# [R] Likelihood ratio test for proportional odds logistic regression

xinyi lin x1lin at ucsd.edu
Sat Jan 5 14:14:23 CET 2008

```Hi,

I want to do a global likelihood ratio test for the proportional odds
logistic regression model and am unsure how to go about it. I am using
the polr() function in library(MASS).

1. Is the p-value from the likelihood ratio test obtained by
anova(fit1,fit2), where fit1 is the polr model with only the intercept
and fit2 is the full polr model (refer to example below)? So in the
case of the example below, the p-value would be 1.

2. For the model in which there is only one independent variable, I
would expect the Wald test and the likelihood ratio test to give
similar p-values. However the p-values obtained from anova(fit1,fit3)
(refer to example below) are very different (0.0002622986 vs. 1). Why
is this so?

> library(MASS)
> fit1 <- polr(housing\$Sat~1)
> fit2<- polr(housing\$Sat~housing\$Infl)
> fit3<- polr(housing\$Sat~housing\$Cont)
> summary(fit1)

Re-fitting to get Hessian

Call:
polr(formula = housing\$Sat ~ 1)

No coefficients

Intercepts:
Value   Std. Error t value
Low|Medium  -0.6931  0.2500    -2.7726
Medium|High  0.6931  0.2500     2.7726

Residual Deviance: 158.2002
AIC: 162.2002
> summary(fit2)

Re-fitting to get Hessian

Call:
polr(formula = housing\$Sat ~ housing\$Infl)

Coefficients:
Value Std. Error      t value
housing\$InflMedium 6.347464e-06  0.5303301 1.196889e-05
housing\$InflHigh   6.347464e-06  0.5303301 1.196889e-05

Intercepts:
Value   Std. Error t value
Low|Medium  -0.6931  0.3953    -1.7535
Medium|High  0.6932  0.3953     1.7536

Residual Deviance: 158.2002
AIC: 166.2002
> summary(fit3)

Re-fitting to get Hessian

Call:
polr(formula = housing\$Sat ~ housing\$Cont)

Coefficients:
Value Std. Error      t value
housing\$ContHigh 0.0001135777  0.4330091 0.0002622986

Intercepts:
Value   Std. Error t value
Low|Medium  -0.6931  0.3307    -2.0956
Medium|High  0.6932  0.3307     2.0960

Residual Deviance: 158.2002
AIC: 164.2002
> anova(fit1,fit2)
Likelihood ratio tests of ordinal regression models

Response: housing\$Sat
Model Resid. df Resid. Dev   Test    Df      LR stat. Pr(Chi)
1            1        70   158.2002
2 housing\$Infl        68   158.2002 1 vs 2     2 -6.375558e-10       1
> anova(fit1,fit3)
Likelihood ratio tests of ordinal regression models

Response: housing\$Sat
Model Resid. df Resid. Dev   Test    Df      LR stat. Pr(Chi)
1            1        70   158.2002
2 housing\$Cont        69   158.2002 1 vs 2     1 -1.224427e-07       1

Thank you,
Xinyi

```