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Bayesian RM ANOVA w/ random slopes

Kia ora,

I was reading the recent pre-print on including random slopes in Bayesian ANOVAs. In the paper, it notes that the specification can be done using the lmBF function from BayesFactor. I would like to implement this in my analyses but am having a bit of trouble with model comparisons and obtaining BFincl for the effects

Say we have a two-way RM ANOVA with the factors F1 & F2, would the formulae for the models (w/ random slopes) be specified as follows?

full <- DV~F1 + F2 + ID + F1*F2 + F1*ID + F2*ID

main <- DV~F1 + F2 + ID + F1*ID + F2*ID

F1 <- DV~F1 + ID + F1*ID + F2*ID

F2 <- DV~F2 + ID + F1*ID + F2*ID

Null <- DV~ID + F1*ID + F2*ID

where the R input is: mdl <- lmBF(formula, data=data, whichRandom="id").

When previously using anovaBF, the mdl output would give a summary of BFincl for the analysis of effects. However, the lmBF outputs give the BF a single model. I imagine we would need to do model comparisons to obtain the BFincl values but I am unsure of this process, or rather, how this is done in JASP.

Generally, it would be nice to have a way to reproduce the model comparison JASP output in R. I would appreciate it greatly if someone could point me in the right direction.

Thanks,

Corey

• Hi Corey,

Happy to hear someone read the preprint 😊

That is correct, you need to use lmBF which only fits one model at a time (anovaBF does the wrong thing, or well, the wrong thing in this scenario). Let's look at an example in the jasp data library, namely Alcohol attitudes.

In jasp 0.16.3 I get this output:

To replicate this in R you can run this code:

```library(tidyr)
library(BayesFactor)
# read the data file directly from GitHub (https://github.com/jasp-stats/jasp-desktop/blob/stable/Resources/Data%20Sets/Data%20Library/3.%20ANOVA/Alcohol%20Attitudes.csv)

# reshape the data to match bayes factor input
dat_long <- dat |> pivot_longer(!participant, names_to = c("Drink", "Imagery"), names_pattern = "(beer|wine|water|)(pos|neg|neu)", values_to = "DV")

# these formulas are indeed the full and null model
full <- DV ~ Drink + Imagery + participant + Drink*Imagery + Drink*participant + Imagery*participant
null <- DV ~                   participant +                 Drink*participant + Imagery*participant

# however, there is an easier way to generate the entire modelspace
# generate all formulas without random slopes
formulas <- BayesFactor::enumerateGeneralModels(DV ~ Drink + Imagery + participant + Drink*Imagery, whichModels = "withmain", neverExclude = "participant")

formulas <- lapply(formulas, function(f) {
update.formula(f, ~ . + Drink:participant + Imagery:participant)
})

# fit all models
bfs <- lapply(formulas, lmBF, data = dat_long)

# reduce the list to vector type inside BayesFactor
bfs_c <- Reduce(c, bfs)

# divide by the best performing model an sort decreasingly, as in the Jasp output
sort(bfs_c / max(bfs_c), decreasing = TRUE)
#> Bayes factor analysis
#> --------------
#> [1] Drink + Imagery + participant + Drink:Imagery + Drink:participant + Imagery:participant : 1            ±0%
#> [2] Drink + Imagery + participant + Drink:participant + Imagery:participant                 : 6.211023e-08 ±3.21%
#> [3] Imagery + participant + participant:Drink + Imagery:participant                         : 3.469623e-12 ±2.89%
#> [4] Drink + participant + Drink:participant + participant:Imagery                           : 2.777261e-41 ±3.49%
#> [5] participant + participant:Drink + participant:Imagery                                   : 6.836036e-42 ±2.72%
#>
#> Against denominator:
#>   DV ~ Drink + Imagery + participant + Drink:Imagery + Drink:participant + Imagery:participant
#> ---
#> Bayes factor type: BFlinearModel, JZS
```

The order of the models is exactly the same as in Jasp. Note that there are small differences in the values of the Bayes factor due to approximation error. If you increase the numerical accuracy in both Jasp and R then the differences should disappear, although that will (significantly) increase the running time.

Let me know if anything is unclear!

Cheers,

Don

• edited June 2022

Hi Don,

That is very helpful! Thanks for sharing the code. My next question would be how to progress from the model comparison output to the analysis of effects output...

• If using a matched models approach would we compute the main and interaction effects as follows?
```mainEff <- c(bfs_c[1]/bfs_c[5], bfs_c[2]/bfs_c[5]) # drink, imagery
interEff <- bfs_c[4]/bfs_c[3] # drink * imagery

effects <- c("drink", "imagery", "drink * imagery")
BFincl <- c(exp(mainEff@bayesFactor[["bf"]][1]), exp(mainEff@bayesFactor[["bf"]][2]), exp(interEff@bayesFactor[["bf"]]))
error <- c(mainEff@bayesFactor[["error"]][1], mainEff@bayesFactor[["error"]][2], interEff@bayesFactor[["error"]])
effTbl <- data.frame(effects,BFincl,error)
```
• These seem to match relatively well with the output in JASP, however, some effects seem to have a large discrepancy. For example, imagery in JASP had a BFincl of 1.32e14 whereas when I ran it in R I got a BFincl of 5.02e29. I understand there is an expectation for some variation but it seems quite large.
• How many iterations would we need to run to get a better match between numerical accuracy? I am currently using 100,000. Is set seed = 1 equivalent for R and JASP?

Also, how would we get the Analysis of Effects table using the 'Across all models' approach? Based on my understanding, it has something to do with summing across models but not sure what the actual order of operations is.

Thanks,

Corey