# 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.

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