## Problem Description:

The GEE Models assignment involves analyzing a Generalized Estimating Equations (GEE) population-averaged 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 population-averaged 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 (auto-regressive 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 population-averaged 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 population-averaged 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 population-averaged 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 gender-specific 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 |

## Logistic-normal random-intercept model:

Results: Fixed effects coefficients differ from the marginal model, emphasizing the impact of individual participant-level random effects on estimates. This variation is crucial in understanding participant-specific effects.

>Question 3b

sigma_u | 5.160847 | .2861546 | 4.629394 | 5.753309 |

rho | .89006 | .0108514 | .866921 | .9095953 |

Likelihood-ratio test of rho=0: chibar2(01) = 2326.41 Prob >= chibar2 = 0.000

## Random Effects:

- Random intercept standard deviation: 5.160847
- Intra-participant correlation (rho): 0.89006

The high intra-participant correlation indicates that participants with similar characteristics tend to have similar outcomes, underscoring the significance of individual-level 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 logistic-normal random-intercept model, highlighting the nuances in associations and emphasizing the importance of considering individual-level factors in understanding smoking behaviour among adolescents.

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