![]() Microbenchmark(your_code(), plyr_code(), base_code(), data.table_code()) Thanks to Samuel Davenport (University of Oxford) for pointing out.If I'm understanding your question correctly, this data.table solution will also work: library(data.table)Īdding to Robert's benchmark above: library(plyr) 07.Jun.2020: Lemma 2 corresponds to the Frisch–Waugh–Lovell theorem.07.Jun.2020: A graphical representation of Lemma 1 can be found in p.40 of Jaromil Frossard PhD thesis (Université de Genève), available here.Permutation inference for the general linear model. Winkler AM, Ridgway GR, Webster MA, Smith SM, Nichols TE.Journal of Business & Economic Statistics 1983 1(4):292. A Nonstochastic Interpretation of Reported Significance Levels. Australian & New Zealand Journal of Statistics of Statistics 2001 43(1):75–88. Journal of Statistical Computation and Simulation 1999 62(3):271–303. An empirical comparison of permutation methods for tests of partial regression coefficients in a linear model. Hence, the hat matrix cancels out, meaning that it is not necessary. Thus, reversing it, we obtain the final result: What is left has the same form as the result of Lemma 2. To see why, remember that multiplying both sides of an equation by the same factor does not change it (least squares solutions may change transformations using Lemma 2 below do not act on the fitted model). Lemma 2 ( Frisch–Waugh–Lovell theorem): Given a GLM expressed as, we can estimate from an equivalent GLM written as. ![]() This is because, hence since is idempotent. Lemma 1: The product of a hat matrix and its corresponding residual-forming matrix is zero, that is. However, in the paper we do not offer any proof of this important result, that allows algorithmic acceleration. add the nuisance variables back in Step 3 is not strictly necessary, and the model can be expressed simply as, implying that the permutations can actually be performed just by permuting the rows of the residual-forming matrix. Where is a permutation matrix (for the -th permutation, is the hat matrix due to the covariates, and is the residual forming matrix the superscript symbol represents a matrix pseudo-inverse. Count how many times was found to be equal to or larger than, and divide the count by the number of permutations the result is the p-value.Repeat the Steps 2-4 many times to build the reference distribution of under the null hypothesis of no association between and.Use the estimated to compute the statistic of interest.Regress the permuted data against the full model, i.e.This is done by pre-multiplying the residuals from the reduced model produced in the previous step,, by a permutation matrix,, then adding back the estimated nuisance effects, i.e. , obtaining estimated parameters and estimated residuals. Regress against a reduced model that contains only the covariates, i.e.Use the estimated parameters to compute the statistic of interest, and call this statistic. Regress against the full model that contains both the effects of interest and the nuisance variables, i.e.The procedure can be performed through the following steps: One of these various methods is the one published in Freedman and Lane (1983), which consists of permuting data that has been residualised with respect to the covariates, then estimated covariate effects added back, then the full model fitted again. (2014) other previous work include the papers by Anderson and Legendre (1999) and Anderson and Robinson (2001). Over the years, many methods have been proposed. Where is a matrix of observed variables, is a matrix of predictors of interest, is a matrix of covariates (of no interest), and is a matrix of the same size as with the residuals.īecause the interest is in testing the relationship between and, in principle it would be these that would need be permuted, but doing so also breaks the relationship with, which would be undesirable. Doing a permutation test with the general linear model (GLM) in the presence of nuisance variables can be challenging.
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