Assumptions And Dependence In Random Groups

Why Random Groups Change the Independence Assumption

In a one-way random-effects study, observations are organized into groups, such as patients within clinics or measurements made by different laboratories. A random group effect represents how each sampled group differs from the population average. Because two observations from the same group share that effect, they can resemble one another even if their individual errors are unrelated.

This distinction matters: the random-effects model assumes independent residual errors, but it generally does not assume that all observed responses are mutually independent. Treating observations within a group as independent can misrepresent the information in the data. To see why, first state the model and its assumptions precisely.

The One-Way Random-Effects Model

Let \(Y_{ij}\) be the response for observation \(j\) in group \(i\), where \(i=1,\ldots,I\) and \(j=1,\ldots,n_i\). The one-way random-effects model is

Here, \(\mu\) is the population mean response, \(A_i\) is the random deviation of group \(i\) from that mean, and \(\varepsilon_{ij}\) is the observation-specific residual error. The group effects are random because the observed groups are treated as a sample from a wider population of possible groups.

The assumptions can be understood in three related parts:

• Normality: the random group effects have a normal distribution, and the residual errors have a normal distribution. Consequently, responses within a group are conditionally normal given that group’s effect.
• Independence: effects for different groups are independent; residual errors are independent across observations; and residual errors are independent of all group effects. This is a model for the random components, not a claim that all observed responses are independent.
• Constant residual variance: every residual has the same variance \(\sigma^2\), regardless of its group or observation index. The random group effects also have a common variance \(\sigma_A^2\) across groups.

The phrase “constant variance” needs care. Conditional on a group effect, every response has residual variance \(\sigma^2\). Across the population of groups, an individual response also varies because its group effect varies. Its marginal variance is therefore \(\sigma_A^2+\sigma^2\), not just \(\sigma^2\).

How the Shared Group Effect Creates Dependence

Consider two distinct observations \(Y_{ij}\) and \(Y_{ik}\) from the same group \(i\), with \(j\ne k\). Both include the same term \(A_i\), while their residual errors are independent. Their covariance follows directly from the model:

The constant \(\mu\) contributes no covariance. The cross-terms involving a group effect and a residual error are zero because the effects and errors are independent. The residual covariance is zero because the residuals for distinct observations are independent. What remains is the variance of the group effect shared by both responses.

For observations from different groups, \(Y_{ij}\) and \(Y_{\ell k}\), where \(i\ne\ell\), the group effects and residuals are independent across those groups. Therefore,

For any one observation, the random group effect and its residual are independent, so its marginal variance is

Thus the model allows observations in a group to be dependent even though their residuals are independent. In this Gaussian model, if \(\sigma_A^2>0\), distinct responses in the same group have positive covariance and are dependent. If \(\sigma_A^2=0\), there is no random group variation, and the shared term does not induce within-group dependence.

Conditional Independence and Marginal Dependence

Conditioning on a group effect means treating its value as given. If \(A_i\) is fixed at a particular value, the responses in group \(i\) have mean \(\mu+A_i\), and their remaining variation comes from their separate residual errors. Under the model, those responses are conditionally independent. But before the group effect is known, two responses from that group share uncertainty about \(A_i\). This shared uncertainty generates their marginal covariance.

The distinction can be seen by comparing a conditional and a marginal distribution. Given \(A_i=a\),

Without conditioning on the group effect, the two independent normal components \(A_i\) and \(\varepsilon_{ij}\) add together. Therefore,

The mean across groups is \(\mu\), but a particular group’s responses are centered around its own \(\mu+A_i\). This is why assessing the spread of individual observations without accounting for group membership can blur together residual variation and variation between groups.

A Further Consequence: Differences Within a Group

A useful way to isolate the effect of dependence is to subtract two responses from the same group. Their common group effect cancels:

Since the two residuals are independent, each with variance \(\sigma^2\), the variance of their difference is

The group effect contributes no variation to this difference because it shifts both observations equally. By contrast, treating the two responses as independent would ignore their shared shift and generally give the wrong variance for comparisons involving observations within the same group.

What the Assumptions Mean in Practice

The model’s assumptions describe different features of a study and should not be collapsed into a single claim that “the data are normal and independent.” Group sampling and the measurement process matter, as does the distinction between residuals and observed responses.

These questions are partly answered from how the data were collected. For example, repeated measurements on the same person may have additional dependence not represented by a single group effect. Similarly, groups that are related or clustered within a higher-level unit may not have independent group effects. A model cannot correct for such structures merely by naming a group effect.

Plots of observations within groups can help reveal strongly different spreads or unusual residual patterns, but a plot cannot prove normality or independence. Independence in particular depends heavily on the design and the subject-matter context. A small sample may also make departures difficult to detect. Assumptions are therefore assessed using both the study design and the observed data, with uncertainty kept in mind.

Common Mistake: Calling All Responses Independent

A common incorrect argument is: “The residual errors are independent, so the observed responses are independent.” This misses the shared group effect. In fact, for two observations in group \(i\), the covariance is \(\sigma_A^2\), which is positive whenever there is nonzero between-group variation.

The correction is to say that the model assumes residual errors are independent, while observations from different groups are independent under the stated assumptions. Observations within the same group are generally dependent because they share \(A_i\). Constant residual variance also does not mean that every observed response has variance \(\sigma^2\): its marginal variance includes the group-effect variance as well.