= Using the Wellek-Welch procedure in R 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 because it has good power in detecting difference in group means under various simulations when group variances are different. 

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
}}}

Having chosen our inputs the Wellek-Welch test can then be implemented in R.

{{{
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)
}}}

__References__

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. 

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