(Bootstrapping) Follow-Up Contrasts for Within-Subject ANOVAs

R

statistics

ANOVA

bootstrap

code

mixed-design

permutation

Author

Mattan S. Ben-Shachar

Published

May 9, 2019

So you’ve run a repeated-measures 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:

I’ve never seen this done before.

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 anovafit_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)

resampled P(>F)

treatment

treatment

179.73

2

228.06

10

89.87

22.81

3.94

0.055

0.055

gender

gender

83.45

1

228.06

10

83.45

22.81

3.66

0.085

0.082

treatment:gender

treatment:gender

130.24

2

228.06

10

65.12

22.81

2.86

0.104

0.104

phase

phase

129.51

2

80.28

20

64.76

4.01

16.13

<0.001

<0.001

treatment:phase

treatment:phase

77.89

4

80.28

20

19.47

4.01

4.85

0.007

0.009

gender:phase

gender:phase

2.27

2

80.28

20

1.14

4.01

0.28

0.757

0.765

treatment:gender:phase

treatment:gender:phase

10.22

4

80.28

20

2.56

4.01

0.64

0.642

0.641

hour

hour

104.29

4

62.50

40

26.07

1.56

16.69

<0.001

<0.001

treatment:hour

treatment:hour

1.17

8

62.50

40

0.15

1.56

0.09

>0.999

>0.999

gender:hour

gender:hour

2.81

4

62.50

40

0.70

1.56

0.45

0.772

0.772

treatment:gender:hour

treatment:gender:hour

7.76

8

62.50

40

0.97

1.56

0.62

0.755

0.755

phase:hour

phase:hour

11.35

8

96.17

80

1.42

1.20

1.18

0.322

0.319

treatment:phase:hour

treatment:phase:hour

6.64

16

96.17

80

0.42

1.20

0.35

0.990

0.990

gender:phase:hour

gender:phase:hour

8.96

8

96.17

80

1.12

1.20

0.93

0.496

0.498

treatment:gender:phase:hour

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.

Rows: 16
Columns: 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, 2.25, -7.…
$ treatment <fct> control, control, control, control, control, A, A, A, A, B, …
$ 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]>

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:

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 phasec_ <-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 estimatesest_names <-c("fup: control - A", "fup: control - B", "fup: A - B","post: control - A", "post: control - B", "post: A - B","pre: control - A", "pre: control - B", "pre: A - B")est_values <-summary(c_)$estimatenames(est_values) <- est_namesest_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 pre: control - A pre: control - B
-2.54166667 -0.12500000 -0.75000000 0.08333333
pre: 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","pre: control - A", "pre: control - B", "pre: A - B") est_values <-summary(c_)$estimatenames(est_values) <- est_names est_values}# test ittreatment_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 pre: control - A pre: control - B
-2.54166667 -0.12500000 -0.75000000 0.08333333
pre: 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
pre: control - A -0.750000 -0.129339 0.63190 -7.0000e-01
pre: control - B 0.083333 -0.099923 1.01857 9.1667e-02
pre: A - B 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
pre: control - A -2.0000000 0.7500000
pre: control - B -2.2500000 2.0416667
pre: A - B -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):

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 pre: control - A pre: control - B
0.0625 0.9375 0.1250 0.9375
pre: 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!