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Bayesian First Aid is an attempt at implementing reasonable Bayesian alternatives to the classical hypothesis tests in R.

This is a straight forward extension of the Bayesian First Aid alternative to the binomial test which can be used to estimate the underlying relative frequency of success given a number of trials and, out of them, a number of successes.

If you were standing up and had an erection, if you looked down at the penis, would it be pointing straight ahead or pointed somewhat to the right or left? I dona€™t know about you, but the first question I had was are right handed people more like to have it on the right, or perhaps the oposite?

Not too bad, we get both a confidence interval on the difference between the groups and maximum likelihood estimates. Pretty similar estimates (and they should be) but now we also get estimates [and credible intervals] for both groups and the group difference.

So, it is most likely that left-handers lean more to the right by around 6-7 percentage points (and Bogaert discusses some reasons why this might be the case).

I dona€™t like Pearsona€™s chi-squared test, it is used as a catch-all analysis for any tables of counts and what you get back is utterly uninformative: a p-value relating to the null hypothesis that the row variable is completely independent of the column variable (which is anyway known to be false a priori most of the time, see here and here for some discussion). When doing in vitro fertilization the egg is fertilized by sperm outside the body and later, if successfully fertilized, reinserted into the uterus. Unfortunately they use chi-square tests to analyze these counts and they dona€™t even report the full p-values, for all but one of the measures all we get to know is NS. Using the concept of a region of practical equivalence (ROPE) we can calculate the probability that the difference between the two procedures is small. The probability that the relative frequency of high quality embryos is practically equivalent between the two procedures is only 20%, thus the probability that there is a substantial difference is 80%. Caveat: I know very little about in vitro fertilization and this is definitely not a critique of the study in any way. As is shown both in the print out and in the plot, group 4 seems to differ slightly from the rest. Another reason to modify the code that is printed out by model.code is in order to change the assumptions of the model. Length: 6 HoursDescription This course that teaches students critical skills needed to respond to and manage a first aid, choking or sudden cardiac arrest emergency in the first few minutes until emergency medical services (EMS) arrives. Correlation does not imply causation, right but, as Edward Tufte writes, a€?it sure is a hint.a€? The Pearson product-moment correlation coefficient is perhaps one of the most common ways of looking for such hints and this post describes the Bayesian First Aid alternative to the classical Pearson correlation test. The classical test of Pearson product-moment correlation coefficient between two paired variables $x_i$ and $y_i$ assumes bivariate normality.

The priors on $\mu_x,\mu_y, \sigma_x,\sigma_y$ and $\nu$ are also the same as in the alternative to the two sample t-test, that is, by peeking at the data we set the hyperparameters and resulting in a, for all practical purposes, flat prior. Wea€™ll leave Hone & McCullougha€™s analysis behind us (their analysis included many more variables, for one thing) and will just look at the strength and digit ratio data. Indeed, 2D:4D ratio and strength seems to be slightly less correlated than for the male group, but to claim there is evidence for no correlation seems a bit unfounded. A great thing with Bayesian data analysis is that it is simple to step away from the well trodden model and explore alternatives. One thing that we might want to change is the prior on rho which is currently $\text{Uniform}(-1,1)$.

Here the top left prior is the uniform, representing little prior information on the correlation between measures of puppy exposure and happiness.

All these worksheets and activities for teaching First aid have been designed by English language teachers.

At Quickworksheets we have 20+ smart worksheet generators for making fun, effective lesson materials. These are among the best phonics worksheets, games, videos and flash cards you will find online. Just looking at those with right or leftward disposition leaves us 1624 cases with 275 right leaners out of 1454 right-handers and 43 right leaners out of 170 non-right-handers. Looking at the estimates it seems like both left and right-handers tend to lean to the left much more often than to the right. Still the posterior of the group difference is pretty wide (and the credible interval kisses the 0.0) and even though the analysis is leaning (he he) towards there being a difference it would be nice to have a few more data points to get a tighter estimate.

The egg and the sperms are usually left together to co-incubate for more than an hour before the egg is separated from the sperm and left to incubate for itself. Just looking at the raw data it seems like there is little difference between the proportions of fertilized eggs in the two groups, but there seems to be a difference in embryo quality with more embryos in the 90 min. There is definitely weak evidence for a€?no differencea€? here but we would need more data to be able state the magnitude of the difference with reasonable certainty.

I dona€™t know what would be considered a region of practical equivalence in this case and I dona€™t know if embryo quality is considered an important outcome.

Here is an example with a dataset from the prop.test help file on the number of smokers in four groups of patients with lung cancer.

The current model does not assume any dependency between the groups and if this is an unreasonable assumption you might want to modify the model code to include such a dependency.

Use of these materials in an educational course does not represent course sponsorship by the American Heart Association, and any fees charged for such a course do not represent income to the Association. Except for being based on Bayesian estimation (a good thing in my book) this alternative is more robust to outliers and comes with a pretty nice default plot. It assumes that the relation is linear and that both $x_i$ and $y_i$ are normally distributed. This is the same trick as in the Bayesian First Aid alternative to the t-test, compared to the normal distribution the wider tails of the t will downweight the influence of stray data points. When modeling correlations it is common to directly put a prior distribution on the covariance matrix (the Inverse-Wishart distribution for example).

It is not clear to me how p = 0.09 is strong evidence that prenatal exposure to testosterone does not influence strength in women at all.

This question could be answered without any data, of course there is some correlation between 2D:4D ratio and strength (no matter how tiny).

At the top we have the posterior distribution for the correlation $\rho$ with a 95% highest density interval.

We could, however, take a look at the posterior difference in correlation between the male and the female group. Given that I have a relatively high (less a€?manlya€?) 2D:4D ratio of 1.0 does it mean I have an excuse for my subpar arm strength?

As described by Barnard, McCulloch & Meng (2000), the beta distribution stretched to the interval [-1,1] is a flexible alternative that facilitates the construction of a more informative prior. The top right prior is skeptical of there being a large correlation but is agnostic with respect to the direction. A multi-level English curriculum featuring cartoon animated videos, engaging games, interactive tests and a progress tracker. Just enter your list of words and this website will create bingo, dominoes, crossword, memory games, etc.

The has everything you need to help a child learn to read through phonics: decodable stories, listening exercises, you name it. We also get to know that the probability that left-handers lean more often to the right compared to right-handers is 97.7%. Sure, the credible interval kisses zero, but the evidence for a small difference, which was hinted at in the original article, is definitely not strong. I have no strong intuition about what would be a small difference in this particular case, so Ia€™m arbitrarily going to go with 5 percentage points, yielding a ROPE of [-5, 5] percentage points (for more about ROPEs see Kruschke, 2011).

However, I still believe that the analysis would have been more informative if they would have used something better than chi-square tests and p-values! A nice example of how to extend the model to assume a hierarchical dependency between the relative frequencies of success of each group can be found on the LingPipe blog.

A prospective study, using sibling oocytes, examining the effect of 30 seconds versus 90 minutes gamete co-incubation in IVF. Ia€™ve already written about a Bayesian alternative to the correlation test here and about how that model can be made more robust here. The model has six parameters: the means of the two marginal distributions ($\mu_x,\mu_y$), the SDs ($\sigma_x,\sigma_y$), the degree-of-freedoms parameter that influences the heaviness of the tails ($\nu$), and finally the correlation ($\rho$). Here we instead do as described by Barnard, McCulloch & Meng (2000) and put separate priors on $\sigma_x,\sigma_y$ and $\rho$, where $\rho$ is given a uniform prior. Stated in a sloppy way, the working hypothesis is that the 2D:4D ratio is a proxy variable for prenatal androgen exposure and could therefore be related to a host of other traits related to a€?manlinessa€? such as aggression, prostate cancer risk, sperm count, etc. In complex systems, such as humans, it would be extremely unlikely that any given trait (such as hair color, shoe size, movie taste, running speed, no of tweets, etc.) does not correlate at all with any other trait. At the bottom we see the original data with superimposed posterior predictive distributions (that is, the distribution we would expect a new data point to have).

The lower left prior is just slightly in favor of a positive correlation, with the lower right being quite optimistic putting more than 99% of the prior probability over rho > 0. A lot of questions out there involves estimating the proportion or relative frequency of success of two or more groups (where success could be a saved life, a click on a link, or a happy baby) and there exists a little known R function that does just that, prop.test.

Bayesian First Aid is a work in progress and Ia€™m grateful for any suggestion on how to improve it!

But the chi-square analysis says NS which is interpreted in the result section as: a€?the two groups were comparablea€?. The Bayesian First Aid alternative is basically the robustified version with slightly different priors.

The advantage with separate priors is that it gives you greater flexibility and makes it easy to add prior information into the mix. The two ellipses show the 50% (darker blue) and 95% (lighter blue) highest density regions.

The red histograms show the marginal distributions of the data with a smatter of marginal densities drawn from the posterior. Looking at this plot we see that the model fits quite well, however, we could be concerned with the right skewness of the ratio_2d4d marginal which is not captured by the model.

This is a straight forward extension of the Bayesian First Aid alternative to the binomial test which can be used to estimate the underlying relative frequency of success given a number of trials and, out of them, a number of successes.

If you were standing up and had an erection, if you looked down at the penis, would it be pointing straight ahead or pointed somewhat to the right or left? I dona€™t know about you, but the first question I had was are right handed people more like to have it on the right, or perhaps the oposite?

Not too bad, we get both a confidence interval on the difference between the groups and maximum likelihood estimates. Pretty similar estimates (and they should be) but now we also get estimates [and credible intervals] for both groups and the group difference.

So, it is most likely that left-handers lean more to the right by around 6-7 percentage points (and Bogaert discusses some reasons why this might be the case).

I dona€™t like Pearsona€™s chi-squared test, it is used as a catch-all analysis for any tables of counts and what you get back is utterly uninformative: a p-value relating to the null hypothesis that the row variable is completely independent of the column variable (which is anyway known to be false a priori most of the time, see here and here for some discussion). When doing in vitro fertilization the egg is fertilized by sperm outside the body and later, if successfully fertilized, reinserted into the uterus. Unfortunately they use chi-square tests to analyze these counts and they dona€™t even report the full p-values, for all but one of the measures all we get to know is NS. Using the concept of a region of practical equivalence (ROPE) we can calculate the probability that the difference between the two procedures is small. The probability that the relative frequency of high quality embryos is practically equivalent between the two procedures is only 20%, thus the probability that there is a substantial difference is 80%. Caveat: I know very little about in vitro fertilization and this is definitely not a critique of the study in any way. As is shown both in the print out and in the plot, group 4 seems to differ slightly from the rest. Another reason to modify the code that is printed out by model.code is in order to change the assumptions of the model. Length: 6 HoursDescription This course that teaches students critical skills needed to respond to and manage a first aid, choking or sudden cardiac arrest emergency in the first few minutes until emergency medical services (EMS) arrives. Correlation does not imply causation, right but, as Edward Tufte writes, a€?it sure is a hint.a€? The Pearson product-moment correlation coefficient is perhaps one of the most common ways of looking for such hints and this post describes the Bayesian First Aid alternative to the classical Pearson correlation test. The classical test of Pearson product-moment correlation coefficient between two paired variables $x_i$ and $y_i$ assumes bivariate normality.

The priors on $\mu_x,\mu_y, \sigma_x,\sigma_y$ and $\nu$ are also the same as in the alternative to the two sample t-test, that is, by peeking at the data we set the hyperparameters and resulting in a, for all practical purposes, flat prior. Wea€™ll leave Hone & McCullougha€™s analysis behind us (their analysis included many more variables, for one thing) and will just look at the strength and digit ratio data. Indeed, 2D:4D ratio and strength seems to be slightly less correlated than for the male group, but to claim there is evidence for no correlation seems a bit unfounded. A great thing with Bayesian data analysis is that it is simple to step away from the well trodden model and explore alternatives. One thing that we might want to change is the prior on rho which is currently $\text{Uniform}(-1,1)$.

Here the top left prior is the uniform, representing little prior information on the correlation between measures of puppy exposure and happiness.

All these worksheets and activities for teaching First aid have been designed by English language teachers.

At Quickworksheets we have 20+ smart worksheet generators for making fun, effective lesson materials. These are among the best phonics worksheets, games, videos and flash cards you will find online. Just looking at those with right or leftward disposition leaves us 1624 cases with 275 right leaners out of 1454 right-handers and 43 right leaners out of 170 non-right-handers. Looking at the estimates it seems like both left and right-handers tend to lean to the left much more often than to the right. Still the posterior of the group difference is pretty wide (and the credible interval kisses the 0.0) and even though the analysis is leaning (he he) towards there being a difference it would be nice to have a few more data points to get a tighter estimate.

The egg and the sperms are usually left together to co-incubate for more than an hour before the egg is separated from the sperm and left to incubate for itself. Just looking at the raw data it seems like there is little difference between the proportions of fertilized eggs in the two groups, but there seems to be a difference in embryo quality with more embryos in the 90 min. There is definitely weak evidence for a€?no differencea€? here but we would need more data to be able state the magnitude of the difference with reasonable certainty.

I dona€™t know what would be considered a region of practical equivalence in this case and I dona€™t know if embryo quality is considered an important outcome.

Here is an example with a dataset from the prop.test help file on the number of smokers in four groups of patients with lung cancer.

The current model does not assume any dependency between the groups and if this is an unreasonable assumption you might want to modify the model code to include such a dependency.

Use of these materials in an educational course does not represent course sponsorship by the American Heart Association, and any fees charged for such a course do not represent income to the Association. Except for being based on Bayesian estimation (a good thing in my book) this alternative is more robust to outliers and comes with a pretty nice default plot. It assumes that the relation is linear and that both $x_i$ and $y_i$ are normally distributed. This is the same trick as in the Bayesian First Aid alternative to the t-test, compared to the normal distribution the wider tails of the t will downweight the influence of stray data points. When modeling correlations it is common to directly put a prior distribution on the covariance matrix (the Inverse-Wishart distribution for example).

It is not clear to me how p = 0.09 is strong evidence that prenatal exposure to testosterone does not influence strength in women at all.

This question could be answered without any data, of course there is some correlation between 2D:4D ratio and strength (no matter how tiny).

At the top we have the posterior distribution for the correlation $\rho$ with a 95% highest density interval.

We could, however, take a look at the posterior difference in correlation between the male and the female group. Given that I have a relatively high (less a€?manlya€?) 2D:4D ratio of 1.0 does it mean I have an excuse for my subpar arm strength?

As described by Barnard, McCulloch & Meng (2000), the beta distribution stretched to the interval [-1,1] is a flexible alternative that facilitates the construction of a more informative prior. The top right prior is skeptical of there being a large correlation but is agnostic with respect to the direction. A multi-level English curriculum featuring cartoon animated videos, engaging games, interactive tests and a progress tracker. Just enter your list of words and this website will create bingo, dominoes, crossword, memory games, etc.

The has everything you need to help a child learn to read through phonics: decodable stories, listening exercises, you name it. We also get to know that the probability that left-handers lean more often to the right compared to right-handers is 97.7%. Sure, the credible interval kisses zero, but the evidence for a small difference, which was hinted at in the original article, is definitely not strong. I have no strong intuition about what would be a small difference in this particular case, so Ia€™m arbitrarily going to go with 5 percentage points, yielding a ROPE of [-5, 5] percentage points (for more about ROPEs see Kruschke, 2011).

However, I still believe that the analysis would have been more informative if they would have used something better than chi-square tests and p-values! A nice example of how to extend the model to assume a hierarchical dependency between the relative frequencies of success of each group can be found on the LingPipe blog.

A prospective study, using sibling oocytes, examining the effect of 30 seconds versus 90 minutes gamete co-incubation in IVF. Ia€™ve already written about a Bayesian alternative to the correlation test here and about how that model can be made more robust here. The model has six parameters: the means of the two marginal distributions ($\mu_x,\mu_y$), the SDs ($\sigma_x,\sigma_y$), the degree-of-freedoms parameter that influences the heaviness of the tails ($\nu$), and finally the correlation ($\rho$). Here we instead do as described by Barnard, McCulloch & Meng (2000) and put separate priors on $\sigma_x,\sigma_y$ and $\rho$, where $\rho$ is given a uniform prior. Stated in a sloppy way, the working hypothesis is that the 2D:4D ratio is a proxy variable for prenatal androgen exposure and could therefore be related to a host of other traits related to a€?manlinessa€? such as aggression, prostate cancer risk, sperm count, etc. In complex systems, such as humans, it would be extremely unlikely that any given trait (such as hair color, shoe size, movie taste, running speed, no of tweets, etc.) does not correlate at all with any other trait. At the bottom we see the original data with superimposed posterior predictive distributions (that is, the distribution we would expect a new data point to have).

The lower left prior is just slightly in favor of a positive correlation, with the lower right being quite optimistic putting more than 99% of the prior probability over rho > 0. A lot of questions out there involves estimating the proportion or relative frequency of success of two or more groups (where success could be a saved life, a click on a link, or a happy baby) and there exists a little known R function that does just that, prop.test.

Bayesian First Aid is a work in progress and Ia€™m grateful for any suggestion on how to improve it!

But the chi-square analysis says NS which is interpreted in the result section as: a€?the two groups were comparablea€?. The Bayesian First Aid alternative is basically the robustified version with slightly different priors.

The advantage with separate priors is that it gives you greater flexibility and makes it easy to add prior information into the mix. The two ellipses show the 50% (darker blue) and 95% (lighter blue) highest density regions.

The red histograms show the marginal distributions of the data with a smatter of marginal densities drawn from the posterior. Looking at this plot we see that the model fits quite well, however, we could be concerned with the right skewness of the ratio_2d4d marginal which is not captured by the model.

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