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I would like to ask you whether the Wald Z test which is used in context of null model for multilevel logistic regression is actually two-tailed or not.

So far, I've found that Wald has chi-square distribution which is one-tailed, but I have not been able to find anything about the situation when there is value for Wald Z. I want to test the null hypothesis whether the variance in second level is actually statistically significant different from zero.

Also, if the Wald-z is two-tailed, can I divide the p-value by two so I get just the right p-value for $H_{0}$ (the variance in second level is actually statistically significant bigger than zero)?

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  1. The Wald test is a bad enough test for fixed effects in multilevel models. There is no Wald test for variance components.
  2. The likelihood ratio can be used to test variance components. The asymptotic test stat follows a a chi-square distribution with 0.5 degrees of freedom.
  3. The test tailedness is set up however you specify. Most software defaults to two tailed, which can be divided by two only when the statistician ensures the correction direction of effect
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  • $\begingroup$ "There is no Wald test for variance components." Uh... isn't that given by $\chi^{2} = \frac{\sigma^{2}}{var(\sigma^{2})}$? For example the MLwiN manuals (multilevel modelling software) give instructions for conducting Wald tests on variance components (although they note that Wald tests make rough approximations, and recommend using MCMC estimation methods to generate interval estimates on variance components). $\endgroup$
    – Alexis
    Feb 5 '20 at 21:32
  • $\begingroup$ @Alexis Hauck and Donner 77 show undesireable traits in the Wald test for hypotheses that lie on the boundary, such as $\sigma^2_0 = 0$. $\endgroup$
    – AdamO
    Feb 5 '20 at 21:53
  • $\begingroup$ I don't disagree that it is not a great test, but people do use it as a test for variance components. $\endgroup$
    – Alexis
    Feb 5 '20 at 21:56
  • $\begingroup$ Thanks for the answers. I do share some of yours concern regarding the use of Wald test for this purpose, however in my current situation there is no other scenario - I have to use this test and report results. I've got output from multilevel logit model from SPSS which reports the results of Wald Z, so my question stands still. Is this Wald Z calculated from normal distribution ergo the p-value is reported for two-tailed hypothesis? If so, the direction is perfectly clear, I'd just assume that variance is stat. bigger than zero. Thanks. $\endgroup$
    – Nothingman
    Feb 6 '20 at 6:56
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    $\begingroup$ @Roman you're asking a lot of questions so it's hard to pinpoint an answer. If you're testing for a variance component why does tailedness matter? Variance is 0 or not 0. $\endgroup$
    – AdamO
    Feb 6 '20 at 16:47

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