# Repeated measurements, multiply assessed exposure with single outcome

We have data points from a prospective study in which participants were assessed 3 times during follow-up for an exposure of interest, and during a 4th follow-up they were assessed for an outcome of interest. All measures are continuously coded. We believe that there is a somewhat complicated mechanism underlying the possible association, however at this point in the analysis, we are curious whether there is any type of marginal association in which the "earlier" exposure is associated with the "later" outcome-- this language is key to not assuming that they have coincident associations.

In this fashion, we could look at any analysis in which the baseline exposure is regressed against the follow-up, or the most recent exposure is regressed against the follow-up. The exposure that we have assessed is an imperfect measure and there is substantial intraclass correlation. Therefore it would be nice to use a modeling approach which can come up with an estimated average of the exposure.

The current analytic approach is to merge the outcome to the three exposure assessments in a long dataset and use a GEE adjusting for age to account for repeated measures. We are interested, however, in inference about the cluster level association. Therefore, a GEE with a linear link is apt at estimating a first order trend.

Does the GEE give appropriate inference on this outcome or must we go about marginalizing estimates in some form?

Somehow, I am worried that we are artificially inflating the sample size since it is the same analytic approach that would be used if the outcome were assessed three separate times. However, since it is the same outcome, my belief is that the correlation between these observations should be higher, and the regression model should therefore "downweight" the highly correlated intracluster observations.

The linear model is of the fashion $$E[Y_{it} | X_{is} : s < t] = \beta_{i0} + \sum_{r=1}^s \beta_r X_{ir}$$
The process is simple: it is a matter of constructing regression models for the outcome adjusting for all relevant measures of the exposure as covariates. Since I only have 3 observations of the 'lagged' covariate in each member, I feel fairly confident I can use all three as adjustment values also adjusting for age at time of collection for the exposure (since a similar amount of study time elapsed between the measures for each participant). For inference, usually one considers multiple df tests. I have used linearHypothesis in the car package with the composite hypothesis argument , hypothesis.matrix=paste0('riskmeas', 1:3, ' = 0'). I have also used forest plots to show the CIs for each risk factor.