Advising on research methods
Selected topics 2014

Advising on research methods
Selected topics 2014

edited by

Herman J. Adèr and Gideon J. Mellenbergh

Johannes van Kessel Publishing
Huizen, The Netherlands

© Herman J. Adèr and Gideon J. Mellenbergh 2014

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, transmitted or utilized in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without permission in writing from the Publisher.

PIC
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ISBN 978-90-79418-38-1
NUR: 840, General Education

Chapter 6
Including prior information in bayesian procedures
by Quentin F. Gronau and Tom H. Oreel

This chapter is concerned with how prior information can be included in Bayesian procedures.
In the first part of the chapter, we will introduce the reader to the fundamental concepts needed to understand Bayesian inference and how they can be applied. We will explain how distributions are used to quantify prior knowledge including a short discussion about possible advantages/disadvantages, e.g., the common critique of subjectivity. We will cover how prior information can be updated via Bayes theorem to yield a so-called posterior distribution which represents the knowledge about the parameter of interest after having seen the data. In this introduction about Bayes theorem, we will also compare Bayesian inference with the frequentist approach, so that the ordinary (frequentist) reader will get a clearer picture of the difference.
In the second part of our chapter we will present some real-life examples that illustrate the technique.

6.1 Why use Bayesian inference?

In the last years, the application of Bayesian methods has become more and more popular in scientific research. To understand the appeal of Bayesian inference, it is best to compare it to the “standard” approach in statistics which is taught to students at most universities: the frequentist approach.

6.1.1 The frequentist approach

Let us consider an example to explain how frequentist reasoning works. Suppose that you want to know whether a correlation between two variables of interest that you found in your sample will hold for the population of interest. In order to investigate this, you will use the so-called sampling distribution of your test statistic, which is in this case the sampling distribution of the correlation under the null hypothesis (i.e., the correlation is equal to zero) and you will evaluate how likely it is to find a correlation that is as least as extreme as you have found in your sample under this sampling distribution. This is exactly what the p value tells you: The probability of finding an at least as extreme test statistic given that the null hypothesis is true. You will then probably also compute the 95% confidence interval of your test statistic, which gives an estimate of the precision of the test statistic (in this case the correlation). To summarize, in order to evaluate your test statistic, you compare it to the distribution that you would obtain when sampling over and over again. This is the reason why it is called the frequentist approach. This approach contains some “indirect” reasoning: Although I am interested in whether my correlation is relevant, I assume that there is no correlation at all to obtain the sampling distribution. When the obtained value is very unlikely under the null hypothesis, it is concluded that the correlation is substantial. Furthermore, this sampling distribution relies on non-observed but imagined data (i.e., what would happen if I would do the experiment over and over again). For an excellent discussion why this is problematic and other issues with p values, the interested reader is referred to Wagenmakers (2007).

There is also a lot of confusion, how to interpret confidence intervals. A common assumption is that the probability that the true test statistic is contained in the 95% confidence interval equals .95. But this is not true: If the experiment has been done and the confidence interval has been calculated, the probability of the test statistic falling in this interval is either zero or one. The 95% correspond to what would happen if I would do the experiment over and over again each time calculating the 95% confidence interval: 95% of these intervals would contain the true test statistic. Thus, the interpretation of results of a frequentist analysis is not as straightforward as a lot of people think.

6.1.2 The Bayesian approach

The Bayesian approach offers a way of analyzing which is more in line with how most people reason. With the help of Bayesian inference, it is possible to make statements about the probability of parameters (for example a correlation, a mean, etc.) given the data that were obtained thus far. So the analysis results are not restricted to making an a priori assumption about the parameter of interest (e.g., the correlation is equal to zero) and then evaluating the likelihood of the data given this assumption, but the direction of the inference step can be “flipped” via the Bayes theorem to evaluate how likely different parameter values are, given the data that were obtained. The interpretation of “confidence” intervals becomes more straightforward, too: The Bayesian analog to the confidence interval is called highest density interval (Kruschke, 2011b), credible set (Carlin & Louis, 2000) or credible interval (Lee & Wagenmakers, 2013) and this can actually be interpreted as the probability that the parameter of interest is located in that region. Another advantage of Bayesian inference is that prior knowledge can elegantly (and formally) be incorporated in the analysis.

The remainder of the chapter is organized as follows: First, the basic principles of Bayesian inference are explained and it is illustrated how prior information can be included in a Bayesian analysis. After that, some examples are presented on ways to use prior knowledge to make inferences.

6.2 Bayesian inference: Natural inclusion of prior Information

This section starts with a disclaimer: Although the basic principle of Bayesian inference is relatively simple, it is a broad field and a lot can and should be said about it to get a thorough understanding of it. This section just sketches the basics so that the reader can get a grasp of the way Bayesian inference works and how prior information can be included in the analysis. For a deeper understanding of the topic, the interested reader is referred to one of the several excellent books that cover Bayesian inference (e.g., Carlin and Louis (2000), Kruschke (2011b), Jackman (2009), Lee and Wagenmakers (2013), or the seminal, but not always easy to read Jeffreys (1961)). Suppose that we have a parameter of interest, let us call it θ. This could be any parameter of interest: A mean, a correlation, a regression coefficient, a rate, etc.

6.2.1 Bayes’ theorem

The basic structure of how Bayesian inference is done is always the same and can be expressed via Bayes’ theorem. Recall that in Bayesian inference we are interested in the conditional probability of the parameter of interest θ given the observed data:

(6.1)

We notice that the joint probability p(data,θ) can be expressed as p(data,θ) = p(data|θ)p(θ). When inserting this in Equation 6.1, we obtain Bayes theorem:

(6.2)

The components of this equation are named as follows:

(6.3)

To understand this equation, it is important to realize that the entities that are called posterior, likelihood, and prior are not single values, but distributions. So the inference about the parameter of interest yields not just a single value, but a distribution of values, the so-called posterior distribution.

The equation can be paraphrased as follows: We have prior knowledge about our parameter of interest which is expressed by a distribution, the so-called prior distribution. Now some data are collected which probably will change our knowledge about the parameter of interest: The prior knowledge about the parameter of interest is updated by the likelihood which corresponds to the probability of the data given the different possible values of the parameter of interest. So the prior and the likelihood are distributions. The denominator of the fraction, the marginal likelihood corresponds to the probability of the observed data. This is not a distribution, but a value, often called normalizing constant, which ensures that the area under the posterior distribution will be equal to one. What we obtain is the posterior, which again is a distribution over the values of the parameter of interest. In most cases, this distribution will look different from the prior distribution which means that by incorporating data, we have learned something about the parameter of interest.

6.2.2 Example: The rate of female PhD students in the Netherlands

To facilitate understanding, we will again use an example. In the following example, the parameter of interest, let us call it θ again, is a rate of a binomial process. Suppose that you are interested in determining the rate of female PhD students in the Netherlands. Further assume that you have no prior knowledge about what the rate of female PhD students is. This can be translated into a prior distribution over θ, the parameter of interest, which in this case is the rate. The rate parameter can take values between zero and one and they are a priori all equally likely since we do not have prior knowledge about it. Hence, our prior distribution which corresponds to our knowledge about the parameter of interest will be a uniform distribution over the interval zero to one (see Figure 6.1, left panel, dashed line). The likelihood function in this case is given by the binomial:

(6.4)

Note that this likelihood function is not a discrete probability mass function, since n and k are fixed and θ is variable. To obtain the posterior distribution, we have to use a trick: We express the uniform prior for θ as a Beta probability density function. This is a very flexible function which is defined in the interval between zero and one which is perfectly suited for our parameter of interest, the rate of female PhD students θ. The shape of the Beta probability density function Beta(θ|α,β) is characterized by the two parameters α and β. If we set both of them equal to one, we obtain the uniform distribution prior in the interval zero to one (see Figure 6.1, left panel, dashed line).

Note that it is extremely important when specifying a prior distribution that all possible values of the parameter of interest get some density (it can be small, but not zero), because otherwise the posterior density for this value will always be also zero.

This would mean we cannot learn anything from the data about this possible value of the parameter of interest; this is known as Cromwell’s rule (Lindley, 1985).

Figure 6.1: Learning about the rate of female PhD students in the Netherlands in three stages: The posterior distribution after the first stage of the data collection (left panel) is used as prior distribution for the second set of data (middle panel), the posterior distribution after the second stage of the data collection (middle panel) is used as prior distribution for the third set of data (right panel).
Legend: Dashed lines correspond to prior distributions, solid lines to posterior distributions.

6.2.3 Why do we want to use a Beta probability density function?

This is because the Beta probability function is, with respect to the likelihood function, a so-called conjugate prior (Raiffa & Schlaifer, 1961). This means that no matter what data we collect, we know beforehand, that the posterior distribution will also be a Beta probability density function (with different α and β). This makes it easy to evaluate the expression in Equation 6.2. The posterior distribution in the example with the female PhD students will have the following Beta probability density function: Beta(θ|α + k,β + n - k), where n is the number of observations (the total number of PhD students we look at) and k corresponds to the number of female PhD students in our sample (if you are interested in why this holds, see for example Wagenmakers (2007, Appendix A)). α and β are the parameters of our prior distribution; so in the case of our uniform prior they are both equal to one. It is highly recommended, when possible, to use a conjugate prior because this enables to evaluate the expression in Equation 6.2 analytically. If this is impossible, obtaining the posterior distribution is more difficult and instead of analytical solutions, resampling methods have to be used. Markov Chain Monte Carlo methods (MCMC) are usually mathematically and computationally elaborate. A good introduction into MCMC is given in Jackman (2009); if the reader wants to deeply get into the topic, the book by Brooks, Gelman, Jones, and Meng (2011) may be interesting. Luckily, nowadays software like WinBUGS (Lunn, Thomas, Best, & Spiegelhalter, 2000) exists that makes it relatively easy to implement these methods. For many examples of how to use this program, the reader should consult Lee and Wagenmakers (2013), who provide a lot of example code for cognitive modeling applications. For the reader that is mostly concerned with doing “classical” analyses like t tests, ANOVA, etc. in a Bayesian way, and knows how to use R, the R package BayesFactor by Richard D. Morey and Jeff N. Rouder may be very interesting.

Let’s resume our example: We have now expressed our prior knowledge about the rate of female PhD students in the Netherlands as a Beta probability density distribution Beta(θ|1,1) which means that we have no prior knowledge about the rate. This prior also ensures that every possible value of θ has non-zero density which is important to be able to accurately learn something about θ (Lindley, 1985, Cromwell’s rule). Furthermore, we know how we can obtain the posterior distribution:

This means we have everything in hand to start collecting data and learn about the rate of female PhD students in the Netherlands. Now assume that we observe three PhD students, one of them being female and two being male. This means that our posterior distribution for the rate parameter θ will be Beta(θ|1 + 1,1 + 3 - 1)= Beta(θ|2,3). This posterior distribution (solid line) as well as the Beta(θ|1,1) prior distribution (dashed line) are displayed in the left panel of Figure 6.1.

If you look at the posterior distribution you will notice that through gathering data, even though only three PhD students, we have learned something about the rate parameter: Now there are regions with higher density. Suppose that we want to collect more data, to learn more about the rate of female PhD students and we obtain a sample of twelve different PhD students, five being female and seven male. What should our prior distribution on θ now be?

We could again choose the so-called uninformative prior (which in this case is a uniform distribution over the interval zero to one), but why? We already “know” something about the rate parameter because we already observed three PhD students. The beauty of Bayesian inference is that we can easily include this prior information in our analysis: The posterior distribution Beta(θ|2,3) now simply becomes our new prior distribution. A famous quote that illustrates this is: “Today’s posterior is tomorrow’s prior” (Lindley, 1972, p.2). Using this prior distribution, the new posterior distribution is Beta(θ|2 + 5,3 + 12 - 5)= Beta(θ|7,10). This is illustrated in the middle panel of Figure 6.1 (again, the dashed line corresponds to the prior distribution and the solid line to the posterior distribution).

It becomes obvious, that, again, we have learned something new about the rate of female PhD students in the Netherlands. This is also expressed through the higher precision (smaller variance) of the posterior distribution compared to the prior distribution. Of all possible values for θ there are clearly some that are more credible than others, expressed by a higher density of the posterior distribution.

To further assess the rate of female PhD students in the Netherlands we look at another 22 PhD students, eleven of them being female and eleven male. We incorporate our prior knowledge about the rate by choosing Beta(θ|7,10) as the prior distribution for θ (dashed line in Figure 6.1, right panel). We obtain the posterior distribution Beta(θ|7 + 11,10 + 22 - 11)= Beta(θ|18,21) which is displayed as solid line in the right panel of Figure 6.1. If we now want to report the results of our analysis, besides plotting the posterior distribution, we can report the mode, median or mean of the posterior distribution and its variance and/or a 95% credible interval. For our example, the mode, median, and mean of our posterior distribution Beta(θ|18,21) are all approximately 0.46, the variance is approximately 0.006. The 95% credible interval ranges from 0.31 to 0.62. When analyzing the data with the frequentist approach, 0.49 is obtained as a maximum likelihood estimate with a 95% confidence interval ranging from 0.33 to 0.65. Note that the frequentist confidence interval is slightly wider than the Bayesian 95% credible interval, so we obtain easier to interpret and even slightly more “precise” information when applying the Bayesian approach.¹
____________________________

¹. The comparison of confidence and credible intervals should usually be avoided since, as pointed out, they have to be interpreted differently.

6.2.4 Critique of Bayesian inference

At this point, the common critique of subjectivity of Bayesian inference due to choosing a prior distribution should be addressed. Critics of Bayesian inference often mention that it is subjective and say that if several researchers analyze the same data, they can all come to a different result due to different priors. It is true that choosing a different prior can lead to different results, but not always; when the data “overwhelm” the prior which means that the data clearly point in one direction, it hardly matters what the prior distribution was — the posterior will always look similar. Furthermore, it can actually be an advantage to be able to manipulate the prior: As we have seen for our female PhD rate example, choosing different priors enables to elegantly incorporate prior information in our analysis, and there is little reason to not use already existing knowledge for the analysis. Lastly, a less subjective way exists if someone refuses to use prior information which is implemented by choosing so-called uninformative priors which we have applied at the first step of our analysis: When no prior knowledge existed about the parameter of interest, we used a uniform prior for θ.²
____________________________

². To be fair, sometimes, it is not that easy to define an uninformative prior.

In our case, when we are interested in learning about the rate of female PhD students, we notice that when using an uninformative prior (at the first step of our analysis), the mode of the posterior distribution is equal to the frequentist maximum likelihood estimate of θ which is given by k n = 1 3.

Two important points to keep in mind when incorporating prior information in Bayesian analysis should be emphasized:

1. If possible, choose a prior that is conjugate with respect to the likelihood function, this enables an analytical solution without having to rely on MCMC methods.
2. When specifying the prior distribution make sure that no possible value of the parameter of interest gets zero density, otherwise the posterior density will always be also zero (“Cromwell’s rule”).

6.2.5 The Bayes Factor

So far, in our example, we were interested in learning about the rate of female PhD students in the Netherlands. Let us now assume that we want to test the hypothesis, whether the proportion of female and male PhD students in the Netherlands is equal, i.e., we want to test whether θ = 0.5. We can test this hypothesis using a quantity called Bayes factor. The Bayes factor is used to compare two hypotheses (models) and yields a number that expresses how much the data support one hypothesis (model) over the other. For our example, the null hypothesis is that θ = 0.5; the alternative hypothesis is that θ can be anything else. Since a single value of a continuous variable, in this case θ = 0.5, has probability zero, we can express the alternative hypothesis as θ = Beta(1,1), which is a uniform distribution over the interval zero to one. The formula for the Bayes factor is given by:

(6.5)

The easiest way to think about the Bayes factor is in terms of odds: Before conducting data, we have the prior odds p(M₀) (i.e., the null hypothesis) over p(M₁) (i.e., the alternative hypothesis). Often this will be equal to one, which means that both models (hypotheses) are equally likely before seeing the data. By learning from the data about the parameter of interest, we obtain the posterior odds p(M₀|data) (i.e., the null hypothesis) over p(M₁|data) (i.e., the alternative hypothesis). The Bayes factor expresses how much our prior beliefs about the two models (hypotheses) have changed through looking at the data or, to state it differently, how much the odds of one model (hypothesis) over the other have changed. It can be interpreted as how much more likely the data are given one model (hypothesis), so for example a Bayes factor BF₀₁ = 3 would indicate that the data are three times more likely given M₀ (i.e., the null hypothesis). Note that whenever the Bayes factor is larger than one, the data are in favor of the model (hypothesis) that is represented in the numerator of the fraction in Equation 6.5. If the Bayes factor is smaller than one, the data are in favor of the model (hypothesis) that is represented in the denominator of Equation 6.5.

To make a statement about how much more likely the data are given the model in the denominator, one has take the inverse of the Bayes factor, e.g., BF₀₁ = means that the data are three times more likely given M₁ (i.e., the alternative hypothesis). To test our hypothesis that the proportion of female and male PhD students in the Netherlands is equal (after having conducted three samples), we first notice that p(data|M₀) (numerator of Equation 6.5) is given by

The denominator of Equation 6.5, p(data|M₁) is given by

In the case of the uniformative Beta(1,1) prior, it can be shown that this integral is equal to 1∕(n + 1). Thus, we can compute the Bayes factor via:

This result indicates that the data are 4.89 times more likely given that θ = 0.5 which means through gathering data, we found evidence in favor of the null hypothesis. Note that this would not have been possible when using the frequentist approach: In frequentist statistics one can only fail to reject the null hypothesis, but cannot gather evidence in favor of it.

After having introduced the reader to the basic concepts of Bayesian inference, the next section will present some applications.

6.3 An application to illustrate how prior information can be used in Bayesian inference

In the previous part, we described the basics of the Bayesian analysis procedure. We emphasized the use of prior information in general. We showed that, when new information becomes available, the prior distribution can be updated into a posterior distribution.

To illustrate the use of prior information in the Bayesian process in research we will, in this section, show an application of the Bayesian procedure in Psychological research. In recent years, the use of Bayesian statistics became more important in the analysis of scientific data. The use of Bayesian statistics seems especially important for Psychological research, where there are often few data points, but relatively large quantities of prior knowledge (Kruschke, 2011a; Wagenmakers, 2007). In these situations, it is possible to use knowledge of previous studies to specify a prior. With this prior we can then estimate a more reliable posterior distribution. As a result of this, we can sometimes draw more reliable conclusions.

6.3.1 Illustrating the Bayesian procedure

To illustrate the Bayesian procedure, we will discuss a study of Plöderl et al. (2013). Using a Bayesian approach, researchers reanalyzed data from (frequentist) studies about suicide attempts and suicide deaths among sexual minorities. Their motivation to reanalyse the studies was the fact that these frequentist studies showed inconclusive results. Several studies about suicide attempts reported that, compared with heterosexuals, sexual minorities have a higher probability of attempting suicide. However, autopsy studies reported no (significant) difference in suicide deaths between sexual minorities and heterosexuals (Shaffer, Fisher, Hicks, Parides, & Gould, 1995; Renaud, Berlim, Begolli, McGirr, & Turecki, 2010). So, while studies reported higher rates of suicide attempts of sexual minorities, autopsy studies reported no difference in number of suicide deaths between sexual minorities and heterosexuals. Thus, there seems to be a clear contrast between proportions of sexual minorities in suicide attempt studies and suicide death studies. This contrast is surprising, given the fact that a suicide attempt gives a high risk of conducting suicide in the future (Owens, Wood, Greenwood, Hughes, & Dennis, 2005). Given this information, one would expect that suicide death would also be more common among sexual minorities than among heterosexuals.

Reanalysing the data of suicide deaths using Bayesian methods, the researchers showed that sexual minorities do indeed have an increased risk of suicide death. This means that, given the proportion of sexual minorities and heterosexuals in the living population, the proportion of sexual minorities in the suicide population is larger than proportion of heterosexuals in the suicide population. By combining the data from two studies of Shaffer et al. (1995) and Renaud et al. (2010), and installing priors distributions based on previous knowledge, researchers computed a posterior distribution of the difference in proportion between the two groups. From this posterior distribution, researchers extracted a credible interval. Furthermore, researchers computed a Bayes Factor for the strength of the alternative hypothesis that sexual minorities are more common among suicide deaths.

Note that the study of Shaffer et al. had a sample size of 55 and the study of Renaud et al. had a sample size of 147. So combining these studies led to a total sample size of 202. The proportional difference of sexual minorities between the living group and the suicide group is investigated.

Because Bayesian statistics work with probability distributions, this difference in proportion between the two groups is translated into a distribution. So the prior and posterior distributions, are in this study a probability distribution of the differences in proportions between the living and suicide group. The first step for the researchers was specifying a prior distribution. This prior distribution stands for the proportional difference in sexual minorities and heterosexuals between the living group and suicide group.

Because there was some debate in the literature about the likely proportions of sexual minorities in the total population, researchers used two prior distributions. One more likely, truncated prior distribution, and one uninformative prior distribution which was highly unlikely, but had no bias of any differences beforehand. Researchers did this, because they wanted to know the possible difference in final conclusions, when having some assumptions beforehand versus having no assumptions. The uninformative prior distribution assumed that there were no differences between the two populations, 50% sexual minority members in the living population and 50% sexual minority members in the suicide death population. This uninformative prior distribution is of course highly unlikely, but rules out critique for subjective bias of priors that could influence the final posterior distribution. The truncated prior distribution assumed that at most 15% of the living population consisted of a sexual minority, and that at most 50% of the suicide death population consisted of a sexual minority. This truncated prior restricts the values of the distributions to maximum proportions of sexual minorities in the two populations. The researchers based these values on earlier studies about proportions of sexual minorities. Because of this, the truncated prior distribution can be seen as a more plausible proportion.

6.3.2 Estimating a credible interval

Once the prior distribution has been specified, one can start estimating a posterior distribution and credible interval, by adding data from the two combined studies. After analysing the data of the two studies combined, the final posterior distribution had a 95% credible interval between 0.01 and 0.08 (Δ = 0.01 - 0.08;M = 0.04); see Figure 6.2. This means that, given the data, the difference in proportion between the two groups (proportion sexual minorities in living group and proportion sexual minorities in suicide group) lies between 1% and 8% with a probability of 95%. So, given the data, one can be 95% sure that, among individuals who died from suicide, sexual minorities are overrepresented with a proportional difference between 1% and 8%.

Figure 6.2: Combining data of two studies.
The posterior distributions of the two studies from Shaffer et al. (1995) and Renaud et al. (2010) are depicted with the grey lines.
The posterior distribution of the combined data of two studies is depicted with the grey shaded area, with an 95% credible interval above. The credible interval shows that it does not include a proportional difference of zero (Plöderl et al., 2013).
Legend: The x-axis stands for the proportional difference between the living and suicide groups; The y-axis stands for the density of the posterior distributions.

Thus, these Bayesian credible interval give us a quantification of uncertainty of the estimated parameters. The previously found (frequentist) confidence intervals did not found any differences in proportions between the two groups; (95%CI = 0.45 - 171.82) in Shaffer et al., and (95%CI = 0.51 - 184.63) in Renaud et al..

These frequentist confidence intervals are also differently interpreted. In a confidence interval, one cannot state that the difference between the two groups falls within an exact range with a certain probability. And there is no exact quantification of the uncertainty about the estimated parameters. In these frequentist confidence intervals, one would say that, given an infinite number of observations, 95% of such (future possible) intervals would contain the true value. In this way of reasoning, the confidence interval is interpreted in contrast with other possible confidence intervals that could exist in future events. Here, the interval is not interpreted in isolation, but in comparison with other (possible) confidence intervals. Because of this, frequentist confidence intervals are more difficult to understand than Bayesian credible interval.

6.3.3 Estimating the Bayes factor

Besides the credible interval, the researchers computed the Bayes Factor for testing the alternative hypothesis (sexual minorities members are proportionally more common in the suicide group than in the living group), given the null hypothesis (no differences in proportions between the living and suicide groups). This Bayes factor quantifies the strength of a hypothesis. Given the fact that the researchers used two prior distributions (uninformative and truncated), the analysis gave two Bayes factors.

First, the more likely truncated prior distribution, had a Bayes factor that gave substantial evidence in favour of the alternative hypothesis. Second, even the unlikely uninformative prior, had a Bayes factor that gave anecdotal evidence in favour of the alternative hypothesis. So, even setting a prior that was highly unlikely and opposite to the alternative hypothesis, gave some evidence in favour of the alternative hypothesis. In contrast with the frequentist hypothesis testing that was used in Shaffer et al. and Renaud et al.. A one-side Fischer’s Exact T-test of the data of the two studies reported a non-significance difference between the two groups of proportion in suicide death (p=.06 in Shaffer et al. (1995) and p=.09 in Renaud et al.. The authors thought this was due to low power of the two studies, because of small sample size.

The example demonstrates how you can make use of prior knowledge in the Bayesian procedure to update the posterior distribution with new or combined data. In contrast with the frequentist confidence interval, the Bayesian credible interval is more easily interpreted. These Bayesian credible interval give us a quantification of the uncertainty of estimated parameters. In Bayesian analysis the conclusion is based on the data only, without reasoning via an probable future event if the sample is taken an infinite number of times.

This leads to a more realistic interpretation of results, and therefor to sounder and more reliable conclusions.

References

   Aragon, T. J., Fray, M. P., & Wollschlaeger, D.  (2014). epiTools: R-package for Epidemiologic data and graphics [R-package version 0.5-7].

   Brooks, S., Gelman, A., Jones, G., & Meng, X.-L.  (2011). Handbook of Markov Chain Monte Carlo. Boca Raton, FL: CRC Press.

   Carlin, B. P., & Louis, T. A.  (2000). Bayes and empirical Bayes methods for data analysis (2nd ed.). London, UK: Chapman & Hall.

   Epskamp, S., Cramer, A. O. J., Waldorp, L. J., Schmittmann, V. D., & Borsboom, D.  (2012). qgraph: Network Visualizations of Relationships in Psychometric Data. Journal of Statistical Software, 48, 1–18.

   Ising, E.  (1925). Beitrag zur theorie des ferromagnetismus. Zeitschrift fur Physik B-Condensed Matter, 31, 253–258.

   Jackman, S.  (2009). Bayesian analysis for the social sciences. New York, NY: Wiley.

   Jeffreys, H.  (1961). Theory of probability (3rd ed.). Oxford, UK: Oxford University Press.

   Kruschke, J. K.  (2011a). Bayesian assessment of null values via parameter estimation and model comparison. Perspectives on Psychological Science, 6, 299–312.

   Kruschke, J. K.  (2011b). Doing Bayesian data analysis: A tutorial with R and BUGS. Burlington, MA: Academic Press.

   Lee, M. D., & Wagenmakers, E.-J.  (2013). Bayesian cognitive modeling: A practical course. Cambridge, UK: Cambridge University Press.

   Lindley, D. V.  (1972). Bayesian statistics, a review. Philadelphia, PA: SIAM.

   Lindley, D. V.  (1985). Making decisions. New York, NY: Wiley.

   Lunn, D. J., Thomas, A., Best, N., & Spiegelhalter, D.  (2000). WinBUGS – a Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing, 10, 325–337.

   McCullagh, P.  (1980). Regression models for ordinal data. Journal of the Royal Statistical Society, Series B, 42, 109–142.

   Owens, D., Wood, C., Greenwood, D. C., Hughes, T., & Dennis, M. (2005). Mortality and suicide after non-fatal self-poisoning: 16-year outcome study. The British Journal of Psychiatry, 187, 470–475.

   Plöderl, M., Wagenmakers, E.-J., Tremblay, P., Ramsay, R., Kralovec, K., Fartacek, C., et al.  (2013). Suicide risk and sexual orientation: A critical review. Archives of Sexual Behavior, 42, 715–727.

   Raiffa, H., & Schlaifer, R.  (1961). Applied statistical decision theory. Cambridge, MA: Harvard University Press.

   Renaud, J., Berlim, M. T., Begolli, M., McGirr, A., & Turecki, G.  (2010). Sexual orientation and gender identity in youth suicide victims: An exploratory study. Canadian Journal of Psychiatry. Revue Canadienne de Psychiatrie, 55, 29–34.

   Revelle, W.  (2014). psych: Procedures for Psychological, Psychometric, and Personality Research [R-package version 1.4.8.11]. Evanston.

   Rosseel, Y.  (2012). lavaan: An R package for Structural Equation Modeling. Journal of Statistical Software, 48, 1–36.

   Shaffer, D., Fisher, P., Hicks, R. H., Parides, M., & Gould, M.  (1995). Sexual orientation in adolescents who commit suicide. Suicide and Life-Threatening Behavior, 25, 64–71.

   van Borkulo, C. D., Borsboom, D., Epskamp, S., Blanken, T. F., Boschloo, L., Schoevers, R. A., et al.  (2014). A new method for constructing networks from binary data. Scientific Reports, 4.

   Wagenmakers, E.-J. (2007). A practical solution to the pervasive problems of p values. Psychonomic Bulletin & Review, 14, 779–804.

   Zhao, T., Liu, H., Roeder, K., Lafferty, J., & Wasserman, L.  (2014). huge: High-dimensional Undirected Graph Estimation [R-package version 1.2.6].