Why Review Random Effects and Covariance?
Random effects describe between-individual variation in model parameters. Their implementation can change the model even when the structural equations and typical parameter values appear correct. A diagonal covariance matrix, for example, asserts that the random effects are uncorrelated; a matrix with nonzero off-diagonal terms allows association. Confusing a variance with a standard deviation changes the amount of variation, while assigning a covariance to the wrong parameter pair changes which parameters are associated.
A review therefore connects three records: the written model specification, the code that generates the random effects, and the reported variance–covariance estimates. The earlier material on comparing model code with its written specification and auditing parameterization provides the context for those comparisons. Here the focus is the random-effects distribution, covariance terms, and associations implied by them. Residual-error implementation is a separate component and is not assessed here.
For two random effects, \(\eta_{1,i}\) and \(\eta_{2,i}\), write their variances as \(v_1\) and \(v_2\), and their covariance as \(c\):
The correlation expression applies when both variances are positive. A covariance matrix must be symmetric and positive semidefinite; for this two-effect matrix, that requires \(v_1 \geq 0\), \(v_2 \geq 0\), and \(c^2 \leq v_1v_2\). If either variance is zero, the corresponding random effect is constant with probability one and its correlation with the other effect is undefined. A reported or coded correlation cannot repair an invalid covariance matrix.
Translate the Model Into a Code-Level Checklist
Before inspecting syntax, write down which effects are random, their distributions, how they enter model parameters, and the intended covariance structure. For example, a model might specify a bivariate normal vector \((\eta_{CL,i},\eta_{V,i})\) with a freely estimated covariance, while defining individual clearance and volume by exponential transformations. Code that instead sets the off-diagonal covariance to zero implements a different model, even if both random effects have the same marginal variances.
The review should distinguish quantities that are often presented using similar labels:
- Variance: the diagonal element \(v_j\) of \(\Omega\), measured in squared random-effect units.
- Standard deviation: \(\sqrt{v_j}\), measured in random-effect units.
- Covariance: an off-diagonal element \(c\), measured in the product of the two random-effect units.
- Correlation: \(c/\sqrt{v_1v_2}\), a dimensionless value between \(-1\) and \(1\), when both variances are positive.
Some software accepts standard deviations and correlations as inputs, then constructs \(\Omega\); other implementations specify variances and covariances directly. The reviewer must follow the software’s documented convention and compare equivalent quantities, not merely compare labels or raw numbers. For example, an estimated standard deviation of \(0.20\) corresponds to variance \(0.20^2=0.0400\), not variance \(0.20\).
Match every random-effect name in the model statement to the code’s effect definition and to the parameter it modifies. Check for omissions, duplicates, and swapped associations.
Compare the stated distribution, mean, and any constraints with the implemented distribution. Confirm whether the code generates a variance, a standard deviation, or another parameterization.
Use the code’s parameterization to form the implied matrix. Check its diagonal, symmetry, off-diagonal terms, and positive-semidefinite constraints.
Trace reported variances, standard deviations, covariances, and correlations to the corresponding effects and scale. Record any transformation used to obtain each reported quantity.
Worked Example: Checking Variance Versus Standard Deviation
In a hypothetical two-parameter model, the specification says that \(\eta_{CL,i}\) and \(\eta_{V,i}\) are independent normal random effects with standard deviations \(0.20\) and \(0.30\). Code declares a diagonal covariance matrix with entries \(0.20\) and \(0.30\). Does the code implement the specification?
The stated standard deviations imply variances \(0.20^2=0.0400\) and \(0.30^2=0.0900\). Independence is represented by zero covariance, so the intended matrix is
If the code’s entries are variances, its matrix is instead
The code therefore implies standard deviations \(\sqrt{0.20}=0.4472\) and \(\sqrt{0.30}=0.5477\), not \(0.20\) and \(0.30\). It does not implement the stated model. This conclusion depends on confirming that the software interprets the entries as variances; if they are standard deviations, the implied matrix must instead be reconstructed according to that convention.
Covariance Constraints and Parameter Associations
Covariance is scale-dependent: its numerical value depends on the units of both random effects. Correlation standardizes covariance by the two standard deviations, making it easier to compare association strength across scales. A valid correlation matrix must also be positive semidefinite. For a two-effect model with positive variances, the determinant condition \(v_1v_2-c^2 \geq 0\) is equivalent to an effect-scale correlation between \(-1\) and \(1\).
A diagonal \(\Omega\) specifies zero covariance among the random effects under the model. A full \(\Omega\) estimates or specifies off-diagonal covariances. A structured covariance matrix imposes a pattern, such as a common correlation or a block structure. These are alternative model assumptions, not interchangeable coding shortcuts. The correct choice is the structure in the stated model; if the specification is ambiguous, document the ambiguity rather than silently choosing a structure.
Worked Example: Verifying a Covariance and Correlation
Suppose a hypothetical report gives random-effect variances \(v_{CL}=0.0400\), \(v_V=0.0900\), and covariance \(c=0.0300\). Verify the implied correlation and check whether the matrix is admissible.
The standard deviations are \(\sqrt{0.0400}=0.2000\) and \(\sqrt{0.0900}=0.3000\). Therefore,
For the matrix check, \(v_{CL}v_V=0.003600\) and \(c^2=0.000900\), so \(c^2 \leq v_{CL}v_V\). Equivalently, its determinant is \(0.003600-0.000900=0.002700 \geq 0\). The matrix is admissible, and the random effects have a positive association on their own scale. The covariance has units equal to the product of the two random-effect units; the correlation is unitless.
Associations After Transforming Parameters
The association between random effects is not always numerically identical to the association between the individual parameters they generate. Consider positive typical values \(TVP_1\) and \(TVP_2\), with parameter definitions \(P_1=TVP_1\exp(\eta_1)\) and \(P_2=TVP_2\exp(\eta_2)\). Assume \((\eta_1,\eta_2)\) is jointly normal with mean zero, variances \(v_1,v_2\), and covariance \(c\). These assumptions are needed for the following moment formulas.
For any normal random variable \(Z\sim N(m,s^2)\), its exponential has expectation \(E[\exp(Z)]=\exp(m+s^2/2)\). Also, the joint normal moment-generating formula gives \(E[\exp(\eta_1+\eta_2)]=\exp((v_1+v_2+2c)/2)\). Thus,
The marginal variances are \(TVP_j^2\exp(v_j)(\exp(v_j)-1)\). Dividing the covariance by the square root of their product yields the parameter-scale correlation:
Both variances must be positive for this correlation formula to be defined. If either variance is zero, the corresponding transformed parameter is constant, its variance is zero, and its correlation with the other parameter is undefined. With positive variances, the formula applies under the stated joint-normal and exponential-transformation assumptions. It also shows why a reported parameter-scale correlation should not automatically be equated with the random-effect correlation.
Worked Example: Comparing Random-Effect and Parameter-Scale Correlations
For a hypothetical model, let \(v_1=0.0900\), \(v_2=0.1600\), and \(c=0.0600\). Both random effects enter positive parameters through exponential transformations. Calculate the correlation on each scale.
First, on the random-effect scale,
The covariance is admissible because \(c^2=0.003600\) and \(v_1v_2=0.01440\), so \(c^2 \leq v_1v_2\). For the transformed parameters,
The induced correlation is approximately \(0.4838\), rather than \(0.5000\). Both are positive, but they refer to different scales. This calculation is relevant only if the written model uses the assumed lognormal transformation and jointly normal effects.
Diagnostics, Alternatives, and Limits
Useful review diagnostics are direct consistency checks, not substitutes for reading the model statement. Reconstruct the matrix from the actual code convention, then check that it is symmetric and admissible. Compare the number and ordering of random effects with the parameter equations. For each reported association, identify the pair, scale, units, and calculation used. If a parameter is fixed at zero variance, flag any reported correlation involving it as undefined rather than treating it as zero.
A diagonal covariance model is simpler and imposes no pairwise covariance terms; it is appropriate only when that independence structure is intended. A full covariance model permits pairwise associations but requires more parameters and a valid positive-semidefinite matrix. A structured model lies between these approaches by imposing constraints on the covariance pattern. This review checks whether the chosen implementation matches the stated model; it does not determine which covariance structure is scientifically preferable or whether the data support its estimation.
Review conclusions also have limits. An admissible matrix does not prove that the code uses the correct effects, that the model is scientifically appropriate, or that estimates are reliable. Conversely, a numerical discrepancy may arise from different reporting conventions, so establish whether a value is a variance, standard deviation, covariance, or correlation before declaring an error. Preserve the evidence: record the relevant model statement, code location, reconstructed matrix, reported quantities, and the basis for each conclusion.
Common Mistake
A frequent error is to read a reported random-effect standard deviation as a variance and compare it directly with a diagonal covariance-matrix entry. For example, calling a code entry of \(0.0400\) inconsistent with a reported standard deviation of \(0.2000\) overlooks that \(0.2000^2=0.0400\). The correction is to convert both quantities to the same parameterization before comparing them. Another error is to assume that a zero covariance means the transformed individual parameters are independent in every possible model; that conclusion requires the distributional and transformation assumptions to be checked. In the jointly normal random-effects model, zero covariance implies independence of the two effects, but interpretations on the parameter scale still depend on the specified transformations.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A random effect has reported standard deviation \(0.15\). What variance should be compared with a covariance-matrix diagonal entry?
- For variances \(0.04\) and \(0.09\), is covariance \(0.07\) admissible? Explain using the determinant or covariance constraint.
- If one random-effect variance is zero, what is the correlation between that effect and another effect?
- Under jointly normal random effects and exponential parameter transformations, must parameter-scale correlation equal random-effect correlation?
Solutions. (1) The variance is \(0.15^2=0.0225\). (2) No: \(c^2=0.0049\), whereas \(v_1v_2=0.0036\); since \(c^2>v_1v_2\), the proposed matrix is not positive semidefinite. (3) It is undefined because the zero-variance effect is constant, so its standard deviation is zero and the correlation denominator is zero. (4) No. The transformed correlation follows the exponential-moment formula and generally differs from the random-effect correlation; both require the stated distributional and transformation assumptions to apply.