Resampling under the null hypothesis instead of case resampling will preserve the correlations and distributional characteristics of the data and allow you to perform hypothesis testing. For more information, please refer to Westfall and Young (1993).

# Resampling under the null hypothesis

## Selecting a null model for resampling

The very first step of this approach is to base the bootstrap on the null distribution, which is the population distribution that represents the null hypothesis.

In the context of ANOVA or ANCOVA designs, the following null distributions may be applicable:

• One-way ANOVA: intercept-only model
• One-way ANCOVA: covariates-only model
• In two-way factorial designs, we are testing three null hypotheses simultaneously based on two main effects and the interaction effect.
• ANOVA: intercept-only model
• ANCOVA: covariates-only model

## Steps

1. Fit the full model.
2. Obtain the observed F-values from the full model.
3. Fit the null model (i.e., intercept-only model for ANOVA or covariate-only model for ANCOVA).
4. Optionally centre and rescale the residuals of the null model based on the leverage values of the observations.
5. Our new dependent variable for the bootstrapping is the sum of the fitted values of the null model and the resampled residuals from the null model, but we use the predictors from the full model.
6. Obtain the resampled F-values.
7. Repeat steps 5-6, times (e.g., 10,000).
8. The bootstrapped p-value is the proportion of resampled F-values that are greater then the observed F-values.

# Function

## Required packages

• `car` to fit the ANOVA and get the F-values
• `boot` to run the bootstrapping and calculate the BCa confidence intervals

# Options

```null.model: Null model of class lm
full.model: Full model of class lm
B: Number of bootstraps (default = 1000)
scaled: Rescale residuals (default = TRUE)
seed: Seed for replication (default = 1234)
ci: Calculate confidence intervals (default = TRUE)
ci.type: Confidence interval type (options = c(“bca”, “perc”)).
cent: Percentile for confidence intervals (default = .95)
dec: Number of decimal places (default = 5)```

## Code

 # Load required packages -------------------------------------------------- if (!require(pacman)) { install.packages("pacman") } pacman::p_load(car, boot) # Function ---------------------------------------------------------------- nullboot.Anova <- function(null.model, full.model, B = 1000, scaled = TRUE, seed = 1234, ci = TRUE, cent = .95, dec = 5, ci.type = c("bca", "perc")) { # Returns error if models are not 'lm' class stopifnot(class(null.model) == "lm" | class(full.model) == "lm") # Set seed set.seed(seed) # Extract data from the two models data_full <- model.frame(full.model) data_null <- model.frame(null.model) # Get residuals from null model and recenter and rescale based on # Davidson and MacKinnon (2004) if requested. if (isTRUE(scaled)) { # Number of observations N <- nrow(data_null) # Number of predictors (from full model) K <- length(labels(terms(full.model))) # Extract and scale residuals (if requested) from null model Er <- residuals(null.model) / sqrt(1 - hatvalues(null.model)) } else { Er <- residuals(null.model) } # Predicted values from null model Yhat <- fitted(null.model) # Get ANOVA statistics for full model mod_ANOVA <- Anova(full.model, type = 3) # Get observed F-values (excluding Intercept and NA [Residuals]) Fobs <- as.vector(na.omit(mod_ANOVA[-1, 3])) # Updated formula for bootstapping with Ystar as DV formula_resample <- update.formula(formula(full.model), Ystar ~ .) # Bootstrap function bs.Anova <- function(data, i) { data_full\$Ystar <- Yhat + Er[i] # Fit linear model model_resample <- lm(formula_resample, data = data_full) # Get F-values from ANOVA Fstar <- as.vector(na.omit(Anova(model_resample, type = 3)[-1, 3])) return(Fstar) } # Run bootstrapping bootAnova <- boot( data_full, statistic = bs.Anova, R = B, parallel = "snow" ) # Bootstrapped F-values Fstar <- bootAnova\$t # The proportion of resampled F-values greater than or equal to # the observed F-values pBoot <- vector() for (i in 1:length(Fobs)) { pBoot[i] <- (sum(Fstar[, i] >= Fobs[i]) + 1) / (length(Fstar[, i]) + 1) } # Create data.frame for outputting varNames <- labels(terms(full.model)) pRaw <- as.vector(na.omit(mod_ANOVA[-1, 4])) # Calculate confidence intervals if requested if (isTRUE(ci)) { ci.type <- match.arg(ci.type) FstarCI <- list() ciLB <- vector() ciUB <- vector() bcaLB <- vector() bcaUB <- vector() for (i in 1:length(Fobs)) { suppressWarnings(FstarCI[[i]] <- boot.ci( bootAnova, type = ci.type, index = i, conf = cent )) # Percentile confidence intervals if (ci.type == "perc") { ci_desc <- "Percentile" ciLB[i] <- FstarCI[[i]]\$perc[, 4] ciUB[i] <- FstarCI[[i]]\$perc[, 5] } # BCa confidence intervals if (ci.type == "bca") { ci_desc <- "BCa" ciLB[i] <- FstarCI[[i]]\$bca[, 4] ciUB[i] <- FstarCI[[i]]\$bca[, 5] } } bootOut <- data.frame( "F.value" = Fobs, "p.value" = pRaw, "p.boot" = pBoot, "CI.LB" = ciLB, "CI.UB" = ciUB, row.names = varNames ) } else { ci_desc <- "Not requested" bootOut <- data.frame( "F.value" = Fobs, "p.value" = pRaw, "p.boot" = pBoot, row.names = varNames ) } cat( "\n", "Bootstrapped ANOVA with Type III tests", "\n", "\n", "Resampling type: ", ifelse(isTRUE(scaled), "rescaled residuals", "residuals" ), "\n", "Number of bootstrap resamples: ", B, "\n", "Bootstrapped confidence interval type: ", ci_desc, "\n", "Confidence interval: ", 100 * cent, "%", "\n", "\n", sep = "" ) # Turn off scientific options(scipen = 999) print(round(bootOut, dec)) # Turn on scientific options(scipen = 0) }

# Example

To prevent issues with missing data I recommend fitting the full model first. Alternatively, you could just omit cases with missing values from your dataset.

`full_model <- lm(dv ~ sex*group, data = df)`

Then, fit the null model using the `update()` function, which is the intercept-only model for this example and using the `model.frame()` of the `full_model`.

`null_model <- update(full_model, . ~ + 1, data = model.frame(full_model))`

Finally, run the function. Note that I have not requested confidence intervals.

`nullboot.Anova(null.model = null_model, full.model = full_model, ci = FALSE)`

## Output

The function will also output the BCa or percentile bootstrapped confidence intervals if requested.

```Resampling type: rescaled residuals
Number of bootstrap resamples: 1000
Bootstrapped confidence interval type: Not requested
Confidence interval: 95%

F.value p.value p.boot
sex         0.42978 0.51425 0.51449
group       0.94369 0.33468 0.32867
group:sex   0.00101 0.97476 0.97203```

### Testing the null hypothesis

As we are resampling the residuals from the null model, the output below is the as above but with the addition of confidence intervals.

`nullboot.Anova(null.model = null_model, full.model = full_model, ci = TRUE)`

#### Output

```Bootstrapped ANOVA with Type III tests

Resampling type: rescaled residuals
Number of bootstrap resamples: 1000
Bootstrapped confidence interval type: BCa
Confidence interval: 95%

F.value p.value p.boot  CI.LB   CI.UB
sex         0.42978 0.51425 0.51449 0.00073 5.23939
group       0.94369 0.33468 0.32867 0.03607 9.48676
group:sex   0.00101 0.97476 0.97203 0.00001 0.00111```

## Model-based resampling

If you are not interested in hypothesis testing, but would like to obtain confidence intervals for the F-statistic, then you should resample the residuals of the full model. This is also known as fixed-x resampling or the parametric bootstrap.

In which case, you would run the function with `full_model` for both the `null.model` and `full.model` options:

`nullboot.Anova(null.model = full_model, full.model = full_model, ci = TRUE)`

### Parametric bootstrapping

Below is the parametric bootstrap where we’re resampling the residuals of the full model.

```Bootstrapped ANOVA with Type III tests

Resampling type: rescaled residuals
Number of bootstrap resamples: 1000
Bootstrapped confidence interval type: BCa
Confidence interval: 95%

F.value p.value p.boot  CI.LB   CI.UB
sex         0.42978 0.51425 0.59540 0.00003 4.28447
group       0.94369 0.33468 0.50649 0.00095 8.67616
sex:group   0.00101 0.97476 0.97602 0.00000 0.00090```

Notice that the p-values and confidence intervals differ depending on whether you’re resampling under the null hypothesis or are using parametric bootstrapping.

## Recommendations

• If you are interesting in hypothesis testing (i.e., obtaining a bootstrapped p-value), then you should resample under the null hypothesis.
• If you are interested in obtaining confidence intervals for the F-values, then you should use parametric bootstrapping.

# Reference

Westfall, P. H., & Young, S. S. (1993). Resampling-based multiple testing: Examples and methods for p-value adjustment (Vol. 279). John Wiley & Sons.