# Testing for trends in partial proportional odds models

I am trying to build a model to explain an ordinal response variable $y$ with 4 levels: $y_0$, $y_1$, $y_2$ and $y_3$. The independent variable in this model is $v$. $v$ is a categorical variable with three categories $v_a$, $v_b$ and $v_c$. The assumption of proportional odds fails for $v$. Therefore, I am using the partial proportional odds (PPO) model. The model gives me 3 ORs for $y_1$, $y_2$ and $y_3$ that correspond to the odds for events $\{y>=y_1\}$, $\{y>=y_2\}$, and $\{y>=y_3\}$.

I would like to perform a test for trend for each of these 3 events such that each test gives a p-value. For example, the test for trend for $y_1$ would explain if there is a trend between the change in level of v and in the odds for $\{y>=y_1\}$.

Is there a way to use the partial PO model directly to compute a trend test for the 3 events?

Here’s a solution I am thinking of:

1. I first convert the categorical variable, v into a continuous variable with values 1,2, and 3 and build a PPO model using this continuous variable - let’s call this model PPOC.
2. For each outcome level $i$, I divide the samples in the data into 2 sets: $\{y>=y_i\}$ and $\{y < y_i\}$ and compute the likelihood of this split using PPO models with a) only an intercept term, and b) both intercept and exposure (i.e. PPOC). The difference in likelihoods of a) and b) gives me a test for this level.

Is this the right approach? Will this test be equivalent to a test for trend?

Are there any other approaches that I can adopt to determine if a trend exists between $v$ and $y$ for each of the 3 levels of $y$?

PS: I posted this question on the MedStats forum last week. However, there have been no responses thus far. I am hoping someone might have an opinion about this problem here.

Unfortunately, I don't know the partial proportional odds model. I do know that Agresti--discussing other situations (eg, PO)--has suggested that people go ahead & make up values for your categories and run them as though they were continuous. I find this somewhat discomfiting, but his argument is that unless you're way off, the induced bias will be small. (N.b., there will be bias induced, as a function of the nature and magnitude of the misrepresentation, but it is likely to be small.) Hope this helps.

• Thank you! Do you think there is way to estimate just how much bias there possibly could be? i.e. a way to tell how much of a departure from the categorical structure, the continuous (1, 2, 3) conversion is? – Ariel May 9 '12 at 8:58
• Not sure if this is a good approach, but even if you do decide to pretend the variable is properly coded for a numeric interval scale analysis the lack of fit in the model revealed by non-PO can still be present in the continuous model in another form. – Frank Harrell May 9 '12 at 12:07
• @FrankHarrell, I'm not sure either. I only offered this suggestion b/c no one had answered, & the OP asked me to. If you know of a productive approach in this situation, I'm sure the OP would appreciate it. – gung May 9 '12 at 13:22
• @Ariel, I really have no idea how much bias will result--it will depend on many factors. If your made up values are exactly correct, there won't be any, but that's not going to happen. 1 thing you could try is to run some simulations. Eg, you think they're really 1,2,3, (arithmetic sequence), but they're actually 1,2,4 (geometric sequence) and see how results differ, etc. – gung May 9 '12 at 13:34
• Peterson and my paper should give a bit of information. Once you fit the model you should be able to test any contrast you want. Also look at stat.auckland.ac.nz/~yee/VGAM – Frank Harrell May 11 '12 at 20:41

I think this is a reasonable way of assessing trend. The key boils down to the interpretation of the coefficient or effect that is estimated by the model. What you estimate is, in effect, an odds ratio for endorsing a unit higher $Y$ response comparing groups differing by 1 unit in $V$.

The proportional odds model usually requires the analyst to inspect the proportional odds assumption by either a significance test or by graphical inspection. I favor the latter. To inspect proportional odds graphically, one can directly fit logistic models for each of the cumulative responses and ensure the 95% CIs for the OR overlap with the "grand" OR estimated in the proportional odds model. If the overlap is consistent, we can be assured that any possible finding in the proportional odds model is not necessarily driven by one particular level of cumulative Y responses.

An alternative to this of course is just fitting the linear regression model with ordinal $Y$ response and ordinal $V$ response. This can be visually inspected with a scatterplot to assess the same assumptions, and the trend line can be compared to a more flexible cubic smoothing spline. To ensure consistent inference due to a possible mean-variance relationship, inference on the trend-line can be made robust using sandwich based standard errors. The similarity between proportional odds models and linear regression with an ordinal outcome has been discussed in The Importance of Normality Assumptions in Large Public Health Datasets by Lumley, Dier, Emerson, and Chen.