A one-way ANOVA can be framed as comparing two linear models:

• Null model: Predicts outcomes using only the grand mean.

• Full model: Predicts outcomes using the grand mean plus the effect of interest (e.g., group, time).

The F-ratio tests whether adding the effect improves model fit enough to suggest a real effect.

Predicts each outcome as:

\[ Y_i = \mu + \epsilon_i \]

• \(\mu\): grand mean

• \(\epsilon_i\): residual error

Predicts each outcome as:

\[ Y_i = \mu + \alpha_j + \epsilon_i \]

• \(\alpha_j\): group effect (between-subjects) or time effect (repeated measures)

• \(\epsilon_i\): residual error

Fit both models using ordinary least squares (or generalized least squares for repeated measures).

Compute residual Sum of Squares for the Null model: \(SS_{\text{residual null}}\)

Compute residual sum of squares for the Full model: \(SS_{\text{residual full}}\)

Calculate:

\[ SS_{\text{effect}} = SS_{\text{residual null}} - SS_{\text{residual full}} \]

Then:

\[ MS_{\text{effect}} = \frac{SS_{\text{effect}}}{df_{\text{effect}}} \]

\[ MS_{\text{residual}} = \frac{SS_{\text{residual full}}}{df_{\text{residual full}}} \]

The F-ratio:

\[ F = \frac{MS_{\text{effect}}}{MS_{\text{residual}}} \]

If the full model does not improve over the null model:

• \(SS_{\text{residual null}} = SS_{\text{residual full}}\)

• \(SS_{\text{effect}} = 0\)

• \(F = 0\)

• Fail to reject the null hypothesis.

If the full model improves substantially:

• \(SS_{\text{residual null}} > SS_{\text{residual full}}\)

• Numerator \(MS_{\text{effect}}\) is large relative to \(MS_{\text{residual}}\).

• \(F\) is large → Evidence of a significant effect.

In repeated measures ANOVA:

\[ MS_{\text{residual}} = \frac{SS_{\text{residual full}}}{df_{\text{residual full}}} \]

However \(SS_{\text{residual full}}\) in repeated measures ANOVA contains subject-related variance in addition to trial-to-trial error.

\[ SS_{\text{residual full}} = SS_{\text{within}} + SS_{\text{subject}} \]

The inclusion of \(SS_{\text{subject}}\) reflects the correlated residuals within subjects assumed under compound symmetry.

In practice GLS fitting with compound symmetry implicitly accounts for this structure, ensuring \(MS_{\text{residual}}\) is appropriate for testing the effect of time.