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On a graph, a binary configuration has probability proportional to $\exp (\alpha {\sum }_v{x}_v+\beta {\sum }_{\{v,w\}}{x}_v{x}_w)$, with each edge counted once. The full conditional probability of a one is logistic with linear predictor $\alpha +\beta {\sum }_{w\sim v}{x}_w$. Local neighbor sums eliminate the need to evaluate the normalizing constant.

The best mean-square linear predictor is orthogonal projection onto the span of the available centered observations, plus the known mean. Orthogonal innovations give a convenient basis of that span. Prediction at a future horizon must use only actually observed variables; it need not be a one-step predictor.

A covariate is an observed explanatory variable used to model the conditional distribution of a response. A vector of covariates can enter a linear predictor, a treatment-effect adjustment or a hazard multiplier. Conditioning on a covariate does not itself establish that its effect is causal.

A generalized additive model has linear predictor $g({\mu }_i)=\alpha +{\sum }_j{s}_j(x_{ij})$. Centering the smooth functions identifies the intercept. The response family and link function remain part of the model; an unspecified software family must not be inferred from the response name.

A generalized linear model specifies independent responses in an exponential dispersion family, their means ${\mu }_i=E(Y_i)$, and a link function connecting those means to a linear predictor. In the common notation,where $a_i$ are known positive weights, $\varphi$ is the common dispersion parameter, and $V({\mu }_i)=b^{\prime \prime }({\theta }_i)$ is the variance function. The systematic part isHere $x_i$ is a row of the known design matrix and $\beta$ contains the unknown regression coefficients. The link function is invertible on the permitted mean domain; the canonical link function takes the mean to the natural parameter. This separates distributional, predictor, and link assumptions: it does not require the response itself to be normally distributed or the mean itself to be linear in the covariates.

This is a generalized linear model with independent Bernoulli distributions for the responses and the logit link. Under treatment coding with colour one as the reference level, the linear predictor isColour is a categorical variable, not a numerical trend. Width has the same slope for every colour, since no interaction term is fitted. The parameters are estimated by maximum likelihood estimation.

The approximate Wald confidence interval for the width coefficient is $\boxed{0.46\pm 1.96(0.106)}$, or approximately $(0.2522,0.6678)$. This uses the approximate normal distribution of the maximum-likelihood estimator, rather than a confidence interval for an individual binary response.

For the requested crab, the fitted linear predictor is $\hat{\eta }=-11.38-0.22+0.46(20)=-2.40$. Applying the inverse logit link gives $\boxed{\hat{p}=(1+e^{2.40}{)}^{-1}\simeq 0.0832}$. The rounded displayed coefficients need not reproduce the reported test statistics exactly.