F-test Calculations
Statistical equivalence testing (or “omnibus non-inferiority testing” as stated by Campbell and Lakens (2021)) for F-tests are special use case of the cumulative distribution function of the non-central F distribution.
As Campbell and Lakens (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 and Lakens (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 (\(H_0: \eta^2_p > 0.35\)).
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")| term | df | sumsq | meansq | statistic | p.value |
|---|---|---|---|---|---|
| spray | 5 | 2668.833 | 533.76667 | 34.70228 | 0 |
| Residuals | 66 | 1015.167 | 15.38131 | NA | NA |
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 \(\eta^2_p\).
##
## Equivalence Test from F-test
##
## data: Summary Statistics
## F = 34.702, df1 = 5, df2 = 66, p-value = 1
## 95 percent confidence interval:
## 0.5806263 0.7804439
## sample estimates:
## [1] 0.724439Based 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).
## effect df1 df2 F.value p.null pes eqbound p.equ
## 1 spray 5 66 34.70228 3.182584e-17 0.724439 0.35 0.9999965Visualize 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.
