= Equivalence test for McNemar's test =

McNemar's test is described in the Categorical Data talk [[StatsCourse2007|here]].

$$\delta$$ represents the true difference $$p_text{10} - p_text{01}$$ where $$p_text{ij}$$ = P(time 1 = i, time 2 = j) for dichotomous responses measured on each subject at times 1 and 2.

H0: $$\delta$$ $$\leq$$ -t or $$\delta$$ $$\geq$$ t

HA:   -t $$\leq$$ $$\delta$$ $$\leq$$ t

||||||||<style="TEXT-ALIGN: center">  || ||||<style="TEXT-ALIGN: center">Time 2 ||
||||||||<25% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center">   || ||  - || + ||
||||||||<25% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> Time 1  || - ||  n00 || n01 ||
||||||||<25% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center">    || + ||  n10 || n11 ||

The R code uses the formulae of Wellek (2003) to test if the observed difference in proportions provides sufficient evidence to say that there is a relationship between times 1 and 2 by comparing the difference in proportions to a specified criterion difference 'tdel' between 0 and 1.

$$p_text{01}$$ and $$p_text{10}$$ are estimated using the inputted observed frequencies $$n_text{01}$$ and $$n_text{10}$$ from a sample of size, n. Type II error is beta (usually 0.05).

If ind equals 1 then we reject nonequivalence so -t $$\leq$$ $$\delta$$ $$\leq$$ t otherwise we accept the null hypothesis for the given type II error, beta.

[TYPE INTO R THE DESIRED INPUTS N, DELTA, N10, N01 AND BETA USING VALUES IN FORM BELOW]. 

{{{
n <- 72
tdel <- 0.20
n10 <- 5
n01 <- 4 
beta <- 0.05
}}}

[THEN COPY AND PASTE THE BELOW INTO R]

{{{
tstat <- (sqrt(n)*abs((n10-n01))) / sqrt(n*(n10+n01) - (n10-n01)^2 ) 
nc <- n^(3)*tdel^2 / (n*(n10+n01) - (n10-n01)^2)
cv <- sqrt(qchisq(p=beta,df=1,ncp=nc))
ind <- 0
if (tstat < cv) ind = 1
print(ind)
}}}

__Reference__

Wellek S (2003) Testing statistical hypotheses of equivalence. Chapman & Hall/CRC Press.