Interpreting Intraclass Correlation

Why quantify similarity within a subject?

Repeated biomarker measurements from one person are usually not independent. A subject whose biomarker level tends to be high may have high measurements on several visits, even after accounting for the population mean trajectory. The intraclass correlation coefficient (ICC) summarizes this within-subject similarity under a specified model.

The ICC is not a universal property of a biomarker. It depends on the mean structure, the random-effects structure, and the relative sizes of the between-subject and measurement-level variances. In this tutorial, the target is the random-intercept model introduced in the earlier material on Random Intercepts for Subject Baselines.

Random-intercept model and assumptions

Let $Y_{ij}$ denote the biomarker measurement for subject $i$ at visit $j$. A fixed-effects mean trajectory, which may include visit time and treatment, is written $x_{ij}^{T}\beta$. The random-intercept model is

Y_{ij}=x_{ij}^{T}\beta+b_i+\varepsilon_{ij}.

Here $b_i$ is subject $i$'s deviation from the population mean trajectory, and $\varepsilon_{ij}$ is the residual deviation for that particular measurement. For the standard Gaussian model, assume

b_i\sim N(0,\tau_0^2),\qquad \varepsilon_{ij}\sim N(0,\sigma^2),

with random intercepts independent across subjects, residuals independent across measurements and subjects, and all residuals independent of all random intercepts. The same $\tau_0^2$ and $\sigma^2$ apply across subjects and visits. The mean model must also adequately account for systematic changes in biomarker level, such as a population trend over time; otherwise a mean difference can be mistaken for residual dependence.

Definition: Under the random-intercept model, the intraclass correlation for two distinct measurements from the same subject is the correlation after accounting for the fixed-effects mean. Under the assumptions above, it equals the between-subject intercept variance divided by the total variance of one measurement: $$ \rho=\frac{\tau_0^2}{\tau_0^2+\sigma^2}. $$

The ratio is between 0 and 1 when both variance components are nonnegative and their sum is positive. If $\tau_0^2=0$, the model implies no residual within-subject correlation. As $\tau_0^2$ becomes large relative to $\sigma^2$, the ICC approaches 1. A zero total variance would leave the correlation undefined, because there would be no variation in measurements.

Deriving the ICC

The fixed mean $x_{ij}^{T}\beta$ can vary between visits, so first consider deviations from that mean. Write $R_{ij}=Y_{ij}-x_{ij}^{T}\beta=b_i+\varepsilon_{ij}$. The variance of one such deviation is

\operatorname{Var}(R_{ij}) =\operatorname{Var}(b_i)+\operatorname{Var}(\varepsilon_{ij}) =\tau_0^2+\sigma^2,

because $b_i$ and $\varepsilon_{ij}$ are independent. For two distinct visits $j\ne k$ on the same subject,

\begin{aligned} \operatorname{Cov}(R_{ij},R_{ik}) &=\operatorname{Cov}(b_i+\varepsilon_{ij},b_i+\varepsilon_{ik})\\ &=\operatorname{Var}(b_i) =\tau_0^2. \end{aligned}

All terms involving residuals vanish because the residuals at distinct visits are independent and independent of $b_i$. Dividing the covariance by the common standard deviation of the two measurements gives

\operatorname{Corr}(R_{ij},R_{ik}) =\frac{\operatorname{Cov}(R_{ij},R_{ik})} {\sqrt{\operatorname{Var}(R_{ij})\operatorname{Var}(R_{ik})}} =\frac{\tau_0^2}{\tau_0^2+\sigma^2}.

The same derivation shows why the ICC concerns distinct measurements: a measurement correlated with itself has correlation 1 by definition. For two different subjects, the model assumes independent random intercepts and residuals, so their deviations have covariance zero. The ICC is therefore a within-subject correlation, not the correlation between a subject's measurement and another person's measurement.

Normality is a convenient distributional assumption for fitting the Gaussian mixed model, but the displayed covariance and correlation formulas follow from the stated independence and finite-variance assumptions; normality is not needed for the algebra itself.

Worked Example: Interpreting a Moderate ICC

Suppose a hypothetical Gaussian random-intercept model for a biomarker, measured in units $U$, estimates $\tau_0^2=36\ U^2$ and $\sigma^2=64\ U^2$. Assume the mean trajectory is correctly specified and the model's independence and constant-variance assumptions are reasonable. Find and interpret the ICC.

The total variance of one measurement around its fixed mean is $36+64=100\ U^2$. Thus

\hat{\rho}=\frac{36}{36+64}=0.3600.

The estimated correlation between two distinct measurements from the same subject, after accounting for the fixed mean, is 0.3600. In this model, 36% of the variance in an individual measurement is due to differences in subject-specific intercepts, while 64% is residual variance. This is a statement about the model's variance decomposition, not a claim that 36% of any particular person's biomarker value is explained by identity.

Reading the variance components in context

The random intercept shifts a subject's measurements together. If $b_i$ is positive, all of that subject's measurements tend to lie above the fixed mean trajectory; if it is negative, they tend to lie below it. Residual variation $\varepsilon_{ij}$, by contrast, is measurement-specific under this model. The ICC describes the balance between these two sources of variation.

An equivalent way to see the interpretation is to compare two measurements from the same subject. The shared intercept contributes to both, while the two independent residuals differ. Subtracting the measurements removes the shared intercept:

Y_{ij}-Y_{ik} =(x_{ij}-x_{ik})^T\beta+\varepsilon_{ij}-\varepsilon_{ik}.

After accounting for their fixed means, the difference has variance $2\sigma^2$. This does not replace the ICC, but it helps explain why high between-subject variation can produce substantial similarity even when measurements still vary from visit to visit.

Worked Example: Using a Fitted ICC to Describe Repeated Measurements

In a hypothetical analysis, the estimated subject-intercept variance is $25\ U^2$, and the estimated residual variance is $75\ U^2$. Suppose the biomarker mean at each visit has been included in the fixed-effects model, and the residual independence and equal-variance assumptions are plausible. Find the estimated ICC and the covariance of two distinct measurements from the same subject.

The total variance for one measurement is $25+75=100\ U^2$, so

\hat{\rho}=\frac{25}{100}=0.2500.

The model-implied within-subject covariance is the estimated intercept variance, $25\ U^2$, and the correlation is $25/100=0.2500$. Thus two measurements from the same subject have a positive but not especially high model-implied correlation after the mean structure is accounted for. The statement does not imply that every pair of observed values will be close: residual variation remains substantial.

As a check, the variance of the difference between two mean-adjusted measurements is $2(75)=150\ U^2$. The shared intercept does not appear in this difference, consistent with the model.

Estimation, diagnostics, and alternatives

In practice, a mixed-model fitting method supplies estimates of the variance components. Substituting those estimates into the ICC formula gives a plug-in estimate. The point estimate alone does not show how precisely the components, or their ratio, have been estimated. When inference about the ICC is needed, an interval or other uncertainty assessment should reflect uncertainty in both variance components, especially when an estimate is near zero.

Check whether the assumptions that make the ICC meaningful are credible. Useful checks include examining residuals for changing spread across visits, looking for remaining time patterns, and comparing observed within-subject dependence with the dependence implied by the fitted model. Sparse follow-up or a small number of subjects can make variance estimates unstable. Estimates of a variance component may also be at the boundary of zero; an ICC of zero can then reflect limited evidence for between-subject variation rather than proof that subjects are identical.

Changing residual variance: if measurement variability differs systematically by visit or biomarker level, one common $\sigma^2$ may be unsuitable, and a single ICC can obscure that pattern.
Residual dependence across visits: if measurements remain associated after accounting for the shared intercept, the assumption of independent residuals is doubtful. The stated ICC no longer summarizes all within-subject dependence.
Changing dependence over follow-up: the random-intercept model implies the same correlation for every pair of distinct visits. If observed association appears to depend on the separation or timing of visits, this constant-correlation implication may be inadequate.
Misspecified mean trajectory: an unmodeled time trend can make residuals at different visits look dependent. Check the mean structure before interpreting the ICC as subject-level similarity.

A raw Pearson correlation between two selected visits is one alternative summary, but it answers a narrower question and may mix within-subject dependence with systematic differences in the visit means. The model-based ICC instead describes the correlation implied for any two distinct visits under the random-intercept assumptions, after accounting for specified fixed effects. A correlation that varies by visit pair, or a more flexible covariance model, may be preferable when the data contradict the constant-correlation implication. The choice depends on the scientific target and the observed dependence pattern; an ICC should not be reported without identifying the model that defines it.

Worked Example: How Measurement Noise Changes the ICC

Consider two hypothetical versions of a biomarker measurement model, each with $\tau_0^2=25\ U^2$. In the first, $\sigma^2=75\ U^2$; in the second, improved measurement precision reduces the residual variance to $25\ U^2$. Assume in each case that fixed means are adequate and the random-intercept assumptions hold. Compare the ICCs.

For the first version,

\hat{\rho}_1=\frac{25}{25+75}=0.2500.

For the second version,

\hat{\rho}_2=\frac{25}{25+25}=0.5000.

The estimated ICC increases from 0.2500 to 0.5000 because the residual variance is smaller relative to the between-subject variance. This does not establish that the subjects became more different: the between-subject variance stayed fixed. It shows that the proportion of total measurement variance attributable to subject intercepts increased.

Common mistake and key limitation

A frequent mistake is to say, “An ICC of 0.36 means 36% of measurements from the same person are identical,” or “the subject explains 36% of each biomarker value.” Neither statement follows. The ICC is a correlation between distinct, mean-adjusted measurements under the model, and equivalently the fraction $\tau_0^2/(\tau_0^2+\sigma^2)$ of the modeled measurement variance associated with subject-level intercept variation.

A second mistake is treating the ICC as an observed raw correlation that must hold for every pair of visits in any data set. It is a model-implied summary. Its usefulness depends on the random-intercept structure and the assumptions behind the variance components. It does not establish a causal effect of subject characteristics, and it does not by itself describe how biomarker levels change over time.

Key takeaway: In a random-intercept model with independent, equal-variance residuals, the ICC for two distinct measurements from one subject is $\tau_0^2/(\tau_0^2+\sigma^2)$. Interpret it as model-based within-subject correlation and as a variance proportion, always stating the model and its assumptions.

Check Your Understanding

Answer each question, then compare with the solutions below.

A hypothetical model estimates $\tau_0^2=18\ U^2$ and $\sigma^2=42\ U^2$. What is the ICC, and how should it be interpreted?
Under the random-intercept assumptions, what is the covariance of two distinct mean-adjusted measurements from the same subject? Give the general expression.
If $\tau_0^2=0$ and $\sigma^2>0$, what ICC does the model imply? What does this say about the modeled dependence?
Why can an unmodeled population time trend make the fitted ICC difficult to interpret?
Two fitted models have the same $\tau_0^2$, but the second has a smaller $\sigma^2$. What happens to the ICC, and why?

Solutions. (1) $\hat{\rho}=18/(18+42)=18/60=0.3000$. Under the model, two distinct measurements from the same subject have correlation 0.3000 after accounting for the fixed mean; 30% of the modeled measurement variance is attributable to subject-intercept variation. (2) The covariance is $\tau_0^2$, since the two measurements share $b_i$ and their residuals are independent. (3) The ICC is $0/(\,0+\sigma^2\,)=0$. The model implies no correlation between distinct measurements after the fixed mean is accounted for. (4) A remaining systematic pattern across visits can appear as residual dependence, although the model attributes within-subject covariance to the shared intercept. The mean structure should be checked before interpreting the ICC. (5) The ICC increases: with $\tau_0^2$ fixed, reducing $\sigma^2$ reduces the denominator relative to the numerator, so the between-subject share of total variance is larger.