Information in Regression Contrasts

Why Information Depends on the Contrast

In a regression model, an analysis may focus on a slope, a treatment coefficient, or a difference between two predicted means. Each target is a particular combination of the model coefficients. Its precision depends not only on the regression design but also on which combination is being estimated. In particular, two coefficients can be correlated, so adding their estimated values can have a different variance from subtracting them.

The inverse information matrix accounts for both coefficient variances and their covariances. This makes it possible to calculate the variance of a specified regression contrast directly, then express its precision as scalar information. In a group sequential setting, that scalar refers to the same contrast at each analysis; it is not automatically proportional to the number of participants enrolled.

Model, Contrast, and Information

Consider the fixed-design linear model \(Y=X\beta+\varepsilon\), where \(Y\) is an \(n\)-vector of outcomes, \(X\) is a known \(n\)-by-\(p\) design matrix, and \(\beta\) is a \(p\)-vector of regression coefficients. For the standard normal linear model, assume that the errors are independent and normally distributed with mean zero and common variance \(\sigma^2\), so \(\varepsilon\sim N(0,\sigma^2 I_n)\). Also assume that \(X\) has full column rank, meaning its columns are linearly independent. These assumptions make the coefficients identifiable and give the usual model-based variance formula.

A linear contrast is a specified scalar combination of coefficients, written \(\theta=c^\mathsf{T}\beta\), where \(c\) is a fixed \(p\)-vector. For example, \(c\) may select one slope, or combine an intercept and slope to express a mean at a particular covariate value. The coefficient vector must be stated in the same order as the columns of \(X\).

The least-squares estimator is \(\hat{\beta}=(X^\mathsf{T}X)^{-1}X^\mathsf{T}Y\). Under the stated model, its covariance matrix is \(\sigma^2(X^\mathsf{T}X)^{-1}=J^{-1}\). The contrast estimate is \(c^\mathsf{T}\hat{\beta}\), and applying the variance rule for a linear combination gives \(c^\mathsf{T}J^{-1}c\). Thus the off-diagonal entries of the inverse information matrix matter whenever more than one coefficient contributes to the contrast.

When \(\sigma^2\) is unknown, the model-based variance is estimated by replacing \(\sigma^2\) with the residual mean square \(s^2\). The resulting estimated standard error is often used for a \(t\)-test or confidence interval. For planning under a specified variance, however, information is calculated using that variance assumption. If the variance is estimated from interim data, the estimated information is the reciprocal of the estimated contrast variance, and its uncertainty should not be mistaken for exact knowledge of the information.

From the Information Matrix to Contrast Information

The formula follows directly from the normal likelihood. Up to a constant, its log-likelihood as a function of \(\beta\), for fixed \(\sigma^2\), is \(-\{(Y-X\beta)^\mathsf{T}(Y-X\beta)\}/(2\sigma^2)\). The negative matrix of second derivatives with respect to \(\beta\) is \(X^\mathsf{T}X/\sigma^2\), which is the information matrix. Inverting it yields the covariance matrix of \(\hat{\beta}\).

For completeness, if \(A\) is the covariance matrix of a random vector \(U\), then the variance of \(c^\mathsf{T}U\) is \(c^\mathsf{T}Ac\). Applying this identity with \(U=\hat{\beta}\) and \(A=J^{-1}\) gives the contrast variance. Taking its reciprocal defines the contrast information. This reciprocal convention reflects that smaller variance means greater information; it does not mean that the full matrix \(J\) can be reduced to one meaningful scalar without specifying a target contrast.

The units provide a useful check. If \(\theta\) is measured in outcome units, its variance is in squared outcome units and \(I_c\) is in inverse squared outcome units. If the contrast is rescaled to \(a\theta\), its variance is multiplied by \(a^2\) and its information is divided by \(a^2\). Consequently, information values are comparable only when they refer to the same contrast scaling and outcome units.

Worked Calculations

Diagnostics, Alternatives, and Limitations

Before interpreting an information calculation, verify that the contrast is identifiable and that the covariance model is appropriate. The following checks catch common failures.

• Check rank and conditioning. If \(X\) is rank deficient, \(X^\mathsf{T}X\) cannot be inverted as written. Near-collinearity can also make the inverse unstable and produce large contrast variances. A generalized inverse alone does not make every contrast estimable; the particular contrast must be identifiable from the design.
• Check the contrast and coefficient order. A misplaced coefficient or a missing contrast multiplier changes the target and its variance. Confirm that \(c\) describes the requested scientific comparison and that its units are understood.
• Check covariance assumptions. Residual plots and knowledge of the design can reveal unequal variances, dependence, or strong departures from the assumed error distribution. If the errors are correlated or have known non-identity covariance \(V\), the model-based information becomes \(X^\mathsf{T}V^{-1}X\); using \(X^\mathsf{T}X/\sigma^2\) without justification can misstate precision.
• Check positivity and scale. With a valid full-rank design and positive variance, the computed contrast variance must be positive. A nonpositive result usually signals an arithmetic, coding, or matrix-inversion error. Check that rescaling the contrast changes information by the reciprocal squared factor.

A convenient alternative is to obtain the standard error of the contrast directly from regression software and calculate information as \(1/SE^2\), provided the standard error uses the same model and contrast. For heteroskedasticity-robust or other sandwich standard errors, this reciprocal can summarize estimated precision, but it is not necessarily the inverse of the model’s Fisher information matrix. Generalized least squares is the model-based alternative when a suitable error covariance matrix \(V\) is specified. These approaches answer related questions but rely on different variance assumptions.

For sequential monitoring, the relevant quantity is the variance or information for the prespecified contrast at each analysis, calculated on a consistent scale. The inverse matrix is not itself a universal measure of study information: a different target contrast can have a different variance under the same fitted design. Furthermore, plugging in an estimated residual variance adds uncertainty, and a changed design or covariance structure can make information from different analyses less directly comparable. The calculation describes model-based precision; it does not by itself establish that the model or the underlying scientific interpretation is correct.

Common Mistake

A frequent error is to read the diagonal entry of the inverse information matrix for one coefficient and use it as the variance of any contrast involving that coefficient. That works only for a contrast that selects exactly that coefficient. For a general contrast, calculate the full quadratic form \(c^\mathsf{T}J^{-1}c\), including the off-diagonal covariance terms and all contrast multipliers. In particular, the variance of a difference is not generally the sum of the two coefficient variances; the covariance contributes as well.