# The trinity of tests in maximum likelihood: what to do when faced with contradicting conclusions?

The Wald, Likelihood Ratio and Lagrange Multiplier tests in the context of maximum likelihood estimation are asymptotically equivalent. However, for small samples, they tend to diverge quite a bit, and in some cases they result in different conclusions.

How can they be ranked according to how likely they are to reject the null? What to do when the tests have conflicting answers? Can you just pick the one which gives the answer you want or is there a "rule" or "guideline" as to how to proceed?

• is this, perhaps, just a case where the asymptotic approximations to the [null] distributions of one or more of the three test statistics are not so good? perhaps the results differ because the actual sizes of the tests are different? how large are your sample sizes? Commented Sep 20, 2010 at 1:56

I do not know the literature in the area well enough to offer a direct response. However, it seems to me that if the three tests differ then that is an indication that you need further research/data collection in order to definitively answer your question.

You may also want to look at this Google Scholar search

Update in response to your comment:

If collecting additional data is not possible then there is one workaround. Do a simulation which mirrors your data structure, sample size and your proposed model. You can set the parameters to some pre-specified values. Estimate the model using the data generated and then check which one of the three tests points you to the right model. Such a simulation would offer some guidance as to which test to use for your real data. Does that make sense?

• Are you referring to a particular paper? I imagine I could find an answer to my question if I researched, studied, read a lot, but so could 95% of the questions other people ask here... Also, in some cases, particularly with macroeconomics data (which is my area), there is no more data to be collected. Data is scarce (the number of observations, I mean), and you just have to live with it. There is no "get more data" solution. I was hoping someone here would know the topic, but it doesn't seem like. Maybe once the website is opened to the general public?
– Vivi
Commented Jul 21, 2010 at 20:07
• I suspect the answer to your question will be domain/model specific and hence I am not sure I can recommend a specific paper.
– user28
Commented Jul 22, 2010 at 1:19
• Sorry for the late reply. I like your suggestion of simulation. That is not really easy, though. The truth is, what I see in practice is that researchers just do the test that is computationally easier or that give them the result they want.
– Vivi
Commented Jul 26, 2010 at 7:41

I won't give a definitive answer in terms of ranking the three. Build 95% CIs around your parameters based on each, and if they're radically different, then your first step should be to dig deeper. Transform your data (though the LR will be invariant), regularize your likelihood, etc. In a pinch though, I would probably opt for the LR test and associated CI. A rough argument follows.

The LR is invariant under the choice of parametrization (e.g. T versus logit(T)). The Wald statistic assumes normality of (T - T0)/SE(T). If this fails, your CI is bad. The nice thing about the LR is that you don't need to find a transform f(T) to satisfy normality. The 95% CI based on T will be the same. Also, if your likelihood isn't quadratic, the Wald 95% CI, which is symmetric, can be kooky since it may prefer values with lower likelihood to those with higher likelihood.

Another way to think about the LR is that it's using more information, loosely speaking, from the likelihood function. The Wald is based on the MLE and the curvature of the likelihood at null. The Score is based on the slope at null and curvature at null. The LR evaluates the likelihood under the null, and the likelihood under the union of the null and alternative, and combines the two. If you're forced to pick one, this may be intuitively satisfying for picking the LR.

Keep in mind that there are other reasons, such as convenience or computational, to opt for the Wald or Score. The Wald is the simplest and, given a multivariate parameter, if you're testing for setting many individual ones to 0, there are convenient ways to approximate the likelihood. Or if you want to add a variable at a time from some set, you may not want to maximize the likelihood for each new model, and the implementation of Score tests offers some convenience here. The Wald and Score become attractive as your models and likelihood become unattractive. (But I don't think this is what you were questioning, since you have all three available ...)