# Is it possible to have a lower AIC value for the smaller of two nested models?

I am testing the parallel lines assumption of an ordered probit model by running a likelihood-ratio test against a model where the coefficients can vary for each step of the ordinal response variable.

As far as I understand the underlying concept, the ordered probit model can be considered as nested into the latter one by restricting all coefficients belonging the same independent variable to be equal.

Now it turns out that the likelihood-ratio test gives a p-value of 0.014, indicating that the latter model is a significant improvement over the ordered probit model and the parallel lines assumption therefore fails, wheareas the AIC value is lower (difference of 9) for the ordered probit model, indicating that the ordered probit model is preferable.

Since the likelihood-ratio statistic and the AIC seem to be generated similarly, I am wondering if and why these contradicting results are plausible?

Thanks in advance for any help

Markus

• LR result --> the restrictions do not hold in population. AIC result --> the restrictions reduce the estimation imprecision so greatly that even if they do not hold, they help predict outcomes better due to the reduced estimation imprecision. – Richard Hardy Mar 23 '18 at 14:02

$\Delta AIC = AIC_S - AIC_B = -2 log \mathcal{L}(\hat{\theta})_S + 2 DF_S -(-2 log \mathcal{L}{(\hat{\theta})}_B + 2 DF_B) = Test Statistic - 2\Delta DF = 47 - 56 = -9$
and $p=Pr(\chi^2_{28}>47)=0.014.$