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A linear regression in the coefficients with a polynomial mean function $\mathbb{E}(Y\mid t)={\sum }_{j=0}^d{\beta }_j{t}^j$. Successive degrees define nested models that can be compared by an F-test.

The unscaled residual deviance of an exponential dispersion family compares its fitted mean with the saturated statistical model at a common dispersion parameter. Its normalization removes the factor $1/\varphi$ in the log-likelihood. In Poisson regression and binomial regression, $\varphi =1$, so it is twice the saturated-minus-fitted log-likelihood. Under suitable regularity, $D/\varphi$ has approximately a chi-squared distribution with the residual degrees of freedom; this is not an automatic accurate approximation for sparse responses. Nested deviance reductions with estimated dispersion lead to approximate F-tests after scaling by the dispersion estimate.

A two-way analysis of variance with the full interaction and one observation in each observed factor cell has one independently varying fitted cell mean per observation. A cell-indicator design matrix has full row matrix rank, even when some combinations are missing, so the model is a saturated statistical model. Its residual sum of squares is zero and its residual degrees of freedom are zero. Missing combinations make some coefficients nonidentifiable; they do not restore a pure-error estimate. Standard errors and an ordinary residual-based F-test for interactions therefore cannot be recovered merely by fitting this larger model.

For an ordered sequence of nested linear regression models $M_0\subset M_1\subset \cdots$, the sequential sum of squares for the $j$th added term is $\mathrm{RSS}(M_{j-1})-\mathrm{RSS}(M_j)$. These are also called type-I sums of squares. They depend on term order when the design is not orthogonal. Dividing by the number of newly added coefficients and the full model's residual mean square gives the sequential F-test. A covariate added first is tested before adjustment for later terms, whereas a treatment added after the covariate is tested with that covariate already included.