Contrasts, ANOVA, and nested models

For group means collected in a vector μ, a contrast is a linear combination whose coefficients sum to zero. The difference between two group means is the simplest example. The zero-sum condition makes the comparison unchanged if every group mean shifts by the same amount. More general regression hypotheses use $C\beta=d$, where the rows of C specify coefficient combinations and d contains their hypothesized values. In a redundant parameterization those combinations must be estimable before an ordinary test has a meaningful target.

The model's parameterization affects coefficient labels but not the fitted group means or valid contrasts. An intercept-plus-indicators coding and a direct group-means coding can describe the same column space. For independent errors with common variance, the variance of a difference of two sample means is σ² times the sum of their reciprocal group sizes. Unequal sample sizes therefore affect precision even though the scientific meaning of the mean difference is unchanged.

Let the reduced design's column space be contained in the full design's column space. Denote their ranks by r and p, with q=p−r positive and n>p. The full residual sum of squares cannot exceed the reduced one, because the full fit can always choose the reduced fit. Their difference is the squared length of the response projected onto the q added orthogonal directions. It is not generally the same as fitting each predictor alone and adding its separate contribution.

Under the null that the true mean belongs to the reduced space, spherical Gaussian errors make that extra sum of squares divided by σ² chi-squared with q degrees of freedom. It is independent of the full-model residual sum of squares, which has n−p degrees of freedom. Dividing the two sums by their respective dimensions removes σ² and gives an exact F statistic. The full model supplies the denominator noise estimate; using the reduced residual there would mix the tested departure into the scale estimate.

For one estimable restriction, the corresponding two-sided t statistic squared equals the nested-model F statistic. With several restrictions, the joint F test asks whether at least one tested component departs from the null; it does not identify which component or how large a practically relevant effect is. A small p value can accompany a tiny effect in a very precise experiment, while a wide interval can leave an important effect unresolved in a small experiment.

In unbalanced or nonorthogonal designs, sequential sums of squares can depend on the order in which terms enter. State the actual reduced and full spaces rather than relying on a table label. Testing many response-chosen contrasts introduces multiplicity and selection issues, and observational group differences do not by themselves establish a causal effect. Random assignment, a prespecified comparison, and a model for dependence address different parts of the argument; none follows merely from the ANOVA decomposition.