lmerPerm - Perform Permutation Test on General Linear and Mixed Linear Regression

We provide a solution for performing permutation tests on linear and mixed linear regression models. It allows users to obtain accurate p-values without making distributional assumptions about the data. By generating a null distribution of the test statistics through repeated permutations of the response variable, permutation tests provide a powerful alternative to traditional parameter tests (Holt et al. (2023) <doi:10.1007/s10683-023-09799-6>). In this early version, we focus on the permutation tests over observed t values of beta coefficients, i.e.original t values generated by parameter tests. After generating a null distribution of the test statistic through repeated permutations of the response variable, each observed t values would be compared to the null distribution to generate a p-value. To improve the efficiency,a stop criterion (Anscombe (1953) <doi:10.1111/j.2517-6161.1953.tb00121.x>) is adopted to force permutation to stop if the estimated standard deviation of the value falls below a fraction of the estimated p-value. By doing so, we avoid the need for massive calculations in exact permutation methods while still generating stable and accurate p-values.