Ordinal Logistic Regression with a Different Link Function

Consider an outcome variable that has four clear, ordered categories to it. This seems like a good use of ordinal logistic regression to estimate Odds Ratios for the effect of covariates on moving a subject one "step" up the ladder.

But the subjects are particularly evenly spread throughout the categories, so a question arises:

• Is the "rare outcome assumption" for an OR to approximate a relative risk still true in ordinal logistic regression?
• If so, is it possible to change the link function to directly estimate a relative risk, and is it still possible to use something like a poisson approximation with robust standard errors to deal with convergence issues in such a case?

I think we first have to ask whether it's necessary to use proportional odds logistic regression to approximate a cumulative relative risk, e.g. the relative risk of reporting a higher outcome. The probabilistic formulation of the proportional odds model relies on observing arbitrary bins of a latent logistic random variable. See my relevant question here. The elegance of this method is that the survival function (1-CDF) of a logistic RV is the inverse logit, e.g. $P(Z > z) = \exp(-z)/(1+\exp(-z))$.

If we are to assume a similar probabilistic derivation of a relative risk model, the desire is to find a latent random variable whose survival function is $P(Z > z) = \exp(-z)$. But that is just an exponential random variable, which is memoryless. Therefore, if we construct the matrix of thresholded outcome variables, $O_{ij} = \mathcal{I}(Y_{i} \ge j)$, (I believe) the cell frequencies are conditionally independent, and thus are amenable to modeling via a log-linear model which is just Poisson regression. This is reassuring because the interpretation of Poisson coefficients is as a relative rate. Modeling the interaction between the response variable as numeric outcome and the regression coefficients leads to the correct interpretation.

That is, fit the log-linear model:

$$\log (N_{ij} | Y_{i}, \mathbf{X}_{i,}) = \eta_0 I(Y_{i} = 0) + \ldots + \eta_j I(Y_i == j) + \vec{\beta} \mathbf{X}_{i,} + \vec{\gamma} \text{diag(Y)} \mathbf{X}_{i,}$$

Using the example from the MASS package: we see the desired effect that the relative risk is much smaller than the OR in all instances:

newData <- data.frame('oy'=oy, 'ny'=as.numeric(y), housing)

## trick: marginal frequencies are categorical but interactions are linear
## solution: use linear main effect and add indicators for remaining  n-2 categories
## equivalent model specifications
fit <- glm(Freq ~ oy.2 + ny*(Infl + Type + Cont), data=newData, family=poisson)
effects <- grep('ny:', names(coef(fit)), value=T)
print(cbind(
coef(summary(fit))[effects, ],
coef(summary(house.plr))[gsub('ny:','', effects), ]
), digits=3)

Gives us:

Estimate Std. Error z value Pr(>|z|)  Value Std. Error t value
ny:InflMedium       0.360     0.0664    5.41 6.23e-08  0.566     0.1047    5.41
ny:InflHigh         0.792     0.0811    9.77 1.50e-22  1.289     0.1272   10.14
ny:TypeApartment   -0.299     0.0742   -4.03 5.55e-05 -0.572     0.1192   -4.80
ny:TypeAtrium      -0.170     0.0977   -1.74 8.21e-02 -0.366     0.1552   -2.36
ny:TypeTerrace     -0.673     0.0951   -7.07 1.51e-12 -1.091     0.1515   -7.20
ny:ContHigh         0.106     0.0578    1.84 6.62e-02  0.360     0.0955    3.77

Where the first 4 columns are inference from the log-linear model and the second 3 columns come from the proportional odds model.

This answers perhaps the most important question: how does one fit such a model. I think it can be used to explore the relative approximation(s) of ORs for rare events to the RRs.

Is the "rare outcome assumption" for an OR to approximate a relative risk still true in ordinal logistic regression?

Not really. You said yourself that your outcomes are evenly spread throughout the four categories, so no category is going to be particularly rare.

If so, is it possible to change the link function to directly estimate a relative risk, and is it still possible to use something like a poisson approximation with robust standard errors to deal with convergence issues in such a case?

You can, but there is a risk that when you use your model to make predictions, the predicted probability of being in a class might be more than 1.

The standard ordered logit model is formulated $$Y_i \sim categorical({\bf{p}}_i);logit({\bf{p}}_i) = X\beta$$ together with the proportional odds assumption. All we are doing is replacing the "logit" with "log", which still produces a valid model with a valid likelihood that produces valid estimates for $\beta$. When you apply these to real data though, it is possible that a component for $\bf{p}_i$ is more than one (and since this is outside the range of the proportional odds assumption, you can't use it to populate the remaining components).

This can't happen if you only use your model to predict on the data it was trained on, and is less likely if

• you have a lot of training data
• your training data covers all possible combinations of covariates (if they are categorical) or the full range of covariates (if they are numeric)
• I don't think you have written the correct standard ordered logit model. Do you have a copy of Agresti or McCullogh & Nelder? Agreed that if the response distribution is even among the $K$ ordered logit categories, the approximation is moot. But what if most participants were clustered in the lowest response category? – AdamO Jan 9 '18 at 15:24
• @AdamO it is probably not the formulation you are used to, but it is equivalent (as long as X includes an intercept). I chose it as it emphasises the points most relevant to the question. (It obviously isn't equivalent once you replace logit with log, but this formulation appears to generalise the best) – JDL Jan 10 '18 at 9:12
• The prop odds model has important constraints, namely the intercept term (not depicted) for each categorical contrast is ordered, furthermore $Y_i$ is the categorical designation, but the modeled probability is the cumulative probability. I think I am safe in saying you have merely written an unconditional logistic model and this is not correct. – AdamO Jan 10 '18 at 18:47
• These constraints are addressed by the proportional odds assumption. (I agree they are not expressed in the equation I have stated) – JDL Jan 11 '18 at 8:36