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I'm beginner in statistics. I would like to know which test statistic is insensitive to the degrees of freedom among Z, T and F statistic. Are all of these sensitive or insensitive? Thanks !

Sang
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1 Answers1

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First, there might be a misconception to correct. Degrees of freedom are not an external quantity of the statistics. It pertains to the test itself. For example whether you consider when doing a t-test that variances in different groups are the same or not, you won't have the same test and won't have the same numbers of degree of freedom even if it's still a t statistic because you have a different number of quantities to be estimated.

One have to remember how much data we have burned off to estimate some parameters. What remains is the number of degrees of freedom. The idea is that the more you estimate quantities, the less you have degrees of freedom, the more your test must punish this. A statistic with $N$ degrees of freedom would then be more extreme than the same value with $N-1$ degrees of freedom. considered less extreme In your question :

is insensitive to the degrees of freedom

could then be replaced by

do not need to be punished because of estimated parameters.

Sometimes you don't need to punish for these estimated quantities, this is the case for the z test.

Degrees of freedom are a confusing concept, for a deeper understanding I would suggest to have a look on this question How to understand degrees of freedom?

Community
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brumar
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    This answer starts well. The remarks about the z test make me think I did not understand, though. Although the z test does not "use" degrees of freedom, the accuracy of its results will depend strongly on the actual degrees of freedom. Moreover, a z test computes both a sample mean and a sample standard deviation, each of which have always been understood as estimates of underlying distributional properties. This suggests that phrases like "use" and "rely on" may be too ambiguous to convey the somewhat subtle ideas you are writing about. – whuber Jun 12 '15 at 16:35
  • @whuber Thanks for your helpful review. My remarks on the z test were dubious to say the least. I tried to improve my answer based on your comment. I know this idea of punishment is not perfect but this is the simplest way I found. – brumar Jun 12 '15 at 19:57
  • +1 I like the "punishment" wording because it conveys an essential idea but is sufficiently imprecise and colloquial to warn us it's not a technical term. I believe another popular term, which might not have quite the same negative connotations, is "penalized"--but that one has acquired some technical meanings. – whuber Jun 12 '15 at 20:03