We realize an intervention design with three conditions. Two if which will be randomly assigned to participants (within person design) - presenting all three would make variation too obvious and might create artifacts. As we assume \(\mu 1\)<\(\mu 2\)<\(\mu 3\) with a small to medium effect (\(d= .3-.4\)), we computed power analyses for two t-tests with a power of \(1-\beta = .90\) (\(=.81\) for both tests). Below find power analyses with stopping rule at \(BF= 10\) or \(BF= \frac{1}{10}\) respectively and

- an effect of \(d= .4\)
- an effect of \(d= .3\)
- no effect (\(d= 0\))

Out of financial and institutional reasons we aim for a \(N_{max}= 250\). Due to expected variations in the BF with low \(n\), we begin observing the data (optional stopping) at \(n=150\).

```
library(BFDA)
s1.4 <- BFDA.sim(expected.ES=0.4,
prior=list("Cauchy", list(prior.location=0, prior.scale=1)),
n.min=20, stepsize=10, n.max=250,
type="t.paired",
design="sequential",
alternative="greater",
B=10000,
cores=2,
verbose=TRUE)
```

```
## [1] "Simulation started at 2019-06-07 12:22:38"
## [1] "Simulation finished at 2019-06-07 13:54:16"
## Duration: Time difference of 1.526999 hours
```

`BFDA.analyze(s1.4, design="sequential", boundary=10)`

```
## outcome percentage
## 1 Studies terminating at n.max (n=250) 0%
## 2 Studies terminating at a boundary 100%
## 3 --> Terminating at H1 boundary 99.1%
## 4 --> Terminating at H0 boundary 0.9%
##
## Of 0% of studies terminating at n.max (n=250):
## 0% showed evidence for H1 (BF > 3)
## 0% were inconclusive (3 > BF > 1/3)
## 0% showed evidence for H0 (BF < 1/3)
##
## Average sample number (ASN) at stopping point (both boundary hits and n.max): n = 59
##
## Sample number quantiles (50/80/90/95%) at stopping point:
## 50% 80% 90% 95%
## 50 80 110 130
```

`plot(s1.4, n.min=150, boundary=c(1/10, 10))`

```
# As we plan to do 2 t-tests that supposed to have
SSD(s1.4, power=.90, boundary=c(1/10, 10)) # .8 power together, we aim for .90 power for each test.
```

```
## A >= 90% (actual: 91.8%) power achieved at n = 120
## This setting implies long-term rates of:
## 8.2% inconclusive results and
## 0% false-negative results.
```

```
s1.3 <- BFDA.sim(expected.ES=0.3,
prior=list("Cauchy", list(prior.location=0, prior.scale=1)),
n.min=20, stepsize=10, n.max=250,
type="t.paired",
design="sequential",
alternative="greater",
B=10000,
cores=2,
verbose=TRUE)
```

```
## [1] "Simulation started at 2019-06-07 13:54:17"
## [1] "Simulation finished at 2019-06-07 15:24:48"
## Duration: Time difference of 1.508403 hours
```

`BFDA.analyze(s1.3, design="sequential", boundary=10)`

```
## outcome percentage
## 1 Studies terminating at n.max (n=250) 2.4%
## 2 Studies terminating at a boundary 97.6%
## 3 --> Terminating at H1 boundary 93.3%
## 4 --> Terminating at H0 boundary 4.3%
##
## Of 2.4% of studies terminating at n.max (n=250):
## 1.2% showed evidence for H1 (BF > 3)
## 1.2% were inconclusive (3 > BF > 1/3)
## 0% showed evidence for H0 (BF < 1/3)
##
## Average sample number (ASN) at stopping point (both boundary hits and n.max): n = 93
##
## Sample number quantiles (50/80/90/95%) at stopping point:
## 50% 80% 90% 95%
## 80 140 180 220
```

`plot(s1.3, n.min=150, boundary=c(1/10, 10))`

`SSD(s1.3, power=.90, boundary=c(1/10, 10))`

```
## A >= 90% (actual: 91.5%) power achieved at n = 220
## This setting implies long-term rates of:
## 8.5% inconclusive results and
## 0% false-negative results.
```

```
s0 <- BFDA.sim(expected.ES=0,
prior=list("Cauchy", list(prior.location=0, prior.scale=1)),
n.min=20, stepsize=10, n.max=250,
type="t.paired",
design="sequential",
alternative="greater",
B=10000,
cores=2,
verbose=TRUE)
```

```
## [1] "Simulation started at 2019-06-07 15:24:49"
## [1] "Simulation finished at 2019-06-07 17:18:10"
## Duration: Time difference of 1.889126 hours
```

`BFDA.analyze(s0, design="sequential", boundary=1/10)`

```
## outcome percentage
## 1 Studies terminating at n.max (n=250) 7.9%
## 2 Studies terminating at a boundary 92.1%
## 3 --> Terminating at H1 boundary 1.6%
## 4 --> Terminating at H0 boundary 90.6%
##
## Of 7.9% of studies terminating at n.max (n=250):
## 0.2% showed evidence for H1 (BF > 3)
## 3% were inconclusive (3 > BF > 1/3)
## 4.7% showed evidence for H0 (BF < 1/3)
##
## Average sample number (ASN) at stopping point (both boundary hits and n.max): n = 79
##
## Sample number quantiles (50/80/90/95%) at stopping point:
## 50% 80% 90% 95%
## 50 120 220 250
```

`plot(s0, n.min=0, boundary=c(1/10, 10))`