In this vignette we’ll cover how to replicate the results of the
*default* *t*-tests developed Rouder, Speckman, Sun, & Morey (2009). We’ll
cover both one-sample/paired *t*-tests and independent samples
*t*-tests, and we’ll cover how specify the models both in terms
of the *t* statistic and in terms of Cohen’s *d*.

One-sample *t*-tests are already covered in the basic
usage vignette, so the example presented here is simply a repeat of
that example. However, we’ll change the order of presentation around a
bit so that the relationship between the one-sample and independent
samples models are a little clearer.

We’ll start with an example from Rouder et al.
(2009), in which they analyse the results if a one sample
*t*-test. Rouder et al. (2009)
report a *t* statistic of 2.03, from a sample size of 80. The
sampling distribution of the *t* statistic, when the null
hypothesis is false is the noncentral
*t*-distribution distribution. Therefore, we can use this
fact to construct a *likelihood function* for our *t*
value from the noncentral *t*-distribution. For this, we’ll need
two parameters—the *t* value itself, and the *degrees of
freedom*. In the one-sample case, the *degrees of freedom*
will just be N - 1.

From plotting our *likelihood* function we can see that values
of *t* at our observation (*t* = 2.03) and most consistent
without our observation while more extreme values of *t* (e.g.,
*t* = -2, or *t* = 6) are less consistent without our
observed data. This is of course, as we would expect.

Now that we’ve defined our likelihood, the next thing to think about
is the *prior*. Rouder et al.
(2009) provide a more detailed justification for their prior, so
we won’t go into that here; however, their prior is based on the
*Cauchy* distribution. A Cauchy distribution is like a very fat
tailed *t* distribution (in fact, it **is** a very
fat tailed *t* distribution). We can see an example of a
*standard Cauchy* distribution below.

The *Cauchy* distribution can be *scaled* so that it is
wider or narrow. You might want to choose a `scale`

parameter
based on the range of *t* values that your theory prediction. For
example, here is a slightly *narrower* *Cauchy*
distribution.

Although scaling the prior distribution by our theoretical
*predictions* makes sense, we also need to factor in another
thing—sample size. Because of how the *t* statistic is calculated
(\(t = \frac{\mu}{\sigma/\sqrt{n}}\)),
we can see that *t* statistics are also dependent on sample size.
That is, for any given effect size (*standardized mean
difference*), the corresponding *t* value will be a function
of sample size—or more, specifically, the square root of the sample
size. That is, for a *given underlying true effect size*, when we
have a large sample size we can expect to see larger values of
*t* that if we have a small sample size. Therefore, we should
also scale our prior by the square root of the sample size. In the
specification below, we’re specifying a *standard Cauchy*, but
further scaling this by \(\sqrt{n}\).

For our *null* we’ll use a point located at 0.

Now we can proceed to compute the Bayes factor in the standard way.

```
bf_onesample_1 <- integral(data_model * alt_prior) / integral(data_model * null_prior)
summary(bf_onesample_1)
#> Bayes factor
#> Using the levels from Wagenmakers et al (2017)
#> A BF of 0.6421 indicates:
#> Anecdotal evidence
```

In the preceding example, we’ve taken account of the fact that the
*same underlying effect size* will produce different values of
*t* depending on the sample, and we’ve scaled our *prior*
accordingly. However, we could also apply the scaling at the other end
by re-scaling our *likelihood*. The `bayesplay`

package contains two additional noncentral *t* likelihoods that
have been rescaled. The first of the these is the
`noncentral_d`

likelihood. This is a likelihood based on the
sample distribution of the *one-sample/paired samples Cohen’s d*.
This is calculated simply as \(d =
\frac{\mu_{\mathrm{diff}}}{\sigma_{\mathrm{diff}}}\).
Alternatively, we can conver the observed *t* value by dividing
it by \(\sqrt{n}\). The
`noncentral_d`

likelihood just takes this effect size and the
sample size are parameters.

Now that we’ve applied our scaling to the *likelihood*, we
don’t need to apply this scaling to the *prior*. Therefore, if we
wanted to use the same prior as the preceding example, we’ll have a
`scale`

value of `1`

, rather than
`1 * sqrt(n)`

.

We can re-use our null prior from before and calculate the Bayes factor the same way as before.

```
bf_onesample_2 <- integral(data_model2 * alt_prior2) / integral(data_model2 * null_prior)
summary(bf_onesample_2)
#> Bayes factor
#> Using the levels from Wagenmakers et al (2017)
#> A BF of 0.6421 indicates:
#> Anecdotal evidence
```

As expected, the two results are identical.

Rouder et al. (2009) also provide an
extension of their method to the two sample case, although they do not
provide a worked example. Instead, we can generate our own example and
directly compare the results from the `bayesplay`

package
with the results from the `BayesFactor`

package.

For this example, we’ll start by generating some data from an independent samples design.

First, let us see the results from the `BayesFactor`

package.

```
Bayes factor analysis
--------------
[1] Alt., r=1 : 0.9709424 ±0%
Against denominator:
Null, mu1-mu2 = 0
---
Bayes factor type: BFindepSample, JZS
```

As with the one-sample case, we can run the analysis in the
`bayesplay`

package using either the *t* statistic or
the *Cohen’s d*. We’ll start by running the analysis using the
*t* statistic. The easiest way to do this, is to simply use the
`t.test()`

function in R.

```
t_result <- t.test(x = group1, y = group2, paired = FALSE, var.equal = TRUE)
t_result
#>
#> Two Sample t-test
#>
#> data: group1 and group2
#> t = -1.7922, df = 29, p-value = 0.08354
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#> -21.411872 1.411872
#> sample estimates:
#> mean of x mean of y
#> 25 35
```

From this output we need the *t* statistic itself, and the
*degrees of freedom*.

With the *t*, and *df* value in hand, we can specify
our likelihood using the same noncentral *t* distribution as the
one sample case.

As with the one-sample case, a *Cauchy* prior is used for the
alternative hypothesis. Again, this will need to be appropriately
scaled. In one sample case we scaled it by \(\sqrt{n}\). In the two-sample case,
however, we’ll scale it by \(\sqrt{\frac{n_1
\times n_2}{n_1 + n_2}}\)

```
alt_prior3 <- prior("cauchy", location = 0, scale = 1 * sqrt((n1 * n2) / (n1 + n2)))
plot(alt_prior3)
```

We’ll use the same point null prior as before, and then compute the Bayes factor in the usual way.

```
bf_independent_1 <- integral(data_model3 * alt_prior3) / integral(data_model3 * null_prior)
summary(bf_independent_1)
#> Bayes factor
#> Using the levels from Wagenmakers et al (2017)
#> A BF of 0.9709 indicates:
#> Anecdotal evidence
```

Appropriately scaling our *Cauchy* prior can be tricky, so an
alternative is instead, as before, to scale our likelihood. The
`bayesplay`

package contains a likelihood that is appropriate
for independent samples *Cohen’s d*, the
`noncentral_d2`

likelihood. To use this, we’ll need the
*Cohen’s d* value, and the two sample sizes.

For independent samples designs the *Cohen’s d* is calculated
as follows:

\[d = \frac{m_1 - m2}{s_\mathrm{pooled}},\]

where \(s_\mathrm{pooled}\) is given as follows:

\[s_\mathrm{pooled} = \sqrt{\frac{(n_1 - 1)s^2_1 + (n_2 - 1)s^2_2}{n_1 + n_2 -2}}\]

This is fairly straightforward to calculate, as shown below.

```
m1 <- mean(group1)
m2 <- mean(group2)
s1 <- sd(group1)
s2 <- sd(group2)
md_diff <- m1 - m2
sd_pooled <- sqrt((((n1 - 1) * s1^2) + ((n2 - 1) * s2^2)) / (n1 + n2 - 2))
d <- md_diff / sd_pooled
d
#> [1] -0.6441105
```

However, it can also be obtained from the `effsize`

package using the following syntax.

With the *d* value in hand, we can how specify a new
likelihood.

```
data_model4 <- likelihood("noncentral_d2", d = d, n1 = n1, n2 = n2)
data_model4
#> Likelihood
#> Family
#> noncentral_d2
#> Parameters
#> d: -0.644110547740848
#> n1: 15
#> n2: 16
#>
```

Because we’ve used the appropriately scaled noncentral *t*
likelihood, the `noncentral_d2`

, we no longer need to scale
the *Cauchy* prior.

And we can now calculate the Bayes factor in the usual way.

```
bf_independent_2 <- integral(data_model4 * alt_prior4) / integral(data_model4 * null_prior)
summary(bf_independent_2)
#> Bayes factor
#> Using the levels from Wagenmakers et al (2017)
#> A BF of 0.9709 indicates:
#> Anecdotal evidence
```

Again, we obtain the same result as the `BayesFactor`

package.

Rouder, J. N., Speckman, P. L., Sun, D., & Morey, R. D. (2009).
Bayesian t tests for accepting and rejecting the null hypothesis.
*Psychonomic Bulletin & Review*, *20*, 225–237. https://doi.org/10.3758/PBR.16.2.225