Problem Description:
The GEE Models assignment involves analyzing a Generalized Estimating Equations (GEE) populationaveraged model to understand the association between parental smoking and adolescent smoking. The model is fitted with different correlation structures, and the results are presented with a focus on Odds Ratios (ORs) and confidence intervals.
Solution:
Question 1
GEE populationaveraged model Number of obs = 7706
Group and time vars: id wave Number of groups = 1502
Link: logit Obs per group: min = 2
Family: binomial avg = 5.1
Correlation: AR(1) max = 6
Wald chi2(3) = 88.22
Scale parameter: 1 Prob > chi2 = 0.0000
regsmoke  Coef.  Robust Std. Err.  z  P>z  95% Conf.  Interval 
_Isex_1  .2831992  .1368981  2.07  0.039  .014884  .5515145

c_wave  .3098144  .0598181  5.18  0.000  .1925732  .4270557 
_IsexXc_wav_1  .0288783  .075397  0.38  0.702  .1188971  .1766537

_cons  2.212193  .1026366  21.55  0.000  2.413357 
2.011029 
GEE (autoregressive order 1 working correlation)  

Coefficient  Standard error  
Constant  2.21  0.103 
Sex(female)  0.28  0.137 
Wave(per year): males  0.31  0.060 
females  0.34  0.046 
Table 1: Analyzing GEE correlation between parental smoking and adolescent
GEE populationaveraged model:
Number of observations: 7706
Number of groups: 1502
Correlation structure: AR(1)
Results: The model reveals associations between smoking and various factors. Notably, when using the autoregressive (AR) working correlation structure, some groups were omitted due to unequal spacing or insufficient data. This omission can affect the accuracy of estimated coefficients and standard errors.
Question 2
GEE populationaveraged model Number of obs = 8498
Group and time vars: id wave Number of groups = 1702
Link: logit Obs per group: min = 1
Family: binomial avg = 5.0
Correlation: unstructured max = 6
Wald chi2(4) = 177.20
Scale parameter: 1 Prob > chi2 = 0.0000
(Std. Err. adjusted for clustering on id)
regsmoke   Odds Ratio  Robust Std. Err.  z  P>z  95% Conf.  Interval 
c_wave  1.395727  .0443211  10.50  0.000  1.311507  1.485355 
_Isex_1  .9387056  .1553567  0.38  0.702  .6786638  1.298387 
parsmk  1.728383  .3118086  3.03  0.002  1.21361  2.461506 
_IsexXparsm_1  1.852877  .449922  2.54  0.011&  1.15121  2.98221 
_cons  .0995425  .0114585  20.04  0.000  .0794375  .124736 
Association within males
regsmoke  Coef.  Std. Err.  z  P>z  95% Conf.  Interval 
(1)  .5471863  .1804048  3.03  0.002  .1935993  .9007733 
Association between females
regsmoke  Coef.  Std. Err.  z  P>z  95% Conf.  Interval 
(1)  1.163926  .1623436  7.17  0.000  .845738  1.482113 
GEE populationaveraged model:
 Number of observations: 8498
 Number of groups: 1702
 Correlation structure: Unstructured
Results: The associations between parental smoking and adolescent smoking are presented as Odds Ratios with 95% confidence intervals. Differences in associations are observed between males and females, emphasizing the importance of considering genderspecific effects.
Question 3a
regsmoke  Coef.  Std. Err.  z  P>z  95% Conf.  Interval 
_Isex_1  .2032646  .3309636  0.61  0.539  .4454121  .8519414 
c_wave  .6957047  .1090732  6.38  0.000  .4819252  .9094843 
_IsexXc_wav_1  .3547225  .1486396  2.39  0.017  .0633943  .6460507 
_cons  6.140336  .3543214  17.33  0.000  6.834793  5.445878 
Logisticnormal randomintercept model:
Results: Fixed effects coefficients differ from the marginal model, emphasizing the impact of individual participantlevel random effects on estimates. This variation is crucial in understanding participantspecific effects.
>Question 3b
sigma_u  5.160847  .2861546  4.629394  5.753309 
rho  .89006  .0108514  .866921  .9095953 
Likelihoodratio test of rho=0: chibar2(01) = 2326.41 Prob >= chibar2 = 0.000
Random Effects:
 Random intercept standard deviation: 5.160847
 Intraparticipant correlation (rho): 0.89006
The high intraparticipant correlation indicates that participants with similar characteristics tend to have similar outcomes, underscoring the significance of individuallevel factors.
Question 3c
Weighted Average Probability:
 Weighted Average Probability of Smoking: 0.4427
 The coefficient associated with the probability: 0.3098
The weighted average probability provides insights into the predicted probability of smoking for men in year zero, considering assigned weights.
In summary, the analysis employs GEE models and a logisticnormal randomintercept model, highlighting the nuances in associations and emphasizing the importance of considering individuallevel factors in understanding smoking behaviour among adolescents.