Analysis Example: Rpt & Mixed ANOVA

This is optional for the DAPR3 course, but may be useful for your dissertations should your field/supervisor prefer the ANOVA framework to that of the linear model.

This walks briefly through these models with the ez package. There are many other packages available, and many good tutorials online should you desire extra resources in the future:

• https://www.datanovia.com/en/lessons/repeated-measures-anova-in-r
• https://www.r-bloggers.com/2021/04/repeated-measures-of-anova-in-r-complete-tutorial/
• https://stats.idre.ucla.edu/r/seminars/repeated-measures-analysis-with-r/
• https://www.datanovia.com/en/lessons/mixed-anova-in-r/

Data: Audio interference in executive functioning

This data is from a simulated study that aims to investigate the following research questions:

How do different types of audio interfere with executive functioning, and does this interference differ depending upon whether or not noise-cancelling headphones are used?

24 healthy volunteers each completed the Symbol Digit Modalities Test (SDMT) - a commonly used test to assess processing speed and motor speed - a total of 15 times. During the tests, participants listened to either no audio (5 tests), white noise (5 tests) or classical music (5 tests). Half the participants listened via active-noise-cancelling headphones, and the other half listened via speakers in the room.

The data is in stored in two separate files - the research administering the tests recorded the SDMT score in one spreadsheet, while details of the audio used in the experiment are held in a separate sheet

1. Information about the audio condition for each trial of each participant is stored in .csv format at https://uoepsy.github.io/data/ef_music.csv. The data is in long format (1 row per participant-trial).

variable	description
PID	Participant ID
trial_n	Trial Number (1-15)
audio	Audio heard during the test (‘no_audio’, ‘white_noise’,‘music’)
headphones	Whether the participant listened via speakers in the room or via noise cancelling headphones

2. Information on participants’ Symbol Digit Modalities Test (SDMT) for each trial is stored in .xlsx format at https://uoepsy.github.io/data/ef_sdmt.xlsx. The data is in wide format (1 row per participant, 1 column per trial).

variable	description
PID	Participant ID
Trial_01	SDMT score in trial 1
Trial_02	SDMT score in trial 2
Trial_03	SDMT score in trial 3
…	SDMT score in trial …
…	SDMT score in trial …
Trial_15	SDMT score in trial 15

The code below will read in both datasets and join them for you:

Code

library(tidyverse)
library(readxl)
download.file(url = "https://uoepsy.github.io/data/ef_sdmt.xlsx",
              destfile = "ef_sdmt.xlsx",
              mode = "wb")
efdata <- 
  left_join(
    read_csv("https://uoepsy.github.io/data/ef_music.csv"),
    read_xlsx("ef_sdmt.xlsx") %>%
      pivot_longer(Trial_01:Trial_15, names_to = "trial_n", values_to = "SDMT")
  )

One-Way Repeated Measures ANOVA

For a repeated measures ANOVA, we have one independent variable that is within group.
This would be appropriate if our research question were the following:

How do different types of audio interfere with executive functioning?

Mapping this to the variables in our dataset, our model is going to be SDMT ~ audio, and we want to account for PID differences. So for now we will ignore the headphones variable.

Code

head(efdata)

# A tibble: 6 × 5
  PID    trial_n  audio       headphones  SDMT
  <chr>  <chr>    <chr>       <chr>      <dbl>
1 PPT_01 Trial_02 no_audio    speakers      31
2 PPT_01 Trial_08 no_audio    speakers      23
3 PPT_01 Trial_11 no_audio    speakers      23
4 PPT_01 Trial_13 no_audio    speakers      24
5 PPT_01 Trial_15 no_audio    speakers      34
6 PPT_01 Trial_01 white_noise speakers      38

The easiest way to conduct a repeated measures ANOVA in R is to use the ez package, which comes with some handy functions to visualise the experimental design.
We can see from below that every participant completed 5 trials for each type of audio interference:

Code

library(ez)
ezDesign(data = efdata, x = audio, y = PID)

The ezANOVA() function takes a few arguments.

The ones you will need for this are:

• data the name of the dataframe
• dv the column name for the dependent variable
• wid the column name for the participant id variable
• within the column name(s) for the predictor variable(s) that vary within participants
• between the column name(s) for any predictor variable(s) that vary between participants

Fit a repeated measures ANOVA to examine the effect of the audio type on SDMT:

Code

ezANOVA(data = efdata, dv = SDMT, wid = PID, within = audio)

$ANOVA
  Effect DFn DFd        F            p p<.05       ges
2  audio   2  46 44.69618 1.647271e-11     * 0.2534633

$`Mauchly's Test for Sphericity`
  Effect         W          p p<.05
2  audio 0.8105961 0.09927715      

$`Sphericity Corrections`
  Effect       GGe      p[GG] p[GG]<.05      HFe        p[HF] p[HF]<.05
2  audio 0.8407573 5.0677e-10         * 0.899603 1.427119e-10         *

Mixed ANOVA

Mixed ANOVA can be used to investigate effects of independent variables that are at two different levels, i.e. some are within clusters and some are between.

Does the effect of audio interference on executive functioning differ depending upon whether or not noise-cancelling headphones are used?

Look at the two lines below. Can you work out what the plots will look like before you run them?

Code

ezDesign(data = efdata, x = headphones, y = PID)
ezDesign(data = efdata, x = headphones, y = audio)

Participants 1-20 are in one condition, and 21-40 are in another.
This should look like a two big blocks on the diagonal.

Code

ezDesign(data = efdata, x = headphones, y = PID)

In each condition, all different types of audio were observed in the same number of trials. This should be a full grid:

Code

ezDesign(data = efdata, x = headphones, y = audio)

Fit a mixed ANOVA to examine the interaction between audio and headphone use on SDMT:

Code

ezANOVA(data = efdata, dv = SDMT, wid = PID, within = audio, between = headphones)

$ANOVA
            Effect DFn DFd         F            p p<.05        ges
2       headphones   1  22  9.815545 4.836945e-03     * 0.26784992
3            audio   2  44 59.615596 2.980503e-13     * 0.32788320
4 headphones:audio   2  44  8.677316 6.657590e-04     * 0.06629911

$`Mauchly's Test for Sphericity`
            Effect         W         p p<.05
3            audio 0.9422531 0.5355001      
4 headphones:audio 0.9422531 0.5355001      

$`Sphericity Corrections`
            Effect       GGe        p[GG] p[GG]<.05      HFe        p[HF]
3            audio 0.9454057 1.196469e-12         * 1.031585 2.980503e-13
4 headphones:audio 0.9454057 8.648057e-04         * 1.031585 6.657590e-04
  p[HF]<.05
3         *
4         *

The ez package also contains some easy plotting functions for factorial experiments, such as ezPlot(). It takes similar arguments to the ezANOVA() function.

• look up the help documentation for ezPlot().
• let’s use ezPlot() to make a nice plot

Code

ezPlot(data = efdata, dv = SDMT, 
       wid = PID, within = audio, between = headphones,
       x = audio, split = headphones)

The same thing in lmer

Code

library(lme4)
library(lmerTest)
mod <- lmer(SDMT ~ 1 + headphones * audio + (1 + audio | PID), 
            data = efdata)
anova(mod, type="III")

Type III Analysis of Variance Table with Satterthwaite's method
                 Sum Sq Mean Sq NumDF DenDF F value    Pr(>F)    
headphones        325.0  325.04     1    22  9.8155  0.004837 ** 
audio            3212.0 1606.01     2    22 48.4976 8.626e-09 ***
headphones:audio  490.1  245.06     2    22  7.4001  0.003486 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Code

library(sjPlot)
plot_model(mod, type="eff", terms=c("audio","headphones"))