A Bayesian Alternative to Parametric Hypothesis Testing
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This seems limited, what if you have a number of counts, say you sell ice cream during a whole week, what to do then? The trick here is that you can add up the counts and the number of time periods and this will be perfectly fine. The code below will still give you an estimate for the underlying rate of ice cream sales per day:. Note that this only works if the counts are well modeled by the same Poisson distribution. If the ice cream sales are much higher on the weekends, adding up the counts might not be a good idea.
As the Bayesian alternative accepts the same input as poisson. The model for the Bayesian First Aid alternative to the one sample possion. In the two sample case the one sample model is just separately fitted to each sample.
Bayesian First Aid: Poisson Test
In the the literature there are two common recommendations for an objective prior for the rate of a Poisson distribution. I feel that seeing zero events should tell the model something and, at least, not cause it to blow up. The difference between these two priors is very small and will only matter when you have very few counts. Therefore Bayesian First Aid alternative to poisson. The bayes.
If you just ran poisson. By saving the output, for example, like fit you can inspect it further using plot fit , summary fit and diagnostics fit.
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To demonstrate the use of bayes. Here is the full data set:.
Bayesian Statistics | Society for Industrial and Applied Mathematics
Here WY stand for woman-years as if woman-years would be different from man-years, or person-years…. While the data is from a relatively old article we are going to replicate a more recent reanalysis of that data from the article Testing the Ratio of Two Poisson Rates by Gu et al. They tested the alternative hypothesis that the rate of breast cancer per person-year would be 1. They tested it like this:.
It is oh-so-easy to interpret this as that there is no evidence that the incidence rate is more than 1. The warning here is nothing to worry about, there is no need to specify what alternative is tested and bayes. That is, rather than just saying that there is not enough evidence we have quantified how much evidence there is, and the evidence actually slightly favors the alternative hypothesis. This is also easily seen in the default plot of bayes.
Back to the ice cream truck, say that you sold 14 ice creams in one day and your competitors Karl and Anna sold 22 and 7 ice creams, respectively. How would you estimate and compare the underlying rates of sold ice creams of these three trucks when bayes. One important benefit of Bayesian statistics is that you can provide relative support for the null hypothesis. Bayes Factors can express relative evidence for the null hypothesis H0 compared to the alternative hypothesis H1 , referred to as a BF 01 , or relative evidence for H1 compared to H0 a BF Table 1 from Wagenmakers et al.
Since a Bayes Factor provides relative support for one over another hypothesis, the choice for an alternative hypothesis is important.
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In Neyman-Pearson Frequentist approaches, researchers have to specify the alternative hypothesis to be able to calculate the power of their test. It then becomes easy to calculate that you need participants in each group of your experiment, if you plan to perform a two-sided independent t -test with an alpha of 0. When calculating the Bayes Factor, the alternative is specified through the r-scale. Thus, the researcher above would choose an r-scale of 0.
The default Bayesian t -test originally used a r-scale of 1 a very large effect , but the updated default test uses a r-scale of 0. You are also completely fine with a null-result: You have submitted your study to a journal that accepts pre-registered reports , and the reviewers there agreed the data will be interesting whether they support the null hypothesis or the alternative hypothesis. What is the probability you will observe support for H0 relative to H1 see this paper by Felix Schonbrodt and colleagues for similar Frequentist calculations of Bayes Factors?
There are no Bayesian power calculators yet, as far as I know. Using the R code below, you can easily calculate this probability yourself. It also provides the probability of finding support for the alternative hypothesis. This means you can run the simulations twice, once setting the true effect size to 0, once setting the true effect size to 0. Note you need quite some simulations to be accurate, and this takes minutes.
Improving the replicability of empirical research
In the histogram of Bayes Factors below, this is the distribution to the left of the left vertical red line. Approximately 1. The simulations also show that the probability of an incorrect conclusion observing support for H0 when H1 is true, or support for H1 when H0 is true is very low. For example, if we set a threshold as low as 1. I think these cut-offs are most important when planning an experiment.
A Bayesian alternative to parametric hypothesis testing
It makes sense to aim for support most people will find convincing. When the data are in hand, you can interpret it any way you like. If you agree, then important questions are which cut-off values are acceptable under which circumstances?
Can we develop formulas for Bayesian power analysis instead of having to rely on simulations? I learned something by playing around with these simulations, and I expect others moving to Bayes Factors will have similar questions as I tried to examine here. Dienes, Z. How Bayes factors change scientific practice. Journal of Mathematical Psychology. Rouder, J. Bayesian t tests for accepting and rejecting the null hypothesis. Wagenmakers, E. Why psychologists must change the way they analyze their data: the case of psi: comment on Bem Joe January 14, at AM.