Using the Wellek-Welch procedure to test if differences between group means in a one-way ANOVA are above a set threshold, eta

Koh and Cribbie recommend using Wellek-Welch to test for equivalence in means in a one-way ANOVA. They quote Wellek (2003) who suggests setting the tolerance (equivalence interval) eta = 0.25 for strict equivalence and eta = 0.50 for liberal equivalence.

The inputs are group sizes, group means, group variances and eta

n <- c(3,4,5)
mu <- c(2,4,6)
var <- c(1,2,1)
eta <- 0.25

k <- length(n)
w <- c(n/var)
xbar <- sum(w*mu) / (sum(w))
ftop <- sum(w*(mu-xbar)*(mu-xbar))/(k-1)
fbot <- 1 + ((2*(k-2)/(k*k-1))*sum(1/(n-1)*(1-(w/sum(w)))*(1-w/sum(w))))
f <- ftop/fbot
phi2 <- f*((k-1)/mean(n))
df <- (k*k-1)/(3*sum(1/(n-1)*(1-w/sum(w))*(1-w/sum(w))))
phicrit <- ( (k-1) / (mean(n)) )*( df(0.05,k-1,df,mean(n)*eta*eta))

# Ho : phi >= eta^2 is rejected if phi2<phicrit (ie if phi2<phicrit is TRUE)

(phi2<phicrit)

Reference

Wellek, S. (2003). Testing statistical hypotheses of equivalence. Boca Raton, FL: Chapman & Hall/CRC.