
stan
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The ` sigma ~ x` part specifies that sigma should be estimated separately for each group. Note that I'm also using scaled data, since then I can go by the Stan team's [prior choice recommendations](https://github.com/standev/stan/wiki/PriorChoiceRecommendations). We also specified ` family = gaussian` , which is telling the model to treat ` y` , the difference between the two variables, as normally distributed. In other words, this is the likelihood! There are [lots](https://cran.rproject.org/web/packages/brms/vignettes/brms_families.html) of "families" in brms. In particular, if you use a Student's t distribution instead, your model will be more robust against outliers!

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Okay so first off, I recommend that you read [this](https://link.springer.com/article/10.3758/s1342301612214) article about "The Bayesian New Statistics", which highlights estimation rather than hypothesis testing from a Bayesian perspective (see Fig. 1, second row, second column). Instead of a ttest, then, we can *estimate the difference* between two groups/variables. If you want to go deeper than JASP etc, I recommend that you use [brms](https://paulbuerkner.github.io/brms/), or, if you want to go even deeper, [Stan](https://mcstan.org/) (brms is a frontend to Stan).

Second, one thing that is often overlooked is that most models can be seen as [variants of linear regression](https://lindeloev.github.io/testsaslinear/), including ttests. To estimate the difference between two variables using linear regression (in R), you use `lm(y ~ x, data = data), where `x` is the group variable (factor coded) and `y` is the variable of interest. If you suppress the intercept, you directly estimate the means of the two variables: `lm(y ~ 0 + x, data = data)`` . Finally, the ttest assumes equal variance between groups, which is often [a weird assumption](https://www.ripsirsp.com/articles/10.5334/irsp.82/). Thus we'll make sure to allow variance to differ.
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