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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 `permuco` package.

```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)```

```## Warning in checkBalancedData(fixed_formula = formula_f, data = cbind(y, :
## The data are not balanced, the results may not be exact.```

`fit_mixed_p`

term SSn dfn SSd dfd MSEn MSEd F parametric P(>F) permutation P(>F)
treatment 179.73 2 228.06 10 89.87 22.81 3.94 0.055 0.055
gender 83.45 1 228.06 10 83.45 22.81 3.66 0.085 0.082
treatment:gender 130.24 2 228.06 10 65.12 22.81 2.86 0.104 0.104
phase 129.51 2 80.28 20 64.76 4.01 16.13 <0.001 <0.001
treatment:phase 77.89 4 80.28 20 19.47 4.01 4.85 0.007 0.009
gender:phase 2.27 2 80.28 20 1.14 4.01 0.28 0.757 0.765
treatment:gender:phase 10.22 4 80.28 20 2.56 4.01 0.64 0.642 0.641
hour 104.29 4 62.50 40 26.07 1.56 16.69 <0.001 <0.001
treatment:hour 1.17 8 62.50 40 0.15 1.56 0.09 0.999 1.000
gender:hour 2.81 4 62.50 40 0.70 1.56 0.45 0.772 0.772
treatment:gender:hour 7.76 8 62.50 40 0.97 1.56 0.62 0.755 0.755
phase:hour 11.35 8 96.17 80 1.42 1.20 1.18 0.322 0.319
treatment:phase:hour 6.64 16 96.17 80 0.42 1.20 0.35 0.990 0.990
gender:phase:hour 8.96 8 96.17 80 1.12 1.20 0.93 0.496 0.498
treatment:gender:phase:hour 14.15 16 96.17 80 0.88 1.20 0.74 0.750 0.753

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 `car` package. 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) %>%

```##   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()

glimpse(obk_mixed_matrixDV)```

```## Observations: 16
## Variables: 5
## \$ id        <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## \$ age       <dbl> -4.75, -2.75, 1.25, 7.25, -5.75, 7.25, 8.25, 2.25, 2...
## \$ treatment <fct> control, control, control, control, control, A, A, A...
## \$ gender    <fct> M, M, M, F, F, M, M, F, F, M, M, M, F, F, F, F
## \$ M         <dbl[,15]> <matrix[16 x 15]>```

### 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. Run 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```

`## Loading required namespace: boot`

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

```##
## Number of bootstrap replications R = 31
##                      original  bootBias  bootSE     bootMed
## fup..control...A    -2.916667  0.044892 0.65137 -2.8333e+00
## fup..control...B    -2.958333 -0.026805 0.82950 -3.0000e+00
## fup..A...B          -0.041667 -0.071697 0.40960 -1.6667e-01
## post..control...A   -2.416667 -0.011444 0.74882 -2.5000e+00
## post..control...B   -2.541667  0.048310 0.94075 -2.4167e+00
## post..A...B         -0.125000  0.059754 0.64484  4.3374e-15
## post..control...A.1 -0.750000 -0.129339 0.63190 -7.0000e-01
## post..control...B.1  0.083333 -0.099923 1.01857  9.1667e-02
## post..A...B.1        0.833333  0.029416 0.89102  8.3333e-01```

`confint(treatment_phase_results, type = "perc") # does include zero?`

```## Bootstrap percent confidence intervals
##
##                          2.5 %     97.5 %
## fup..control...A    -4.0000000 -1.8000000
## fup..control...B    -4.3571429 -1.4000000
## fup..A...B          -0.8571429  0.7083333
## post..control...A   -4.0000000 -1.3000000
## post..control...B   -4.0000000 -0.7500000
## post..A...B         -1.3809524  1.0000000
## post..control...A.1 -2.0000000  0.7500000
## post..control...B.1 -2.2500000  2.0416667
## post..A...B.1       -0.9666667  2.2083333```

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)) {
# Based on:
# https://blogs.sas.com/content/iml/2011/11/02/how-to-compute-p-values-for-a-bootstrap-distribution.html
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
##            0.0625            0.0625            0.6875            0.0625
## post: control - B       post: A - B post: control - A post: control - B
##            0.0625            0.9375            0.1250            0.9375
##       post: A - B
##            0.3750```

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!