## Relations

The tests in Section 2 are fine for detecting evidence against the null hypothesis in the direction of a positive trend. However, they do not lend much insight about the form of the relationship.

A model-based perspective is superior for this purpose. A good-fitting model describes the nature of the association, provides parameters for describing the strength of the relationship, provides predicted probabilities for the response categories at any dose, and helps us to determine the optimal dose. As a by-product, it also yields tests for the hypothesis of no effect. In fact, some tests presented in the previous section have natural connections with models. In this section, we again focus on the first question posed by Ruberg11 - that is, whether an effect exists; however, the model-based approach is also well-suited for pursuing the other three questions.

The models we discuss are generalizations of logistic regression models that handle ordinal response categories. For further discussion of these and other models for ordinal responses, see Agresti20 (Chapters 8 and 9) and McCullagh.44

### 3.1. Proportional odds models

Currently, the most popular model for ordinal responses uses logits of cumulative probabilities. A J-category response has (J — 1) non-redundant cumulative probabilities, P(Y;^j), j = 1, ..., J — 1. For the dose-response problem, consider the model where logit[P(Yf = log[P(Y; ^j)/P(Yi > jj]. This model adds effects {/?,} of the drug dosages on the response to the null model that contains only parameters {«,} pertaining to the logit of each cumulative probability. It treats the effects {/?;} as identical for each cumulative probability; that is, the effect does not depend on j in the model formula.

This form of model is called proportional odds44 Independence of dose and response is equivalent to /?! = • • • = each cumulative probability then being identical for all doses. Using a minus sign before the effect of dose in equation (5) implies that the higher the value of ft relative to other {ft,}, the lower the cumulative probability tends to be at dose i, and hence the higher the response tends to be at dose i compared to other doses. The response distributions are stochastically ordered according to {ft}. The case of a monotone relationship with direction (2) corresponds to ft ^ ••• ^ ft.

A monotone relation in which the trend is linear in dose scores {¿¡} has the simpler model form with fi > 0 implying (2). The ordinary logistic regression model with a linear dose effect is the special case J = 2. For this ordinal model, the odds that the response falls above any given category are multiplied by exp(ft for each unit increase in dose.

The ML fit of any model of this type yields estimated cumulative probabilities at each dose, and hence predicted numbers of observations (fitted values) in the cells of the table. One can test the fit using Pearson or likelihood-ratio chi-squared statistics that compare the observed cell counts to the model's fitted values. The adequacy of these goodness-of-fit tests improves as the cell counts increase in size, the Pearson test being preferred if the cell counts are relatively small.

Model (6) treats the doses as ordinal, whereas model (5) treats them as nominal. To increase power for testing independence when one expects a monotone trend, it is better to use model (6) as the alternative rather than (5). This leads to single degree-of-freedom chi-squared tests for testing independence QS = 0 in this model).

The likelihood-ratio test has chi-squared statistic given by double the difference in maximized log-likelihoods between the fit of model (6) and the simpler independence model having ft = 0.

logit [P(Y; <;)] = a, - ft, i = 1, ..., /, j = 1, ..., J - 1

logit[P(Y, </)] = «j - ßdh I = 1, ...,/,; = 1, ..., J - 1

Table IV. Example of part of SAS output (using PROC LOGISTIC) for fitting proportional odds model (6)

to Table I

Model Fitting Information and Testing Global Null Hypothesis

Table IV. Example of part of SAS output (using PROC LOGISTIC) for fitting proportional odds model (6)

to Table I

Model Fitting Information and Testing Global Null Hypothesis

 BETA = 0
0 0