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Suppose $Y$ is an ordinal response variable with 4 levels (i.e. $Y$ can take on the values $1,2,3$ or $4$). We want to model $Y$ with $X_1, X_2$ using a proportional odds model. If $Y$ takes on the values of $1,2,3$ and $4$ an equal number of times (e.g. $Y = 1$ 4 times, $Y = 2$ 4 times, $Y = 3$ 4 times, and $Y= 4$ 4 times), will the estimates of some of the coefficients be 0?

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Not necessarily - it depends on the values of $X_1, X_2$. – Macro Feb 12 at 21:37
@Macro: Suppose $X_1$ and $X_2$ are binary and take on 1 and 0 an equal number of times. – Tony J Feb 12 at 22:13
Whether or not you will get 0 coefficients depends on how those outcomes correspond to the values of $Y$. If all possible combinations of $X_1,X_2$ occur equally often for each value of $Y$, then the coefficients for $X_1,X_2$ will be 0. – Macro Feb 12 at 22:53
@Macro: How would you interpret the regression coefficients if they are 0? What if the y intercept is 0? – Tony J Feb 12 at 23:08
If you're fitting a 4-level ordinal response using a proportional odds model then there should be three "threshold" values that are returned, not just a single intercept (ordinal logistic regression is equivalent to a regression model of a logistic random variable that is thresholded to produce ordinal levels - the thresholds returned denote the changeover points on that latent scale). In a binary logistic regression model, a '0' intercept and '0' values for all coefficients tells me half of all samples a 0s and half are 1s, and that predictors are distributed exactly the same in both halves. – Macro Feb 13 at 1:01
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up vote 3 down vote

Macro's answered this pretty well in the comments but to illustrate, and in case there were any doubt remaining that the answer is "not necessarily, indeed it is unlikely":

> library(MASS)
> x1 <- sample(0:1, 1000, replace=T)
> x2 <- sample(0:1, 1000, replace=T)
> y <- ordered(sample(1:4, 1000, replace=T))
> polr(y~x1+x2)
Call:
polr(formula = y ~ x1 + x2)

Coefficients:
      x1       x2 
 0.16223 -0.05669 

Intercepts:
     1|2      2|3      3|4 
-1.04553  0.05007  1.19741 

Residual Deviance: 2769.64 
AIC: 2779.64 
> polr(y~x1*x2)
Call:
polr(formula = y ~ x1 * x2)

Coefficients:
     x1      x2   x1:x2 
 0.2466  0.0277 -0.1638 

Intercepts:
     1|2      2|3      3|4 
-1.00048  0.09521  1.24313 

Residual Deviance: 2769.12 
AIC: 2781.12
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