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, B 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`)

# 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 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 = FALSE
)
```

### 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.

```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
```

### 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.

# References

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

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