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Comments to question in last exercise session

Comments to question in last exercise session

par Jan Stöcker,
Nombre de réponses : 1
In the last exercise session, the question was raised, whether the null-hypothesis for testing random-slope in a LME has to be formulated as

H0: d22 = d21 = 0  (like in the lecture slides) or H0: d22 = 0 (like in the exercise session)

where d22 is the variance of the random slope and d21 is the covariance of random slope and random intercept.

I replied that both is okay, since d22 = 0 already implies d21 = 0. However, to clearify things I want to add two comments:

1. The above is correct - you can use both formulations and procede like we did in the exercise and like it is desribed in Chapter 5 Section 'LRT for the variance components' in the lecture slides.

2. Still, it is not correct that the fact, that also d21 = 0, does not inflict the testing like I suggest it: it is accounted for in the second part of the mixture distribution of the test statistic, the chi-squared distribution with one less degree of freedom. But this is already part of the procedure, so for application, you don't have to consider anything but comment 1. 

Best regards,

Almond

En réponse à Jan Stöcker

Re: Comments to question in last exercise session

par Sonja Greven,
Maybe to further clarify on both points:
1. While d_22 implies d_21, as the covariance cannot be different from zero if the variance is zero, is it clearer to write H_0 : d_22 = d_21 = 0, as this makes it more explicit that we are testing two parameters.
2. Of these two parameters,
  • d_22 is on the boundary of the parameter space under the null and only testing H_0: d_22 =0 would result in a mixture distribution of a point mass at zero and a chi-square distribution with one degree of freedom as explained on the slides for d_11.
  • d_21 is not on the boundary of the parameter space under the null and just testing this parameter would result in a chi-square distribution with one degree of freedom as usual.
The limiting distribution when testing both at the same time is the sum of this chi-square-1 and this mixture distribution, which is the mixture distribution of a chi-square-1 and a chi-square-2 given on the slides. (Similarly to the usual chi-square-2 distribution that we would get without the boundary issue corresponding to the sum of two independent chi-square-1 distributed variables.)