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I just learned that there are three ways of constructing test statistics based on likelihood: likelihood ratio test, Wald test, and score test. I think it nicely categorizes various tests, and I liked it very much

  1. I would like to know if the three can cover all or most of the specific tests.

    What are some commonly-seen examples of specific test tasks and test rules that belong to any of the three?

    Are there some test task and test rule which doesn't belong to any of them? Especially, do the nonparametric tests belong to any of them?

  2. What are some other ideas or heuristics to construct test statistics, besides the three likelihood-based ones? By "other", they can be either likelihood-based or not.

Thanks and regards!

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The likelihood ratio test is considered the gold standard as it measures the difference in likelihood between the null value and the ML estimate. It also only requires you to calculate the value of the likelihood (or log-likelihood), though you will often calculate or estimate the 1st and/or second derivatives to find the ML estimate.

The Wald statistic only requires you to calculate the likelihood at the ML value, you do not need to calculate it at the null hypothesis value, though you need to calculate/estimate the 2nd derivative at the ML value.

The Wald statistic works well when the likelihood function is symmetric about the ML estimate, but can be misleading if the likelihood is very non-symmetric.

The score test only requires you to calculate the slope at the null hypothesis value, you do not need to find the ML estimate or calculate the likelihood at the ML estimate.

Both the Wald and Score methods have problems if there is a section of the likelihood that flattens out. One of the more common cases of this is the Hoek Donner effect in logistic regression where a very significant coefficient shows up as non-significant due to the flattening.

For your 2nd question, test statistics based on resampling (permutation and bootstrap) tests focus more on the way the data was collected and pretty much ignore the likelihood of the distribution of the data.

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  • $\begingroup$ Thanks, Greg! Why is "The Wald statistic works well when the likelihood function is symmetric about the ML estimate, but can be misleading if the likelihood is very non-symmetric"? $\endgroup$
    – Tim
    Commented May 28, 2013 at 0:33
  • $\begingroup$ @Tim, The Wald test and interval estimates the variance and then measures the distance relative to the variance in both directions. If the likelihood is symmetric then the distance in either direction will be about the same. But if the likelihood is not symmetric then a value a distance in one direction can have a very different likelihood than the value the same distance in the either direction. The likelihood ratio test would show them as very different in significance but the Wald test would give them equal significance. $\endgroup$
    – Greg Snow
    Commented May 28, 2013 at 1:33

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