Number Of Clusters Versus Cluster Size In PQL

Why Cluster Count and Cluster Size Are Different

In a binary generalized linear mixed model, a large total number of responses does not tell the whole story. One design might collect a few observations from each of many independent clusters; another might collect many observations from only a few clusters. Both designs can have the same total sample size, but they do not provide the same information about the population of clusters or the same protection against the approximation error in penalized quasi-likelihood (PQL).

The distinction is especially important when interpreting a PQL fit. More clusters can make estimates based on the PQL criterion less variable across repeated samples. But if every cluster remains small, each cluster still supplies limited binary information, and PQL still relies on a local approximation to a nonlinear mean. Replicating that approximation many times does not make it exact.

Definition: Let \(J\) be the number of independent clusters, \(n_j\) the number of Bernoulli observations in cluster \(j\), and \(N=\sum_{j=1}^{J} n_j\) the total number of observations. Adding clusters increases \(J\), usually by sampling more independent realizations of the random effects. Increasing cluster size increases one or more \(n_j\), adding observations that share a cluster’s random effect.

Model and Information Sources

For clarity, consider a random-intercept logistic model. For observation \(i\) in cluster \(j\), let \(Y_{ij}\) equal 1 for an event and 0 otherwise. Conditional on the cluster random effect \(b_j\), assume

Y_{ij}\mid b_j \ \text{are independent Bernoulli}(p_{ij}),\qquad \operatorname{logit}(p_{ij})=\mathbf{x}_{ij}^{\mathsf T}\boldsymbol{\beta}+b_j,\qquad b_j\overset{\mathrm{iid}}{\sim}N(0,\tau^2).

Here the clusters are independent, the covariates are treated as observed, and the conditional Bernoulli model and normal random-effect distribution are assumed correctly specified. The random intercept induces dependence among responses in the same cluster after \(b_j\) is integrated out. PQL replaces the nonlinear conditional mean by a local working approximation and fits a corresponding penalized working model.

Each additional cluster provides another draw from the random-effect distribution. That replication is central to estimating \(\tau^2\) and to learning how cluster-level heterogeneity is distributed. More observations within an existing cluster can reveal that cluster’s event probability more precisely, but they are not new independent draws of \(b_j\). In particular, increasing \(n_j\) does not turn \(J\) observed random effects into more than \(J\) independent draws from their population distribution.

The added observations are not useless or equivalent to duplicates. They can improve information about fixed effects and the particular cluster’s conditional event probability. Their marginal contribution is often smaller than that of observations from new clusters because responses within a cluster are correlated. The balance depends on the model, covariate pattern, event frequency, and random-effect variance; there is no universal conversion of a within-cluster observation into a fraction of a new cluster.

A Design Calculation With Equal Total Sample Sizes

A simple calculation makes the design distinction concrete. Suppose clusters are equally sized and outcomes have common marginal probability \(p\) and within-cluster intraclass correlation \(\rho\). For the overall observed proportion \(\bar{Y}\), independence between clusters and exchangeability within a cluster imply

\operatorname{Var}(\bar{Y}) =p(1-p)\left\{\frac{\rho}{J}+\frac{1-\rho}{N}\right\}, \qquad N=Jn.

This is a variance formula for the overall proportion under the stated exchangeable structure, not a general standard-error formula for every fixed-effect estimate in a GLMM. It shows why, at fixed \(N\), reallocating observations toward more independent clusters can reduce the contribution associated with within-cluster dependence.

Worked Example: Same Total, Different Number of Clusters

Suppose two hypothetical designs each collect \(N=120\) binary responses, with marginal event probability \(p=0.20\) and within-cluster correlation \(\rho=0.15\). Design A has \(J=60\) clusters of size \(n=2\); Design B has \(J=12\) clusters of size \(n=10\). Assume equal cluster sizes and independent clusters.

For Design A, the variance formula gives

\operatorname{Var}(\bar{Y}_A) =(0.20)(0.80)\left\{\frac{0.15}{60}+\frac{0.85}{120}\right\} =0.16(0.002500+0.007083) =0.001533.

Thus \(SE(\bar{Y}_A)=\sqrt{0.001533}=0.0392\). For Design B,

\operatorname{Var}(\bar{Y}_B) =(0.20)(0.80)\left\{\frac{0.15}{12}+\frac{0.85}{120}\right\} =0.16(0.01250+0.007083) =0.003133,

so \(SE(\bar{Y}_B)=\sqrt{0.003133}=0.0560\). In this particular comparison, the design with 60 clusters has a smaller standard error for the overall proportion, despite the same total number of responses. The calculation does not prove that Design A is always preferable for every GLMM parameter: covariates, costs, cluster-size variation, and the target of inference also matter. It does show why counting individual responses alone can overstate how much independent information a clustered sample contains.

Why More Small Clusters Do Not Remove PQL Approximation Bias

PQL’s central approximation is local: it uses a working response and weight based on the mean and its derivative at the current linear predictor. For a binary logit model, if \(\eta_{ij}=\mathbf{x}_{ij}^{\mathsf T}\boldsymbol{\beta}+b_j\) and \(p_{ij}=\operatorname{logit}^{-1}(\eta_{ij})\), the usual working response and weight have the forms

z_{ij}=\eta_{ij}+\frac{Y_{ij}-p_{ij}}{p_{ij}(1-p_{ij})}, \qquad w_{ij}=p_{ij}(1-p_{ij}).

These quantities support a locally weighted linear mixed-model update. They do not make the logistic mean linear over the full distribution of \(b_j\). When clusters are small, each cluster contributes only a small number of binary outcomes to that local approximation. Increasing \(J\) repeats the approximation across more clusters; it does not increase the number of observations used to form the approximation within any one cluster.

A Taylor expansion illustrates the curvature involved. Write \(g(\eta)=\operatorname{logit}^{-1}(\eta)\), and consider the marginal probability for a random-intercept observation with fixed linear predictor \(\eta\). For small \(\tau^2\), expand \(g(\eta+b)\) around \(b=0\), where \(b\sim N(0,\tau^2)\). Taking expectations uses \(E(b)=0\), \(E(b^2)=\tau^2\), and the vanishing odd normal moments:

E\{g(\eta+b)\} =g(\eta)+\frac{1}{2}g''(\eta)\tau^2+O(\tau^4).

For the logistic function, \(g'(\eta)=g(\eta)\{1-g(\eta)\}\) and \(g''(\eta)=g(\eta)\{1-g(\eta)\}\{1-2g(\eta)\}\). Therefore the second-order term is generally nonzero. A first-order treatment that ignores curvature cannot be made exact merely by taking more independent clusters. This expansion is an illustration of the nonlinear integration issue, not an exact formula for the bias of every PQL estimator.

Worked Example: Curvature Does Not Depend on the Number of Clusters

Consider a hypothetical random-intercept model at a covariate value with \(p=g(\eta)=0.20\) and \(\tau^2=0.64\). The logistic second derivative is

g''(\eta)=0.20(1-0.20)(1-2(0.20)) =0.20(0.80)(0.60)=0.0960.

The second-order term in the marginal-mean expansion is

\frac{1}{2}\tau^2g''(\eta) =\frac{1}{2}(0.64)(0.0960)=0.03072.

Thus the second-order expansion is \(E\{g(\eta+b)\}\approx 0.20+0.03072=0.23072\), with higher-order terms omitted. This is a local approximation, not an exact marginal probability. Its purpose is to show that curvature can matter at these parameter values. Changing \(J\) from 20 to 200 does not change this term: it depends on the mean location and random-effect variance, not on how many clusters were sampled. More clusters can make the data reveal the systematic discrepancy more clearly, but cannot remove the omitted curvature from the PQL working approximation.

What Larger Clusters Can and Cannot Supply

Increasing cluster size changes the information available about each cluster’s event probability. For example, with a fixed conditional probability \(q\), the likelihood of observing all successes in a cluster is \(q^{n_j}\). If \(q=0.80\), the likelihood ratio for all successes under \(q=0.80\) versus \(q=0.50\) is \(1.6^{n_j}\): it is \(1.6^2=2.56\) for two observations and \(1.6^{10}\approx109.95\) for ten. More observations can therefore distinguish conditional probabilities more sharply within a cluster.

That gain does not mean that a few very large clusters substitute fully for many independent clusters. With only a small \(J\), the sample still contains few independent random-effect realizations. Large \(n_j\) can make the realized cluster probabilities clearer, while uncertainty about the population distribution of those probabilities remains. Furthermore, PQL’s local approximation and the nonlinear relationship between the random effects and response probabilities continue to require scrutiny.

Worked Example: Many Observations Within Few Clusters

Suppose a hypothetical study has 12 clusters with 10 binary responses each, so \(N=120\). Increasing each cluster to 20 responses would produce \(N=240\), but would still leave only 12 independent draws \(b_1,\ldots,b_{12}\) from the random-effect distribution.

The second design can provide more precise information about the conditional event probability in each of those 12 clusters. It does not provide 240 independent random effects; it provides 12. If the scientific goal includes estimating \(\tau^2\), the number of independent clusters remains a key constraint. If instead the design adds 12 new clusters of size 10, it has 24 independent clusters, which adds replication of the random-effect distribution even though the original clusters have not become more informative individually.

Neither allocation automatically fixes PQL bias. The first increases within-cluster information while retaining a small number of random-effect draws; the second increases independent replication but retains small clusters and their local approximation limitations. Which design is preferable depends on the target parameters and practical costs, and should not be decided from \(N\) alone.

Design Comparisons, Diagnostics, and Limitations

The main comparison is not “many observations versus few observations,” but “more within-cluster information versus more independent clusters.” Likelihood-based integration methods target the nonlinear random-effect model without replacing the conditional mean by the same PQL working linearization. They are alternatives to consider when approximation error is a concern, although they do not remove limitations caused by a weak design or a misspecified model. The issue here is not that PQL is always unusable; it is that a large number of small clusters should not be mistaken for a guarantee that its approximation bias is negligible.

Useful design and fit checks include:

Count independent clusters, not just responses. A very large \(N\) can still arise from a small \(J\), limiting replication for random-effect variation.
Inspect the cluster-size distribution. A few large clusters may dominate within-cluster information, while many tiny clusters may contribute little detail about their individual event probabilities.
Consider event information within clusters. With binary outcomes, small clusters can easily contain all zeros or all ones. This limits how much each cluster reveals about its conditional probability.
Assess whether the approximation is under strain. Strong random-effect variation, fitted probabilities near zero or one, and sparse within-cluster outcomes make local linearization more consequential.

These are reasons to interpret a PQL result cautiously, not standalone tests that establish bias or diagnose its exact size. The expansion above is local, and actual PQL performance depends on the covariates, random-effect distribution, cluster sizes, event rate, and estimand. Increasing cluster size can reduce some sampling limitations and may improve behavior in particular settings, but it does not guarantee negligible approximation error. Increasing cluster count can reduce sampling variability around the approximate target, but it does not generally move that target onto the exact model parameter.

Common Mistake

Common mistake: “There are hundreds of clusters, so PQL’s bias must be negligible.” A large \(J\) supplies independent replication, but if each \(n_j\) remains small, the same local nonlinear approximation is still being applied to small binary clusters. More replication can reduce random sampling fluctuation without eliminating systematic approximation error. Conversely, a very large \(n_j\) does not create additional independent draws from the random-effect distribution.

Key takeaway: \(J\) and the \(n_j\) answer different design questions. More clusters primarily add independent cluster-level replication; larger clusters add more response information conditional on existing random effects. Neither dimension, considered alone, guarantees that PQL’s approximation error is negligible.

Check Your Understanding

Answer each question using the distinction between independent clusters and observations within clusters.

A study increases from 20 clusters of size 5 to 20 clusters of size 10. What increases, and what does not increase in the random-effects sampling structure?
At fixed \(N\), why might 50 clusters of size 4 provide a different amount of information from 10 clusters of size 20?
In the expansion \(E\{g(\eta+b)\}\approx g(\eta)+\frac{1}{2}g''(\eta)\tau^2\), which quantities determine the displayed curvature term? Does \(J\) appear in it?
Is a large number of small clusters sufficient to conclude that PQL approximation bias is negligible? Explain.

Solutions. (1) The number of Bernoulli observations per existing cluster increases, so conditional cluster probabilities can be informed by more outcomes. The number of independent random-effect draws remains 20. (2) Both designs have \(N=200\), but the first has more independent clusters and therefore more cluster-level replication; the second has more observations informing each of fewer cluster probabilities. Their information depends on the target and within-cluster dependence. (3) The term depends on the mean location through \(g''(\eta)\) and on the random-effect variance \(\tau^2\); \(J\) does not appear. (4) No. Large \(J\) can reduce sampling variability, but if clusters remain small the local PQL approximation is still applied to small binary clusters and its systematic error need not disappear.