# Do small odds ratios observed at multiple time points equate to a large odds ratio when averaged across time points?

I am testing the relative odds of two groups, placebo vs active treatment, guessing that they received active treatment. These guesses were made at four time points, 4, 8, 12, and 24 weeks into a clinical trial.

This is a graph of the percentage of people in each group who guessed they received active treatment over time When I perform a series of logistic regressions, regressing guessed treatment on actual treatment at each of these time points in isolation, using this sort of syntax

glm(guess ~ group, data = df, family = binomial, subset = week == 4)


I get these coefficients and odds ratios

  week     coef       or
1    4 1.345286 3.839286
2    8 2.063441 7.873016
3   12 1.652497 5.220000
4   24 2.197225 9.000000


Consistent with the graph, the odds of a person guessing that they received the active treatment are consistently higher in the active group than the placebo group (a failure of blinding basically). The odds ratio starts at around 4 on week 4 but by week 24 the odds ratio is 9.

When I perform a longitudinal logistic regression, with participant id as the random factor, and using simple coding so that the estimates for each factor, group and week (the time factor), represent the effect of that factor on log-odds averaged across all levels of the other factor (see here), like this...

# write function for creating simple coding matrix
simpMatFunct <- function (nLevels) {
k <- nLevels
mat <- matrix(rep(-1/k, k*(k-1)), ncol = k-1) + contr.treatment(k)
return(mat)
}

# run the glmer, creating simple coding matrix for each factor by using the matrix function above
glmer(guess ~ group*week + (1|id),
data = w24,
family = binomial,
contrasts = list(group = simpMatFunct(2), week = simpMatFunct(4)),
control = glmerControl(optimizer = "bobyqa"))


...these are the coefficients

Fixed Effects:
(Intercept)           group2         weekFac2         weekFac3
10.2474           5.0411           2.8542          -1.8699
weekFac4  group2:weekFac2  group2:weekFac3  group2:weekFac4
0.7396           7.8657           0.8067           9.5187


The group2 coefficient, which represents the between-group difference in odds of participants guessing they recieved the active treatment, averaged across all time points, when exponentiated..

exp(5.0411)
 154.64


yields an odds ratio of 150!!

Could this estimate be correct? Does the cumulative effect of consistent smaller odds ratios over time translate to a much larger odds ratio when all time points are considered together? or am I doing something wrong?

• Doesn't the glmer model produce subject-specific odds ratios? You would need to compute marginal effects for what you need. Also, if there is an interaction between group and week, doesn't it make more sense to report the marginal effect of group separately at each time point? See this post for some pointers: stats.stackexchange.com/questions/397578/…. – Isabella Ghement Mar 15 '19 at 22:48
• If you're averaging things out across time points, you're missing out on the fact that different things are happening at different time points - at least that's what it looks like to me! – Isabella Ghement Mar 15 '19 at 22:51
• This link might come in handy in terms of understanding subject-specific odds ratios versus marginal odds ratios : pdfs.semanticscholar.org/bc38/…. The glmer models are notorious for providing different interpretations whether you look at subject-specific effects or marginal effects! – Isabella Ghement Mar 16 '19 at 1:44
• Thank you @Isabelle Ghement. I'm a bit concerned that it took me this long to learn this. However I think I've managed to avoid reporting an incorrect GLMM simply because the odds ratios have always seemed wrong. I will take solace from the fact that at least now I have beem set upon the right path. – llewmills Mar 16 '19 at 1:47
• Can you fit your model just using the default dummy coding available in R? – Isabella Ghement Mar 16 '19 at 1:55