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I am concerned about a lack of attention among researchers towards whether (or how) nuisance parameters affect degrees of freedom. For our purposes here we are considering

$$\underbrace{\text{df}}_{\text{Residual Degrees of Freedom}} = \underbrace{m}_{\text{Sample Degrees of Freedom}} - \underbrace{k}_{\text{Model Degrees of Freedom}}$$

rather than other notions such as corrections similar to Bessel's correction or effective degrees of freedom. I often see worded definitions of Model Degrees of Freedom taken to mean "the number of estimated parameters in a model". This worded definition seems to imply that we would include nuisance parameters in this calculation if they are estimated, but often this is not what is done in practice.

Example: DeSeq2 uses a model in which certain dispersion parameters are introduced and estimated, but one of the authors of this package has implied that these parameters are ignored in the calculation of degrees of freedom.

Please (mathematically and with references, if possible) explain why nuisance parameters should or should not be considered in computing model degrees of a freedom.

DifferentialPleiometry
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    I think the answers must depend on how these "degrees of freedom" are used. What do you have in mind? In the context you describe, one would expect it to be related to calculation of p-values. But in that case the proof is in the pudding: the p-values are either correct (or at least approximately so) or they are not. To the extent those p-values are computed using some "degree of freedom" quantity, arguably that DF is correct if and only if the associated p-values are going to be correct. – whuber Dec 01 '21 at 18:46
  • @whuber By "correct" in the description of a p-value, you mean "unbiased"? Are you alluding to *ad hoc* usages of degrees of freedom other than model identification (e.g. in linear regression) and obtaining unbiased estimators (e.g. Bessel's correction)? – DifferentialPleiometry Dec 23 '21 at 20:00
  • I don't really mean "unbiased." The p-value will be *erroneously* calculated when one uses the wrong DF in a chi-squared test, for instance. This calculation of a p-value is not usually viewed as an estimator (but see https://stats.stackexchange.com/questions/181611 for some counterpoint). I am not alluding to *ad hoc* usages, but rather to *wrong* usages, as in the example I criticize towards the end of https://stats.stackexchange.com/a/17148/919. – whuber Dec 23 '21 at 22:01

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