p value from likelihood ratio test is STILL not the same as p value from maximum likelihood estimate

In yesterday's post, I described two ways for finding a p value for a parameter, and pointed out that the two ways lead to different p values. As an example, I considered the slope parameter in simple linear regression. One way to get a p value for the slope parameter is with a standard likelihood ratio test, for linear model with free slope versus intercept-only model. The other way to get a p value is to generate a sampling distribution of the MLE slope parameter itself, from a null hypothesis that consists of the MLE intercept-only model. Through remarks from commenters, especially Arash Khodadadi and Noah Silbert, I've been prodded to clarify the differences between these approaches. This post points out

• The likelihood ratio test compares model fits relative to the variance, while the sampling distribution of the MLE slope parameter is on the absolute scale of the data.
• An example in which the slope parameter has p < .05 two-tailed in the sampling distribution of the MLE slope parameter but has p > .05 (one-tailed) in the sampling distribution of the likelihood ratio.

First, the data (a little different from the previous posts):

The header of the plot, above, indicates the MLE of the linear model, and -2log(LR), a.k.a. G2, rounded to three significant digits.

The null hypothesis is an intercept-only model (beta1=0) with beta0 and sigma set to their MLE values when beta1 is fixed at zero. (Which means, in this case, that beta0 is the mean of y and sigma is the sd of y using N not N-1 in the denominator.) I generated sample data from the null hypothesis using the x values in the actual data. For each sample I computed the MLE of the full model and -2log(LR). The resulting marginal sampling distributions are shown here:

In the left panel, above, the one-tailed p value of MLE beta1 is displayed; multiply it by 2 to get the two-tailed p value. Notice it is different than the p value of the likelihood ratio.

Below is the joint sampling distribution, where each point is a sample from the null hypothesis. There is a new twist to this figure: Each point is color coded for the magnitude of MLE sigma, where blue is the largest MLE sigma in the distribution and red is the smallest MLE sigma in the distribution.

The joint sampling distribution also shows the thresholds for p=.05 (one-tailed or two-tailed), and the actual data statistics are plotted as a "+".

You can see from the joint sampling distribution that MLE beta1 can be large-ish even when -2log(LR) is small-ish when the sample MLE sigma is large-ish (blue points). But the opposite can happen when the sample MLE sigma is small-ish (red points). Thus, a key difference between the measures of the slope parameter is how they deal with the variance. The likelihood ratio compares the free-slope against intercept-only models relative to the variance, while the MLE beta1 considers the slope on the absolute scale of the data, not relative to the variance.

As discussed in yesterday's post, I don't think either test is inherently better than the other. They just ask the question about the slope parameter in different ways. As mentioned yesterday, posing the question in terms of absolute MLE beta1 has direct intuitive interpretation. It's also much easier to use when defining confidence intervals as the range of parameter values not rejected by p<alpha (which is, for me, the most coherent way to define confidence intervals). But that's a topic for another day!