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In his book on Applied Longitudinal Data Analysis for Epidemiology, page 60 there is an equation that describes a generalized estimating equations (GEE) model.

" This equation models the relationship between an outcome variable ($Y$) and a set of covariates ($X_1$, $X_2$ ... $X_j$) for $i$ persons at different timepoints $t$. It adjusts for repeated measurements (i.e. within-subject correlation) by assuming a 'working' correlation structure for the repeated measurements of the outcome variable $Y$ (i.e. the correlation structure $CORR$).

With GEE analysis the relationships between the variables of the model at different time-points are analyzed simultaneously.The estimated $\beta_1$ reflects the longitudinal relationship between the outcome variable $Y$ and the corresponding covariates $X$:

$$Y_{it} = \beta_0 + \sum_{j=1}^J \beta_{1j}\chi_{itj} + ... + CORR_{it} + \epsilon_{it}$$

where $Y_{it}$ are observations for subject i at time t, $\beta_0$ is the intercept, $X_{ijt}$ is the covariate j for subject i at time t, $\beta_1$ is the regression coefficient for covariate j, J is the number of covariates, $CORR_{it}$ is the working correlation structure, and $\epsilon$ is the “error” for subject i at time t. "

Is the 1 in beta1 just there to denote that it is related to vector of the first covariate? If so, why does it not also say X1?

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  • $\begingroup$ I'm a little puzzled by $CORR_{it}$ shown as an additive term in the mean model. You wouldn't, in a GEE, come up with any such term. It's an interesting mathfact that exchangeable and AR-1 correlation structures provides a broader class of models than random effects. And those effects would be partialled out in the GEE anyway. $\endgroup$
    – AdamO
    Jul 6, 2021 at 13:56
  • $\begingroup$ Thanks for your comment. That is how it is in the book, which does not focus at all on math or algebra, so probably that is why it is not correct. In Fitzmaurice (2014) the generalized estimating equation is vastly different, but also I really don't understand it: $$\sum_{i=1}^N D'_i V_i^{-1}(y_i - \mu_i = 0)$$ Where $V_i$ is the working covariance matrix and $D_i$ the gradient/derivative matrix. The way that Twisk describes the equation and concept behind it is a lot easier to grasp :) $\endgroup$
    – tcvdb1992
    Jul 6, 2021 at 15:22
  • $\begingroup$ you've got an error in your formula. The parens should close after the $\mu_i$ term. Anyway, only the $\mu_i$ term contains the mean model. The $V_i$ and its particulars are considered nuisance parameters. $\endgroup$
    – AdamO
    Jul 6, 2021 at 15:29

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Yes, basically. The 1 subscript just means that vector of (time independent) betas comprise the "slope" terms in the mean model of the GEE. Why not subscript with $X_1$? Then you'd have a double subscript, and it's hard to read in printed text if not impossible to typeset.

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  • $\begingroup$ I don't think I completely follow. Am I correct in assuming that $X_{itj}$ is, for $j=1$, a vector of the covariate $X_{j=1}$ 'over time' (i.e. measured at each timepoint)? Then in the denotation of the equation, why not just lose the '1' in $\beta_1$? What does it add if the $j$ subscript already associates it with the corresponding covariate? In other words my question is: am I right in assuming that every covariate $X_1, X_2 ... X_j$ is linked to $Y$ by $\beta_1, \beta_2 ... \beta_j$? $\endgroup$
    – tcvdb1992
    Jul 6, 2021 at 15:08
  • $\begingroup$ @tcvdb1992 ah, no $X_{itj}$ is the observed covariate value for subject $i$ at time $t$ for variable $j$. $\endgroup$
    – AdamO
    Jul 6, 2021 at 15:27
  • $\begingroup$ Thanks by the way for your responses. I see now that it's not a vector indeed, as there is a subscripted $t$ for each timepoint. Still I don't see why you wouldn't just lose the subscripted 1 in $\beta_{1j}$, as by the subscripted $j$ you already know that this is the beta-value for the $j$th covariate... right? $\endgroup$
    – tcvdb1992
    Jul 7, 2021 at 15:30
  • $\begingroup$ @tcvdb1992 you're right. there is no loss of generality to express $Y$ as an atomic, or vector, or matrix. It's just a matter of preference. The most critical point is clarifying the structure of $\Sigma$ (or $\epsilon$). $\endgroup$
    – AdamO
    Jul 7, 2021 at 15:36
  • $\begingroup$ @ AdamO Thanks again. I think I got confused about the fact that $\beta_{1j}$ is a vector, while $X$ is not. Thanks to your replies I now see that this is because for all timepoints $t$ for covariate $X$ there is one regression coefficient $\beta$, so that is why it gets subscripted with a 1 and there is no subscript $t$. Thanks again! $\endgroup$
    – tcvdb1992
    Jul 8, 2021 at 6:25

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