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# Analysis of Longitudinal Data (SoSe 2016)  #
#                                            #
# Solution for lab session 7                 #  
# Datum: 12.07.2016                          #
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# Exercise 1#
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# a) 

# read data set
leprosy <- read.table('../Vorher_online/leprosylong.txt')

# overview
str(leprosy)
leprosy$Drug <- factor(leprosy$Drug, levels = c("C", "A", "B"))
table(leprosy$Drug) # number of measurements for each drug separately

# --> We need to specify them marginal expectation, the marginal variance, and the association/correlation
#     between measurements of the same subject. For more details, see lecture slides and lab session.

# b)

# Generalized as now the estimation equations are also depending on \alpha.

# c)

# Estimate a marginal model using GEE with unstructured covariance.

library(gee)  # load package
gee_indep <- gee(count ~ time + time:Drug, data = leprosy, family = poisson(link = "log"),
                 id = id, corstr = "independence")
summary(gee_indep)

# d)

# naive covariance matrix
gee_indep$naive.variance
diag(gee_indep$naive.variance)

# robust covariance matrix
gee_indep$robust.variance
diag(gee_indep$robust.variance)


# --> There is quite a large difference between the robust and the naive estimates.
#     This indicates that the assumption of uncorrelated observations is not appropriate.
#     When both estimates are similar, the working covariance is close to the observed covariance 
#     (compare lecture slides and lab session).


# e)

# --> Robustness is an asymptotical property. It relies on many replications.
#     It is less efficient than an model-based estimator that approximates the truth well. 

# --> It is preferable to use the model-based estimator instead of the robust estimator when
#     - N is small (relatively to $n_i$)
#     - there are few subjects for each covariate combination
#     - data are strongly unbalanced