So you’ve run an ANOVA and found that your residuals are neither normally distributed, nor homogeneous, or that you are in violation of any other assumptions. Naturally you want to run some a-parametric analysis… but how?

In this post I will demonstrate how to run a permutation test ANOVA (easy!) and how to run bootstrap follow-up analysis (a bit more challenging) in a mixed design (both within- and between-subject factors) ANOVA. I’ve chosen to use a mixed design model in this demonstration for two reasons:

1. I’ve never seen this done before.
2. You can easily modify this code (change / skip some of these steps) to accommodate purely within- or purely between-subject designs.

Permutation ANOVA

Running a permutation test for your ANOVA in R is as easy as… running an ANOVA in R, but substituting aov with aovperm from the permucopackage.

library(permuco)
data(obk.long, package = "afex") # data from the afex package 

# permutation anova
fit_mixed_p <-
  aovperm(value ~ treatment * gender * phase * hour + Error(id / (phase * hour)),
          data = obk.long)

fit_mixed_p

## 
## Permutation test using Rd_kheradPajouh_renaud to handle nuisance variables and 5000 permutations.
##                                 SSn dfn    SSd dfd    MSEn   MSEd        F
## treatment                   179.730   2 228.06  10 89.8652 22.806  3.94049
## gender                       83.448   1 228.06  10 83.4483 22.806  3.65912
## treatment:gender            130.241   2 228.06  10 65.1206 22.806  2.85547
## phase                       129.511   2  80.28  20 64.7557  4.014 16.13292
## treatment:phase              77.885   4  80.28  20 19.4713  4.014  4.85098
## gender:phase                  2.270   2  80.28  20  1.1351  4.014  0.28278
## treatment:gender:phase       10.221   4  80.28  20  2.5553  4.014  0.63660
## hour                        104.285   4  62.50  40 26.0714  1.563 16.68567
## treatment:hour                1.167   8  62.50  40  0.1458  1.563  0.09333
## gender:hour                   2.814   4  62.50  40  0.7035  1.562  0.45027
## treatment:gender:hour         7.755   8  62.50  40  0.9694  1.562  0.62044
## phase:hour                   11.347   8  96.17  80  1.4183  1.202  1.17990
## treatment:phase:hour          6.641  16  96.17  80  0.4151  1.202  0.34529
## gender:phase:hour             8.956   8  96.17  80  1.1195  1.202  0.93129
## treatment:gender:phase:hour  14.155  16  96.17  80  0.8847  1.202  0.73594
##                             parametric P(>F) permutation P(>F)
## treatment                          5.471e-02            0.0586
## gender                             8.480e-02            0.0916
## treatment:gender                   1.045e-01            0.1084
## phase                              6.732e-05            0.0002
## treatment:phase                    6.723e-03            0.0056
## gender:phase                       7.566e-01            0.7644
## treatment:gender:phase             6.424e-01            0.6480
## hour                               4.027e-08            0.0002
## treatment:hour                     9.992e-01            0.9996
## gender:hour                        7.716e-01            0.7638
## treatment:gender:hour              7.555e-01            0.7614
## phase:hour                         3.216e-01            0.3260
## treatment:phase:hour               9.901e-01            0.9910
## gender:phase:hour                  4.956e-01            0.5100
## treatment:gender:phase:hour        7.496e-01            0.7590

The results of the permutation test suggest an interaction between Treatment (a between subject factor) and Phase (a within-subject factor). To fully understand this interaction, we would like to conduct some sort of follow-up analysis (planned comparisons or post hoc tests). Usually I would pass the model to emmeans for any follow-ups, but here, due to our assumption violations, we need some sort of bootstrapping method.

Bootstrapping with car and emmeans

For bootstrapping we will be using the Boot function from the carpackage. In the case of within-subject factors, this function requires that they be specified in a multivariate data structure. Let’s see how this is done.

1. Make your data WIIIIIIIIIIDEEEEEEEE

library(dplyr)
library(tidyr)

obk_mixed_wide <- obk.long %>%
  unite("cond", phase, hour) %>%
  spread(cond, value)

head(obk_mixed_wide)

##   id treatment gender   age fup_1 fup_2 fup_3 fup_4 fup_5 post_1 post_2
## 1  1   control      M -4.75     2     3     2     4     4      3      2
## 2  2   control      M -2.75     4     5     6     4     1      2      2
## 3  3   control      M  1.25     7     6     9     7     6      4      5
## 4  4   control      F  7.25     4     4     5     3     4      2      2
## 5  5   control      F -5.75     4     3     6     4     3      6      7
## 6  6         A      M  7.25     9    10    11     9     6      9      9
##   post_3 post_4 post_5 pre_1 pre_2 pre_3 pre_4 pre_5
## 1      5      3      2     1     2     4     2     1
## 2      3      5      3     4     4     5     3     4
## 3      7      5      4     5     6     5     7     7
## 4      3      5      3     5     4     7     5     4
## 5      8      6      3     3     4     6     4     3
## 6     10      8      9     7     8     7     9     9

This is not enough, as we also need our new columns (representing the different levels of the within subject factors) to be in a matrix column.

obk_mixed_matrixDV <- obk_mixed_wide %>%
  select(id, age, treatment, gender)
obk_mixed_matrixDV$M <- obk_mixed_wide %>%
  select(-id, -age, -treatment, -gender) %>%
  as.matrix()

str(obk_mixed_matrixDV)

## 'data.frame':    16 obs. of  5 variables:
##  $ id       : Factor w/ 16 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
##  $ age      : num  -4.75 -2.75 1.25 7.25 -5.75 7.25 8.25 2.25 2.25 -7.75 ...
##  $ treatment: Factor w/ 3 levels "control","A",..: 1 1 1 1 1 2 2 2 2 3 ...
##  $ gender   : Factor w/ 2 levels "F","M": 2 2 2 1 1 2 2 1 1 2 ...
##  $ M        : num [1:16, 1:15] 2 4 7 4 4 9 8 6 5 8 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : NULL
##   .. ..$ : chr  "fup_1" "fup_2" "fup_3" "fup_4" ...

2. Fit your regular model

fit_mixed <- aov(M ~ treatment * gender, obk_mixed_matrixDV)

Note that the left-hand-side of the formula (the M) is a matrix representing all the within-subject factors and their levels, and the right-hand-side of the formula (treatment * gender) includes only the between-subject factors.

3. Define the contrast(s) of interest

For this step we will be using emmeans. But first, we need to create a list of the within-subject factors and their levels (I did say this was difficult - bear with me!). This list needs to correspond to the order of the multi-variate column in our data, such that if there is more than one factor, the combinations of factor levels are used in expand.grid order. In our case:

colnames(obk_mixed_matrixDV$M)

##  [1] "fup_1"  "fup_2"  "fup_3"  "fup_4"  "fup_5"  "post_1" "post_2"
##  [8] "post_3" "post_4" "post_5" "pre_1"  "pre_2"  "pre_3"  "pre_4" 
## [15] "pre_5"

rm_levels <- list(hour = c("1", "2", "3", "4", "5"),
                  phase = c("fup", "post", "pre"))

Make sure you get the order of the variables and their levels correct! This will affect your results!

Let's use emmeans to get the estimates of the pairwise differences between the treatment groups within each phase of the study:

library(emmeans)
# get the correct reference  grid with the correct ultivariate levels!
rg <- ref_grid(fit_mixed, mult.levs = rm_levels)
rg

## 'emmGrid' object with variables:
##     treatment = control, A, B
##     gender = F, M
##     hour = multivariate response levels: 1, 2, 3, 4, 5
##     phase = multivariate response levels: fup, post, pre

# get the expected means:
em_ <- emmeans(rg, ~ phase * treatment)
em_

##  phase treatment emmean    SE df lower.CL upper.CL
##  fup   control     4.33 0.551 10     3.11     5.56
##  post  control     4.08 0.628 10     2.68     5.48
##  pre   control     4.25 0.766 10     2.54     5.96
##  fup   A           7.25 0.604 10     5.90     8.60
##  post  A           6.50 0.688 10     4.97     8.03
##  pre   A           5.00 0.839 10     3.13     6.87
##  fup   B           7.29 0.461 10     6.26     8.32
##  post  B           6.62 0.525 10     5.45     7.80
##  pre   B           4.17 0.641 10     2.74     5.59
## 
## Results are averaged over the levels of: gender, hour 
## Confidence level used: 0.95

# run pairwise tests between the treatment groups within each phase
c_ <- contrast(em_, "pairwise", by = 'phase')
c_

## phase = fup:
##  contrast    estimate    SE df t.ratio p.value
##  control - A  -2.9167 0.818 10 -3.568  0.0129 
##  control - B  -2.9583 0.719 10 -4.116  0.0054 
##  A - B        -0.0417 0.760 10 -0.055  0.9983 
## 
## phase = post:
##  contrast    estimate    SE df t.ratio p.value
##  control - A  -2.4167 0.931 10 -2.595  0.0634 
##  control - B  -2.5417 0.819 10 -3.105  0.0275 
##  A - B        -0.1250 0.865 10 -0.144  0.9886 
## 
## phase = pre:
##  contrast    estimate    SE df t.ratio p.value
##  control - A  -0.7500 1.136 10 -0.660  0.7911 
##  control - B   0.0833 0.999 10  0.083  0.9962 
##  A - B         0.8333 1.056 10  0.789  0.7177 
## 
## Results are averaged over the levels of: gender, hour 
## P value adjustment: tukey method for comparing a family of 3 estimates

# extract the estimates
est_names <- c("fup: control - A", "fup: control - B", "fup: A - B",
               "post: control - A", "post: control - B", "post: A - B",
               "post: control - A", "post: control - B", "post: A - B")
est_values <- summary(c_)$estimate
names(est_values) <- est_names
est_values

##  fup: control - A  fup: control - B        fup: A - B post: control - A 
##       -2.91666667       -2.95833333       -0.04166667       -2.41666667 
## post: control - B       post: A - B post: control - A post: control - B 
##       -2.54166667       -0.12500000       -0.75000000        0.08333333 
##       post: A - B 
##        0.83333333

4. Running the bootstrap

Now let's wrap this all in a function that accepts the fitted model as an argument:

treatment_phase_contrasts <- function(mod){
  rg <- ref_grid(mod, mult.levs = rm_levels)

  # get the expected means:
  em_ <- emmeans(rg, ~ phase * treatment)

  # run pairwise tests between the treatment groups within each phase
  c_ <- contrast(em_, "pairwise", by = 'phase')

  # extract the estimates
  est_names <- c("fup: control - A", "fup: control - B", "fup: A - B",
                 "post: control - A", "post: control - B", "post: A - B",
                 "post: control - A", "post: control - B", "post: A - B")
  est_values <- summary(c_)$estimate
  names(est_values) <- est_names
  est_values
}

# test it
treatment_phase_contrasts(fit_mixed)

##  fup: control - A  fup: control - B        fup: A - B post: control - A 
##       -2.91666667       -2.95833333       -0.04166667       -2.41666667 
## post: control - B       post: A - B post: control - A post: control - B 
##       -2.54166667       -0.12500000       -0.75000000        0.08333333 
##       post: A - B 
##        0.83333333

Finally, we will use car::Boot to get the bootstrapped estimates!

library(car)

treatment_phase_results <-
  Boot(fit_mixed, treatment_phase_contrasts, R = 50) # R = 599 at least

summary(treatment_phase_results) # original vs. bootstrapped estimate (bootMed)

## 
## Number of bootstrap replications R = 27 
##                      original   bootBias  bootSE  bootMed
## fup..control...A    -2.916667  0.0342593 0.58002 -3.05000
## fup..control...B    -2.958333 -0.0246914 0.73110 -2.96667
## fup..A...B          -0.041667 -0.0589506 0.35745 -0.16667
## post..control...A   -2.416667 -0.1728395 0.65088 -2.75000
## post..control...B   -2.541667 -0.1425926 0.77952 -2.66667
## post..A...B         -0.125000  0.0302469 0.58006 -0.11667
## post..control...A.1 -0.750000  0.0067901 0.83692 -0.56667
## post..control...B.1  0.083333 -0.0169753 0.89481  0.33333
## post..A...B.1        0.833333 -0.0237654 0.73113  1.08333

confint(treatment_phase_results, type = "perc") # does include zero?
## Bootstrap percent confidence intervals
## 
##                         2.5 %    97.5 %
## fup..control...A    -4.000000 -1.750000
## fup..control...B    -4.300000 -1.500000
## fup..A...B          -0.750000  0.750000
## post..control...A   -3.500000 -1.333333
## post..control...B   -4.250000 -1.333333
## post..A...B         -1.416667  0.875000
## post..control...A.1 -2.600000  0.700000
## post..control...B.1 -2.000000  1.500000
## post..A...B.1       -0.600000  1.833333

Results indicate that the Control group is lower than both treatment groups in the post and fup (follow -up) phases.

If we wanted p-values, we could use this little function (based on this demo):

boot_pvalues <- function(x, side = c(0, -1, 1)) {
  side <- side[1]
  x <- as.data.frame(x$t)

  ps <- sapply(x, function(.x) {
    s <- na.omit(.x)
    s0 <- 0
    N <- length(s)

    if (side == 0) {
      min((1 + sum(s >= s0)) / (N + 1),
          (1 + sum(s <= s0)) / (N + 1)) * 2
    } else if (side < 0) {
      (1 + sum(s <= s0)) / (N + 1)
    } else if (side > 0) {
      (1 + sum(s >= s0)) / (N + 1)
    }
  })

  setNames(ps,colnames(x))
}

boot_pvalues(treatment_phase_results)

##  fup: control - A  fup: control - B        fup: A - B post: control - A 
##        0.07142857        0.07142857        0.71428571        0.07142857 
## post: control - B       post: A - B post: control - A post: control - B 
##        0.07142857        0.85714286        0.42857143        0.85714286 
##       post: A - B 
##        0.35714286

These p-values can then be passed to p.adjust() for the p-value adjustment method of your choosing.

Summary

I've demonstrated how to run permutation tests on main effects / interactions, with follow-up analysis using the bootstrap method. Using this code as a basis for any analysis you might have in mind gives you all the flexibility of emmeans, which supports many (many) models!