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I'm currently studying a paper by Winkler et. al on permutation inference for the general linear model and am confused by their reasoning and notation.

They first introduce a formulation for the general linear model:

Y = M ψ + ϵ (Eq. 1)

where Y is the N × 1 vector of observed data, M is the full-rank N × r design matrix that includes all effects of interest as well as all modelled nuisance effects, ψ is the r × 1 vector of r regression coefficients, and ϵ is the N × 1 vector of random errors.

Regarding the null hypothesis, it is stated that:

Our interest is to test the null hypothesis that an arbitrary combination (contrast) of some or all of these parameters is equal to zero, i.e., H0 : C′ψ = 0, where C is a r × s full-rank matrix of s contrasts, 1 ≤ s ≤ r.

The general formulation (Eq. 1) in the first quote can be transformed into a partioned one:

Y = X β + Z γ + ϵ (Eq. 2)

where X is the matrix with regressors of interest, Z is the matrix with nuisance regressors, and β and γ are the vectors of regression coefficients.

When discussing the Freedman-Lane procedure for permutation inference in the following it is stated that:

The rationale for this permutation method is that, if the null hypothesis is true, then β = 0 [...]

These paragraphs pose two questions to me:

  1. What does C′ in H0 : C′ψ = 0 denote mathematically speaking, given that C is a matrix?

  2. How does C′ψ = 0 imply β = 0? Is the reasoning C′ψ = 0, therefore Y = M0 + ϵ = X0 + Z0 + ϵ = ϵ valid?

Thanks a lot in advance, I'd be very grateful for help or advice on how to approach/present this problem in a better way.

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The C matrix just indicates which slopes, or linear combinations of slopes, is equal to 0. If you want to test that the first slope is 0 (but the remaining can be anything) then C will just be a vector with the first element $1$ and the remaining elements are $0$. If you want to test that the first 3 slopes are all 0, then C is a 3x3 identity matrix with the remaining rows equal to $0$. This is the simple case which leads to $\beta=0$ for a set of the slopes. But is also more general in that you could test if $\beta_1 = \beta_2$ by setting the first column of C to $1$ and $-1$.

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  • $\begingroup$ Thanks a lot for that answer! I do however have a basic follow up question: If C is a r x s matrix that indicates which slopes are equal to zero and ψ a r x 1 vector that contains the slopes/regression coefficients themselves, how can C′ψ result in a scalar as stated in the null hypothesis (C′ψ = 0). Shouldn't the result always be of form of a r x 1 vector? $\endgroup$
    – Adrian Mak
    Oct 9, 2023 at 18:58
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    $\begingroup$ @AdrianMak, the null can be a scalar if you only have one contrast ($s=1$), but in general the right side of the null hypothesis is a vector of $0$'s, each contrast is equal to 0. $\endgroup$
    – Greg Snow
    Oct 10, 2023 at 22:47
  • $\begingroup$ Okay thanks, allow me one last question: In C′ψ = 0, what does ` ′ ` indicate? Because from all of your answers, it just seems like Cand ψ are just multiplied in the null hypothesis. $\endgroup$
    – Adrian Mak
    Oct 11, 2023 at 13:59
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    $\begingroup$ The prime "'" indicates the transpose of the C matrix. $\endgroup$
    – Greg Snow
    Oct 11, 2023 at 22:34

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