Appropriateness of a GLM to model binomial data over time

Question

I have (for example) 2,000 observations from around 600 individuals measured at 5 time points. At each time point, the observation is associated with a value of 1 or 0.

I'm trying to model the change in the probability of an observation being a 1 over time.

The way I've tried to do this is to use a GLM with a binomial response distribution and a logit link function; the results are an intercept and slope, both in log-odds units.

Is this a viable way to model binomial data over time (longitudinally)? In particular, I have in mind are:

• Some of the issues related to the use of repeated measures ANOVA for modeling longitudinal data with normally distributed response data - that the variance of the values must be nearly equal at different time points.
• GLM-specific issues, like over-dispersion.

EDIT:

Here's an example of some output for a possible model fit using glm() in R:

Call:
glm(formula = code_0 ~ key_num, family = "binomial", data = dd)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-0.7052  -0.5697  -0.4567  -0.3643   2.5269  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -0.79330    0.16177  -4.904 9.40e-07 ***
key_num     -0.47150    0.06455  -7.305 2.78e-13 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 1401.5  on 1881  degrees of freedom
Residual deviance: 1342.2  on 1880  degrees of freedom
AIC: 1346.2

Number of Fisher Scoring iterations: 5

Here is the output from using glmer() (also in R):

Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
 Family: binomial  ( logit )
Formula: code_0 ~ key_num + (1 | student_ID)
   Data: dd

     AIC      BIC   logLik deviance df.resid 
  1325.1   1341.7   -659.6   1319.1     1879 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-0.9391 -0.2742 -0.2074 -0.1368  4.4166 

Random effects:
 Groups     Name        Variance Std.Dev.
 student_ID (Intercept) 2.881    1.697   
Number of obs: 1882, groups:  student_ID, 845

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -1.24170    0.27151  -4.573  4.8e-06 ***
key_num     -0.57235    0.08629  -6.633  3.3e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
        (Intr)
key_num -0.390

Answer 1

You need to analyse this as a repeated measures model. Use lmer from the lme4 package and include a random effect for the subject -- the entity you have 1000 of, and which you observe at 5 points.

With a binomial response, the variance of the values will be a function of the probability, but the link function takes care of that.

Yes ... there are conditions to take into account, but the model will be fairly robust against them. At least, if you just want to test for a change in time.

If the fit fails to converge, that probably means that the random effect is not significant. In that case, you can drop it and return to the GLM model you have already tried.