# Surprisingly large difference between conditional and marginal effects estimates

I am performing a logistic mixed effects regression on some data I have. There are 201 participants answering a question over 6 time points. The model includes 4 fixed effects: X1, X2, X3, and Time. It also includes a random intercept and a random slope for time, under grouping variable id.

I am interested in the marginal effect of time on the response y. Since I have a random slope, I am using package GLMMadaptive that allows specification of a mixed effects model (thus allowing specification of a random slope) and then allows estimation of the marginal effect as well.

I know in general the conditional and marginal effects can be quite different. However, when I run the code the differences are surprisingly(?) large. For example, whereas the conditional OR for a 1 unit change in time is over 13% increase, the corresponding marginal OR is less than 1% increase. Although I understand why the estimates usually differ, I find such large difference heavily unintuitive. It is very odd that for each specific person the estimated likelihood of yes will increase between the first and last (6th) time point by around ~85%, whereas the corresponding average population likelihood will increase by less than 5%.

Can someone point me to an explanation? What can be the source of such large discrepancy?

Here is my code. First the conditional estimates:

library(GLMMadaptive)
mm = mixed_model(fixed = Y ~ X1 + X2 + X3 + time, random = ~1+time|id, family = "binomial", data = dat )
summary(mm)

Call:
mixed_model(fixed = Y ~ X1 + X2 + X3 + time, random = ~1 + time |
id, data = dat, family = "binomial")

Data Descriptives:
Number of Observations: 1206
Number of Groups: 201

Model:
family: binomial

Fit statistics:
log.Lik  AIC  BIC
-1259 2533 2560

Random effects covariance matrix:
StdDev    Corr
(Intercept)  13.3281
time          1.2752 -0.4208

Fixed effects:
Estimate Std.Err z-value p-value
(Intercept)  -11.497  0.7559 -15.211 <0.0001
X1             0.475  0.1377   3.452  0.0006
X2             0.334  0.0868   3.849  0.0001
X3             0.194  0.2026   0.958    0.34
time           0.123  0.0640   1.925    0.05

Integration:

Optimization:
method: hybrid EM and quasi-Newton
converged: TRUE


And the marginal effects:

marginal_coefs(mm, std_errors = TRUE)

Estimate Std.Err z-value p-value
(Intercept)  -1.4855  0.1595  -9.313 <0.0001
X1            0.0629  0.0195   3.220   0.001
X2            0.0454  0.0118   3.844  0.0001
X3            0.0245  0.0271   0.904   0.366
time          0.0089  0.0147   0.602   0.547


The reason why you see this big difference is because the variance of your random effects is quite big. To see why is this happening, check slide 334 in my course notes for Repeated Measurements. That is, in the case you have only random intercepts, you have the relation $$\beta^M = \frac{\beta^{SS}}{\sqrt{1 + 0.346 \sigma_b^2}},$$ where $$\beta^M$$ are the coefficients with a marginal interpretation, $$\beta^{SS}$$ the coefficients with conditional on the random effects interpretation, and $$\sigma_b^2$$ the variance of the random intercepts. Hence, the bigger the variance, the larger the difference between the two sets of coefficients.
• Have you got a source for the above adjustment factor of $(1+0.346\sigma^2)^{-1/2}$? In Agresti's book on categorical analysis, an adjustment factor of $(1+0.6\sigma^2)^{-1/2}$ is given instead, see math.ntnu.no/emner/TMA4315/2019h/agresti.pdf, p. 499, eq. 12.8. Nov 9, 2022 at 15:06