# Proportional odds assumption violated but not sure what to do

I run an ordinal regression model and I wanted to check the proportional odds assumption. In order to do that I used the VGAM package and I run olr twice, the first under the assumption and the second without the assumption. Below is the code and the results

> fit1 <- vglm(stage ~Ki67+Cyclin_E,family=cumulative(parallel=T))
> summary(fit1)

Call:
vglm(formula = stage ~ Ki67 + Cyclin_E, family = cumulative(parallel = T))

Pearson Residuals:
Min       1Q  Median      3Q    Max
logit(P[Y< = 1]) -3.1177 -0.43593 0.37246 0.53111 1.4927
logit(P[Y< = 2]) -3.8479  0.14119 0.18785 0.28679 1.9903

Coefficients:
Estimate Std. Error  z value
(Intercept):1  2.2414705   1.091225  2.05409
(Intercept):2  3.2164214   1.178916  2.72829
Ki67          -0.1157273   0.039889 -2.90124
Cyclin_E       0.0085266   0.028626  0.29786

Number of linear predictors:  2

Names of linear predictors: logit(P[Y< = 1]), logit(P[Y< = 2])

Dispersion Parameter for cumulative family:   1

Residual deviance: 50.82946 on 62 degrees of freedom

Log-likelihood: -25.41473 on 62 degrees of freedom

Number of iterations: 5

> summary(fit2)

Call:
vglm(formula = grade ~ Ki67 + Cyclin_E, family = cumulative(parallel = F),
maxit = 50)

Pearson Residuals:
Min       1Q   Median      3Q    Max
logit(P[Y< = 1]) -1.1870 -0.65271 -0.23199 0.44910 3.4798
logit(P[Y< = 2]) -2.6235 -0.70599  0.27305 0.72691 2.8544

Coefficients:
Estimate Std. Error   z value
(Intercept):1 -0.059702   0.928078 -0.064328
(Intercept):2  2.687277   1.050909  2.557097
Ki67:1        -0.100832   0.047754 -2.111483
Ki67:2        -0.101817   0.036567 -2.784377
Cyclin_E:1     0.018768   0.022708  0.826474
Cyclin_E:2    -0.015416   0.022927 -0.672390

Number of linear predictors:  2

Names of linear predictors: logit(P[Y< = 1]), logit(P[Y< = 2])

Dispersion Parameter for cumulative family:   1

Residual deviance: 78.93318 on 78 degrees of freedom

Log-likelihood: -39.46659 on 78 degrees of freedom

Number of iterations: 34


In order to check if the difference of the two models is significant I run the next command

pchisq(deviance(fit2)-deviance(fit1),df=df.residual(fit2)-df.residual(fit1),lower.tail=FALSE)
 0.03072927


As you see the result is that the 2 models differ and so the proportional odds assumption isn't true. But if you see the coefficients about the Ki67 (Cyclin is not significantly important so i guess i can skip it) they are almost the same. In that case what should I do? I believe that I could stick with the model under the po assumption but I'd like to know what others think

• If you don't think the proportional odds assumption is reasonable, then you could fit a multinomial logistic model instead of an ordinal logistic model. There are more parameters but fewer assumptions... – Macro Apr 10 '12 at 12:54
• On the contrary, I believe that the proportional odds holds despite the results of the test since the coefficients don't differ almost at all.Regarding the use of a multinomial logistic model isn't it wrong to use it when you know the response variable is ordered? – Nick Apr 10 '12 at 15:03
• It's not "wrong" to use the multinomial logistic model. True, it doesn't take advantage of the ordinal structure in the data but, as I said, the ordinal model is a submodel of the multinomial model. Therefore, any fit achievable with the ordinal model is achievable with the multinomial model. If the proportional odds assumption does hold, you're sacrificing parsimony by using the multinomial model. – Macro Apr 10 '12 at 15:23

• @Peter Flom- I'll check your idea about the scatterplot. In the meantime I did the olr is SPSS and it didn't reject the assumption. Moreover I found on the net a pdf by Agresti where he faces a similar problem and his answer is the following The improvement in fit is statistically significant, but perhaps not substantively significant; effect of dose is moderately negative for each cumulative probability which I think is the case in my problem since in both models the Ki67 coef. are similar. Also could you explain to me what do you mean by saying rescaling of the the parameter estimates? – Nick Apr 11 '12 at 13:36
• Sorry I forgot to tell you the name of the pdf in case you or someone else is interested. Well, it's called Examples of Using R for Modeling Ordinal Data. – Nick Apr 12 '12 at 10:43