Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 28 / 5

Use the index order in the probability specification: influence $i$, accommodation type $j$, contact $k$ and satisfaction $l$. Put $N_{ijk}=n_{ijk+}$, which is conditioned on. The independent multinomial likelihood isThus the relevant sample is a collection of conditional satisfaction distributions for fixed covariate strata; it is not a model for the marginal distribution of the three covariates. In the published dataset there are three influence, four accommodation, two contact and three satisfaction levels, giving 24 strata and 72 cells; these dimensions are documented in stat.ethz.ch/R-manual/R-devel/library/MASS/html/housing.html .

The missing output does not identify its two actual formulas. Two standard canonical models can nevertheless be derived explicitly, and the following general result identifies the sufficient statistics for any specified baseline-logit model. The first standard model has no response/covariate association: $p_{ijkl}=q_l$ for every stratum. Its likelihood factors asTherefore the pooled satisfaction totals $n_{+++l}$ are sufficient, with only two independent totals because the grand total is fixed. The maximum-likelihood estimator is ${\hat{q}}_l=n_{+++l}/1681$, and fitted counts are $N_{ijk}{\hat{q}}_l$. A Poisson surrogate formula with all covariate-stratum interactions and only a satisfaction main effect produces this same conditional model. It has two conditional parameters and 46 residual degrees of freedom when all 24 strata are present.

For the standard additive multinomial logistic regression, use the final satisfaction category as baseline and writeSet a reference effect to zero for each covariate factor. Define ${\eta }_{ijkl}={\alpha }_l+a_{il}+b_{jl}+c_{kl}$ for $l<L$, and ${\eta }_{ijkL}=0$. The probabilities areSubstitution into the conditional log-likelihood givesThe second term depends on the parameters and the fixed stratum totals, not on any additional within-stratum cell allocation. The Fisher-Neyman factorization theorem therefore proves thatThese margins contain linear redundancies; after reference constraints an equivalent independent statistic has dimensionHence its residual degrees of freedom are $24(3-1)-14=34$. In the equivalent Poisson surrogate for a conditional multinomial model, include a separate stratum intercept ${\gamma }_{ijk}$ and response interactions with each individual covariate. The 24 nuisance intercepts plus 14 response parameters give 38 parameters in 72 Poisson cells, again 34 residual degrees of freedom. Since the stratum totals are fixed in the conditional experiment, they need not be included as random sufficient statistics there; they do belong to the unconditional Poisson model's sufficient margins.

The derivation extends to either actual model once its formula is known. If $x_{ijk}$ is its covariate design vector and the baseline logits are $x_{ijk}^T{\beta }_l$, thenThus the conditional multinomial sufficient statistics are $T_l={\sum }_{ijk}{n}_{ijkl}{x}_{ijk}$, for $l<L$. For an identifiable unrestricted canonical parameter space they are also minimal up to linear equivalence: the ratio of two data likelihoods is parameter-independent exactly when the differences of all these design-weighted sums vanish. This follows because that ratio contains $\exp \{{\sum }_l{\beta }_l^T(T_l(n)-T_l(n^{\prime }))\}$.

Consequently an influence-by-type response interaction adds the margins $n_{ij+l}$; influence-by-contact adds $n_{i+kl}$; type-by-contact adds $n_{+jkl}$. A saturated model requires the individual cells, subject to the fixed stratum totals. A hierarchical log-linear model leads to the same response/covariate margins and the fixed covariate margin. There is no need to guess sufficient statistics from the number of displayed coefficients: they follow from the actual design columns and the likelihood.

To read a printed fit, check whether it is a Poisson surrogate, a baseline-category multinomial model or an ordinal cumulative model, then inspect its reference levels, signs of response contrasts, standard errors, residual deviance, degrees of freedom, nested comparisons and predicted probabilities. A baseline-logit coefficient changes odds of one satisfaction level against the baseline, not necessarily cumulative odds of high satisfaction. Normalize Poisson fitted counts within each covariate stratum to get fitted response probabilities. Empty or sparse cells can cause boundary fits or invalidate a naive deviance reference distribution.

A proportional-odds model instead specifies $\mathrm{logit}P(Y\le l\mid x)={\theta }_l-x^T\beta$. It uses the response ordering, with shared slopes across cutpoints. Its category probabilities are differences of cumulative logistic probabilities, so its log-likelihood is not the canonical linear expression above. The baseline-category margins just derived are not generally sufficient for that model; the full conditional cell table is always sufficient. Without the missing formulas it is not possible to certify which two specific models were fitted or their actual sufficient-margin lists and test results. The null, additive and general-design derivations above state precisely which model each answer belongs to.

A direct grouped-binomial logistic regression can dichotomize satisfaction, for example high versus low-or-medium. In each covariate stratum use the high count as successes and the sum of the other counts as failures, with total $N_{ijk}$. This models $P(Y=\mathrm{high}\mid i,j,k)$ and is appropriate if that binary contrast is substantively intended, but discards distinctions among the merged categories. Another cutpoint gives a different binary question.

There is also an exact sequential binomial factorization preserving all three categories. With $p_1,p_2,p_3$ their probabilities in one stratum,Multiplying these two conditional binomial likelihoods gives the multinomial likelihood: the combinatorial coefficients multiply to $N!/(n_1!n_2!n_3!)$, and the probability factors reduce to $p_1^{n_1}{p}_2^{n_2}{p}_3^{n_3}$. Continuation-ratio logits can therefore fit two logistic regressions, using the remaining stratum total as the second denominator. These are conditional stage models, not two independent counts with the same fixed denominator. Their regression restrictions need not equal those of the additive baseline-category model.

Finally, an ordinal analysis may fit cumulative binary contrasts and examine whether their slopes can be shared in a proportional-odds model. The binary indicators at different cutpoints from the same person are dependent. Do not stack them as independent binomial observations and multiply their marginal likelihoods; fit the joint ordinal likelihood or use a method with the appropriate dependence correction. This distinction separates a convenient descriptive collection of binary fits from a coherent multinomial or ordinal model.