# Generalized linear mixed models, extracting effects at all time points using summary, for each group./ [closed]

A bit of lenghty title, but summarizes my problem well enough. Currently I am part of longitudinal-data classes and one of my tasks consists of extract the treatment effects of all the time points for two drugs (old, and a new one) from a given R summary, based on lme4 package (in particular glmer model). To describe the data better:

1. y is response, i.e 1- improvement in condition, 0 - lack of said improvement,
2. id as can id of subject
3. severe is 0-1 factor, informing abou severity of condition (0 is not severe)
4. drug 0-1 factor, where 0 denotes old drug
5. time is indicator of measurment time (0-baseline, 1-time 1, 2- time 2)

Then in R it corresponds to model:

model.glmer <- glmer(y ~ severe + drug * time + (1 | id), family = binomial(), data = model.dat)


which in mathematical terms reads as: $$logit[Pr(Y_{ij}= 1|b_{i1})]=η_{ij}=β_1+β_2severe_i+β_3drug_i+β_4time_{ij}+β_5drug_i∗time_j+b_{ij}$$

Where $$\beta_1$$ is intercept, so in our case I suppose it's log odds ratio at time 0, with mild condition, old drug. Then remaining $$\beta$$'s are regression parameters and $$b_{ij}$$ is a random intercept that allows a different baseline probability of illness for each subject

Then using summary we get following output: From which I want to derive the treatment effects of all the time points for each drug, so to specify in mathematical terms for instance:

1. Treatment effect at time 1

For drug=0 and time=1,

$$logit[Pr(Y_{i1}= 1|b_{i1})]=β_1+β_2severe_i+β_4+b_{i1}$$

For drug=1 and time=1,

$$logit[Pr(Y_{i1}= 1|b_{i1})]=β_1+β_2severe_i+β_3+β_4+β_5+b_{i1}$$

Thus, difference will be equal to $$β_3+β_5$$.

So if I am correct, given summary it would be just: -0.05967+1.01817, so to name just the estimated regression parameters from summary? And for the following times, when value increases it will be $$β_3+2β_5$$ equal to -0.05967+2*1.01817 for each and every subject respectively at time 1 and 2? Am I right, or am I getting something totaly wrong? Because I am kind of lost, and quite can't wrap my mind around this model so far. Also how would I get from this summary time trend for new/standard treatment? As I don't see any "estimate" of random intercepts, only estimated variance...

And how can I later get the estimated odds ratio (confidence interval) of remission comparing a patient on the new treatment to a patient on the standard treatment with the same random intercept and severity of initial diagnosis?

I know those questions are probably very basic and probably I'll get downvoted for them, but if any kind soul would teach how to read those damn outputs, maybe explain a little bit this model, and derive things from them, I will be more than gratefull for it.

Edit: to clarify I'll link dataset, as well as all the operations which lead to summary:

depress.dat <- read.table('C:/Users/Someone/Documents/R/Datasets/depress.txt',na.strings=".")
names(depress.dat) <- c("id", "y", "severe", "drug", "time")
depress.glmer <- glmer(y ~ severe + drug*time + (1|id),family = binomial,data=depress.dat)
summary(depress.glmer)


## closed as off-topic by Michael Chernick, Siong Thye Goh, Peter Flom♦May 26 at 12:23

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• Can you confirm that your time variable (well, actually all of your variables) are numeric and not factor variables in R? It will make a difference in the explanation and analysis. – StatsStudent May 25 at 16:52
• All of them are integer variables, and none of them was assumed in R as factor. In eddit I'll add dataset and all the functions I've used. – Kiwi May 25 at 16:55
• do you really have so many random effect $b_{ij}$. The # of random effect = # of measured response variable? – user158565 May 26 at 6:57
• I am pretty sure, that there is one $b_{i1}$ for each subject, as they allows for diffrent baselines. So assuming that each has 3 observations, # of random effect should equatl to # of measured response variables divided by 3. – Kiwi May 26 at 7:13

Based on your model

• The coefficient for severe is the log odds ratio for severe condition versus not severe for the same time and drug.
• The coefficient for drug is the log odds ratio between the two drugs for the same severity and at time 0.
• The coefficient for time is the log odds ratio for a unit increase of the time variable for the same severity and for the old drug.
• The coefficient for drug:time is the difference between the log odds ratios for a unit increase of the time variable of the new and the old drug.

A couple of other notes:

• You should be aware of the fact that the interpretation of the fixed-effects coefficients in mixed effects logistic regression is conditional on the random effects; for more info check here.
• It is better to fit the model using the adaptive Gaussian quadrature rather than the Laplace approximation.
• I am very grateful, Mr. Dimitris, as your answer cleared a lot for me. I'll try to go through your lecture notes linked in your response to the other question, you have so kindly shared. Nevertheless, If I can, I will ask a few more questions to you: - First of all as I understand, interpretation of each of those is based on subject-specific random effect, on which we condition, as you've said. Accounting for that, relating to my original question, if I want to compare treatment effect at said time, is my reasoning above valid? And how can I get odds ratio for remission, comparing treatments? – Kiwi May 25 at 20:17
• The difference between the two drugs at the same time and for the same severity will be the sum of the coefficient for drug and the coefficient of the interaction term. You could have a look at the emmeans package that streamlines such calculations. – Dimitris Rizopoulos May 26 at 10:05