**A valid statistical test is one where the amount of type-I
errors (rejecting the null when it should not) does not exceed the test
threshold (often 5%).** We say it in bold because it seems to be
commonly forgotten. Logistic regressions applied to proportions deviates
massively from this rule, becoming very liberal tests.

ANOPA respect very closely this rule when the corrected statistics are consulted.

The following code shows how to test the type-I error rate for ANOPA.It uses simulated data with no differences. The only limitations herein is that no correlations is added when repeated measures are used.

We suggests here to shut down error and feedback messages:

```
options("ANOPA.feedback" = 'none')
library(ANOPA)
library(testthat)
nsim <- 1000 # increase for more reliable simulations.
theN <- 20 # number of simulated participants
```

Note that the simulations are actually not run in this vignette, as they take times. We wished to provide code in case you wished to test type-I error rate by yourself. The present code is also not optimized for speed (and in particular, is not parallelized); we wished to keep the code as simple as possible for readability. In all the following, the number of simulated participants per group is small (20) but can be varied.

```
frm <- s ~ grp # the formula
BSDesign <- list(grp = c("ctrl","plcbo")) #one factor, two groups
thePs <- c(0.3, 0.3) # the true proportions, equal
# test type-I error rate when no effect as is the case for factor 2
set.seed(41)
res <- c()
for (i in 1:nsim) {
smp <- GRP( thePs, theN, BSDesign )
w <- anopa(frm, smp[,2:3] )
res <- c(res, if(summarize(w)[1,4]<.05) 1 else 0)
}
typeI <- mean(res)
cat( "Design B, testing B: ", typeI, "\n")
# tolerance is large as the number of simulations is small
expect_equal( typeI, .05, tolerance = 0.035)
```

```
frm <- cbind(s.early, s.middle, s.late) ~ .
WSDesign <- list(moment = c("early","middle","late"))
thePs <- c(0.3, 0.3, 0.3)
# test type-I error rate when no effect as is the case for factor 2
set.seed(42)
res <- c()
for (i in 1:nsim) {
smp <- GRP( thePs, theN, NULL, WSDesign )
w <- anopa(frm, smp[,2:4] , WSFactors = "M(3)" )
res <- c(res, if(summarize(w)[1,4]<.05) 1 else 0)
}
typeI <- mean(res)
cat( "Design W, testing W: ", typeI, "\n")
# tolerance is large as the number of simulations is small
expect_equal( typeI, .05, tolerance = 0.035)
```

```
frm <- s ~ grp * eta
WSDesign <- list()
BSDesign <- list(eta = c("repue","ajun"), grp = c("early","middle","late"))
thePs <- c(0.3, 0.3, 0.5, 0.5, 0.7, 0.7)
# test type-I error rate when no effect as is the case for factor 2
set.seed(41)
res <- c()
for (i in 1:nsim) {
smp <- GRP( thePs, theN, BSDesign )
w <- anopa(frm, smp )
res <- c(res, if(summarize(w)[2,4]<.05) 1 else 0)
}
typeI <- mean(res)
cat( "Design BxB, testing B: ", typeI, "\n")
# tolerance is large as the number of simulations is small
expect_equal( typeI, .05, tolerance = 0.035)
```

```
frm <- cbind(s.repue.early, s.ajun.early,
s.repue.middle, s.ajun.middle,
s.repue.late, s.ajun.late) ~ .
BSDesign <- list()
WSDesign <- list(eta = c("repue","ajun"), moment = c("early","middle","late"))
thePs <- c(0.3, 0.3, 0.5, 0.5, 0.7, 0.7)
# thePs <- c(0.3, 0.7, 0.3, 0.7, 0.3, 0.7) # or no effect on factor 1
# test type-I error rate when no effect as is the case for factor 2
set.seed(41)
res <- c()
for (i in 1:nsim) {
smp <- GRP( thePs, theN, NULL, WSDesign )
w <- anopa(frm, smp, WSFactors = c("e(2)", "m(3)") )
res <- c(res, if(summarize(w)[2,4]<.05) 1 else 0)
}
typeI <- mean(res)
cat( "Design WxW, testing W: ", typeI, "\n")
# tolerance is large as the number of simulations is small
expect_equal( typeI, .05, tolerance = 0.035)
```

```
frm <- cbind(s.early, s.middle, s.late) ~ grp
BSDesign <- list(grp = c("ctrl","plcbo"))
WSDesign <- list(moment = c("early","middle","late"))
thePs <- c(0.3, 0.3, 0.5, 0.5, 0.7, 0.7)
# thePs <- c(0.3, 0.7, 0.3, 0.7, 0.3, 0.7) # or no effect on factor 1
# test type-I error rate when no effect as is the case for factor 2
set.seed(41)
res <- c()
for (i in 1:nsim) {
smp <- GRP( thePs, theN, BSDesign, WSDesign )
w <- anopa(frm, smp[,2:5] , WSFactors = "M(3)")
res <- c(res, if(summarize(w)[2,4]<.05) 1 else 0)
}
typeI <- mean(res)
cat( "Design WxB, testing B: ", typeI, "\n")
# tolerance is large as the number of simulations is small
expect_equal( typeI, .05, tolerance = 0.035)
```

```
frm <- s ~ grp * eta * a
BSDesign <- list(eta = c("repue","ajun"),
grp = c("early","middle","late"), a = c("1","2","3","4"))
thePs <- rep(0.3, 24)
# test type-I error rate when no effect as is the case for factor 2
set.seed(41)
res <- c()
for (i in 1:nsim) {
smp <- GRP( thePs, theN, BSDesign )
w <- anopa(frm, smp )
res <- c(res, if(summarize(w)[2,4]<.05) 1 else 0)
}
typeI <- mean(res)
cat( "Design BxBxB, testing B: ", typeI, "\n")
# tolerance is large as the number of simulations is small
expect_equal( typeI, .05, tolerance = 0.035)
```

```
frm <- cbind(s.repue.early.1, s.ajun.early.1, s.repue.middle.1,
s.ajun.middle.1, s.repue.late.1, s.ajun.late.1, s.repue.early.2,
s.ajun.early.2, s.repue.middle.2, s.ajun.middle.2, s.repue.late.2,
s.ajun.late.2, s.repue.early.3, s.ajun.early.3, s.repue.middle.3,
s.ajun.middle.3, s.repue.late.3, s.ajun.late.3, s.repue.early.4,
s.ajun.early.4, s.repue.middle.4, s.ajun.middle.4, s.repue.late.4,
s.ajun.late.4 ) ~ .
WSDesign <- list(eta = c("repue","ajun"), grp = c("early","middle","late"), a = c("1","2","3","4"))
thePs <- rep(0.3, 24)
# test type-I error rate when no effect as is the case for factor 2
set.seed(43)
res <- c()
for (i in 1:nsim) {
smp <- GRP( thePs, theN, NULL, WSDesign )
w <- anopa(frm, smp, WSFactors = c("e(2)","g(3)", "a(4)") )
res <- c(res, if(summarize(w)[2,4]<.05) 1 else 0)
}
typeI <- mean(res)
cat( "Design WxWxW, testing W: ", typeI, "\n")
# tolerance is large as the number of simulations is small
expect_equal( typeI, .05, tolerance = 0.035)
```

```
frm <- cbind(s.repue.early, s.ajun.early, s.repue.middle,
s.ajun.middle, s.repue.late, s.ajun.late ) ~ a
BSDesign <- list( a = c("1","2","3","4") )
WSDesign <- list(eta = c("repue","ajun"), grp = c("early","middle","late") )
thePs <- rep(0.3, 24)
# test type-I error rate when no effect as is the case for factor 2
set.seed(43)
res <- c()
for (i in 1:nsim) {
smp <- GRP( thePs, theN, BSDesign, WSDesign )
w <- anopa(frm, smp, WSFactors = c("e(2)","g(3)") )
res <- c(res, if(summarize(w)[1,4]<.05) 1 else 0)
}
typeI <- mean(res)
cat( "Design BxWxW, testing W: ", typeI, "\n")
# tolerance is large as the number of simulations is small
expect_equal( typeI, .05, tolerance = 0.035)
```

```
frm <- cbind(s.repue.early, s.ajun.early, s.repue.middle,
s.ajun.middle, s.repue.late, s.ajun.late ) ~ a
BSDesign <- list( a = c("1","2","3","4") )
WSDesign <- list(eta = c("repue","ajun"), grp = c("early","middle","late") )
thePs <- rep(0.3, 24)
# test type-I error rate when no effect as is the case for factor 2
set.seed(42)
res <- c()
for (i in 1:nsim) {
smp <- GRP( thePs, theN, BSDesign, WSDesign )
w <- anopa(frm, smp, WSFactors = c("e(2)","g(3)") )
res <- c(res, if(summarize(w)[3,4]<.05) 1 else 0)
}
typeI <- mean(res)
cat( "Design BxWxW, testing B: ", typeI, "\n")
# tolerance is large as the number of simulations is small
expect_equal( typeI, .05, tolerance = 0.035)
```