One-way normal linear model

A one-way normal model assigns each factor level its own mean while assuming independent Gaussian errors with a common variance.

This Gaussian model is the setting for a one-way instance of analysis of variance.

A cell-means model uses one coefficient per factor level and no intercept, so each coefficient directly equals a group mean.

Suppose a normal linear model gives an unrestricted mean to each cell and has $m$ independent observations per cell, with common error variance ${\sigma }^2$. Each fitted cell mean is its sample mean, with variance ${\sigma }^2/m$; different fitted cell means are independent. Estimate ${\sigma }^2$ by the pooled within-cell residual sum of squares divided by $n-q$, where $q$ is the number of cells. Thus a cell mean has standard error $s/\sqrt{m}$, a difference of two cell means has standard error $s\sqrt{2/m}$, and a two-by-two difference-of-differences has standard error $2s/\sqrt{m}$.

// Target: probability-and-statistics.bigb

A linear contrast is a coefficient-weighted sum of cell means whose coefficients total zero, used to test interpretable differences among groups.

A treatment contrast is a linear contrast of cell means comparing treatment responses, with coefficients adding to zero. Pairwise differences and marginal factorial differences are examples. Its variance depends on the true allocation units and their covariance matrix.

At equally replicated quantitative treatment levels, evaluations of orthogonal polynomials generate pairwise orthogonal treatment contrasts, using the replication-weighted inner product. Five equally spaced levels give four contrast directions representing degrees one through four, besides the constant direction. This decomposes treatment variation without assuming the response has low polynomial degree.

For a vector of estimated treatment means $\hat{\mu }$, the variance of a treatment contrast with coefficient vector $c$ is $c^T\mathrm{Cov}(\hat{\mu })c$. Averaging subsamples from a common experimental unit does not make their shared variance component disappear.

An interaction contrast compares a treatment difference across levels of another factor. A nonzero difference of differences shows that an additive representation of those cell means is inadequate. Its standard error must use the error stratum in which that interaction term is randomized.

For binary predictors $A,B$, a logistic regression with linear predictor $a+bA+cB+\delta AB$ has conditional odds ratios $e^b$ at $B=0$ and $e^{b+\delta }$ at $B=1$. Their ratio is $e^{\delta }$. The same interaction can be computed from the four cell logit link values as $\delta ={\eta }_{11}-{\eta }_{10}-{\eta }_{01}+{\eta }_{00}$. Independent positive event and nonevent counts in each cell give an estimated variance for this contrast equal to the sum of all eight reciprocal counts. An interaction on the odds ratio scale is not an interaction on the absolute-risk scale.

For genotypes $aa,Aa,AA$, full dominance of allele $A$ imposes ${\mu }_{Aa}={\mu }_{AA}$.

With allele counts $0,1,2$, no dominance makes the genotype mean affine in count and imposes $2{\mu }_{Aa}={\mu }_{aa}+{\mu }_{AA}$.