Table 2.

A number of methods are available to obtain parameter estimates and construct a reference distribution in the presence of nuisance variables.

Method	Model
Draper–Stonemana	Y = PXβ + Zγ + ϵ
Still–Whiteb	PRZY = Xβ + ϵ
Freedman–Lanec	(PRZ + HZ)Y = Xβ + Zγ + ϵ
Manlyd	PY = Xβ + Zγ + ϵ
ter Braake	(PRM + HM)Y = Xβ + Zγ + ϵ
Kennedyf	PRZY = RZXβ + ϵ
Huh–Jhung	PQ′RZY = Q′RZXβ + ϵ
Smithh	Y = PRZXβ + Zγ + ϵ
Parametrici	Y = Xβ + Zγ +ϵ, ϵ ∼ N(0, σ2I)

^aDraper and Stoneman (1966). This method was called “Shuffle Z” by (Kennedy, 1995), and using the same notation adopted here, it would be called “Shuffle X”.

^bGail et al. (1988); Levin and Robbins (1983); Still and White (1981). Still and White considered the special anova case in which Z are the main effects and X the interaction.

^cFreedman and Lane (1983).

^dManly (1986); Manly (2007).

^eter Braak (1992). The null distribution for this method considers ${\displaystyle {\widehat{\mathit{\beta }}}_{\mathit{j}}^{\ast }}=\widehat{\mathit{\beta }}$, i.e., the permutation happens under the alternative hypothesis, rather than the null.

^fKennedy (1995); Kennedy and Cade (1996). This method was referred to as “Residualize both Y and Z” in the original publication, and using the same notation adopted here, it would be called “Residualize both Y and X”.

^gHuh and Jhun (2001); Jung et al. (2006); Kherad-Pajouh and Renaud (2010). Q is a N′ × N′ matrix, where N′ is the rank of R_Z. Q is computed through Schur decomposition of R_Z, such that R_Z = QQ′ and ${\mathbf{I}}_{{\mathit{N}}^{\prime }\times {\mathit{N}}^{\prime }}={\mathbf{Q}}^{\mathbf{\prime }}\mathbf{Q}$. For this method, P is N′ × N′. From the methods in the table, this is the only that cannot be used directly under restricted exchangeability, as the block structure is not preserved.

^hThe Smith method consists of orthogonalization of X with respect to Z. In the permutation and multiple regression literature, this method was suggested by a referee of O'Gorman (2005), and later presented by Nichols et al. (2008) and discussed by Ridgway (2009).

ⁱThe parametric method does not use permutations, being instead based on distributional assumptions. For all the methods, the left side of the equations contains the data (regressand), the right side the regressors and error terms. The unpermuted models can be obtained by replacing P for I. Even for the unpermuted models, and even if X and Z are orthogonal, not all these methods produce the same error terms ϵ. This is the case, for instance, of the Kennedy and Huh–Jhun methods. Under orthogonality between X and Z, some regression methods are equivalent to each other.