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In A Comparison of Cluster-Specific and Population-Averaged Approaches for Analyzing Correlated Binary Data, Neuhas, Kalbfleisch, and Hauck state:

"With the cluster-specific approach, the probability distribution of $Y_{ij}$ is modelled as a function of the covariates $X_{ij}$ and parameters $\alpha_{i}$ specific to the $i$th cluster."

I am having trouble getting an intuitive sense of what this means when coefficients are expressed as a single coefficient in a regression output.

For example in the analysis referred to in this post, where I tested the effect of week in treatment (measured at 4 time points per individual, 4, 8, 12, and 24 weeks) and experimental group (two levels: placebo vs active) on the odds of people guessing that they had been allocated to the active group, specified in a binomial generalised linear mixed effects model in the lme4 package in R like so:

glmer(guess ~ group * week + (1 | id), 
      data = w24, family = binomial())

The clusters in this model are participant id. The coefficients for the fixed effects were

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

I just fundamentally don't understand how you can get a single estimate that is "specific to the $i$th cluster", when there are multiple clusters/participants.

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The point that is made in this paper is with regard to the conditional versus marginal interpretation of the regression coefficients. Namely, because of the nonlinear link function used in the mixed effects logistic regression, the fixed effects coefficients have an interpretation conditional on the random effects. Most often this is not the desirable interpretation that relates to groups of individuals. You may find more information regarding this issue here and here.

On the contrary, in linear mixed models and because the link function is the identity, you do not have this problem.

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  • $\begingroup$ Thank you @Dimitris Rizopoulos, given your expertise in longitudinal models I was hoping you would weigh in on this one. Neuhaus et al. say "Population-averaged comparisons on the other hand make no specific use of within-cluster comparisons for cluster-varying covariates and substantially underestimate within-cluster risks. Related to this, population averaged models cannot provide estimates of changes within individuals over time; these are often quantities of central interest in longitudinal studies." Does this mean in my model above: (i) the coefficients for the between-subject factor... $\endgroup$ – llewmills Jan 13 at 19:54
  • $\begingroup$ ...should be interpreted after being re-calculated with the marginal interpretation (using, for example, the marginal_coefs() function in your GLMMadaptive package) but that the within-subjects factor (week in my model above) can (and should) be interpreted in the original cluster-specific model? $\endgroup$ – llewmills Jan 13 at 19:58
  • $\begingroup$ @llewmills It depends on what you’re interested in. If you want to speak about how the outcome changes for specific subjects, then you want the subjects-specific fixed effects for week. But if instead you want to quantify the difference in the log odds for the group of subjects at week $k$ versus the group of subjects at week $k+1$, then you need the marginal coefficient for week. $\endgroup$ – Dimitris Rizopoulos Jan 13 at 22:04
  • $\begingroup$ Ok thank you. So if you want to make inferences for a group of people the time coefficient still needs to be made marginal. That makes life easier then. $\endgroup$ – llewmills Jan 14 at 1:48
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I agree that this can be a little confusing. Some authors avoid setting it up in this way. The important point is that the $\alpha_{i}$ are not estimated individually, instead they are subsumed into a general model and the usual assumption is that they are normally distributed, with an unknown variance, which is to be estimated.

Focusing on the main point:

parameters $\alpha_{i}$ specific to the $i$th cluster

and translating this to something a bit more usual:

$$ y_i = X_i \beta + Z_i b_i + \epsilon_i, \text{ }\text{ }\text{ }\text{ } i=1,...,N $$

where $b_i$ is a vector of random effects and $Z_i$ is the design matrix for the $i$th cluster, we then combine vectors $y_i$ and matrices $X_i$ into the $\Sigma n_i \times 1$ vector $y$ and $\Sigma n_i \times m$ matrix $X$, and letting $Z = \text{diag}(Z_1,...,Z_N)$ the model can be written as

$$ y = X \beta + Z b + \epsilon$$

which is the usual mixed model equation.

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  • $\begingroup$ Thank you @Robert Long. So how then do you report the results in a paper? What do you say? If the coefficients were marginal we could say "There was a exp(5.0411) = 154.64-fold increase in the odds that participants in the active group guessed that they had been given the active drug". But with a cluster-specific I don't really understand what this number means, what use it is, and therefore how to report it. $\endgroup$ – llewmills Jan 13 at 19:42
  • $\begingroup$ Later on in their paper, Neuhaus et al. say "the cluster-specific model presupposes the existence of latent risk groups indexed by $\alpha_i$ and parameter interpretation is with reference to these groups. No empirical verification of this statement can be available from the data unless the latent risk groups can be idenitifed". So from what I gather I could say "There was a exp(5.0411) = 154.64-fold increase in odds that participants in the same latent risk group guessed that they had been given the active drug". But I can't identify what that risk group is, so what use is the coefficient? $\endgroup$ – llewmills Jan 13 at 20:01
  • $\begingroup$ I would have to read the paper to properly answer these other questions. There is a correspondence between mixed models and latent variable models in longitudinal/growth models in the sense that we don't actually observe the intercept and slope or other random effects, hence they can be considered "latent". I don't know if that is what is meant in your quotation from the paper, but it might be. $\endgroup$ – Robert Long Jan 13 at 20:05
  • $\begingroup$ Thank you @Robert Long but my first question, how to report the coefficients from cluster-specific model is independent from the paper. How does one interpret these coefficients, in a paper for example? $\endgroup$ – llewmills Jan 13 at 20:23
  • $\begingroup$ Coefficients from a mixed model are just the fixed effects (and the variances of the random effects, if you consider everything that is estimated to be a coefficient) $\endgroup$ – Robert Long Jan 13 at 20:37

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