knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )

```
library(bayesplay)
```

In this vignette we'll cover how to replicate the results of the *default* *t*-tests developed @rouderbf. 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 @rouderbf, in which they analyse the results if a one sample *t*-test. @rouderbf 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.

t <- 2.03 n <- 80 data_model <- likelihood("noncentral_t", t = t, df = n - 1) plot(data_model)

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*. @rouderbf 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.

plot(prior("cauchy", location = 0, scale = 1))

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.

plot(prior("cauchy", location = 0, scale = .707))

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 $\sqrt{n}$.

alt_prior <- prior("cauchy", location = 0, scale = 1 * sqrt(n)) plot(alt_prior)

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

null_prior <- prior("point", point = 0) plot(null_prior)

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)

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.

d <- t / sqrt(n) data_model2 <- likelihood("noncentral_d", d = d, n = n) plot(data_model2)

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)`

.

alt_prior2 <- prior("cauchy", location = 0, scale = 1) plot(alt_prior2)

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)

As expected, the two results are identical.

@rouderbf 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.

set.seed(2125519) group1 <- 25 + scale(rnorm(n = 15)) * 15 group2 <- 35 + scale(rnorm(n = 16)) * 16

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

package.

BayesFactor::ttestBF(x = group1, y = group2, paired = FALSE, rscale = 1)

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

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

t <- t_result$statistic t

df <- t_result$parameter df

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

data_model3 <- likelihood("noncentral_t", t = t, df = df)

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 $\sqrt{\frac{n_1 \times n_2}{n_1 + n_2}}$

n1 <- length(group1) n2 <- length(group2)

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)

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

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

package using the following syntax.

effsize::cohen.d(group1, group2, paired = FALSE, hedged.correction = FALSE)

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

```
plot(data_model4)
```

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

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

alt_prior4 <- prior("cauchy", location = 0, scale = 1) plot(alt_prior4)

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)

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

package.

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