---
title: "Equivalence Testing for F-tests"
author: "Aaron R. Caldwell"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: True
editor_options: 
  chunk_output_type: console
bibliography: references.bib
vignette: >
  %\VignetteIndexEntry{Equivalence Testing for F-tests}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
---

For an open access tutorial paper explaining how to set equivalence bounds, and how to perform and report equivalence for ANOVA models see @Campbell_2021. These functions are meant to be omnibus tests, and additional testing may be necessary^[Russ Lenth's emmeans R package has some capacity for equivalence testing on the marginal means (i.e., a form of pairwise testing). See the emmeans package vignettes for details]. 

# F-test Calculations

Statistical equivalence testing (or "omnibus non-inferiority testing" as stated by @Campbell_2021) for *F*-tests are special use case of the cumulative distribution function of the non-central *F* distribution.

As @Campbell_2021 state, these type of questions answer the question: "Can we reject the hypothesis that the total proportion of variance in outcome Y attributable to X is greater than or equal to the equivalence bound $\Delta$?"

$$
H_0: \space 1 > \eta^2_p \geq \Delta
\\
H_1: \space 0 \geq \eta^2_p < \Delta
$$

In `TOSTER` we go a tad farther and calculate a more generalization of the non-centrality parameter to allow for the equivalence test for *F*-tests to be applied to variety of designs.

@Campbell_2021 calculate the *p*-value as:

$$
p = p_f(F; J-1, N-J, \frac{N \cdot \Delta}{1-\Delta})
$$

However, this approach could not be applied to factorial ANOVA and the paper only outlines how to apply this approach to a one-way ANOVA and an extension to Welch's one-way ANOVA.

However, the non-centrality parameter (ncp = $\lambda$) can be calculated with the equivalence bound and the degrees of freedom:

$$
\lambda_{eq} = \frac{\Delta}{1-\Delta} \cdot(df_1 + df_2 +1)
$$

The *p*-value for the equivalence test ($p_{eq}$) could then be calculated from traditional ANOVA results and the distribution function:

$$
p_{eq} = p_f(F; df_1, df_2, \lambda_{eq})
$$

## An Example

Using the `InsectSprays` data set in R and the base R `aov` function we can demonstrate how this omnibus equivalence testing can be applied with `TOSTER`.

From the initial analysis we an see a clear "significant" effect (the p-value listed is zero but it just very small) of the factor spray. However, we *may* be interested in testing if the effect is practically equivalent. I will arbitrarily set the equivalence bound to a partial eta-squared of 0.35 ($H_0: \eta^2_p > 0.35$).

```{r warning=FALSE, message=FALSE}
library(TOSTER)
# Get Data
data("InsectSprays")
# Build ANOVA
aovtest = aov(count ~ spray,
              data = InsectSprays)

# Display overall results
knitr::kable(broom::tidy(aovtest),
            caption = "Traditional ANOVA Test")

```

We can then use the information in the table above to perform an equivalence test using the `equ_ftest` function. This function returns an object of the S3 class `htest` and the output will look very familiar to the the t-test. The main difference is the estimates, and confidence interval, are for partial $\eta^2_p$.

```{r}
equ_ftest(Fstat = 34.70228,
          df1 = 5,
          df2 = 66,
          eqbound = 0.35)
```

Based on the results above we would conclude there is a significant effect of "spray" and the differences due to spray are *not* statistically equivalent. In essence, we reject the traditional null hypothesis of "no effect" but accept the null hypothesis of the equivalence test.

The `equ_ftest` is very useful because all you need is very basic summary statistics. However, if you are doing all your analyses in R then you can use the `equ_anova` function. This function accepts objects produced from `stats::aov`, `car::Anova` and `afex::aov_car` (or any anova from `afex`).

```{r}
equ_anova(aovtest,
          eqbound = 0.35)
```

### Visualize partial $\eta^2$

Just like the standardized mean differences, `TOSTER` also has a function to visualize $\eta^2_p$. 

The function, `plot_pes`, operates in a fashion very similar to `equ_ftest`. In essence, all you have to do is provide the F-statistic, numerator degrees of freedom, and denominator degrees of freedom. We can also select the type of plot with the `type` argument. Users have the option of producing a consonance plot (`type = "c"`), a consonance density plot (`type = "cd"`), or both (`type = c("cd","c")`). By default, `plot_pes` will produce both plots.

```{r, fig.width=6, fig.height=6}
plot_pes(Fstat = 34.70228,
         df1 = 5,
         df2 = 66)
```


# Power Analysis for *F*-tests

Power for an equivalence *F*-test can be calculated with the same equations supplied by @Campbell_2021. I have included these within the `power_eq_f` function.

```{r}
power_eq_f(df1 = 2, 
            df2 = 60,
            eqbound = .15)
```


# References
