Longitudinal
Longitudinal Data Analysis
 Longitudinal Data Analysis by Patrick Heagerty
 Core Guide: Longitudinal Data Analysis by Duke Global Health
 Chapter 4 Models for Longitudinal Data lme4
 **A review of Longitudinal Data Analysis in R. Very nice!
 Chapter 6: Multilevel Modeling R for Researchers. GEE & Mixed Effects.
 Longitudinal Data Analysis for the Behavioral Sciences Using R by Jeffrey Long.
 Methods and Applications of Longitudinal Data Analysis Xian Liu
Mixed Effect Model
 Paper by Laird and Ware 1982
 Random effects vs fixed effects model: There may be factors related to country/region (random variable) which may result in different patients’ responses to the vaccine, and, not all countries are included in the study.
 John Fox's Linear Mixed Models Appendix to An R and SPLUS Companion to Applied Regression. Very clear. It provides 2 typical examples (hierarchical data and longitudinal data) of using the mixed effects model. It also uses Trellis plots to examine the data.
 Chapter 10 Random and Mixed Effects from Modern Applied Statistics with S by Venables and Ripley.
 (Book) lme4: Mixedeffects modeling with R by Douglas Bates.
 (Book) Mixedeffects modeling in S and SPlus by José Pinheiro and Douglas Bates.
 Simulation and power analysis of generalized linear mixed models
 Linear mixedeffect models in R by poissonisfish
 Dealing with correlation in designed field experiments: part II
 Mixed Models in R by Michael Clark
 Mixed models in R: a primer
 Chapter 4 Simulating Mixed Effects by Lisa DeBruine
 Linear mixed effects models (video) by Clapham. Output for y ~ x + (xgroup) model.
y ~ x + (1group) # random intercepts, same slope for groups y ~ x + (xgroup) # random intercepts & slopes for groups y ~ color + (colorgreen/gray) # nested random effects y ~ color + (colorgreen) + (colorgray) # crossed random effects
 linear mixed effects models in R lme4
 Using Random Effects in Models by rcompanion
library(nlme) lme(y ~ 1 + randonm = ~1  Random) # oneway random model lme(y ~ Fix + random = ~1  Random) # twoway mixed effect model # https://stackoverflow.com/a/36415354 library(lme4) fit < lmer(mins ~ Fix1 + Fix2 + (1Random1) + (1Random2) + (1Year/Month), REML=FALSE)
 Random effects and penalized splines are the same thing
 Introduction to linear mixed models
 Fitting 'complex' mixed models with 'nlme'
 A random variable has to consist of at least 5 levels. An Introduction to Linear Mixed Effects Models (video)
 A Guide on Data Analysis Mike Nguyen
 FAQs on mixedeffects models, Mixedeffects models in R, and a new tool for data simulation by Pablo Bernabeu.
Repeated measure

R Tutorial: Linear mixedeffects models part 1 Repeated measures ANOVA
words ~ drink + (1subj) # random intercepts

A review of Longitudinal Data Analysis in R.
 Oneway randomeffect ANOVA model. [math]\displaystyle{ Y_{ij} = \beta_0 + u_{0i} + \epsilon_{ij} }[/math], where [math]\displaystyle{ \beta_0 }[/math] is the intercept, [math]\displaystyle{ u_{0i} \sim N(0, \sigma^2_{u0}) }[/math] the random effect, and [math]\displaystyle{ \epsilon_{ij} \sim N(0, \sigma^2) }[/math] is the residual error terms.
lin_0 < lmer(distance ~ 1 + (1  id), data = dental_long) ranova(lin_0) # Testing betweenindividual heterogeneity
 Test whether the mean response is constant over time; we include time effect in the model.
lin_age < lmer(distance ~ measurement + (1  id), data = dental_long) Anova(lin_age)
 Test Group/sex effect
lin_agesex < lmer(distance ~ measurement + sex + (1  id), data = dental_long) summary(lin_agesex) # show the pvalue for the sex variable
 Interaction between time and group
lin_agesexinter < lmer(distance ~ measurement*sex + (1  id), data = dental_long) Anova(lin_agesexinter, type = 3)
 Prediction random effects
lin_agec < lmer(distance ~ age + (1  id), data = dental_long)
 Random intercept and slope
lin_agecr < lmer(distance ~ age + (age  id), data = dental_long) VarCorr(lin_agecr)
 GEE
gee_inter_exch < geeglm(distance ~ sex*age, data = dental_long, id = id, family = gaussian, corstr = "exchangeable") summary(gee_inter_exch) gee_inter_ind < geeglm(distance ~ sex*age, data = dental_long, id = id, family = gaussian, corstr = "independence") QIC(gee_inter_exch, gee_inter_ind
 Oneway randomeffect ANOVA model. [math]\displaystyle{ Y_{ij} = \beta_0 + u_{0i} + \epsilon_{ij} }[/math], where [math]\displaystyle{ \beta_0 }[/math] is the intercept, [math]\displaystyle{ u_{0i} \sim N(0, \sigma^2_{u0}) }[/math] the random effect, and [math]\displaystyle{ \epsilon_{ij} \sim N(0, \sigma^2) }[/math] is the residual error terms.
 Repeated measures with perennial crops, Repeated measures and subsampling with perennial crops
Graphical representation of the model
A review of Longitudinal Data Analysis in R.
Trajectories
 In longitudinal data analysis, trajectories refer to the patterns of change over time in a variable or set of variables for an individual or group of individuals1. These patterns can be used to identify typical trajectories in longitudinal data using mixed effects models1. The result is summarized in terms of an average trajectory plus measures of the individual variations around this average
 Identifying typical trajectories in longitudinal data: modelling strategies and interpretations 2020
 Visualise longitudinal data
 An example: a model that there are random intercepts and slopes for each subject:
[math]\displaystyle{ y_{ij} = \beta_0 + \beta_1 x_{ij} + b_{0i} + b_{1i} x_{ij} + e_{ij}
}[/math]
 [math]\displaystyle{ y_{ij} }[/math] is the reaction time for subject i at time j
 [math]\displaystyle{ \beta_0, \beta_1 }[/math] are fixed intercept and slope
 [math]\displaystyle{ x_{ij} }[/math] is the number of days since the start of the study for subject i at time j
 [math]\displaystyle{ b_{0i}, b_{1i} }[/math] are the random intercept & slope for subject i
 [math]\displaystyle{ e_{ij} }[/math] is the residual error for subject i at time j
> library(lme4) > data(sleepstudy) # from lme4 > # Fit a mixed effects model with random intercepts and slopes for each subject > model < lmer(Reaction ~ Days + (Days  Subject), sleepstudy) > > # Plot the fitted model > plot(model, type = "l") > summary(model) Linear mixed model fit by REML ['lmerMod'] Formula: Reaction ~ Days + (Days  Subject) Data: sleepstudy REML criterion at convergence: 1743.6 Scaled residuals: Min 1Q Median 3Q Max 3.9536 0.4634 0.0231 0.4634 5.1793 Random effects: Groups Name Variance Std.Dev. Corr Subject (Intercept) 612.10 24.741 Days 35.07 5.922 0.07 Residual 654.94 25.592 Number of obs: 180, groups: Subject, 18 Fixed effects: Estimate Std. Error t value (Intercept) 251.405 6.825 36.838 Days 10.467 1.546 6.771 Correlation of Fixed Effects: (Intr) Days 0.138
 The fixed effect estimate for Days is 10.4673, which means that on average, reaction times decrease by 10.4673 milliseconds per day.
 The random intercept variance is 612.09, which means that there is considerable variation in the baseline reaction times between subjects.
 The random slope variance is 35.07, which means that there is considerable variation in the rate of change of reaction times over time between subjects.
 It seems no pvalues are returned. See Three ways to get parameterspecific pvalues from lmer & How to obtain the pvalue (check significance) of an effect in a lme4 mixed model?
 To exclude the random slope term, we can use (1  Subject)
 To exclude the random intercept term, we can use (0 + Days  Subject)
 To exclude the fix intercept term, we can use y ~ 1 + x + (Days  Subject)
library(nlme) model2 < lme(Reaction ~ Days, random = ~ Days  Subject, data = sleepstudy) > summary(model2) Linear mixedeffects model fit by REML Data: sleepstudy AIC BIC logLik 1755.628 1774.719 871.8141 Random effects: Formula: ~Days  Subject Structure: General positivedefinite, LogCholesky parametrization StdDev Corr (Intercept) 24.740241 (Intr) Days 5.922103 0.066 Residual 25.591843 Fixed effects: Reaction ~ Days Value Std.Error DF tvalue (Intercept) 251.40510 6.824516 161 36.83853 Days 10.46729 1.545783 161 6.77151 pvalue (Intercept) 0 Days 0 Correlation: (Intr) Days 0.138 Standardized WithinGroup Residuals: Min Q1 Med Q3 3.95355735 0.46339976 0.02311783 0.46339621 Max 5.17925089 Number of Observations: 180 Number of Groups: 18
 To exclude the random slope term, we can use random = ~1  Subject.
 To exclude the random intercept, we can use random = ~Days  1  Subject.
Graphical display using facet_wrap().
 lmerTest  Tests in Linear Mixed Effects Models.
The ranova() function in the lmerTest package is used to compute an ANOVAlike table with tests of randomeffect terms in a mixedeffects model. Paper in JSS 2017.
# Fixed effect > anova(model) Type III Analysis of Variance Table with Satterthwaite's method Sum Sq Mean Sq NumDF DenDF F value Pr(>F) Days 30031 30031 1 17 45.853 3.264e06 ***  Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > anova(model2) numDF denDF Fvalue pvalue (Intercept) 1 161 1454.0766 <.0001 Days 1 161 45.8534 <.0001 # Random effect > ranova(model) ANOVAlike table for randomeffects: Single term deletions Model: Reaction ~ Days + (Days  Subject) npar logLik AIC LRT Df <none> 6 871.81 1755.6 Days in (Days  Subject) 4 893.23 1794.5 42.837 2 Pr(>Chisq) <none> Days in (Days  Subject) 4.99e10 ***  Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
variancePartition
variancePartition Quantify and interpret divers of variation in multilevel gene expression experiments
Growth curve model
GEE/generalized estimating equations
 https://en.wikipedia.org/wiki/Generalized_estimating_equation
 Literatures:
 Longitudinal Data Analysis for Discrete and Continuous Outcomes Scott L. Zeger and KungYee Liang Biometrics 1986
 Longitudinal data analysis using generalized linear models. Liang, K., and S. L. Zeger (1986) Biometrika.
 Analysis of Longitudinal Data (Oxford Statistical Science Series) 2nd edition by Peter Diggle, Patrick Heagerty, KungYee Liang, and Scott Zeger
 Youtube
 12.1  Introduction to Generalized Estimating Equations from PennState
 12.2  Modeling Binary Clustered Responses which shows the "exchangeable" correlation structure
 gee package. First version 1999.
 geepack package. First version 2002.
 Getting Started with Generalized Estimating Equations.
 The Working correlation matrix output from the gee() function when "exchangeable" correlation was used. The dimension of "working correlation" is the number of observations per subject/id. Question: why the diagonal elements of the working correlation is sometimes 0 instead of 1 in my data.
 The Estimated Scale Parameter is the dispersion parameter.
 **A review of Longitudinal Data Analysis in R.
 **DEM 7283: Longitudinal Models for Change using Generalized Estimating Equations
 Chapter 15 GEE, Binary Outcome: Respiratory Illness from Encyclopedia of Quantitative Methods in R, vol. 5: Multilevel Models.
 SAS/STAT 14.1 User’s Guide The GEE Procedure. We can download the data and compare it with the results from R.
 Fast Pure R Implementation of GEE: Application of the Matrix Package
 One example
library(geepack) # load data data("cbpp", package = "lme4") cbpp[1:3, ] # herd incidence size period # 1 1 2 14 1 # 2 1 3 12 2 # 3 1 4 9 3 with(cbpp, table(herd, period)) period herd 1 2 3 4 1 1 1 1 1 2 1 1 1 0 3 1 1 1 1 4 1 1 1 1 5 1 1 1 1 6 1 1 1 1 7 1 1 1 1 8 1 0 0 0 9 1 1 1 1 10 1 1 1 1 11 1 1 1 1 12 1 1 1 1 13 1 1 1 1 14 1 1 1 1 15 1 1 1 1 # fit GEE model with logit link fit < geeglm(cbind(incidence, size  incidence) ~ period, id = herd, family = binomial("logit"), corstr = "exchangeable", data = cbpp) # summary of the model summary(fit) Coefficients: Estimate Std.err Wald Pr(>W) (Intercept) 1.270 0.276 21.24 4.0e06 *** period2 1.156 0.436 7.04 0.0080 ** period3 1.284 0.491 6.84 0.0089 ** period4 1.754 0.375 21.85 2.9e06 ***  Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Correlation structure = exchangeable Estimated Scale Parameters: Estimate Std.err (Intercept) 0.134 0.0542 Link = identity Estimated Correlation Parameters: Estimate Std.err alpha 0.0414 0.129 Number of clusters: 15 Maximum cluster size: 4 summary(fit)$coefficients Estimate Std.err Wald Pr(>W) (Intercept) 1.27 0.276 21.24 4.04e06 period2 1.16 0.436 7.04 7.99e03 period3 1.28 0.491 6.84 8.90e03 period4 1.75 0.375 21.85 2.94e06 # extract robust variancecovariance matrix vcov_fit < vcovHC(fit, type = "HC") # compute confidence intervals for coefficients library(sandwich) vcov_fit < vcovHC(fit, type = "HC") ci < coef(fit) + cbind(qnorm(0.025), qnorm(0.975)) * sqrt(diag(vcov_fit)) colnames(ci) < c("2.5 %", "97.5 %") print(ci)
Compare with gee()
summary(gee(cbind(incidence, size  incidence) ~ period, id = herd, family = binomial("logit"), corstr = "exchangeable", data = cbpp)) # Coefficients: # Estimate Naive S.E. Naive z Robust S.E. Robust z # (Intercept) 1.27 0.214 5.94 0.275 4.61 # period2 1.16 0.424 2.74 0.437 2.66 # period3 1.29 0.455 2.83 0.492 2.62 # period4 1.76 0.600 2.94 0.379 4.65 # # Estimated Scale Parameter: 2.18 # Number of Iterations: 2 # # Working Correlation # [,1] [,2] [,3] [,4] # [1,] 1.0000 0.0274 0.0274 0.0274 # [2,] 0.0274 1.0000 0.0274 0.0274 # [3,] 0.0274 0.0274 1.0000 0.0274 # [4,] 0.0274 0.0274 0.0274 1.0000
[math]\displaystyle{ \operatorname{logit}\left(\frac{p_{ij}}{1  p_{ij}}\right) = \beta_0 + \beta_1 \cdot \text{period}_{ij} + u_j }[/math] where [math]\displaystyle{ p_{ij} }[/math] is the probability of a positive outcome (i.e., an incidence) for
 the [math]\displaystyle{ i }[/math]th animal/observation in herd(group/subject) [math]\displaystyle{ j }[/math], and
 [math]\displaystyle{ \text{period}_{ij} }[/math] is a categorical predictor variable indicating the period of observation for herd/subject [math]\displaystyle{ j }[/math] and observation [math]\displaystyle{ i }[/math].
 The parameters [math]\displaystyle{ \beta_0 }[/math] and [math]\displaystyle{ \beta_1 }[/math] represent the intercept and the effect of the period variable, respectively.
 The [math]\displaystyle{ u_j }[/math] is the random subjectspecific effect. The random effects [math]\displaystyle{ u_j }[/math] are assumed to follow a multivariate normal distribution with mean 0 and covariance matrix [math]\displaystyle{ \Sigma }[/math].
 The correlation structure of the random effects is specified as "exchangeable", which means that any two random effects from the same subject have the same correlation.
R packages
Sort by sample ID & time first
 No effect of corstr argument on correlation structure in gee
 The issue applies to both the gee and geepack packages.
 If observations were not ordered according to cluster and time within cluster we would get the wrong result. See geepack vignette.
 See the section 4 for a case that data is ordered according to subject but the time ordering within subject is random.
 For the depression data from Getting Started with Generalized Estimating Equations, data are sorted by id & time. It is also true for dietox data included in geepack: all(order(dietox$Pig, dietox$Time) == 1:nrow(dietox)) = TRUE.
 Both geeglm() or gee() does not require a 'time' parameter because it assumes the time is sorted within a cluster. But how about data has missing values?
 Do unequal time intervals for waves parameter in gee analysis matter? (geeglm in R) no answer.
 Use the "waves" parameter. waves specifying the ordering of repeated mesurements on the same unit. Also used in connection with missing values. See the vignette.
 waves names a positive integervalued variable that is used to identify the order and spacing of observations within groups in a GEE model. This argument is crucial when there are missing values and gaps in the data. See examples on Lecture 23 (lab 8)—Friday, March 2, 2012.
 "waves" is only available in geeglm(). It is not available in gee().
Compare models
 015. GEE in Practice: Example on COVID19 Test Positivity Data. QIC is available in the geepack package.
 geeglm was used.
 simple scatterplot by plot() with y=positivity_rate, x=time
 lattice::xyplot()
xyplot(positivity_rate ~ timeID, groups = ID, subset = ID %in% sampled_idx, data= covid, panel = function(x, y) { panel.xyplot(x, y, plot='p') panel.linejoin(x, y, fun=mean, horizontal = F, lwd=2, col=1) })
 create a factor
covid$size_factor < cut(covid$estimat_pop, breaks = quantile(unique(covid$estimated_pop)), include.lowest = TRUE, lables = FALSE) > as.factor() xyplot(positivity_rate ~ timesize_factor, # size_factor is like a facet groups = ID, data= covid, panel = function(x, y) { panel.xyplot(x, y, plot='p') panel.linejoin(x, y, fun=mean, horizontal = F, lwd=2, col=1) })
 Residual vs predict plot from plot(fit) can be used to evaluate which correlation structure is better.
 QIC() output and what metric can be used for [model selection. QIC & CIC are most useful for selecting correlation structure. QICU is most useful for comparing models with the same correlation structure but different mean structure.
 Analysis of panel data in R using Generalized Estimating Equations which uses the MESS package
anova()
anova() method works on geeglm() object but not on gee() object.
Simulation
 How to simulate data for generalized estimating equations (GEE) with logistic link function?
 We generate some example data with repeated measurements. In this simulated data, we have 100 subjects, each measured at 5 time points.
# Load the geepack package library(geepack) # Generate some example data with repeated measurements set.seed(123) n < 100 # Number of subjects time < rep(1:5, each = n) # Time points subject < rep(1:n, times = 5) # Subject IDs y < rnorm(n * 5, mean = time * 0.5 + subject * 0.2) # Simulated response variable # Create a data frame data < data.frame(subject, time, y) data < data[order(data$subject, data$time), ] data[1:4, ] # subject time y # 1 1 1 0.1395 # 101 1 2 0.4896 # 201 1 3 3.8988 # 301 1 4 1.4848 # Fit a GEE model using geeglm # In this example, we assume an exchangeable correlation structure model < geeglm(y ~ time, id = subject, data = data, family = gaussian, corstr = "exchangeable") # Summary of the GEE model summary(model) # Coefficients: # Estimate Std.err Wald Pr(>W) # (Intercept) 10.104 0.594 289 <2e16 *** # time 0.510 0.033 240 <2e16 *** #  # Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 # # Correlation structure = exchangeable # Estimated Scale Parameters: # # Estimate Std.err # (Intercept) 35.1 3.07 # Link = identity # # Estimated Correlation Parameters: # Estimate Std.err # alpha 0.972 0.00333 ===> corr estimate is 0.972 # Number of clusters: 100 Maximum cluster size: 5 model2 = geeglm(y ~ time, id = subject, data = data, family = gaussian, corstr = "ind") summary(model2) # Coefficients: # Estimate Std.err Wald Pr(>W) # (Intercept) 10.104 0.594 289 <2e16 *** # time 0.510 0.033 240 <2e16 *** #  # Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 # # Correlation structure = independence # Estimated Scale Parameters: # # Estimate Std.err # (Intercept) 35.1 3.07 # Number of clusters: 100 Maximum cluster size: 5 # gee() from gee package summary(gee(y ~ time, id = subject, data = data, family = gaussian, corstr = "exchangeable")) # Coefficients: # Estimate Naive S.E. Naive z Robust S.E. Robust z # (Intercept) 10.10 0.5945 17.0 0.594 17.0 # time 0.51 0.0327 15.6 0.033 15.5 # # Estimated Scale Parameter: 35.2 # Number of Iterations: 1 # # Working Correlation # [,1] [,2] [,3] [,4] [,5] # [1,] 1.00 0.97 0.97 0.97 0.97 # [2,] 0.97 1.00 0.97 0.97 0.97 # [3,] 0.97 0.97 1.00 0.97 0.97 # [4,] 0.97 0.97 0.97 1.00 0.97 # [5,] 0.97 0.97 0.97 0.97 1.00
gee vs glm
# See the next subsection about dietox dataset glmEx < glm(mf, data=dietox, family=gaussian) > summary(glmEx) Call: glm(formula = mf, family = gaussian, data = dietox) Coefficients: Estimate Std. Error t value Pr(>t) (Intercept) 15.073 0.709 21.27 < 0.0000000000000002 *** Time 6.948 0.069 100.66 < 0.0000000000000002 *** EvitEvit100 2.081 0.589 3.54 0.00043 *** EvitEvit200 1.113 0.582 1.91 0.05619 . CuCu035 0.789 0.583 1.35 0.17638 CuCu175 1.777 0.589 3.02 0.00264 **  Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for gaussian family taken to be 48.62) Null deviance: 536592 on 860 degrees of freedom Residual deviance: 41571 on 855 degrees of freedom AIC: 5796 Number of Fisher Scoring iterations: 2
In this particular instance, the application of linear regression yields a lower standard error (SE) for each variable when compared to the Generalized Estimating Equations (GEE) method. In other words, linear regression assumes that all observations are independent. If this assumption is violated, as is often the case with longitudinal data where measurements from the same subject over time are likely to be correlated, the standard errors produced by linear regression may be underestimated. This could lead to overly optimistic pvalues and confidence intervals that do not cover the true parameter value as often as they should.
gee vs geepack packages
GENERALIZED ESTIMATING EQUATIONS (GEE) by Practical Statistics (Rebecca Barter).
Subjectspecific versus Populationaveraged.
GEE estimates populationaveraged model parameters and their standard errors.
library("geepack") data(dietox) dietox$Cu < as.factor(dietox$Cu) dietox$Evit < as.factor(dietox$Evit) mf < formula(Weight ~ Time + Evit + Cu) geeEx < geeglm(mf, id=Pig, data=dietox, family=gaussian, corstr="ex") summary(geeEx) Call: geeglm(formula = mf, family = gaussian, data = dietox, id = Pig, corstr = "ex") Coefficients: Estimate Std.err Wald Pr(>W) (Intercept) 15.0984 1.4206 112.96 <2e16 *** Time 6.9426 0.0796 7605.79 <2e16 *** EvitEvit100 2.0414 1.8431 1.23 0.27 EvitEvit200 1.1103 1.8452 0.36 0.55 CuCu035 0.7652 1.5354 0.25 0.62 CuCu175 1.7871 1.8189 0.97 0.33  Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Correlation structure = exchangeable Estimated Scale Parameters: Estimate Std.err (Intercept) 48.3 9.31 Link = identity Estimated Correlation Parameters: Estimate Std.err alpha 0.766 0.0326 Number of clusters: 72 Maximum cluster size: 12
Note that the correlation estimate obtained from geeglm() is 0.766 = summary(geeEx)$geese$correlation$estimate.
gee() can show the complete correlation structure in the output. The correlation estimate is 0.761 =summary(geeEx2)$working.correlation[1, 2].
geeEx2 = gee::gee(Weight ~ Time + Evit + Cu, id=Pig, data=dietox, family=gaussian, corstr="exchangeable") summary(geeEx2) Beginning Cgee Sfunction, @(#) geeformula.q 4.13 98/01/27 running glm to get initial regression estimate (Intercept) Time EvitEvit100 EvitEvit200 CuCu035 CuCu175 15.073 6.948 2.081 1.113 0.789 1.777 GEE: GENERALIZED LINEAR MODELS FOR DEPENDENT DATA gee Sfunction, version 4.13 modified 98/01/27 (1998) Model: Link: Identity Variance to Mean Relation: Gaussian Correlation Structure: Exchangeable Call: gee::gee(formula = Weight ~ Time + Evit + Cu, id = Pig, data = dietox, family = gaussian, corstr = "exchangeable") Summary of Residuals: Min 1Q Median 3Q Max 34.396 3.245 0.903 4.127 20.157 Coefficients: Estimate Naive S.E. Naive z Robust S.E. Robust z (Intercept) 15.098 1.6955 8.905 1.4206 10.628 Time 6.943 0.0337 205.820 0.0796 87.211 EvitEvit100 2.041 1.7991 1.135 1.8431 1.108 EvitEvit200 1.110 1.7813 0.623 1.8452 0.602 CuCu035 0.765 1.7813 0.430 1.5354 0.498 CuCu175 1.787 1.7991 0.993 1.8189 0.982 Estimated Scale Parameter: 48.6 Number of Iterations: 2 Working Correlation [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [1,] 1.000 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 [2,] 0.761 1.000 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 [3,] 0.761 0.761 1.000 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 [4,] 0.761 0.761 0.761 1.000 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 [5,] 0.761 0.761 0.761 0.761 1.000 0.761 0.761 0.761 0.761 0.761 0.761 0.761 [6,] 0.761 0.761 0.761 0.761 0.761 1.000 0.761 0.761 0.761 0.761 0.761 0.761 [7,] 0.761 0.761 0.761 0.761 0.761 0.761 1.000 0.761 0.761 0.761 0.761 0.761 [8,] 0.761 0.761 0.761 0.761 0.761 0.761 0.761 1.000 0.761 0.761 0.761 0.761 [9,] 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 1.000 0.761 0.761 0.761 [10,] 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 1.000 0.761 0.761 [11,] 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 1.000 0.761 [12,] 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 1.000
and we need to calculate pvalues manually.
> format.pval(2*(1pnorm(abs(summary(geeEx2)$coefficients[, "Robust z"])))) [1] "< 2e16" "< 2e16" "0.26804" "0.54737" "0.61823" "0.32585"
GEE vs. mixed effects models
 When to use generalized estimating equations vs. mixed effects models?
 GEE fits a marginal regression model to the longitudinal data.
 Instead of attempting to model the withinsubject covariance structure, GEE models the average response. The goal is to make inferences about the population when accounting for the withinsubject correlation. For every oneunit increase in a covariate across the population, GEE tells us how much the average response would change.
 GEE can take into account the correlation of withinsubject data (longitudinal studies) and other studies in which data are clustered within subgroups. Failure to take into account correlation would lead to the regression estimates (Bs) being less efficient, meaning they would be more widely scattered around the true population value.
 GEE fits a marginal model, which means it models the average response over time, rather than modeling each individual's response over time. This is different from other methods like mixedeffects models, which model both the population average and individual deviations from that average.
 The advantage of using a marginal model is that it provides a simpler interpretation of the results. The regression coefficients from a GEE model represent the average change in the response for a oneunit change in the predictor, averaged over all individuals in the population. This can be easier to interpret and communicate than individuallevel effects.
 Moreover, GEE is less sensitive to misspecification of the correlation structure compared to other methods. This means that even if you don't correctly specify how the repeated measurements are correlated, you can still get valid estimates of the average effects. That is, GEE is robust to misspecification of the withinsubject correlation.
 Mixedeffects models, on the other hand, provide an individuallevel approach by adopting random effects to capture the correlation between the observations of the same subject. They give an interpretation conditional on the random effects.
 The fundamental difference between GEE and mixedeffects models lies in the interpretation of the coefficients. GEEs produce populationaveraged effects, while mixedeffects models produce subjectspecific effects. In other words, GEEs give you the more usual interpretation of comparing groups of subjects, while mixedeffects models give an interpretation conditional on the random effects.
 For example, consider a binary outcome with a logit link. The GEE would give you the logodds ratio between groups (e.g., males vs. females), which is a populationaveraged effect. On the other hand, a mixedeffects model would give you an estimate that accounts for individualspecific random effects.
 In summary, whether to use GEE or a mixedeffects model depends on your research question and whether you are interested in populationaveraged or subjectspecific effects.
 https://stats.stackexchange.com/a/16415
 Getting Started with Generalized Estimating Equations
 It uses "gee", "lme4", "geepack" packages for model fittings
dep_gee2 < gee(depression ~ diagnose + drug*time, data = dat, id = id, family = binomial, corstr = "exchangeable")
 How does the GEE model with the exchangeable correlation structure compare to a Generalized MixedEffect model. We allow the intercept to be conditional on subject. This means we’ll get 340 subject_specific models with different intercepts but identical coefficients for everything else. The model coefficients listed under “Fixed effects” are almost identical to the GEE coefficients. ?glmer for GLMMs/Generalized Linear MixedEffects Models.
library(lme4) # version 1.131 dep_glmer < glmer(depression ~ diagnose + drug*time + (1id), data = dat, family = binomial) summary(dep_glmer, corr = FALSE)
dep_gee4 < geeglm(depression ~ diagnose + drug*time, data = dat, id = id, family = binomial, corstr = "exchangeable")
 "ggeffects" package for plotting. The emmeans function calculates marginal effects and works for both GEE and mixedeffect models.
library(ggeffects) library(emmeans) plot(ggemmeans(dep_glmer, terms = c("time", "drug"), condition = c(diagnose = "severe"))) + ggplot2::ggtitle("GLMER Effect plot") data.frame(ggemmeans(dep_glmer, terms = c("time", "drug"), condition = c(diagnose = "severe")) ) x predicted std.error conf.low conf.high group 1 0 0.207 0.175 0.156 0.269 standard 2 0 0.197 0.187 0.146 0.262 new 3 1 0.297 0.119 0.251 0.348 standard 4 1 0.525 0.126 0.463 0.586 new 5 2 0.407 0.156 0.335 0.482 standard 6 2 0.832 0.217 0.764 0.883 new filter(dat, diagnose == "severe") %>% group_by(drug, time) %>% summarize(mean = mean(depression)) drug time mean <fct> <int> <dbl> 1 standard 0 0.21 2 standard 1 0.28 3 standard 2 0.46 4 new 0 0.178 5 new 1 0.5 6 new 2 0.833 plot(ggemmeans(dep_gee4, terms = c("time", "drug"), condition = c(diagnose = "severe"))) + ggplot2::ggtitle("GEE Effect plot")
 While both methods are used for modeling correlated data, there are some key differences between them.
 Assumptions: The key difference between GEE and nlme is that GEE makes fewer assumptions about the underlying data structure. GEE assumes that the mean structure is correctly specified, but the correlation structure is unspecified. In contrast, nlme assumes that both the mean and the variancecovariance structure are correctly specified.
 Model flexibility: GEE is a more flexible model than nlme because it allows for modeling the mean and variance of the response separately. This means that GEE is more suitable for modeling data with nonnormal errors, such as count or binary data. In contrast, nlme assumes that the errors are normally distributed and allows for modeling the mean and variance jointly.
 Inference: Another key difference between GEE and nlme is the type of inference they provide. GEE provides marginal inference, which means that it estimates the populationaveraged effect of the predictor variables on the response variable. In contrast, nlme provides conditional inference, which means that it estimates the subjectspecific effect of the predictor variables on the response variable.
 Computation: The computation involved in fitting GEE and nlme models is also different. GEE uses an iterative algorithm to estimate the model parameters, while nlme uses a maximum likelihood estimation approach. The computational complexity of GEE is lower than that of nlme, making it more suitable for large datasets.
 In summary, GEE and nlme are two popular models used for analyzing longitudinal data. The main differences between the two models are the assumptions they make, model flexibility, inference type, and computational complexity. GEE is a more flexible model that makes fewer assumptions and provides marginal inference, while nlme assumes normality of errors and provides conditional inference.
Independent correlation
If you assume an independent correlation structure in a Generalized Estimating Equation (GEE) model, the GEE estimates are the same as those obtained from a Generalized Linear Model (GLM) or Ordinary Least Squares (OLS) if the dependent variable is normally distributed and no correlation within responses are assumed².
However, the standard errors in a GEE model will be different from those in a GLM or OLS model. This is because GEE takes into account the withinsubject correlation, which can lead to more efficient estimates¹. If this correlation is not taken into account, the regression estimates would be less efficient, meaning they would be more widely scattered around the true population value¹.
It's important to note that while the point estimates might be the same, the interpretation of these estimates differs between GEE and GLM/OLS models. In a GEE model, you're making inferences about the population average response, whereas in a GLM/OLS model, you're making inferences about individuallevel effects¹.
References:
 GEE: choosing proper working correlation structure.
 GEE for Repeated Measures Analysis  Columbia Public Health.
exchangeable correlation
 It seems geeglm() and gee() can return different estimate of correlation.
 For geeglm(), we can find the estimate from the summary output. See an example (0.3285) below.
Estimated Correlation Parameters: Estimate Std.err alpha 0.3285 0.08381
SAS
The GEE procedure  Example 50.1 Comparison of the Marginal and Random Effect Models for Binary Data. The data (& the code) can be downloaded from this link. A copy of the data is available at github.
Note the data has been sorted by subject ("centerid") and time ("visit") within each subject. The data has no missing values. So there is no need to use the waves parameter.
library(geepack) library(gee) resp < read.table("~/Downloads/resp.txt", header = T) > dim(resp) [1] 111 10 > resp[1:2, ] center id treat sex age baseline visit1 visit2 visit3 visit4 1 1 1 P M 46 0 0 0 0 0 2 1 2 P M 28 0 0 0 0 0 library(tidyr) resp.long < pivot_longer(resp, starts_with("visit"), names_to = 'n', values_to = 'v')) resp.long < mutate(resp.long, centerid = center*id) R> resp.long[1:3, ] # A tibble: 3 × 9 center id treat sex age baseline n v centerid <int> <int> <chr> <chr> <int> <int> <chr> <int> <int> 1 1 1 P M 46 0 visit1 0 1 2 1 1 P M 46 0 visit2 0 1 3 1 1 P M 46 0 visit3 0 1 resp.long < mutate(resp.long, center = as.factor(center), treat = as.factor(treat), sex = as.factor(sex), baseline = factor(baseline)) > mod = geeglm(v~ treat + center + sex + age + baseline, family=binomial, data=resp.long, id=centerid, corstr="exch") > summary(mod) Call: geeglm(formula = v ~ treat + center + sex + age + baseline, family = binomial, data = resp.long, id = centerid, corstr = "exch") Coefficients: Estimate Std.err Wald Pr(>W) (Intercept) 0.54603 0.72348 0.570 0.450416 treatP 1.26536 0.34668 13.322 0.000262 *** center2 0.64949 0.35322 3.381 0.065952 . sexM 0.13678 0.44025 0.097 0.756035 age 0.01876 0.01296 2.093 0.147974 baseline1 1.84572 0.34598 28.460 9.57e08 ***  Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Correlation structure = exchangeable Estimated Scale Parameters: Estimate Std.err (Intercept) 1.002 0.1842 Link = identity Estimated Correlation Parameters: Estimate Std.err alpha 0.3285 0.08381 Number of clusters: 111 Maximum cluster size: 4 > mod2 = gee(v~ treat + center + sex + age + baseline, family=binomial, data=resp.long, id=centerid, corstr="exchangeable") Beginning Cgee Sfunction, @(#) geeformula.q 4.13 98/01/27 running glm to get initial regression estimate (Intercept) treatP center2 sexM age baseline1 0.54603 1.26536 0.64949 0.13678 0.01876 1.84572 > summary(mod2) Coefficients: Estimate Naive S.E. Naive z Robust S.E. Robust z (Intercept) 0.54603 0.6585 0.8292 0.72348 0.7547 treatP 1.26536 0.3333 3.7962 0.34668 3.6499 center2 0.64949 0.3379 1.9219 0.35322 1.8388 sexM 0.13678 0.4161 0.3288 0.44025 0.3107 age 0.01876 0.0125 1.5000 0.01296 1.4467 baseline1 1.84572 0.3394 5.4386 0.34598 5.3348 Estimated Scale Parameter: 1.016 Number of Iterations: 1 Working Correlation [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [1,] 1.000 0.327 0 0 0 0 0 0 [2,] 0.327 0.327 0 0 0 0 0 0 [3,] 0.327 1.000 0 0 0 0 0 0 [4,] 0.327 0.327 0 0 0 0 0 0 [5,] 0.327 0.327 0 0 0 0 0 0 [6,] 1.000 0.327 0 0 0 0 0 0 [7,] 0.327 0.327 0 0 0 0 0 0 [8,] 0.327 1.000 0 0 0 0 0 0