# R code for equivalence test in one-way ANOVA

For an effect size in an one-way anova $$\theta^{2 }$$:

$$\theta^{2 }$$ = [ \eta^{2 } (N-k)]/[(1 - $$\eta^{2 } ) \bar(N)]$$

we can formulate hypotheses of form

H0: $$\theta^{2 } ≥ d and HA : $$\theta^{2 } < d $$.

If ind equals 1 then we reject nonequivalence concluding $$\theta^{2 } < d for given $$ $$\eta^{2 }, group sizes and type II error$$.

[TYPE INTO R THE DESIRED INPUTS RSQ, N, K, DCRIT AND BETA USING VALUES IN FORM BELOW].

RSQ represents partial $$eta^text{2}$$ using one of Cohen's rules of thumb, n is a vector of group sizes, dcrit represents the criterion for d and beta is the type II error.

Koh and Cribbie (2013) show that Wellek's test is not robust to differences in group variances whereas their proposed Wellek–Welch test was insensitive to differences in group variances. Wellek-Welch can be used for equivalence testing in R (see to test hypotheses of form H0: phi >= eta^2 vs H1: phi < eta^2.

rsq <- 0.006 n <- c(10,12,13,15) dcrit <- 0.5 beta <- 0.05

[THEN COPY AND PASTE THE BELOW INTO R]

k <- length(n) ns <- sum(n) psi2 <- (rsq/(1-rsq))*k*((ns-k)/ns) cstats <- (k*(k-1)/ns)*qf(beta,k-1,ns-k,(ns/k)*dcrit*dcrit) ind <- 0 if (psi2 < cstats) ind = 1 print(ind)

Reference

Koh A and Cribbie R (2013) Robust tests of equivalence for k independent groups. *British Journal of Mathematical and Statistical Psychology* **66(3)**, 426–434.