[R] Likelihood ratio test for proportional odds logistic regression

[Prof Brian Ripley]

On Sat, 5 Jan 2008, xinyi lin wrote:

> 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.

There is no improvement in fit, as the near-zero coefficients show.  You 
are not calling polr correctly on this example: 'why is this so?' given 
that it *is* the example on the help page and all you had to do was to 
read the help (or the book for which this is support software).

> 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?

Because you compared a t-value to a p-value, not at all the same thing.

>
>
>> 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
>
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>

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595