New York: Springer.
14. Therneau TM, Grambsch PM (2000). Modeling Survival Data: Extending the Cox Model. New York: Springer.
15. Advanced survival analysis topics.
   Interval-Censored Time-to-Event Data Methods and Applications Chapman and Hall/CRC 2012 (esp Chap 14--glrt).
   Recurrent Events: Chapter 9 of Kalbfleisch and Prentice (2nd edition), "Modeling and Analysis of Recurrent Event Data".
      Cook, R. J. and Lawless, J. F. (2007).  The Statistical Analysis of Recurrent Events. (Stanford access) Springer, New. York.
    Joint Models for Longitudinal and Time-to-Event Data. With Applications in R. Dimitris Rizopoulos. Chapman and Hall/CRC 2012(Stanford access)    Book website

Additional Specialized Resources
Harvey Goldstein. The Design and Analysis of Longitudinal Studies: Their Role in the Measurment of Change (1979). Elsevier
  Amazon page    Goldstein Chap 6 Repeated measures data      Multilevel Statistical Models by Harvey Goldstein with data sets   
David Roxbee Cox, Peter A. W. Lewis The statistical analysis of series of events. Chapman and Hall, 1966
  Google books    Poisson process computing program
David J Bartholomew. Stochastic Models for Social Processes, Chichester 3rd edition: John Wiley and Sons.
   David J Bartholomew web page


Grading, Exams, and Credit Units
Stat222/Ed351A is listed as Letter or Credit/No Credit grading for 2-units
For Autumn 2020 grading for the 2-units will be based on a 'take home'(i.e. do at home) Problem Set.
  Each week are posted a few exercises for that week's content--towards the end of the qtr I will identify a subset of those exercises to be turned in.
   Those selected problems will constitute the graded Problem Set.
   Also as you will see, for each week's content a number of Review Questions with Solutions are posted.

Course Problem Set 2020    partial posting Oct 30; complete posting Nov 10

Statistical computing
Class presentation will be in, and students are encouraged to use, R (occasionally, some references to SAS and Mathematica).
Current version of R is R version 4.0.2 (Taking Off Again) released 2020-06-22.
    For references and software: The R Project for Statistical Computing   Closest download mirrors in the past, UCLA and Berkeley, seem no longer avaliable, pick your fave anywhere in the world.
The CRAN Task View: Statistics for the Social Sciences provides an overview of some relevant R packages. Also the new CRAN Task View: Psychometric Models and Methods and CRAN Task View: Survival Analysis and CRAN Task View: Computational Econometrics.
A good R-primer on various applications (repeated measures and lots else). Notes on the use of R for psychology experiments and questionnaires Jonathan Baron, Yuelin Li.   Another version
A Stat209 text, Data analysis and graphics using R (2007) J. Maindonald and J. Braun, Cambridge 2nd edition 2007. 3rd edition 2010   has available a short version in CRAN .
According to Peter Diggle: "The best resource for R that I have found is Karl Broman's Introduction to R page."



                  Course Content: Files, Readings, Examples

Week 1. First class: Longitudinal Research Overview, Analysis of Individual Trajectories.
Lecture slides, week 1 (pdf)
Audio companion, week 1
parta   partb   partc
Lecture Topics
A.    Longitudinal research overview
B.     Examples, illustrations for longitudinal research overview, taken from course resources above:
         Laird,Ware (#1) slides 1-16;    Diggle (#5) slides 4-14, 22-28    Verbeke (#9) slides from Ch 2 and Sec3.3
C.     Data Analysis Examples of Model Fitting for Individual Trajectories and Histories.
    Motto: Individual trajectories are the proper starting point for longitudinal data analysis
         ascii version of class handout     annotated version       pdf version with plots     datasets
               Starting up R-addendum: installing packages and obtaining data (sleepstudy in lme4)

  Additional materials for the trajectory examples
            For Count Data (glm) example. Link functions for generalized linear mixed models (GLMMs), Bates slides (pdf pages 11-18)
     AIDS in Belgium example, (from Simon Wood) single trajectory, count data using glm. Rogosa R session for aids data
        aditional expositions of AIDS data, Poisson regression:  Duke   Kentucky
    A very comprehensive introduction to analysis of count data Regression Models for Count Data in R Achim Zeileis Christian Kleiber Simon Jackman (Stanford University)
        Non-linear models, esp logistic. From week 1, also week 3 Self-Starting Logistic model      SSlogis help page, do ?SSlogis   post of annotated logistic curve with SSlogis arguments   
           Trend in Proportions: College fund raising example     prop.trend.test help page ?prop.trend.test in R-session.
          Trend in proportions, group growth, Cochran-Armitage test. Expository paper: G. Salanti and K. Ulm (2003): Tests for Trend in Binary Response (SU access)


WEEK 1 Review Questions
1. For the straight-line (constant rate of change) fit example to subj 372 in the sleepstudy data. Obtain a confidence interval for the rate of change from the OLS fit. Now compare the OLS fit with day-to-day differences. Under the constant rate of change model these 9 day to day differences also estimate the rate of change. Obtain a estimate of the mean and a confidence interval for rate of change from these first differences. Compare with OLS results.
Solution for question 1
2. Revisit the Belgium Aids data example (counts of new cases by year). Use the parameter estimates for am2 (quadratic in time glm fit) to compute by hand (or calculator) the values of the glm fit at year = 5 and year = 9. Compare those values with results from the model am2 using predict
Solution for question 2
3. Paul Rosenbaum has a little data set on growth in vocabulary that I grabbed from his Wharton coursesite. Following the chicks class example, plot these data and try to fit a logistic growth curve to these data. What is the estimate of the final vocab level (asymptote)? Compare the data and the fits from the logistic growth curve.
For reference,       Self-Starting Logistic model      SSlogis help page, do ?SSlogis   post of annotated logistic curve with SSlogis arguments       additional tools in the grofit package
Solution for question 3
4. More on autocorrelation[extension/enrichment].   In standard regression courses you may have seen in addition to Durbin-Watson test for AR(1) (dwtest()), versions of the Cochrane-Orcutt procedure for remediation. Uses a first difference transformation of the data with an estimate of the autocorrelation (therefore hopeless when you have 3,4 5 observations per unit). To illustrate the statements in class and the similarities to OLS result, the solution to this problem does the straight-line and polynomial examples from the Week 1 class handout using the R-package orcutt
Solution for question 4
WEEK 1 Exercises
1. Straight-line fits for NC Fem data: North Carolina Achievement Data (see Williamson, Applebaum, Epanchin, 1991). These education data are eight yearly observations on achievement test scores in math (Y), for 277 females each followed from grade 1 to grade 8, with a verbal ability background measure (W).
North Carolina, female math performance (also in Rogosa-Saner)    North Carolina data (wide format);         NC data (long)
a. Here we will use the 8 yearly observations on female ID 705810, which you can obtain from either the long form or wide form of these data.
For that female, what is the rate of improvement over grades 1 through 8? Compare the observed improvement for grades 1 through 8 (the difference score) with the amount of improvement indicated by the model fit. Obtain a 95% confidence interval for each (if possible).
b. More on OLS and the difference score. Refer to an old publication: A growth curve approach to the measurement of change. Rogosa, David; Brandt, David; Zimowski, Michele Psychological Bulletin. 1982 Nov Vol 92(3) 726-748 APA record   direct link;  Equation 4, page 728, shows a useful form for the OLS slope. (actually reading the first three pages of that pub is a decent intro to the growth curve topic.) For equally spaced data, that Eq (4) gives a useful equivalence between difference scores (amounts of change) and OLS slopes (multiply rates of change by time interval). For the part a NC data show that the OLS slope can be expressed as a weighted sum of the four differences: { 8-1,7-2,6-3,5-4}. [to say that better {score at time 8 minus score at time 1; score at time 7 minus score at time 2; ...} and so forth]
Seperately, consider three observations at taken at equally spaced time intervals: What is a simple expression for the OLS slope (rate of change)?

2. Revisit the Berkeley Growth Data example from week 1 lecture. Consider the quadratic (polynomial degree 2) fit to these data, and also a (innapropriate?) constant-rate-of-change (straight-line) fit to these data. Then refer to Seigel, D. G. Several approaches for measuring average rates of change for a second degree polynomial. The American Statistician, 1975, 29, 36-37. JStor Link for equivalences for the slope of the straight-line fit to an average rate of change for the quadratic fit. Compare Seigel 'Approach 3" to 'Approach 1'.


Week 2. Analysis of collections of growth curves (Mixed-effects Models, lmer)   Constant rate of change models

Lecture slides, week 2 (pdf)
Audio companion, week 2
parta   partb   partc
Lecture Topics. Analyses of collections of growth curves.
1. Plots, description and SFYS (smart first year student) analyses.
2. Mixed effects models using lmer .     Growth modelling handout

Class Examples
1. Data frame sleepstudy available in lme4 package.
           Music to accompany long-distance truck driver data: Flying Burrito Brothers "Six Days on the Road"
   a. Published Treatments, Sleepstudy example
Source Publication: Belenky, G., Wesensten, N. J., Thorne, D. R., Thomas, M. L., Sing, H. C., Redmond, D. P., Russo, M., & Balkin, T. (2003). Patterns of performance degradation and restoration during sleep restriction and subsequent recovery: A sleep dose-response study. Journal of Sleep Research, 12(1), 1-12.
Sleepstudy data analysis from Doug Bates lme4 book lme4: Mixed-effects modeling with R February 17, 2010 (draft chapters) Chapter 4: Sleepstudy example or a set of Bates slides for the sleepstudy example
Why lmer (lme4) does not provide p-values for fixed effects : Doug Bates    lmer, p-values and all that
   A number of add-on packages seek to provide lmer p-values.(We show afex package in Review Question 1 with 2020 update).
   b. Class Materials, Sleepstudy example
     Individual plots (frame-by-frame)     Plot of straight-line fits     Initial descriptive analyses (SFYS)
        Sleepstudy, Bates Ch 4, lme4 analyses, ascii      Sleepstudy class handout, pdf scan
     more Doug Bates Slides (pdf pages 8-28)    
    Individual effects: fixed and random (and BLUPs).    sleepstudy Rsession    session plots

2.    North Carolina, female math performance (also in Rogosa-Saner)
   North Carolina data (wide format);     NC data (long)    plots for NC data
       Data formatting: wide to long    North Carolina data (wide format);     making the "Long" version
     The UCLA archive has a tutorial using built-in reshape function (rather than the reshape package).
     North Carolina example. wide-form descriptives, background, plots         Initial SFYS analyses of NC data, ascii
     Model Comparisons for North Carolina, female math performance     ascii version      NC class handout, pdf scan
     model ncCon2 without redundent model term      NC bootstrap results (SAS)
  North Carolina example (week 2 handouts),   plotting residuals ,   ascii session.

3. Brain Volume Data, in-class modeling exercise: analyses from "Variation in longitudinal trajectories of regional brain volumes of healthy men and women (ages 10 to 85 years) measured with atlas-based parcellation of MRI"     cartoon plot of Lateral Ventricles data;     actual data plot of Lateral Ventricles data;    development of lmer (random effect) growth models


Background and Resources

Technical Formulation and extensions
      Estimation in lmer.
Fitting linear mixed-effects models using lme4, Journal of Statistical Software Douglas Bates Martin Machler Ben Bolker    also Rnews_2005 pp.27-30
Bates book, Chapter 5, Computational Methods.   Bates talk slides:    Mixed models in R using the lme4 package Part 4: Theory of linear mixed models
      Extensions and Alternatives, lmer.
Plots and diagnostics:   Package Influence.me    RJournal intro    Package merTools  An Introduction to merTools   Also, Prediction Intervals
Non-Gaussian modelling. Hierarchical Generalized Linear Models, Package hglm   Hierarchical Generalized Linear Models, R Journal December 2010.
Extensions of lme4 modeling: Package npmlreg Nonparametric Maximum Likelihood (NPML) estimation;
Package robustlmm: An R Package for Robust Estimation of Linear Mixed-Effects Models
    Package RLRsim Title Exact (Restricted) Likelihood Ratio Tests for Mixed and Additive Models

Data Examples
North Carolina Data also in (with full development of the modelling) Longitudinal Data Analysis Examples with Random Coefficient Models. David Rogosa; Hilary Saner . Journal of Educational and Behavioral Statistics, Vol. 20, No. 2, Special Issue: Hierarchical Linear Models: Problems and Prospects. (Summer, 1995), pp. 149-170. Jstor
Douglas Bates class resource item #3, Texts and Resources. Other Doug Bates materials: Three packages, "SASmixed", "mlmRev" and "MEMSS" with examples and data sets for mixed effect models
North Carolina Data also in (with full development of the modelling) Longitudinal Data Analysis Examples with Random Coefficient Models. David Rogosa; Hilary Saner . Journal of Educational and Behavioral Statistics, Vol. 20, No. 2, Special Issue: Hierarchical Linear Models: Problems and Prospects. (Summer, 1995), pp. 149-170. Jstor    Data sets for Rogosa-Saner
Additional talk materials: An Assortment of Longitudinal Data Analysis Examples and Problems 1/97, Stanford biostat.      Overview and Implementation for Basic Longitudinal Data Analysis CRESST Sept '97.    Another version (short) of the expository material is from the Timepath '97 (old SAS progranms) site: Growth Curve models ;    Data Analysis and Parameter Estimation ; Derived quantities for properties of collections of growth curves and bootstrap inference procedures

WEEK 2 Review Questions
1. More sleepstudy. Confidence interval and p-values. Add on, extension to class example.
I start by fitting the lmer model for the collection of growth curves: sleeplmer = lmer(Reaction ~ Days + (1 + Days|Subject), sleepstudy).
Then try out confint from lme4 (link to manual using likelihood profile or bootstrap methods.
Then look at the pvalues entry in the manual and try out add-on packages, esp for p-values for the fixed effects.       
 Solution for Review Question 1       2017 redo/update using 3.3.3 (barebones)     2020 afex update
2. Ramus Data example. Example consists of 4 longitudinal observations on each of 20 cases. The measurement is the height of the mandibular ramus bone (in mm) for boys each measured at 8, 8.5, 9, 9.5 years of age. These data, which have been used by a number of authors (e.g., Elston and Grizzle 1962), can be found in Table 4.1 of Goldstein (1979).      Ramus data example      long form for Ramus data   tutorial on creating long form data manipulation   and   2017 redo/check of widetolong.     Use lmList to obtain the 20 OLS fits, with the initial time set to 8 years of age, i.e. intercepts are fits for the time of initial measurement (not t=0). Fit the lmer model for the collection of growth curves (using initial time = 8); verify that fixed effects are the sample means (over persons) of the lmList intercepts and slopes. Verify that the random effects variance for "age" (i.e. slopes) is the method-of-moments estimate for Var(theta). Compare the random effect estimates (ranef) which borrow strength for each subject with the OLS estimates from lmList (c.f. Bates Chap 4 discussion of sleepstudy data)       
 Solution for Review Question 2
3.   Artificial data example (used in Myths chapter to illustrate time-1,time-2 data analysis)    Two part artificial data example.   The bottom frame (the X's) is 40 subjects each with three equally spaced time observations (here in wide form).For these the fallible "X" measurements (constructed by adding noise to the Xi measurements). Follow the class examples 'wide-to-long' and obtain the plot showing each subject's data and straight-line fit. Use lmList to obtain the 40 slopes for the straight-line fits.       
 Solution for Review Question 3
4. More with North Carolina data
a. identify the fastest and slowest growth among the 277 females. Compare medians of growth rates for females with verbal ability (Z) at or above 106 with that for females with verbal ability below 106. Show side-by-side boxplots.
b. In the class handout version of the NC analyses (and other postings, but not all) the first thing to do was make the 'time' variable have intitial value = 0 (making the intercept of a straight line fit correspond to level at initial time): i.e. 1 to 8 becomes 0 to 7. Obtain lmList results and fit the ncUnc lmer model (straight-line growth, no Z) using time 1 to 8. Comment on differences of these analyses with those using timeInt in the class handout. In particular, look at the correlation of change and initial status. The correlation between observed change and observed initial status using timeInt was .279 from lmer (Correlation of Fixed Effects) and also from lmList (you should confirm that). What is the result you obtain using time rather than timeInt? The mle of of the correlation of 'true' change and 'true' initial status is  .651 using timeInt. What do you obtain using time?.       
 Solution for Review Question 4

5. xyplot with large sample sizes.
North Carolina data has 277 subjects, a frame-by-frame display of individuals requires subsampling. Construct a plot for 24 (arbitrary) individuals data trajectories.       
 Solution for Review Question 5


WEEK 2 Exercises
1. Tolerance data [note: 10/12/17 data location updated]
A subsample of data from the National Youth Survey is obtained in long-form by
read.table("https://stats.idre.ucla.edu/wp-content/uploads/2016/02/tolerance1_pp.txt", sep=",", header=T)
and in wide form by
read.table("https://stats.idre.ucla.edu/wp-content/uploads/2016/02/tolerance1.txt", sep=",", header=T)
Yearly observations from ages 11 to 15 on the tolerance measure (tolerance to deviant behavior e.g. cheat, drug, steal, beat; larger values indicates more tolerance on a 1to4 scale). Also in this data set are gender (is_male) and an exposure measure obtained at age 11 (self report of close friends involvement in deviant behaviors). note: the time measure is age - 11.
i. obtain individual OLS fits (tolerance over time) and plot the collection of those straight-lines. Provide descriptive statistic summaries for the rate of change in tolerance and initial level.
ii. fit a mixed effects model for tolerance over time (unconditional) for this collection of individuals. Obtain interval estimates for the fixed and random effects. Show that the fixed effects estimates correspond to quantities obtained in part i. Explain.
iii. Investigate whether the exposure measure is a useful predictor of level or rate of change in tolerance. What appears to be the best fitting mixed model for these data using these measures? Show specifics.
2. lmer/lme vs lm
 Consider the sleepstudy and Ramus examples, collections of growth trajectories with no exogenous variable. Ramus Data example. Example consists of 4 longitudinal observations on each of 20 cases. The measurement is the height of the mandibular ramus bone (in mm) for boys each measured at 8, 8.5, 9, 9.5 years of age. These data, which have been used by a number of authors (e.g., Elston and Grizzle 1962), can be found in Table 4.1 of Goldstein (1979).      Ramus data example      long form for Ramus data   tutorial on creating long form data manipulation .   
  Fitting the lmer models with Formula: Reaction ~ 1 + Days + (1 + Days | Subject) or Formula: ramus ~ I(age - 8) + (age | subj) has motivated the student question, what is going on here beyond what lm would do? So let's look at what lm would do in these examples. Verify (or disprove) the assertion that the fixed effects from lmer, which we have seen are the averages of the individual fit parameter estimates (i.e. lmList), and therefore the coefficients of the average growth curve are identical to the fit from lm (which ignores the existence of individual trajectories). Compare the results of the lm and lmer analyses for these two data sets.
3. Early Education data (From Bates and Willett-Singer).
Data on early childhood cognitive development described in Doug Bates talk materials (pdf pages 49-52). Obtain these data from the R-package "mlmRev" or the Willett-Singer book site (in our week 1 intro links). Data are in long form and consist of 3 observations 58 treatment and 45 control children; see the Early entry in the mlmRev package docs. Produce the plot of individual trajectories shown pdf p.49, Bates talk. (note:Bates does connect-the-dots, we have done straight-line fit, your choice). Show five-number summaries of rates of impovement in cognitive scores for treatment and control groups. Develop and fit the fm12 lmer model shown in Bates pdf p.50 (note fm12 allows trt to effect rates of improvement but not level;). Interpret results. Note: this moves us into the comparing groups topics, where the individual attribute is group membership.
4.   Standardizing is always a bad idea is a good motto for life, especially with longitudinal data.
   Artificial data example from Review Question 3 (used in Myths chapter to illustrate time-1,time-2 data analysis)    Start out with the "X" data, and standardize (i.e. transform to mean 0, var 1) at each of the 3 time points. Note "scale" will do this for you (in wide form). For the standardized data obtain the plot showing each subject's data and straight-line fit. What do you have here? Compare the results the mixed-effects models fitting the collection of straight-line growth curves for the measured and standardized data.


Week 3  Collections of growth curves continued: linear and non-linear mixed-effects models
Lecture slides, week 3 (pdf)
Audio companion, week 3
parta   partb   partc
Lecture Topics
1. Review model formulation, North Carolina example (week 2 handouts),
lm and gee alternatives (ignore individual growth). ascii session
2. General formulation of mixed effects model in terms of growth trajectories pdf pages 7-8, handout in An Assortment of Longitudinal Data Analysis Examples and Problems , Stanford biostat (pp.7-8).   Also Ware-Laird ALA slides 234-240.
   also resource item #3, Douglas M. Bates lme4: Mixed-effects modeling with R, section 1.4
3. lmList does logistic (respiration data week5); introducing glmer      lmList, glmer for respiration data
          note: you may need to install HSAUR3 package (instead of 2) for the respiratory data set
4.  Quick look at Ordered time factor.
5.   Beyond Straight-line Growth: Polynomial and Non-linear Models.
Polynomial examples: The book by Mirman, resource item 7   Growth Curve Analysis and Visualization Using R   not surprisingly has some good data examples; see RQ#4.
Logistic Curve Example: Orange Tree growth.     Data from MEMSS package Data sets and sample analyses from Pinheiro and Bates, Mixed effects Models in S and S-PLUS (Springer, 2000).
   Plots and nlmer analysis, Orange tree data
   Doug Bates Slides Orange trees analysis (pdf pages 8-16), Logistic SS (pdf p.6), pharmacokinetics ex (pdf pages 7, 17-24)   Bates NLMM.Rnw      From week 1 SSlogis (Self-Starting Logistic model)  links and materials.        another analysis of Orange Trees in the ASReml package manual section 8.9
Also LDA book Chapter 5. Chapter 5. Non-linear mixed-effects models Marie Davidian
    additional tools in the grofit package and nlmeODE package Title Non-linear mixed-effects modelling in nlme using differential equations

WEEK 3 Review Questions
1. Constrained models.
A somewhat common practice (which I'm not that fond of) is to constrain individuals differences in an lmer model--e.g. force all individuals to have the same rate of change. For purposes here use the sleepstudy data to fit a mixed-model with all individuals having the same time gradient. Compare to the model in class allowing slopes and levels to differ.       
 Solution for Review Question 1
2. Orange tree extras. Take the fixed effects from the orange tree nlmer model, "m1" in the class materials, as the parameters of the "average" growth curve for this group of trees. Plot that logistic growth curve (either use a formula for logistic or the growfit package has a simple function). Compare the fixed effects from nlmer to the results from nls for these data. More challenging Try to superimpose the group logistic curve (above) onto the plots of the individual tree trajectories (you may want to refer to the plots week1 Aids data).       
 Solution for Review Question 2
3. Asymptotic regression, SSasymp slide (pdf p.5 of Bates slides, Nonlinear mixed models talk linked in Week 3, Topic 4). Data are from Neter-Wasserman text in file CH13TA04.txt. The outcome variable is manufacturing relative efficiency (RelEff) over 90 weeks duration for two different locations. Plot the RelEff outcome against week for the two locations. Use the SSasymp function for a nlmer fit (or nls if needed) to see whether the asymptote differs for the two locations.       
 Solution for Review Question 3
4. Quadratic (polynomial) Trends.   The book by Mirman resource item 7   Growth Curve Analysis and Visualization Using R   not surprisingly has some good data examples (primarily psychological learning experiments). Here we use the Chapter 3 data set (sec 3.4) Word Learning. Data at http://www.danmirman.org/gca/WordLearnEx.txt. Use the subset TP == Low. How many subjects in that subset? How many observations on each? Accuracy is the outcome measure, the time ordered measure is Block (see Fig 3.7). Investigate a linear trend versus a quadratic trend using mixed effect models.       
 Solution for Review Question 4

WEEK 3 Exercises
1. Teen age drinking. [note: data location updated 10/12/17]
The UCLA data archive has a comma delimited file (access by read.table("https://stats.idre.ucla.edu/stat/r/examples/alda/data/alcohol1_pp.txt", header=T, sep=",")  .
Measurements on 82 adolescents (initial age 14) included 3 time-ordered observations on alcohol use and two background (exogenous) variables: dichotomous coa (child of an alcoholic) and measured variable peer (alcohol use by target's peers). Describe the collection of time trajectories in alcohol use. Fit an unconditional mixed model to this collection of time-trajectories and obtain interval estimates for the random and fixed effects. Show a plot for the random effects (subjects) and interpret the fixed effects. Now consider the two exogenous variables. Using conditional models, identify the best fitting model. Interpret the fixed effects for the best fitting model.
2.  Vocabulary learning data from test results on file in the Records Office of the Laboratory School of the University of Chicago. Source D R Bock, MSMBR. The data consist of scores, obtained from a cohort of pupils at the eigth through eleventh gade level on alternative forms of the vocabulary section of the Cooperative Reading Tests." There are 64 students in all, 36 male, 28 female (ordered) each with four equally spaced observations (test scores). Wide form of these data are in BOCKwide.dat and I kindly also made a long-form version BOCKlong.dat . Construct the usual collection of individual trajectory displays (either connect-the-dots or compare to a straight-line). Obtain the means (over persons) and plot the group growth curve. Does there appear to be curvature (i.e. deceleration in vocabulary skill growth)?
a. Construct an lmer model with the individual growth curve a quadratic function of grade (year), most convenient to use uncorrelated predictors grade - mean(grade) and (grade - mean(grade))^2. Fit the lmer model and interpret the fixed and random effects you obtain. Compare the results with a lmer model in which the individual trajectories are straight-line. Use the anova model comparison functionality in R (e.g. anova(modLin, modQuad) to test whether the quadratic function for individual growth produces a better model fit.
b. Investigate (via lmer model) gender differences (isMale) in vocabulary growth. Fit appropriate lmer models and interpret results,
3. Data on the growth of chicks on different diets. Hand and Crowder (1996), Table A.2, p. 172 Hand, D. and Crowder, M. (1996), Practical Longitudinal Data Analysis, Chapman and Hall, London. The dataset is available as a .R file; easiest to bring this page down to your machine and then load into your R-session (or try to load remotely). Here we consider the 20 chicks on Diet 1. (select these). Construct the plots analogous to those for the class example Orange trees: individual chicks frame-by-frame and all chicks on one plot. Fit a nlmer model that allows final weight (asymptote) to differ over chicks (other params fixed). Use ranef (individual estimates) to identify the largest asymptote value and smallest value. Plot the "average" growth curve under diet 1. Compare that nlmer model with a model that does not allow asymptotes to differ. What is your conclusion. Also compare with a nls model that ignores repeated measurements structure (i.e. ignores individual chicks). Compare the average growth curves.



Week 4. Special case of time-1, time-2 data; Traditional measurement of change and more
Lecture slides, week 4 (pdf)
Audio companion, week 4
parta   partb

Lecture Topics
1. Properties of Collections of Growth Curves. class handout
2. Time-1, time-2 data. (paired data)
     The R-package PairedData has some interesting plots and statistical summaries for "before and after" data;
          here is a McNeil plot for Xi.1, Xi.5 in data example
     Paired dichotomous data, McNemar's test (in R, mcnemar.test {stats}), Agresti (2nd ed) sec 10.1
      Also see R-package PropCIs       Prime Minister example
3. Issues in the Measurement of Change. Class lecture covers Myths 1-6+.
     Slides from Myths talk    . Class Handout, Companion for Myths talk
4. Examples for Exogenous Variables and Correlates of Change (use of lagged dependent variables)
   Time-1,time-2 data analysis examples    Measurement of change: time-1,time-2 data
      data example for handout    scan of regression handout      ascii version of data analysis handout    
   Extra material for Correlates and predictors of change: time-1,time-2 data
    Rogosa R-session to replicate handout, demonstrate wide-to-long data set conversion, and descriptive fitting of individual growth curves. Some useful plots from Rogosa R-session
        Technical results: Section 3.2.2 esp Equation 27 in Rogosa, D. R., & Willett, J. B. (1985). Understanding correlates of change by modeling individual differences in growth. Psychometrika, 50, 203-228.      Talk slides
5. Comparing groups on time-1, time-2 measurements: repeated measures anova vs lmer OR the t-test
Comparative Analyses of Pretest-Posttest Research Designs, Donna R. Brogan; Michael H. Kutner, The American Statistician, Vol. 34, No. 4. (Nov., 1980), pp. 229-232.   JSTOR link
     urea synthesis, BK data       data, long-form
    BK plots (by group)     BK overview
    2017 Analysis handout     Extended BK lmer analysis
Additional stuff
     BK repeated measures analysis      pdf version
    Stat141 analysis
    archival example analyses. SAS and minitab

Background Readings and Resources
Myths Chapter. Rogosa, D. R. (1995). Myths and methods: "Myths about longitudinal research," plus supplemental questions. In The analysis of change, J. M. Gottman, Ed. Hillsdale, New Jersey: Lawrence Erlbaum Associates, 3-65.
Myths Talk. Rogosa, D. R. (1983)
More stuff (if you don't like the ways I said it)   
I noticed John Gottman did a pub rewriting the myths: Journal of Consulting and Clinical Psychology 1993, Vol. 61, No. 6,907-910 The Analysis of Change: Issues, Fallacies, and New Ideas
Also John Willett did a rewrite of the Myths 'cuz I didn't want to reprint it again (or write a new version): Questions and Answers in the Measurement of Change REVIEW OF RESEARCH IN EDUCATION 1988 15: 345
Reliability Coefficients: Background info. Short primer on test reliability    Informal exposition in Shoe Shopping and the Reliability Coefficient    extensive technical material in Chap 7 Revelle text
A growth curve approach to the measurement of change. Rogosa, David; Brandt, David; Zimowski, Michele Psychological Bulletin. 1982 Nov Vol 92(3) 726-748 APA record   direct link
Rogosa, D. R., & Willett, J. B. (1985). Understanding correlates of change by modeling individual differences in growth. Psychometrika, 50, 203-228.
available from John Willet's pub page
Demonstrating the Reliability of the Difference Score in the Measurement of Change. David R. Rogosa; John B. Willett Journal of Educational Measurement, Vol. 20, No. 4. (Winter, 1983), pp. 335-343. Jstor
Maris, Eric. (1998). Covariance Adjustment Versus Gain Scores--Revisited. Psychological Methods, 3(3) 309-327. apa link  
A good R-primer on repeated measures (a lots else). Notes on the use of R for psychology experiments and questionnaires Jonathan Baron, Yuelin Li.   Another version
Multilevel package   has behavioral scienes applications including estimates of within-group agreement, and routines using random group resampling (RGR) to detect group effects.
More repeated measures resources: Background primer on analysis of variance (with R); see sections 6.8, 6.9 of Notes on the use of R for psychology experiments and questionnaires Jonathan Baron, Yuelin Li.   Pdf version    The ez package provides extended anova capabities.   Examples (blog notes) : Repeated measures ANOVA with R (functions and tutorials)   Repeated Measures ANOVA using R    Obtaining the same ANOVA results in R as in SPSS - the difficulties with Type II and Type III sums of squares
Application publications, time-1, time-2 Experimental Group Comparisons:
a.  Mere Visual Perception of Other People's Disease Symptoms Facilitates a More Aggressive Immune Response Psychological Science, April 2010   Pre-post data and difference scores (see Table 1)
b. Guns and testosterone. Guns Up Testosterone, Male Aggression
Guns, Testosterone, and Aggression: An Experimental Test of a Mediational Hypothesis Klinesmith, Jennifer; Kasser, Tim; McAndrew, Francis T,   Psychological Science. Vol 17(7), Jul 2006, pp. 568-571.


WEEK 4 Review Questions
1. Time1-time2 regressions; Class example
Repeat the handout demonstration regressions using the fallible measures (the X's) from the bottom half of the linked data page. The X's are simply error-in-variable versions of the Xi's: X = Xi + error, with error having mean 0 and variance 10. Compare 5-number summaries for the amount of change from the earliest time "1" to the final observation "5" using the "Xi" measurements (upper frame) and the fallible "X" observations (lower frame).       
 Solution for Review Question 1
2. (more challenging). Use mvrnorm to construct a second artificial data example (n=100) mirroring the week 4 myths data class handout BUT with the correlation between true individual rate of change and W set to .7 instead of 0. Carry out the corresponding regression demonstration.        
 Solution for Review Question 2
3. Reliability versus precision demonstration
  Consider a population with true change between time1 and time2 distributed Uniform [99,101] and measurement error Uniform [-1, 1]. If you used discrete Uniform in this construction then you could say measurement of change is accurate to 1 part in a hundred.
Calculate the reliability of the difference score.
Also try error Uniform [-2,2], accuracy one part in 50.
A similar demonstration can be found in my Shoe Shopping and the Reliability Coefficient
      
 Solution for Review Question 3
4. Revisit Brogan-Kutner data analysis.
a. Demonstrate the Brogan-Kutner Section 5 equivalences (from paper, shown in class) for repeated measures anova and/or BK lmer analyses.
b. Is amount of gain/decline related to initial status? For the 8 new procedure patients and for the 13 old procedure patients, seperately, estimate the correlation between change and initial status and obtain a confidence interval if possible.
c. Analysis of Covariance. For the Brogan-Kutner data carry out an analysis of covariance (using premeasure as covariate) for the relative effectiveness of the surgery methods. Compare with class analyses.
Slides 203-204 in the Laird-Ware text materials purport to demonstrate that analysis of covariance produces a more precise treatment effect estimate than difference scores (repeated measures anova). What very limiting assumption is slipped into their analysis? Can you create a counter-example to their assertion/proof?       
 Solution for Review Question 4
                                                 part c. Solution Notes on the ALA (Laird-Ware) assertion
5.  Repeat Brogan-Kutner lmer analyses from lecture. Just another repitition of BK class handout.
Use lmer (or lme) to determine the comparative efficacy of the surgical methods on liver function. Investigate whether a model allowing for pretest differences is helpful.       
 Solution for Review Question 5


WEEK 4 Exercises
1. Captopril and Blood pressure
The file captopril.dat contains the data shown in Section 2.2 of Verbeke, Introduction to Longitudinal Data Analysis, slides. Captopril is an angiotensin-converting enzyme inhibitor (ACE inhibitor) used for the treatment of hypertension.
a. Smart First Year Student analyses. Use the before and after Spb measurements to examine the improvement (i.e. decrease) in blood pressure. Obtain a five-number summary for observed improvement. What is the correlation between change and initial blood pressure measurement? Obtain a confidence interval for the correlation and show the corresponding scatterplot. What special challenges are present in this analysis?
b. lmer analyses. Try to obtain a good confidence interval for the amount of decline. Obtain a point and interval estimate for the correlation beween initial status and change in Spb.
2. Regression toward the mean? Galton's data on the heights of parents and their children
In the "HistData" or "psych" packages reside the "galton" dataset, the primordial regression toward mean example.
Description: Galton (1886) presented these data in a table, showing a cross-tabulation of 928 adult children born to 205 fathers and mothers, by their height and their mid-parent's height. A data frame with 928 observations on the following 2 variables. parent Mid Parent heights (in inches) child Child Height. Details: Female heights were adjusted by 1.08 to compensate for sex differences. (This was done in the original data set)
Consider "parent" as time1 data and "child" as time2 data and investigate whether these data indicate regression toward the mean according to either definition (metric or standardized)? Refer to Section 4 of the Myths chapter supplement (pagination 61-63) for an assessment of regression toward the mean (i.e. counting up number of subjects satisfying regression-toward-mean).
Aside: if you like odd plots, look at the sunflowerplot code in the docs for the galton data.
3. Paired and unpaired samples, continuous vs categorical measurements.
Let's use again the 40 subjects in the Review Question 1 "X" data.
a. Measured data. Take the time1 and time5 observations and obtain a 95% Confidence Interval for the amount of change. Compare the width of that interval with a confidence interval for the difference beween the time5 and time1 means if we were told a different group of 40 subjects was measured at each of the time points (data no longer paired).
b. Dichotomous data. Instead look at these data with the criterion that a score of 50 or above is a "PASS" and below that is "FAIL". Carry out McNemar's test for the paired dichotomous data, and obtain a 95% CI for the difference between dependent proportions. Compare that confidence interval with the "unpaired" version (different group of 40 subjects was measured at each of the time points) for independent proportions.
4. Beat the Blues from Chap 12 of HSAUR 2nd ed (resource # 4).
Data in wide form: data("BtheB", package = "HSAUR2"). Chap. 12 describes the cognitive behavioural program and conducts various analyses. We will use the pretest and the two-month followup (additional followups have lots of missing data).
Investigate the effectiveness of Beat the Blues from these 2-wave data. Follow the various descriptive and modelling strategies shown in the BK class example.
5. From 2017 In the news
The 3 billion dollar (and counting) change score
(items below clipped from 2017 various press reports; we do not have the data)

Sage Therapeutics (NASDAQ:SAGE) surged in response to its announcement of positive results from a Phase 2 clinical trial assessing SAGE-217 for the treatment of adult patients with moderate-to-severe major depressive disorder (MDD), a Fast Track indication in the U.S.[2020 note: name, Zuranolone]
It is estimated that there are around 16 million people in the United States with MDD.
SAGE-217, a neuroactive steroid, is next-generation GABA modulator. The GABA system, the major inhibitory signaling pathway in the brain and central nervous system (CNS), plays a key role in regulating CNS function. The company intends to advance the program into Phase 3 development.
The phase 2 looked at the effect of the positive allosteric modulator of the gamma-aminobutyric acid (GABA) receptor as compared to placebo in 89 patients with MDD.

About the Placebo-controlled Phase 2 trial of SAGE-217 in MDD:
In the randomized, double-blind, parallel-group, placebo-controlled trial, 89 eligible patients (with a minimum total score of 22 on the Hamilton Rating Scale for Depression) were stratified based on use of antidepressant treatment (current/stable or not treated/withdrawn >= 30 days) and randomized in a 1:1 ratio to receive SAGE-217 Capsules (30mg) (n=45) or matching placebo (n=44). All doses of study drug were administered at night with food. The study consisted of a 14-day treatment period, and a 4-week follow-up period. The mean HAM-D total scores at baseline were 25.2 for the SAGE-217 group and 25.7 for the placebo group (overall range 22-33), representing patients with moderate to severe MDD. Approximately 90 percent of patients in each group completed the study.
Sage Therapeutics (NASDAQ: SAGE), a clinical-stage biopharmaceutical company developing novel medicines to treat life-altering central nervous system (CNS) disorders, today announced positive top-line results from the Phase 2, double-blind, placebo-controlled clinical trial of SAGE-217 in the treatment of 89 adult patients with moderate to severe major depressive disorder (MDD). In the trial, treatment for 14 days with SAGE-217 was associated with a statistically significant mean reduction in the Hamilton Rating Scale for Depression (HAM-D) 17-Item total score from baseline to Day 15 (the time of the primary endpoint) of 17.6 points for SAGE-217, compared to 10.7 for placebo (p<0.0001). Statistically significant improvements were observed in the HAM-D compared to placebo by the morning following the first dose through Week 4 and the effects of SAGE-217 remained numerically greater than placebo through the end of follow-up at Week 6. At Day 15, 64 percent of patients who received SAGE-217 achieved remission, defined as a score of 7 or less on the HAM-D scale, compared with 23 percent of patients who received placebo (p=0.0005).
The 89-subject study met its primary endpoint of a statistically significant average reduction in HAM-D score from baseline to day 15 (p<0.0001) versus placebo. HAM-D is a rating scale for depression. At day 15, 64% of patients in the treatment group achieved remission compared to 23% for placebo (p=0.0005).
There were a total of 89 patients recruited into the study who were either given SAGE-217 or a placebo compound. Patients were treated for a 14 day period and were then measured for clinical outcome using the Hamilton Rating Scale for Depression or HAM-D 17-item total score from baseline. It was shown that SAGE-217 achieved a statistically significant improvement over placebo according to the HAM-D scale. Patients that took SAGE-217 were shown to achieve a 17.6 point improvement at day 15, compared to only a 10.7 point improvement for placebo. That meant that the drug achieved a statistically significant p-value of p < 0.0001. It was also noted that 64% of patients who took SAGE-217 achieved MDD remission, compared to only 23% of placebo patients. MDD remission was classified as patients having a HAM-D score of 7 or less. This was the secondary endpoint of the study, which was also achieved.
Investigators saw a statistically significant improvement in SAGE-217 patients on a depression scale the day after the first dose. By the time the two-week treatment period came to an end, the mean score in the SAGE-217 arm had dropped 17.6 points, as compared to a 10.7 point decline in the control group. That seven-point placebo-adjusted improvement was enough for the trial to hit its primary endpoint with a p value of less than 0.0001.
The positive results continued beyond the end of the treatment period. The mean reduction on the depression scale in the treatment arm remained statistically superior to that of the placebo group two weeks after dosing stopped.
--------------------------------------------------------
Questions
Consider the remission outcome (secondary) at day 15 (after 14 days of dose).
part a. For these time1-time2 dichotomous data (remission or not), explain what I did below to approximate the results reported by SAGE.
part b. In week 4 (time1-time2 data) materials we introduced some more advanced capabilities for time1-time2 dichotomous data, such as mcnemar.test from base R and diffpropci.mp from package PropCIs. Comment on the applicability of those functions to the remission study and whether those are preferable here to the basic analysis in part a.
---------------
> sage2 = matrix(c(29, 10, 16,  34), nr=2)  # remission counts for the two groups
> sage2
     [,1] [,2]
[1,]   29   16
[2,]   10   34
> prop.test(sage2)
        2-sample test for equality of proportions with continuity correction
data:  sage2
X-squared = 14.078, df = 1, p-value = 0.0001754
alternative hypothesis: two.sided
95 percent confidence interval:
 0.2079003 0.6264431
sample estimates:
   prop 1    prop 2 
0.6444444 0.2272727 

> chisq.test(sage2)
        Pearson's Chi-squared test with Yates' continuity correction
data:  sage2
X-squared = 14.078, df = 1, p-value = 0.0001754
-----------------------------------------------------------
part c.  Consider the primary outcome, change in depression score (HAM-D).
In weeks 4 and 5 we conducted analysis of time1- time2 (and multiwave) outcome data for comparisons of experimental groups. For the SAGE study pretend we have long form data, with time coded 0 for baseline and 1 for Day 15 endpoint, and outcome HAM-D score at the timepoints (0,1) and group indicating T/P. So we have 178 rows, and columns HAM-D group time subj.
If we fit the model in R syntax
sagelmer = lmer(outcome ~ time + time:group + (time|subj), data = sage, control = lmerControl(check.nobs.vs.nRE = "warning"),
from the information you have, give the point estimate for the fixed effects, time and time:group .
Write out the level 1, level 2 model corresponding to the combined model in the lmer statement.

Week 5.  Experimental Protocols and Comparing Group Growth

Lecture slides, week 5 (pdf)
Audio companion, week 5
parta   partb   partc
From the Longitudinal in the news archive

Time 1, Time 2 and Multiwave Experiments.  
When Adolescents Give Up Pot, Their Cognition Quickly Improves
From 2017.    Stents?   A Controversial Experiment Upends The Conventional Wisdom On Heart Stents    Publication: Percutaneous coronary intervention in stable angina (ORBITA): a double-blind, randomised controlled trial The Lancet.

Crossover Designs in the news
1. Does nutrition science know anything?     Is white or whole wheat bread 'healthier?' Depends on the person    Publication: Bread Affects Clinical Parameters and Induces Gut Microbiome-Associated Personal Glycemic Responses Cell Metabolism, Korem et al DOI: 10.1016/j.cmet.2017.05.002
2. This time with 3 conditions   For Exercise, Nothing Like the Great Outdoors   Publication: Niedermeier M, Einwanger J, Hartl A, Kopp M (2017) Affective responses in mountain hiking-- randomized crossover trial focusing on differences between indoor and outdoor activity. PLoS ONE 12(5): e0177719. https://doi.org/10.1371/journal.pone.0177719
3. One thing at a time. Why listening to a podcast while running could harm performance    Publication: A trade-off between cognitive and physical performance, with relative preservation of brain function  Scientific Reports 7, Article number: 13709 (2017) nature.com.
4. Another crossover design (from Stat266). RCT (cross-over design). Damn right! The secret of success is swearing: How shouting four letter words can help make you stronger    Swearing can help you boost your physical performance    The full power of swearing is starting to be discovered.   Publication:  Stephens, R, Spierer, DK and Katehis, E (2018) Effect of swearing on strength and power performance. Psychology of Sport and Exercise, 35. pp. 111-117. ISSN 1469-0292 .


Lecture Topics
1. Cross-over designs (usually time-1, time-2). Laird-Ware text slides (pdf pages 135-150). Crossover design data from slide 137,    anova for crossover design ex       ascii version, anova for crossover design ex   
   R-resources for crossover designs. package Crossover     package crossdes   see Rnews Vol. 5/2, November 2005
        also see slides 5-14 Repeated Measures Design Mark Conaway

2. Multi-wave growth in measured outcome, experimental protocol. Example: Effect of transitional probability (TP) on novel word learning from Mirman text Ch3 (uses orthogonal polynomials). Data from Week3 Review Question 4.

3.  Group Comparisons for Longitudinal Experimental Designs. Group growth and Experimental comparisons for count and dichotomous outcomes  (examples From HSAUR 2ndEd, Ch.13).
Link functions for generalized linear mixed models (GLMMs), Bates slides (pdf pages 11-18)
Note: glmer supercedes lmer
A Handbook of Statistical Analyses Using R, Second Edition Torsten Hothorn and Brian S . Everitt Chapman and Hall/CRC 2009. Analysing Longitudinal Data II -- Generalised Estimation Equations and Linear Mixed Effect Models: Treating Respiratory Illness and Epileptic Seizures (Stanford access)
     Data sets etc Package 'HSAUR2' August 2014, Title A Handbook of Statistical Analyses Using R (2nd Edition)
Note: epilepsy available in all vintages of HSAUR
  A.    Analysis of Count data.      Epilepsy example, group comparisons, collection of individual trajectories. HSAUR chap 13    Rogosa R-session using gee and lmer     class handout
   Recap Group Comparisons, Epilepsy example. Comparison of lmer models
For SAS (and GEE) fans another analysis
  B.    Binary Response, dichotomous outcomes. Respiratory Illness Data from HSAUR package. Data and description also at the ALA (Laird-Ware) site   Rogosa R-session using lmer     class handout
        
4. Study Design: Power Calculations for Longitudinal Group Comparsions.
   R-package longpower Vignettes found by "browseVignettes(package = "longpower")" .    Functions in MBESS package--ss.power.pcm.
   R-package: powerlmm
   Background pubs:  Power for linear models of longitudinal data with applications to Alzheimer's Disease Phase II study design Michael C. Donohue, Steven D. Edland, Anthony C. Gamst
Sample Size Planning for Longitudinal Models: Accuracy in Parameter Estimation for Polynomial Change Parameters Ken Kelley Notre Dame Joseph R. Rausch Psychological Methods 2011
        basic R analogues, power.t.test   power.anova.test

5. Missing Data Concerns.
   Nontechnical overviews:
  Phil Lavori et al. Psychiatric Annals, Volume 38, Issue 12, December 2008 Missing Data in Longitudinal Clinical Trials, Part A    Part B
   Robin Henderson,   Missing Data in Longitudinal Studies pdf pages 89-93
We probably won't get to this (usually defer to DW)
Missing data wide-form imputation: mice multiple regression example, nhanes data in package mice     R-session using mice package
New package: hmi: hierarchical multiple imputation,   vignette
Vignette from merTools package (Stat196):  Analyzing Imputed Data with Multilevel Models and merTools
          Rogosa R-session for vignette
Additional resources.
Technical review: Missing data methods in longitudinal studies: a review   Joseph G. Ibrahimcorresponding author and Geert Molenberghs
More on Missing data and imputation, including mice week 10 topic.     Flexible Imputation of Missing Data. Stef van Buuren Chapman and Hall/CRC 2012. Chapter 9, Longitudinal Data Sec 3.8 Multilevel data. He is the originator of mice
Multiple Imputation. van Buuren S and Groothuis-Oudshoorn K (2011). mice: Multivariate Imputation by Chained Equations in R. Journal of Statistical Software, 45(3), 1-67. see also multiple imputation online    Flexible Imputation of Missing Data. Stef van Buuren Chapman and Hall/CRC 2012. Chapter 9, Longitudinal Data Sec 3.8 Multilevel data. He is the originator of mice      book extras      R resources.  Multivariate Analysis Task View, Missing data section, esp packages mice   see also multiple imputation online
          CHAPTER 17 Incomplete data: Introduction and overview. Longitudinal Data Analysis Edited by Geert Verbeke , Marie Davidian , Garrett Fitzmaurice , and Geert Molenberghs Chapman and Hall/CRC 2008. Also CHAPTER 21 Multiple imputation Michael G. Kenward and James R. Carpenter and CHAPTER 22 Sensitivity analysis for incomplete data. online supplement for LDA book  . van Buuren S (2010). Multiple Imputation of Multilevel Data. In JJ Hox, K Roberts (eds.), The Handbook of Advanced Multilevel Analysis, chapter 10, pp. 173{196. Routledge, Milton Park, UK
Handling drop-out in longitudinal studies (pages 1455-1497) Joseph W. Hogan, Jason Roy and Christina Korkontzelou, Statistics in Medicine 15 May 2004 Volume 23, Issue 9. (SAS implementations)
Bayesian approach. Missing Data in Longitudinal Studies. Strategies for Bayesian Modeling and Sensitivity Analysis Joseph W . Hogan and Michael J . Daniels Chapman and Hall/CRC 2008 Ch 5 Missing Data Mechanisms and Longitudinal Data     Corresponding talk, A Brief Tour of Missing Data in Longitudinal Studies Mike Daniels
Overview and applications paper: Assessing missing data assumptions in longitudinal studies: an example using a smoking cessation trial Xiaowei Yanga, Steven Shoptawb. Drug and Alcohol Dependence Volume 77, Issue 3, 7 March 2005, Pages 213-225
R resources.  Multivariate Analysis Task View, Missing data section, esp packages mice and mi   R-package pan Multiple imputation for multivariate panel or clustered data. Schafer tech report   Schafer talk: Missing Data in Longitudinal Studies: A Review   Efficient ways to impute incomplete panel data. Kristian Kleinke ¬∑ Mark Stemmler ¬∑ Jost Reinecke ¬∑Friedrich Losel AStA Adv Stat Anal (2011) 95:351-373 DOI 10.1007/s10182-011-0179-9


WEEK 5 Review Questions
1. Power (sample size) calculations for experimental group comparisons.
a. Longpower package (vignette). Reconstruct the sample size calculation for the Alzheimer's disease trial (7 waves) on p.4 of the vignette.
b. MBESS package. Recreate the sample size calculation for width of confidence interval for differential growth using ss.aipe.pcm function in the example used in Kelley and Rausch appendix (and MBESS manual)       
 Solution for Review Question 1
2. Revisit Respiration example.
a. try to do lmList on these data to get odds(good) for each of the each 111 subjects. Investigate effectiveness of treatment.
b Use lmer analyses to compare treament and placebo. Obtain a confidence interval for effectiveness of treament. Investigate gender differences in response to the intervention (i.e. the treatment)
c. Extend the lmer model in part b by adding the age and baseline measurements to the level 2 model. Compare with part b results.       
 Solution for Review Question 2
3. Revisit Epilepsy example.
To supplement the longitudinal texts (HSAUR, ALA etc) full model for the epilepsy data, lets try to build up the analysis from basic description comparing placebo vs drug up through some basic some basic glmer models.
A somewhat similar effort was made in the second class posting "Recap group comparisons (epcomp)" linked above. In this exercise treat period as a time measurement (1,2,3,4) rather than an ordered factor.
How many subjects in placebo and drug groups? Use lmList to obtain slopes and intercepts for fits of time trends to seizures for each subject and compare drug and placebo groups.
Fit and compare glmer models with treatment as the only level 2 predictor (for intercept) without and with a time trend. Compare.
Add the baseline to the glmer models above (in level 2 model for intercept; is effect of the drug significant (use confint)? Does adding age help this model?       
 Solution for Review Question 3
4. Extensions for the epilepsy example: residuals, diagnostics
Revisit and extend analysis of ep3 model in RQ3 and lecture.
Obtain confidence interval for effect of drug, look at residaul plots and identify any anomolous individuals, try out merTools package for additional plots and evaluations of uncertainty.       
 Solution for Review Question 4
5. Revisit cross-over design, class example, Lecture item 1. The class example used repeated measures analysis of variance for estimation the effect of the drug in the dialysis example, (I messed up the medical context in class). Repeat that analysis using lmer and show identical results to class example analysis. Also examine the effectiveness, increase in precision, resulting from each subject functioning as their own comparison, rather than having two separate (randomly assigned) treatment and control groups.       
 Solution for Review Question 5

WEEK 5 Exercises
1. We use a subset of the Baumann data from the car package, which I was nice enought to put in longform at http://rogosateaching.com/stat222/readlongdat .
These data are from a study of reading from Purdue. We use the data to compare two methods: Basal, traditional method of teaching; DRTA, an innovative method; coded 1 and 2 respectively in the data. Random assignment placed twenty-two students in each group; reading test measures were obtained pre and post instruction.
The Directed Reading Thinking Activity (DRTA) is a strategy that guides students in asking questions about a text, making predictions, and then reading to confirm or refute their predictions. The DRTA process encourages students to be active and thoughtful readers, enhancing their comprehension.
Use descriptive and inferential statistical methods to assess the relative efficacy DRTA method.
2.Treatment of Lead Exposed Children (TLC) Trial. Data (wide form) and description: data here
Start out by just using the subset of the longitudinal data Lead Level Week 0 and Week 6. Carry out the repeated measures anova for the relative effectiveness of chelation treatment with succimer or placebo (A,P). Show the three equivalences in the Brogan-Kutner paper between the repeated measures anova results and simple t-tests for these data. Next compare with a lmer fit following the B-K class example (posted). Finally use all 4 longitudinal measures (weeks 0,1,4,6) for a Active vs Placebo comparison using lmer. Compare with the results that use only 2 observations.
3. Crossover Design. The dataset consists of safety data from a crossover trial on the disease cerebrovascular deficiency. The response variable is not a trial endpoint but rather a potential side effect. In this two-period crossover trial, comparing the effects of active drug to placebo, 67 patients were randomly allocated to the two treatment sequences, with 34 patients receiving placebo followed by active treatment, and 33 patients receiving active treatment followed by placebo. The response variable is binary, indicating whether an electrocardiogram (ECG) was abnormal (Y=1) or normal (Y=0). Each patient has a bivariate binary response vector.
Data set is available at http://www.hsph.harvard.edu/fitzmaur/ala/ecg.txt (needs to be cut-and-paste into editor). Carry out the basic analysis of variance for this crossover design following week 5 Lecture topic 2. You may want to use glm to take into account the binary outcome. Does the treatment increase the probability of abnormal ECG? Give a point estimate and significance test for the treatment effect.
4. Data on Amenorrhea from Clinical Trial of Contracepting Women. Source: Table 1 (page 168) of Machin et al. (1988). Reference: Machin D, Farley T, Busca B, Campbell M and d'Arcangues C. (1988). Assessing changes in vaginal bleeding patterns in contracepting women. Contraception, 38, 165-179.
Data in long form  and   a wide-form version
Description: The data are from a longitudinal clinical trial of contracepting women.In this trial women received an injection of either 100 mg or 150 mg of depot-medroxyprogesterone acetate (DMPA) on the day of randomization and three additional injections at 90-day intervals. There was a final follow-up visit 90 days after the fourth injection, i.e., one year after the first injection.
Throughout the study each woman completed a menstrual diary that recorded any vaginal bleeding pattern disturbances. The diary data were used to determine whether a women experienced amenorrhea, the absence of menstrual bleeding for a specified number of days. A total of 1151 women completed the menstrual diaries and the diary data were used to generate a binary sequence for each woman according to whether or not she had experienced amenorrhea in the four successive three month intervals.
In clinical trials of modern hormonal contraceptives, pregnancy is exceedingly rare (and would be regarded as a failure of the contraceptive method), and is not the main outcome of interest in this study. Instead, the outcome of interest is a binary response indicating whether a woman experienced amenorrhea in the four successive three month intervals. A feature of this clinical trial is that there was substantial dropout. More than one third of the women dropped out before the completion of the trial. In the linked data, missing data are designated by "."  [note: in the week 6 terminology consider the dropouts to be missing at random, not necessarily a correct assumption.]
The purpose of this analysis is to assess the influence of dosage on the risk of amenorrhea and any individual differences in the risk of amenorrhea.
Show your model for these data and the results. Provide significance tests and/or interval estimates for the odds of amenorrhea as a function of dose. Display and interpret individual differences in response by showing the random effects within each experimental group.
5. Chick Data, finale. One more use of the chick data (week 3, problem 2; week 1 class lecture). Use the data for all 4 Diets to construct a nlmer model that allows asymptotes to differ across the four diets. Do the diets produce significantly different results? Which diet produces the heaviest 'mature' chick weight?
6. Missing Data. Wide-form longitudinal data
   Artificial data example from week 2 RQ3 and Week 4 Lecture item 4 (used in Myths examples to illustrate time-1,time-2 data analysis)    Two part artificial data example.   The top frame (the Xi's) is 40 subjects each with three equally spaced time observations (here in wide form). For these these perfectly measured "Xi" measurements each subject's observation fall on a straight-line.
   a. Use data set W6prob1a , for which about 15% of the observations have been made missing. Use these data (with lm) to recreate the multiple regression demonstration in Week 4 lecture, part 4: "Correlates and predictors of change: time-1,time-2 data" . Compare with the results for the full data on 40 subjects. What does lm do with missing data?
   b. Repeat part a with data set W6prob1b. Can you find any reason to doubt a "missing at random" assumption for this data set?
Note: if we don't get to it in Week 5, then in Week 10 (DW) we will demonstrate multiple imputation procedures (mice) for wide-form data, at least.



Week 6. Comparing Group Growth, continued. Observational Studies, Cohort Designs.
Lecture slides, week 6 (pdf)
Audio companion, week 6
parta   partb   partc

Lecture Topics
1. Observational Studies: Group Comparisons in Longitudinal Observational (non-experimental,  "quasi"-experimental) Designs
  A. Regression adjustments in quasi-experiments.
     Technical resource: Weisberg, H. I. Statistical adjustments and uncontrolled studies. Psychological Bulletin, 1979, 86, 1149-1164.    class handout
  B. Lord's paradox; pre-post group comparisons. Lord notes
     Publications: Lord, F. M. (1967). A paradox in the interpretation of group comparisons. Psychological Bulletin, 68, 304-305.
        Wainer, H. (1991). Adjusting for differential base rates: Lord's Paradox again. Psychological Bulletin, 109, 147-151.
  C. Economist's differences in differences (or diffs in diffs with matching) for observational studies.  class slide
    A very popular subject these days. Pretty good Wiki page     LSE slides     
      Austin Nichols slides. Causal inference with observational data A brief review of quasi-experimental methods July 2009
         Angrist Ch 5, MHE. Card and Krueger (1994) data, minumimum wage ex
        paper On the Use of Linear Fixed Effects Regression Models for Causal Inference (sec 3.2)
        R-package did
  D. Interrupted time-series.
    Intros: Interrupted Time Series Quasi-Experiments Gene V Glass Arizona State University.
      Assessing impact of stewardship: the why, when, and how of interrupted time series
     Time Series Analysis with R section 4.6 December 2012 Handbook of Statistics 30(1):661-712
    ITSpages for closing time example Class example: Closing time (glm kludge)
    Rogosa R-session
Original publication (ozone data):
Box, G. E. P. and G. C. Tiao. 1975. Intervention Analysis with Applications to Economic and Environmental Problems." Journal of the American Statistical Association. 70:70-79. SAS example for ozone data     
Applications:
Did fertility go up after the Oklahoma City bombing? An analysis of births in metropolitan counties in Oklahoma, 1990-1999. Demography, 2005.
Box-tiao time series models for impact assessment Evaluation Quarterly 1979
Interrupted time-series analysis and its application to behavioral data Donald P. Hartmann, John M. Gottman, Richard R. Jones, William Gardner, Alan E. Kazdin, and Russell S. Vaught J Appl Behav Anal. 1980 Winter; 13(4): 543-559.
Segmented regression analysis of interrupted time series studies in medication use research. By: Wagner, A. K.; Soumerai, S. B.; Zhang, F.; Ross-Degnan, D.. Journal of Clinical Pharmacy & Therapeutics, Aug2002, Vol. 27 Issue 4, p299-309,
R-packages:
tscountvignette       BayesSingleSub: Computation of Bayes factors for interrupted time-series designs
New resource,   Package Wats        Oklahoma City Fertility analyses
  E. Longitudinal Treatments (exposures)
Propensity Scores for Repeated Treatments  A Tutorial for the iptw Function in the TWANG Package
Using cobalt with Longitudinal Treatments
  F. Value-added analysis
J.R. Lockwood, Harold Doran, and Daniel F. McCaffrey. Using R for estimating longitudinal student achievement models. R News, 3(3):17-23, December 2003.
    Value-added does New York City. New York schools release 'value added' teacher rankings
   from the unions: THIS IS NO WAY TO RATE A TEACHER
    Value-Added Models to Evaluate Teachers: A Cry For Help H Wainer, Chance, 2011.
    American Statistical Association Statement on Using Value-Added Models for Educational Assessment

2. Cohort effects. Cohort-sequential, Accelerated longitudinal designs.
Robinson, K., Schmidt, T. and Teti, D. M. (2008) Issues in the Use of Longitudinal and Cross-Sectional Designs, in Handbook of Research Methods in Developmental Science (ed D. M. Teti), Blackwell Publishing Ltd, Oxford, UK

3. Econometric Approaches to Longitudinal Panel Data.
vignette for Panel Data Econometrics in R: The plm Package Yves Croissant Giovanni Millo (esp. section 7. "plm versus nlme/lme4" ).
  R-package plm   Class handout     Maybe more in Week 10.


WEEK 6 Review Questions
1. Interrupted Time Series example, redux
Create a version of the its 'closing time' example presented in class (example linked above) with the 50 months before intervention having mean fatality = 1 and after intervention mean fatality = 2.
Carry out the glm approximation to the time series analysis.       
 Solution for Review Question 1
2. Observational Studies: Lord's Paradox.   
    Part 1. Lord's paradox example
a. construct a two-group pre-post example with 20 observations in each group that mimics the description in Lord (1967):
statistician 1 (difference scores) obtains 0 group effect
statistician 2 (analysis of covariance) obtains large group effect for the group higher on the pre-existing differences in pretest
b. construct second example for which
statistician 1 (difference scores) obtains large group effect
statistician 2 (analysis of covariance) obtains 0 group effect
c. construct a third example (if possible) for which
statistician 1 (difference scores) obtains large postive group effect
statistician 2 (analysis of covariance) obtains large negative group effect
    Part 2. Group Comparisons by repeated measures analysis of variance or lmer
For the examples in part 1, (a and c), carry out the group comparison (i.e. is there differential change?) for the artificial data using a repeated measures anova (one within, one between factor) or lmer equivalent.
Demonstrate the equivalence from Brogan-Kutner paper that testing the groupXtime interaction term is equivalent to a t-test between groups on individual improvement (i.e. a statistician 1 analysis).
Solution to Question 2 RQ2 solution
3. Observational Studies: Regression Adjustments.  more in week 10
The display from lecture of the regression adjustments also has a numerical example (page 2 of pdf). Recreate the results shown for the Anderson et al Head Start example. Also for lecture materials, Regression Adjustments with Non-equivalent groups Week 6, show the Belson adjustment procedure (using control group slope) is equivalent to evaluating the vertical distance between the within-group regression fits at the mean of the treatment group. written out proof.

4. Time 1 Time 2 observational data, Differences in Differences analysis.
We reuse some time-1, time-2 observational data generated to illustrate Lord's paradox (RQ2) -- gender differences in weight gain. (The 'paradox' is solved by Holland, Wainer, Rubin using potential outcomes.) The set up for these artificial data is females gain, males no change
  corr .7 within gender, equal vars time1 time 2 within gender
means
                M               F
X (t1)         170            120
Y (t2)         170            130 
comparison of "gains" 170 - 170 - (130 - 120) = -10    negative effect males (females gain more).
ancova: 170 - 130 - .7*(170 - 120) = 5 positive male effect
So: does being male cause a student to gain weight or lose weight?   Illustrate forms of diffs-in-diffs analyses.
wide form for these data      long form for these data       
 Solution for Review Question 4

WEEK 6 Exercises
1. Smoking and Lung Function. Data and description available   here  (from ALA main page Datasets/ Vlagtwedde-Vlaardingen Study ). From these (rather meager) data, what do you make of the effect of being a (self-selected) smoker vs non-smoker throughout the 19 years of this study.
2. Longitudinal Observational Study: Wages for High School Drop-outs. Data obtained from the National Longitudinal Survey on Youth can be used to look-at the labor-market experiences of high school drop-outs. The subset of these data we will use is available at UCLA-- it's a csv file here's the appropriate read.table statement.
read.table("http://stats.idre.ucla.edu/stat/r/examples/alda/data/wages_pp.txt", header=T, sep=",")
'data.frame':   6402 obs. of  15 variables:
 $ id           : int  31 31 31 31 31 31 31 31 36 36 ...
 $ lnw          : num  1.49 1.43 1.47 1.75 1.93 ...
 $ exper        : num  0.015 0.715 1.734 2.773 3.927 ...
 $ ged          : int  1 1 1 1 1 1 1 1 1 1 ...
 $ postexp      : num  0.015 0.715 1.734 2.773 3.927 ...
 $ black        : int  0 0 0 0 0 0 0 0 0 0 ...
 $ hispanic     : int  1 1 1 1 1 1 1 1 0 0 ...
 $ hgc          : int  8 8 8 8 8 8 8 8 9 9 ...
 $ hgc.9        : int  -1 -1 -1 -1 -1 -1 -1 -1 0 0 ...
 $ uerate       : num  3.21 3.21 3.21 3.29 2.9 ...
 $ ue.7         : num  -3.79 -3.79 -3.79 -3.71 -4.11 ...
 $ ue.centert1  : num  0 0 0 0.08 -0.32 ...
 $ ue.mean      : num  3.21 3.21 3.21 3.21 3.21 ...
 $ ue.person.cen: num  0 0 0 0.08 -0.32 ...
 $ ue1          : num  3.21 3.21 3.21 3.21 3.21 ... 
Variables we will use are id, log-wage (hourly) lnw for each observation, exper time in labor force to the nearest day (in years), black (isblack = 1), hcg (highest grade completed; note hgc.9 is hgc - 9)
a. How many individuals in this data set? Give a five-number summary of the number of observations per person. How many of the individuals in these data have black = 1?
b. SFYS descriptive analyses. We are interested in wages (measured by lnw) as a function of experience (lnw ~ exper). Show a five-number summary of the gradient (slope; i.e. change in log-wage for unit change in exper) and level (here fit for exper = 0, initial status) for the set of individuals. Then stratify on black = 1 vs black = 0 (combining the white and hispanic high school drop-outs). Also show side-by-side boxplots for gradient and initial level stratifying on black. What do these displays indicate. Also show a plot of the lnw ~ exper fits seperately for black = 1 and black = 0. What do these analyses and displays indicate?
c. Use a formal mixed-effects model analysis to obtain random and fixed effects for the lnw~ exper individual level model. Obtain a point estimate and confidence interval for the variance of gradients. Does the bootstrap CI differ from the profile CI?
d. Are there differences in the lnw ~ exper relation for students black = 1 vs black = 0? Show by estimates and confidence intervals from mixed-effects models.
e. Does inclusion of hgc information confirm or alter your indications in part d?
3. Observational study: Gender differences in Vocabulary learning data-- see Week 3 problem 2-- from test results on file in the Records Office of the Laboratory School of the University of Chicago. Source D R Bock, MSMBR. The data consist of scores, obtained from a cohort of pupils at the eigth through eleventh gade level on alternative forms of the vocabulary section of the Cooperative Reading Tests." There are 64 students in all, 36 male, 28 female (ordered) each with four equally spaced observations (test scores). Wide form of these data are in BOCKwide.dat and I kindly also made a long-form version BOCKlong.dat .
For this problem consider gender differences in Vocabulary growth. Obtain the means (over persons) and plot the group growth curves, separately by gender. Does there appear to be curvature (i.e. deceleration in vocabulary skill growth) for both males and females? Construct an lmer model with the individual growth curve a quadratic function of grade (year), most convenient to use uncorrelated predictors grade - mean(grade) and (grade - mean(grade))^2. In the level II model allow each of the three parameters of the individual quadratic curves to differ by gender. Fit the lmer model and interpret the fixed and random effects you obtain. Compare the results with a lmer model in which the individual trajectories are straight-line. Use the anova model comparison functionality in R (e.g. anova(modLin, modQuad) to test whether the quadratic function for individual growth produces a better model fit.
4. Observational Studies: Regression Adjustments.
The class handout on regression adjustments shown in class and linked in RQ3 above contained summary statistics for the Head Start data considered in Anderson et al (1980) Statistical Methods for Comparative Studies. I constructed a corresponding data set located at W6prob3dat
Try out the various regression adjustments described on the handout for these pretest-posttest data. (Handout shows some approximate estimates). Also show the result for the basic diffs-in-diffs estimator from Week 6.


Week 7. Analysis of Durations: Introduction to Survival Analysis (aka event history) Methods

Lecture slides, week 7 (pdf)
Audio companion, week 7
parta   partb   partc
Longitudinal in the news
Hazard functions, Life tables.
Lifespan is continuing to increase regardless of socioeconomic factors    Publication: Advancing front of old-age human survival Wenyun Zuo, Sha Jiang, Zhen Guo, Marcus W. Feldman, and Shripad Tuljapurkar PNAS October 30, 2018 115 (44) 11209-11214;

        Useful introductions to Survival Analysis (mostly with R)
John Fox tutorial: Cox Proportional-Hazards Regression for Survival Data
Survival analysis text by Rupert G. Miller (Ch 2,3,4,6). Available as Stanford Tech Report
Cox Regression exposition from NCSU.   handout source Ch.6 st745
A concise, thorough Q&A overview from Columbia:   Time-To-Event Data Analysis
CHAPTER 11 (9) Survival Analysis: Glioma Treatment and Breast Cancer Survival A handbook of statistical analyses using R (second edition). Brian Everitt, Torsten Hothorn CRC Press, Complete version (through Stanford access)   
An Introduction to Survival Analysis  Mark Stevenson EpiCentre, IVABS, Massey University.   Author R-package   epiR  
From Lausanne a very nice survival analysis intro   plus analyses of Rossi recidivism Data
An R overview from openintro.org
Quick overview Survival analysis in clinical trials: Basics and must know areas   Perspect Clin Res. 2011 Oct-Dec; 2(4): 145-148. Ritesh Singh and Keshab Mukhopadhyay
CHAPTER 11 Survival Analysis: Retention of Heroin Addicts in Methadone Maintenance Treatment. Handbook of Statistical Analyses Using Stata, Second Edition. (Stanford access) Sophia Rabe-Hesketh Chapman and Hall/CRC 2000.
Event History Analysis with R. (Stanford access) Goran Brostrom CRC Press 2012. R-package   eha
Slides on renewal processes and hazard functions
Set of Slides    An introduction to survival analysis, Geert Verbeke

Main R-package: survival; Terry Therneau, Stanford Stat Ph.D
CRAN Task View: Survival Analysis . Survival analysis, also called event history analysis in social science, or reliability analysis in engineering, deals with time until occurrence of an event of interest. However, this failure time may not be observed within the relevant time period, producing so-called censored observations. This task view aims at presenting the useful R packages for the analysis of time to event data.
KM bootstrap in Hmisc package, bootkm. Exact tests, coin package: logrank_test (formerly surv_test).

      Class handouts (scanned) week 7
Class Data examples:
1. Miller leukemia data (Kaplan-Meier); pdf p.42 in online version     class example in R, data in package survival
     extensions of leukemia example (week 7)   2017 update of exact tests
    Cox fits, zph plot
   Using eha package for aml
          Legacy versions SAS    Minitab
2. Heroin (addict) data. Source: D.J. Hand, (et al.) Handbook of Small Data Sets. Properly formatted version   Analyses in Stevenson and Stata expositions above.   Lou Reed, heroin
  ascii version Rogosa R-session class handout
       Additional analyses for heroin: Bootstrapping, Math 159 Pomona    
Publication Source: Caplehorn, J., Bell, J. 1991. Methadone dosage and the retention of patients inmaintenance treatment. The Medical Journal of Australia,154,195-199.
Additional survival data.
3. Recidivism data from John Fox tutorial.   Source: Rossi PH, Berk RA, Lenihan KJ (1980). Money, Work, and Crime: Some Experimental Results. Academic Press, New York.   Another data location (UCLA)
4. Kalbfleisch and Prentice (1980) rat survival Data and description plus SAS analysis (Cox regression). Also best subsets Cox regression example, myeloma
5. R Textbook Examples. Applied Survival Analysis Chapter 3: Regression Models for Survival Data

WEEK 7 Review Questions
1. Part a. In file teacha.dat   are 75 "survival times" (variable name 'career') indicating actual length of teachers careers (in years) in a rather rough school district. What is the median survival time? what proportion of teachers are still in the district after 2 years? 4 years? 6 years? Plot a survival curve from this complete set of times.
Part b. In file teachb.dat   are the more realistic data: censored versions of the 75 "survival times" in part a. Column 1 has the times (career) and Column 2 has the censoring indicator (Note these data have status = 1 if censored). Compute naive answers (ignoring censoring) to the questions in part a: what is the median survival time? what proportion of teachers are still in the district after 2 years? 4 years? 6 years?
Use the Kaplan-Meier product-limit estimate to answer the questions in part a for these censored data: what is the median survival time? what proportion of teachers are still in the district after 2 years? 4 years? 6 years? Plot a survival curve with 95% confidence intervals. Obtain bootstrap (percentile) confidence intervals for the median survival time, and for the lower quartile (25th) of the survival time distribution.       
 Solution for Review Question 1
2. Days to vaginal Cancer Mortality in Rats. The link for data example 4 above has these data and description and an assortment of (gratuitous) SAS analyses. From that file make yourself an R data set. Plot the Kaplan-Meier survival curves for the two groups (with the point-wise condidence intervals for each curve). Carry out the (asymptotic) log-rank test of identical survival curves. Compare those results with the exact (permutation) test. What are the median survival times in the two groups? Obtain a bootstrap estimate of the confidence interval for the difference of median survival times in the two groups (95% is a good default or 90%). How does this confidence interval compare with the tests for differences between the survival done above? One more thing...Redo the group comparsion of survival using Cox regression with predictor (covariate) Group membership (pretreatment regimes). Do the results agree with the previous analyses. Obtain a confidence interval for the hazard ratio (ratio of the hazard functions) between the two groups.       
 Solution for Review Question 2
3. Replicate (some of) the analyses in the John Fox survival analysis tutorial for the Recidivism data (sec 3.2), links above. The experimental variable is fin indicating financial aid (or not). Start with a Kaplan Meier 2-group comparison, with plot and significance test. Then fit the 'full' model (mod.allison) and assess the significance of the experimental manipulation (fin). Plot the survival function from the cox regression (Fox Fig 1). Carry out the comparison (Fig 2) of estimated survival functions for those receiving (fin = 1) and not receiving (fin = 0) financial aid, with other covariates are fixed at their average values. For the model diagnostics in Section 5: use the cox.zph function to assess the proportional hazards assumptions, and plot the scaled Schoenfeld residuals (Fox figure 3).       
 Solution for Review Question 3
4. (multiple groups, no censoring). Among the contributions of Carnegie-Mellon Univ is the Data and Story Library (DASL). The Cancer Survival dataset shown in lecture ( presentation slides here)  is located at DASL . Compare survival times for the 5 types of cancer (Organ in the dataset) with relevant plots and inference procedures.       
 Solution for Review Question 4

5. nicer graphics, survminer package
try ggsurvplot as substitute for plot survfit plot cox.zph

WEEK 7 Exercises
1. Fun with hazards.
Part a. Social Security Life Tables. Use the 2007 Actuarial Life Table, useful discussion on benefits. Plot the hazard functions for males and females. Do these hazard functions appear to be exponential? Also plot the corresponding survival curves. Can you verify (approximately numerically) the relation between surival curve and integrated hazard from the week 7 handout-- S(t) = exp(-H(t)) ?
Part b. Refer to the hazard function shown in class for Alcohol and Incidence of Total Stroke (  Publication: Alcohol Consumption and Risk of Stroke in Women, Stroke, March 2012.  Nurses' Health Study). (figure underneath Table 2).
What is the increase in hazard between 2 drinks/day and 3 drinks/day?
2. Melanoma data. In package ISwR data melanom {ISwR} Survival after malignant melanoma Description: The melanom data frame has 205 rows and 7 columns. It contains data relating to the survival of patients after an operation for malignant melanoma, collected at Odense University Hospital by K.T. Drzewiecki.
 > str(melanom)
'data.frame':   205 obs. of  6 variables:
 $ no    : int  789 13 97 16 21 469 685 7 932 944 ...
 $ status: int  3 3 2 3 1 1 1 1 3 1 ...
 $ days  : int  10 30 35 99 185 204 210 232 232 279 ...
 $ ulc   : int  1 2 2 2 1 1 1 1 1 1 ...
 $ thick : int  676 65 134 290 1208 484 516 1288 322 741 ...
 $ sex   : int  2 2 2 1 2 2 2 2 1 1 ... , 
We are interested in
days: time on study after operation for malignant melanoma
status: the patient's status at the end of study
Documentation shows the possible values of status are: 1: dead from malignant melanoma 2: alive at end of study 3: dead from other causes. Consider 'dead from other causes' as censored (along with alive). Thus, status vector should be status == 1 and the survival object is Surv(days, status == 1) (to do some of the problem for you).
a. How many survival times are censored? Obtain an estimate of the survival curve at each event time (along with CI) using the Kaplan-Meier estimate and plot the survival curve and confidence interval.
b. Does survival differ in men and women? Compare asymptotic (log-rank) and exact tests for gender differences? Compare the exact test with a bootstrap approximation. Plot the male and female survival curves.
c. Use Cox regression to carry out the gender comparison of the survival curves in part b. Obtain a confidence interval for the effect of gender on the hazard.
3. Survival with HIV. The data set obtained from   read.table("http://stats.idre.ucla.edu/stat/r/examples/asa/hmohiv.csv", sep=",", header = TRUE)   contains 100 members of a Health Maintenance Organization (HMO). All of the members are HIV positive. The HMO wants to examine their survival times. Subjects were enrolled in the study from 1/1/89 to 12/31/91. Study ended 12/31/95. After HIV diagnosis members were followed until death due to AIDS, until end of the study, or until lost to followup. In the dataset the survival time is the variable time is in months elapsed between entrydata and enddata. Each member's age in years at the start of followup is recorded in a variable called age. The ages range from 20 to 54 with a median of 35. The censor indicator has value 1 for death due to AIDS and value 0 for lost to followup or alive. In the data set intravenous drug use is recorded as a binary variable called drug, where 1 is a history of IV drug use and 0 otherwise.
For example subject 1 died from AIDS 5 months after being seen in the HMO clinic while subject 2 was not known to have died from AIDS at the conclusion of the study and had been followed for 6 months.
a. Give a point and interval estimate of the survival times of these AIDS patients. Compare survival times for IV drug users and non-IV drug users.
b. Use Cox regression to investigate age and drug use as predictors of survival time. Check the proportional hazards assumptions and interpret the coefficients from the fit.
c. Repeat part b using the age information recoded into four categories    age, "20:29='A'; 30:34='B'; 35:39='C';40:54='D'".
4. Dataframe gastric in the coxphw package
is comprised of survival times of patients with locally advanced, nonresectable gastric carcinoma. The patients were either treated with chemotherapy plus radiation or chemotherapy alone.
Usage: data(gastric) A data frame with 90 observations on 4 variables.
Source: Stablein DM, Carter J, Novak JW. (1981) Analysis of Survival Data with Nonproportional Hazard Functions. Controlled Clinical Trials 2:149-159.
References Gastrointestinal Tumor Study Group. (1982) A Comparison of Combination Chemotherapy and Combined Modality Therapy for Locally Advanced Gastric Carcinoma. Cancer 49:1771-7.
What proportion of the observations are right-censored in this experiment? For this experiment, provide description including graphics and then inferential analyses on the relative efficacy (in terms of survival) for chemotherapy alone versus chemotherapy plus radiation.

Week 8. More survival analysis and analysis of durations

Lecture slides, week 8 (pdf)
Audio companion, week 8
parta   partb   partc
Class Topics
1. Continue Cox Regression examples. class handout from Ch.6 st745
Recidivism data from John Fox tutorial.   .   Another data location (UCLA)
  Fox recidivism example (Rossi data)    Rogosa R-session(ascii)
  (c.f. week 7, review question 3 and week 8, review question 1a for full output).
See John Fox tutorial (week 7); also in his Sociology 761 notes.
  From Lausanne a very nice survival analysis intro plus analyses of Rossi Data
  Source: Rossi PH, Berk RA, Lenihan KJ (1980). Money, Work, and Crime: Some Experimental Results. Academic Press, New York.

2. Interval Censoring; breast cancer data. interval package. Class analysis handout.
Note: to run the interval package you have to fetch package Icens from Bioconductor. 2018 install verification deprecated. See lecture 8 slides.
R Package for Analyzing Interval-Censored Survival Data. Exact and Asymptotic Weighted Logrank Tests for Interval Censored Data: The interval R Package
   Interval Censoring: Tutorial on methods for interval-censored data and their implementation in R Statistical Modelling 2009; 9(4): 259-297.
   Interval-Censored Time-to-Event Data Methods and Applications Chapman and Hall/CRC 2012 (esp Chap 14--glrt).
Finkelstein-Wolfe dataset is bcos in package interval or cosmesis in package MLEcens
package icenReg provides classic survival regression models for interval-censored data.
   Also intcox {intcox}Cox proportional hazards model for interval censored data
another new package: FHtest   FHtest: An R Package for the Comparison of Survival Curves with Censored Data
   Extension: Bayesian Regression Models for Interval-censored Data in R (R Journal Dec 2017)

3. Discrete-time survival analysis. Teacher and first-sex examples (ascii) from Willet and Singer (Chap 10,11). Note data locations at UCLA repository have moved from www.ats.ucla.edu to stats.idre.ucla.edu
      [links in text resource number 2, text data examples]      Willet-Singer presentation version Slides for Ch.11

4. Background: renewal processes and hazard functions. Slides on renewal processes and hazard functions
       Multistate models: Package 'etm' Empirical Transition Matrix     etm presentation
       tutorial Computing Cumulative Incidence Functions with the etmCIF Function, with a view Towards Pregnancy Applications .

5. Design: sample size and power calculations for survival analysis. See Power Analysis section of Survival Analysis Task View.
    Function lrSS in package survMisc [deprecated 2018]. Package powerSurvEpi powerCT family of functions.
6. Parametric Survival Models,  survreg in survival package. Application to Rossi data in Lausanne link.
   Also package flexsurv    vignette for flexsurv


WEEK 8 Review Questions
1. Fox Recidivism data and Additional Diagnostics for Cox Regression.
Use dx function in package survMisc for another look at the Fox Recidivism data from Week 7 RQ 3 and Week 8 Class Topics.
a. From the lecture materials for the Cox Regression try out the reduced model that omits the wexp variable (which showed some PH problems). Compare models and results.
b. try out the diagnostic plots from dx on this reduced model.       
 Solution for Review Question 1
2. More on Week 7 class example, heroin data.
a. In the class example, after seeing a problem with proportional hazards assumption for the clinic variable, we used strata(clinic) which allows different baseline hazards in each clinic with predictors prison and dose. Look at this further by fitting the coxhernSt model seperately within each clinic (i.e. dropping the strata(clinic) term). Are the effects of clinic and dose similar within each clinic?
b. In class question asked whether some model-modification (e.g. interaction terms) in the coxhern model might mitigate the proportional hazards violation in the coxhern model. Try out some augmented models using interactions between the predictors in coxhern.
c. The eha package which goes along with the Event History Analysis book linked in week 7 has an alternative to coxph, coxreg which allows bootstrap replications for the Cox regression. Try out coxreg with B=1000 (bootstrap reps) for the coxhern model. Compare standard errors for coefficients from the coxreg or coxph fits with bootstrap standard errors.       
 Solution for Review Question 2
3. firstsex example from Willet-Singer. Discrete-time analyses. Part 3 week 8 lecture materials.   Note data locations at UCLA repository have moved from www.ats.ucla.edu to stats.idre.ucla.edu
Data Example: Grade at First Intercourse. Research Question: Whether, and when, adolescent males experience heterosexual intercourse for the first time? Citation: Capaldi, et al. (1996). Sample: 180 high-school boys. Research Design: Event of interest is the first experience of heterosexual intercourse. Boys tracked over time, from 7th thru 12th grade. 54 (30% of sample) were virgins (censored) at end of data collection.
The Willet-Singer displays show lifetables and logistic regression estimates for survival analyses.
a. investigate time-to-event as a function of parental transitions (pt = 1, 1 or more transitions) using Kaplan-Meier and cox regression methods shown in class (not really correct for these interval censored data). Compare with the logistic regression results in the Willet-Singer materials.
b. clearly the firstsex data are really interval censored, rather than inately discrete-time. Sex was had sometime during the reported grade. Indicate how you would set up these data for the proper interval censored analysis. See class examples, week 8.       
 Solution for Review Question 3
4. Fox Recidivism data, once more.
The Week 8 class example and RQ1 did crude stratification on age (to adress violations of proportional hazards). As an exercise, carry out the Cox Regression, stratifying on age in thirds (more carefully than I did in the class examples) and also stratifying on age split in fifths. Check the proportional hazards assumptions and compare results. It would be ugly/concerning if small changes in the age stratification kludge for proportional hazards determined statistical significance of the intervention variable (fin).       
 Solution for Review Question 4
5. Parametric survival analysis. In lecture 8 brief description was given to an parametric alternative to the Cox Regression model (which leaves the baseline hazard unspecified). The survreg function in the survival package allows specifying and fitting a parametric model for the baseline hazard. See Fox tutorial p.2 or Lausanne intro pdf pages 21 and 32: Rossi data pdf pages 33-35. In Lausanne the data are seen to be consistent with a Weibull model (see plots of cumulative hazards). Replicate that survival model analogous to the Cox Regression in class for the assumption of a Weibull hazard.       
 Solution for Review Question 5
6. Power calculations. [note: the basic lrSS function in survMisc is no more in 2018]. Try out the power claculations in package powerSurvEpi function powerCT: use powerCT.default example, pages 12-13 of manual. See how number of events drives power by modifying the manual example to have probability .5 then .7 of event in Treatment and Control groups.       
 Solution for Review Question 6

WEEK 8 Exercises
1. Data frame pbc in the survival package: Mayo Clinic Primary Biliary Cirrhosis Data, a randomized placebo controlled trial of the drug Dpenicillamine. Refer to the documentation. As the helpfile tells you: "The first 312 cases in the data set participated in the randomized trial and contain largely complete data. The additional 112 cases [which have value NA for the trt variable] did not participate in the clinical trial...". So pick out the 312 cases that are the D-penicillmain and placebo groups.
For these data 'status': 0=censored, 1=liver transplant, 2=death; so status = 2 represents observed values of time; otherwise censored.
a. Use Kaplan-Meier methods to carry out a simple two-group comparison of the effectiveness of the drug, along with any useful plots.
b. Extend the two-group comparison with a Cox regression using additional predictors (chosen as you wish) age edema log(bili) log(protime) log(albumin). Identify a useful model and Interpret results.  Make sure you check the proportional hazards assumptions for your model.
2. melanom data from week 7, exercise 2. Define the censoring as was done in that problem. I found it useful to make a 0,1 variable isMale from the integer sex designation and make a 0,1 variable isUlcer from the ulceration variable (careful there).
a. Repeat the gender comparison in parts b or c in Ex 2, week 7, stratifying on ulceration of the tumor (or not). Compare with the result in Ex 2 week 7 and interpret.
b. Carry out a Cox regression using predictors log(thick) and the gender indicator, stratifying on ulceration. Interpret the results. Check the viability of the proportional hazards assumption for this cox model.
3. Interval Censored Data. Consider the breast cancer data shown in Week 8 lecture; interval-censored breast cosmesis data set of Finkelstein and Wolfe (1985). The data are from a study of two groups of breast cancer patients, those treated with radiation therapy with chemotherapy (treatment = "RadChem") and those treated with radiation therapy alone (treatment = "Rad"). The response is time (in months) until the appearance of breast retraction, and the data are interval-censored between the last clinic visit before the event was observed (left) and the first visit when the event was observed (right) (or Inf if the event was not observed). One location of these data is:
R> library("interval")
R> data("bcos", package = "interval")
Class examples show parametric and non-parametric survival analyses for these interval censored data. Before these methods were available, various Kludges (imputations) existed. One is to take the midpoint of the interval for any observed event in [left, right] or if right is NA (censored) treat as left+ and carry out a survival analysis for right censored data. Repeat the breast cancer example Cox regression using this strategy and compare with the results from week 8 using the interval censoring.
4. More firstsex data. Follow on part b of Review Question 3 and carry out an (appropriate) interval censoring analysis for these data using the procedures from Class Topic 5 of week 8. Again, investigate time-to-event as a function of parental transitions (pt = 1, 1 or more transitions).
5. The uis data are from a set of trials for residential treatment for drug abuse. Obtain the data from the UCLA archive uis = read.table("http://stats.idre.ucla.edu/stat/r/examples/asa/uis.csv", sep=",", header = TRUE). There are 628 subjects with a small amount of missing data. The time variable in the dataset is the outcome of interest: time (in days) to return to drug use (self report) measured from admission. The censor variable is in a convenient form with value 1 indicating return to drug use and 0 otherwise (censored observation). The dichotomous race variable is coded 0 for White and 1 for non-White (other). The variable age is age at enrollment (years). The ivhx variable is an indicator of IV drug use history and is coded in the datset (1=never, 2=previous, 3=recent). For use here make a dichotomous variable drug that combines previous and recent versus never as in uis$drug = as.numeric(uis$ivhx==1) (be careful about the direction here as the code makes drugNever = 1). [note function recode in the car package makes this operation easy].
Use these variables in the uis dataset to investigate influences on time to relapse. Formulate a useful Cox regression model, check the proportional hazards assumption, and provide interpretations of the parameter estimates.
6.   Home town data. I live on Channing Ave so these data are irresistible.
Channing House Data. The channing data frame has 462 rows and 5 columns.
Channing House is a retirement centre in Palo Alto, California. These data were collected between the opening of the house in 1964 until July 1, 1975. In that time 97 men and 365 women passed through the centre. For each of these, their age on entry and also on leaving or death was recorded. A large number of the observations were censored mainly due to the resident being alive on July 1, 1975 when the data was collected. Over the time of the study 130 women and 46 men died at Channing House. Differences between the survival of the sexes, taking age into account, was one of the primary concerns of this study.
      sex   A factor for the sex of each resident ("Male" or "Female").
      entry The residents age (in months) on entry to the centre
      exit  The age (in months) of the resident on death, leaving the centre or 
                 July 1, 1975   whichever event occurred first.
      time Length of time (in months) that the resident spent at Channing
                  House.(time=exit-entry)
      cens Indicator of right censoring. 1 indicates that the resident died at Channing
                   House, 0 indicates that they left the house prior to July 1, 1975 or 
                   that they were still alive and living in the centre at that date.
> head(channing)
   sex entry exit time cens
1 Male   782  909  127    1
2 Male  1020 1128  108    1
3 Male   856  969  113    1
4 Male   915  957   42    1
5 Male   863  983  120    1
6 Male   906 1012  106    1
> 780/12 [1] 65 # in years just for reference

#some quick descriptives
> attach(channing)
> table(cens, sex)
    sex
cens Female Male
   0    235   51
   1    130   46
> tapply(time, sex, quantile)
$Female                            $Male                       
  0%  25%  50%  75% 100%             0%  25%  50%  75% 100%  
   0   36   87  137  137              0   33   72  120  137  
> tapply(entry, sex, quantile)
$Female                            $Male                   
  0%  25%  50%  75% 100%             0%  25%  50%  75% 100%
 733  851  899  948 1140            751  863  919  967 1073

#Kaplan-Meier fits and tests
> kmfit = survfit(Surv(time, cens) ~ sex, data = channing)
> kmfit
Call: survfit(formula = Surv(time, cens) ~ sex, data = channing)
           records n.max n.start events median 0.95LCL 0.95UCL
sex=Female     365   365     365    130     NA     124      NA
sex=Male        97    97      97     46    108      82      NA
> plot(kmfit, conf.int=TRUE)                                         ## SEE PLOT  

> survdiff(Surv(time, cens) ~ sex, data = channing)
Call: survdiff(formula = Surv(time, cens) ~ sex, data = channing)
             N Observed Expected (O-E)^2/E (O-E)^2/V
sex=Female 365      130    143.1      1.19      6.41
sex=Male    97       46     32.9      5.18      6.41
 Chisq= 6.4  on 1 degrees of freedom, p= 0.0113 
> surv_test(Surv(time, cens) ~ sex, data = channing, distribution=approximate(B = 4000))
        Approximative Logrank Test
data:  Surv(time, cens) by sex (Female, Male)
Z = -2.4251, p-value = 0.015
alternative hypothesis: two.sided

#Cox regressions
> coxfit1 = coxph(Surv(time, cens) ~ sex, data = channing)
> coxfit1
Call: coxph(formula = Surv(time, cens) ~ sex, data = channing)
         coef exp(coef) se(coef)    z     p
sexMale 0.432      1.54    0.172 2.52 0.012
Likelihood ratio test=5.89  on 1 df, p=0.0153  n= 462, number of events= 176 
> summary(coxfit1)
Call: coxph(formula = Surv(time, cens) ~ sex, data = channing)
  n= 462, number of events= 176 
          coef exp(coef) se(coef)     z Pr(>|z|)  
sexMale 0.4321    1.5406   0.1718 2.516   0.0119 *
---
        exp(coef) exp(-coef) lower .95 upper .95
sexMale     1.541     0.6491       1.1     2.157

Concordance= 0.541  (se = 0.016 )
Rsquare= 0.013   (max possible= 0.985 )
Likelihood ratio test= 5.89  on 1 df,   p=0.01526
Wald test            = 6.33  on 1 df,   p=0.01187
Score (logrank) test = 6.43  on 1 df,   p=0.01123

> coxfit2 = coxph(Surv(time, cens) ~ sex + entry, data = channing)
> coxfit2 Call: coxph(formula = Surv(time, cens) ~ sex + entry, data = channing)
           coef exp(coef) se(coef)    z       p
sexMale 0.38004      1.46  0.17191 2.21 2.7e-02
entry   0.00724      1.01  0.00105 6.90 5.4e-12
Likelihood ratio test=50.9  on 2 df, p=8.67e-12  n= 462, number of events= 176 

> summary(coxfit2)
Call: coxph(formula = Surv(time, cens) ~ sex + entry, data = channing)
  n= 462, number of events= 176 
            coef exp(coef) se(coef)     z Pr(>|z|)    
sexMale 0.380045  1.462350 0.171914 2.211   0.0271 *  
entry   0.007239  1.007266 0.001050 6.896 5.36e-12 ***
---
        exp(coef) exp(-coef) lower .95 upper .95
sexMale     1.462     0.6838     1.044     2.048
entry       1.007     0.9928     1.005     1.009

Concordance= 0.65  (se = 0.023 )
Rsquare= 0.104   (max possible= 0.985 )
Likelihood ratio test= 50.94  on 2 df,   p=8.671e-12
Wald test            = 53.05  on 2 df,   p=3.02e-12
Score (logrank) test = 54.2  on 2 df,   p=1.703e-12

> anova(coxfit1, coxfit2)
Analysis of Deviance Table
 Cox model: response is  Surv(time, cens)
 Model 1: ~ sex
 Model 2: ~ sex + entry
   loglik  Chisq Df P(>|Chi|)    
1 -972.57                        
2 -950.04 45.056  1 1.915e-11 ***

> cox.zph(coxfit2)
            rho chisq     p
sexMale -0.0304 0.162 0.687
entry    0.0481 0.400 0.527
GLOBAL       NA 0.553 0.759
> plot(cox.zph(coxfit2))

> k = 1:462   > channing$ID = k
> head(channing)
   sex entry exit time cens ID
1 Male   782  909  127    1  1
2 Male  1020 1128  108    1  2
3 Male   856  969  113    1  3
4 Male   915  957   42    1  4
5 Male   863  983  120    1  5
6 Male   906 1012  106    1  6

> coxfit4 = coxme(Surv(time, cens) ~ sex + entry + (1|ID), data = channing)
> print(coxfit4)
Cox mixed-effects model fit by maximum likelihood
  Data: channing
  events, n = 176, 462
  Iterations= 100 503 
                    NULL Integrated    Fitted
Log-likelihood -975.5084  -950.0293 -947.0135

                  Chisq   df          p   AIC   BIC
Integrated loglik 50.96 3.00 4.9937e-11 44.96 35.45
 Penalized loglik 56.99 4.97 4.8849e-11 47.05 31.29

Model:  Surv(time, cens) ~ sex + entry + (1 | ID) 
Fixed coefficients
               coef exp(coef)    se(coef)    z       p
sexMale 0.382989266  1.466662 0.173129158 2.21 2.7e-02
entry   0.007286551  1.007313 0.001058856 6.88 5.9e-12

Random effects
 Group Variable  Std Dev    Variance  
 ID    Intercept 0.13172664 0.01735191

> fixed.effects(coxfit4)                         > exp(fixed.effects(coxfit4))   
    sexMale       entry                           sexMale    entry               
0.382989266 0.007286551                          1.466662 1.007313               
> stem(exp(ranef(coxfit4)[[1]]))
  The decimal point is 2 digit(s) to the left of the |
   96 | 34
   96 | 8
   97 | 1234
   97 | 6689
   98 | 01111222244
   98 | 5555566666666677777778888888999999999999
   99 | 00000000000011111111111111112222222222222222223333333333333333333444+4
   99 | 55555555555555555666666666666666666667777777777777777777777778888888+33
  100 | 00000000000000000000000000000000000011222223334444
  100 | 555666677777778888999999999
  101 | 00000000111111111111111222222222222222233333333333333344444444444444
  101 | 5555555555555555556666666666666667777777777777777777777777

> quantile(exp(ranef(coxfit4)[[1]]))
       0%       25%       50%       75%      100% 
0.9633244 0.9928323 0.9986529 1.0106724 1.0174679 
QUESTION 6
a. Compare median survival times for males and females obtained by ignoring censoring, with median survival times from Kaplan-Meier. Comment on similarities or differences. Why does KM show NA for female median survival time?
b. Are gender differences in survival statistically significant? Can differences in survival be attributed to gender or Channing House?
c. For the Cox regressions, cox2 adds entry age to the model in cox1. Is entry age a significant predictor in this model?
d. Does the proportional hazards assumption appear to be satisfactory for cox2?
e. What is is the point estimate of the proportional increase or decrease in hazard (give direction) for an increase of 1 year in entry age in the cox2 model?
f. Explain what I am doing in the formulation of the cox4 model.
g. Does cox4 appear to improve the fit compared to cox2? Give a rough test statistic.
h. For the cox4 fit, what is the magnitude of the individual effects--i.e. how much do individual effects impact the estimated hazards?


Week 9. Advanced topics for Time-to-event data.
   Time-dependent Covariates, Mixed Effects Cox models (Nested data, Recurrent Events, Frailty Models)
   Joint Modelling of Longitudinal and Time-to-Event Data

Lecture slides, week 9 (pdf)
Audio companion, week 9
parta   partb   partc

Longitudinal in the news
1. Cross-sectional observational lifespan research.   Regular Exercise May Keep Your Body 30 Years 'Younger'    NYTimes.   Publication: Cardiovascular and Skeletal Muscle Health with Lifelong Exercise  Journal of Applied Physiology, 2018.
2. That Daily Glass Of Wine May Lead To An Earlier Death    Publication: Daily Drinking Is Associated with Increased Mortality Alcoholism: Clinical and Experimental Research, October 2018.
3. Longitudinal Reciprocal Associations Between Anxiety, Depression, and Alcohol Use in Adolescent Girls   Alcoholism: Clinical and Experimental Research, November 2018.

Lecture Topics
1. Time-dependent covariates in survival analysis.
a. Using Time Dependent Covariates and Time Dependent Coefficients in the Cox Model Terry Therneau Cindy Crowson Mayo Clinic October 2018.      function tmerge in survival package
b. Recidivism data, weekly employment measure (Section 4, Fox tutorial).
      week 9 class handout (ascii), recidivism data         longer version shown in class (ascii)   John Fox script  from Soc761
        fold/unfold functions (wide-to-long): Package RcmdrPlugin.survival    web posting of function        older version for Rossi data Fox tutorial script

2. Mixed effects (Frailty) survival models and Recurrent Events.
a. Package coxme Terry Therneau Jan 2020: Cox proportional hazards models containing Gaussian random effects, also known as frailty models coxme manual. Maintainer Terry Therneau. Mixed Effects Cox Models Terry Therneau Mayo Clinic, Jan 2020
           coxme Class handout      plain text version
Additional materials.   frailtyHL: A Package for Fitting Frailty Models with H-likelihood by Il Do Ha, Maengseok Noh and Youngjo Lee;  R Journal December 2012
b. Recurrent events.   (background, week 8, lecture item 4)
Comparing Survivival curves newTestSurvRec package: Statistical tests to compare survival curves of two groups with recurrent events. [replaces TestSurvRec]
     Class example (ascii)  R-session, tumor recurrence in patients with bladder cancer. plots for bladder recurrence
additional R-packages, coxme frailtypack, parfm, survrec, gmrec,, bivrec package etm empirical transition matrix
Recurrent Events: Chapter 9 of Kalbfleisch and Prentice (2nd edition), "Modeling and Analysis of Recurrent Event Data"
Cook, R. J. and Lawless, J. F. (2007).  The Statistical Analysis of Recurrent Events. (Stanford access) Springer, New. York.

Additional examples: Frailty models (individual differences, random effects) and Recurrent events (observe multiple on/off transitions and timing). Asthma data example from Duchateau et al (2003). Evolution of Recurrent Asthma Event Rate over Time in Frailty Models  Journal of the Royal Statistical Society. Series C (Applied Statistics) 355-363.   see also Ch 3 in Frailty Models in Survival Analysis  Andreas Wienke Chapman and Hall/CRC 2010
Additional R-packages:   reda   reReg


3. Joint Modelling of Longitudinal and Time-to-Event Data
     Class handout. JM for aids example (ascii).
R-package JM, Dimitris Rizopoulos Maintainer: Shared parameter models for the joint modeling of longitudinal and time-to-event data. Expository paper JSS 2010, JM: An R Package for the Joint Modelling of Longitudinal and Time-to-Event Data.
Book (Stanford access) Joint Models for Longitudinal and Time-to-Event Data. With Applications in R. Dimitris Rizopoulos. Chapman and Hall/CRC 2012.   Book Table of Contents     Book website  (scripts)
joineR package    joineRML package    JoineR website    presentation

      Survival summary, a remarkable overview of advanced survival analysis topics. Multiple and Correlated Events Terry M. Therneau Mayo Clinic Spring 2009

WEEK 9 Review Questions
1. Fox Recidivism data, once more.
Use the fox.long dataset created and shown in Week 9 lecture and run the original coxph in the Fox tutorial and class example. Show that the same results are obtained in the original wide form and long form.       
 Solution for Review Question 1
2. Mixed effects Cox regression models, hierarchical (nested) survival data
a. Class example; eortc breast cancer data
Compare results ignoring center, stratifying on centers, and using coxme (in class example) for patients nested within centers.
b. Rat data (see R Journal December 2012).
The data set presented by Mantel et al. (1977) is based on a tumorigenesis study of 50 (q = 50) litters of female rats. For each litter, one rat was selected to receive the drug and the other two rats were placebo treated controls (ni = 3). Here each litter is treated as a cluster. The survival time (time) is the time to development of tumor, measured in weeks. Death before occurrence of tumor yields a right-censored observation; forty rats developed a tumor, leading to censoring of about 73%. The survival times for rats in a given litter may be correlated due to a random effect representing shared genetic or environmental effects.
Compare Cox regression models that investigate effect of the drug (rx)
(i) ignoring litters and (ii) mixed effects models incorporating the nested structure of these data (rats within litters)       
 Solution for Review Question 2
3. More on eortc class example, coxme analysis from Terry Therneau vignette.
Class questions motivated extending the anlaysis allowing treatment effects to vary over the 37 centers (random effects in the survival analysis). In the class example model efit2 included a random effect for centers but set the treatment effect constant (fixed effect) over centers. Therneau's model efit3 allows treatment effects to vary over centers by nesting treatment within centers with the (1 | center/trt) term. Class question asks why not instead use (1 + trt | center) analogous to a two sample t-test within each center, i.e. a survdiff within each center. A further question asked for more detail on random effect for the efit3 model (where the class handout left off.       
 Solution for Review Question 3

WEEK 9 Exercises
1. Dialysis Hemodialysis Data from Brazil.
Data on 6805 hemodialysis patients in all federally funded clinics (67 centers) in Rio de Janeiro State, Brazil.
data(Dialysis)   package RcmdrPlugin.survival
Format
A data frame with 6805 observations on the following 7 variables.
center a numeric code indicating in which of 67 centers the patient was treated.
age of the patient.
begin The month in which treatment began, with 1 representing January 1998.
end The month in which observation terminated, either because of death or censoring. The study ended in month 44 (August, 2000).
event 1, death, or 0, censoring.
time the difference between end and begin.
disease a factor with levels congen, (congenital); diabetes; hypert (hypertension); other; and renal.
Use Cox regression with age and disease as predictors to study survival time of patients. Follow the in-class analysis of the eortc data to compare results ignoring centers and using a mixed-effects models with random factor center.

2. Time-dependent covariate, Stanford Heart Transplant data.
Data are available in the survival package in various formats: see heart listing. In the Stanford Heart Transplant study, all patients start on medical treatment. When a heart becomes available, the selected patient 'converts' to the transplant treatment arm. Thus the intervention variable transplant is time-dependent. Use these survival data (including the additional covariates if desirable) in a format of your choosing to evaluate the effect of transplant on survival. Additional discussion of these data is contained in the Thernau overview of advanced survival analysis topics, Multiple and Correlated Events, linked in week 9 materials (pdf page 26 onward) and in the Therneau et al primer Using Time Dependent Covariates and Time Dependent Coefficients in the Cox Model.


Week 10, Dead Week: Special Topics for Longitudinal Data

Lecture slides, week 10 (pdf)
Audio companion, week 10
parta   

Lecture topics: (pick some)
1. Observational Studies, Survival analysis. Marriage and Cancer Survival example (from weeks 5 and 6, Stat266 2016)
2. Stability over time (Myth 8). Change and Sameness
3. Structural equation models for longitudinal data (don't do it; Myth 7) (handout in week 10 handout collection)
4. Reciprocal effects ( Myth 9) Rogosa, Encyclopedia of Social Science
5. Longitudinal Network Data


Resources
Applications of Structural Equation Models (LISREL, path analysis, Myth 7)
David Rogosa. Casual Models Do Not Support Scientific Conclusions: A Comment in Support of Freedman. Journal of Educational Statistics, Vol. 12, No. 2. (Summer, 1987), pp. 185-195. Jstor link       Theme Song Ballad of the casual modeler    http://www.stanford.edu/class/ed260/ballad.mp3
Rogosa, D. R. (March 1994). Longitudinal reasons to avoid structural equation models, UC Berkeley.
Rogosa, D. R. (1993). Individual unit models versus structural equations: Growth curve examples. In Statistical modeling and latent variables, K. Haagen, D. Bartholomew, and M. Diestler, Eds. Amsterdam: Elsevier North Holland, 259-281.
  original publication on the longitudinal path analysis:   Some Models for Analysing Longitudinal Data on Educational Attainment. Harvey Goldstein       Journal of the Royal Statistical Society. Series A (General), Vol. 142, No. 4. (1979), pp. 407-442.  Jstor link
Rogosa, D. R., & Willett, J. B. (1985). Satisfying a simplex structure is simpler than it should be.      Journal of Educational Statistics, 10, 99-107. Jstor link      Follow-up paper: Two Aspects of the Simplex Model: Goodness of Fit to Linear Growth Curve Structures and the Analysis of Mean Trends. Frantisek Mandys; Conor V. Dolan; Peter C. M. Molenaar. Journal of Educational and Behavioral Statistics, Vol. 19, No. 3. (Autumn, 1994), pp. 201-215.    Jstor link

Stability: Consistency, Change and Sameness (Myth 8)
J.H. Ware Tracking in S. Kotz, N.L. Johnson (Eds.), The Encyclopedia of Statistical Sciences (13th Edn.), Vol. 9 John Wiley, New York (1988)
Rogosa, D. R., Floden, R. E., & Willett, J. B. (1984). Assessing the stability of teacher behavior. Journal of Educational Psychology, 76, 1000-1027. APA link also available from John Willet's pub page
Rogosa, D. R., & Willett, J. B. (1983). Comparing two indices of tracking. Biometrics, 39, 795-6.      JStor link
Rogosa, D. R. Stability section of Individual unit models versus structural equations (link above)
Rogosa, D. R. Stability of school scores from educational assessments: Confusions about Consistency in Improvement David Rogosa, June 2003 ; Education Writers Association April 2004
A non-parametric index of tracking James A. Koziol Pages 181-191 Published online: 02 Aug 2010 Journal of Applied Statistics Volume 25, 1998 - Issue 2
Personality research. Stability versus change, dependability versus error: Issues in the assessment of personality over time David Watson Journal of Research in Personality 38 (2004) 319-350.
Some applications: A Stochastic Model for Analysis of Longitudinal AIDS Data J.M.G. Taylor, W.G. Cumberland, Sy J.P.; Journal of the American Statistical Association, Vol. 89, 1994
Body mass index from childhood to middle age: a 50-y follow-up V A Casey J T Dwyer K A Coleman I Valadian The American Journal of Clinical Nutrition, Volume 56, Issue 1, 1 July 1992, Pages 14-18
Tracking and its applicability to Physical Education and Sport
Tracking of objectively measured physical activity from childhood to adolescence: The European youth heart study. Scandinavian Journal of Medicine & Science in SportsVolume 18, Issue 2, 2007.
Factors Associated With Tracking of BMI: A Meta-Regression Analysis on BMI Tracking. Obesity (2011) 19 5, 1069-1076. doi:10.1038/oby.2010.250
Long-term tracking of cardiovascular risk factors among men and women in a large population-based health system The Vorarlberg Health Monitoring and Promotion Programme. European Heart Journal (2003) 24, 1004-1013.
Journal of Traumatic Stress. Reliability of Reports of Violent Victimization and Posttraumatic Stress Disorder Among Men and Women With Serious Mental Illness Volume 12 Issue4 587 - 599 1999-10-01 Lisa A. Goodman Kim M. Thompson Kevin Weinfurt Susan Corl Pat Acker Kim T. Mueser Stanley D. Rosenberg
Computing: Foulkes-Davis gamma (not in R). A GAUSS program for computing the Foulkes-Davis tracking index for polynomial growth curves   TRACK: A FORTRAN program for calculating the Foulkes-Davis tracking index Gerard E. Dallal Computers in Biology and Medicine Volume 19, Issue 5, 1989, Pages 367-371

Reciprocal effects (Myth 9).
Rogosa, D. R. (1980). A critique of cross-lagged correlation. Psychological Bulletin, 88, 245-258. APA site version
An old review of reciprocal effects. Rogosa, D. R. (1985). Analysis of reciprocal effects. In International Encyclopedia of Education, T. Husen and N. Postlethwaite, Eds. London: Pergamon Press, 4221-4225. (reprinted in Educational Research,Methodology & Measurement: An international handbook, J. P. Keeves Ed. Oxford: Pergamon Press, 1988.)
Granger Causality. Nobel 2003. Complete Granger
Relationships--and the Lack Thereof--Between Economic Time Series, with Special Reference to Money and Interest Rates. David A. Pierce Journal of the American Statistical Association, Vol. 72, No. 357. (Mar., 1977), pp. 11-26. Jstor

Longitudinal Networks
  R-package RSiena manual,   RSiena Resource page
Huisman, M. E. and Snijders, T. A. B. (2003). Statistical analysis of longitudinal network data with changing composition. Sociological Methods and Research, 32:253-287.
Application: Kids' friends influence physical activity levels   Publication: The Distribution of Physical Activity in an After-school Friendship Network Sabina B. Gesell, Eric Tesdahl, Eileen Ruchman, Pediatrics; originally published online May 28, 2012.



WEEK 10 Review Questions
1. Stability of Individual Differences
For the Ramus data (week 2 exercise, 20 individuals, 4 time points), the Foulkes-Davis (gamma) index of tracking has point estimate .83 and (bootstrap) standard error .06 for the 18-month time interval 8yrs to 9.5 years. Compare that estimate of consistency of individual differences with time1-time2 correlations for the time intervals [8, 9.5] and [9, 9.5].
2. Path Analysis. Replicate the 3-wave path analysis example based on the Goldstein model for UK reading scores. Use the perfectly measured data Xi(t) in Week 1 Myths chapter data examples and description. Repeat with the fallible X(t) and compare results.
3. Try a path analysis on the 4-waves of the Ramus data. Compare with the growth curve analyses in week 2 exercises. (Solution: see pages 52-54 of 1994 Myths chap on class memory stick)
Week 10 solutions