A short taxonomy of Bayes factors

[Update Oct 2014: Due to some changes to the Bayes factor calculator webpage, and as I understand BFs much better now, this post has been updated …]

I started to familiarize myself with Bayesian statistics. In this post I’ll show some insights I had about Bayes factors (BF).

What are Bayes factors?

Bayes factors provide a numerical value that quantifies how well a hypothesis predicts the empirical data relative to a competing hypothesis. For example, if the BF is 4, this indicates: “This empirical data is 4 times more probable if H₁ were true than if H₀ were true.”
. Hence, evidence points towards H₁. A BF of 1 means that data are equally likely to be occured under both hypotheses.

More formally, the BF can be written as:

BF_{10} = \frac{p(D|H_1)}{p(D|H_0)}

where D is the data. Hence, the BF is a ratio of probabilities, and is related to larger class of likelihood-ratio test.

What researchers usually are interested in is not p(Data | Hypothesis), but rather p(Hypothesis | Data). Using Bayes’ theorem, the former can be transformed into the latter by assuming prior probabilities for the hypotheses. The BF then tells one how to update one’s prior probabilities after having seen the data, using this formula (Berger, 2006):

 p(H_1 | D) = \frac{BF}{BF + [(1-p(H1)]/p(H1)}

Given a BF of 1, one does not have to update his or her priors. If one holds, for example, equal priors (p(H1) = p(H0) = .5), these probabilities do not change after having seen the data of the original study.

The best detailed introduction of BFs I know of can be be found in Richard Morey’s blog posts [1] [2][3]. Also helpful is the ever-growing tutorial page for the BayesFactor package. (For other introductions to BFs, see Wikipedia, Bayesian-Inference, the classic paper by Kass and Raftery, 1995, or Berger, 2006).

Although many authors agree about the many theoretical advantages of BFs, until recently it was complicated and unclear how to compute a BF even for the simple standard designs (Rouder, Morey, Speckman, & Province, 2012). Fortunately, over the last years default Bayes factors for several standard designs have been developed (Rouder et al., 2012; Rouder, Speckman, Sun, Morey, & Iverson, 2009; Morey & Rouder, 2011). For example, for a two-sample t test, a BF can be derived simply by plugging the t value and the sample sizes into a formula. The BF is easy to compute by the R package BayesFactor (Morey & Rouder, 2013), or by online calculators [1][2].

Flavors of Bayes factors

When I started to familiarize myself with BFs, I was sometimes confused, as the same number seemed to mean different things in different publications. And indeed, four types of Bayes factors can be distinguished. “Under the hood”, all four types are identical, but you have to be aware which type has been employed in the specific case.

The first distinction is, whether the BF indicates “H_0 over H_1” (=BF_{01}), or “H_1 over H_0” (=BF_{10}). A BF_{01} of 2 means “Data is 2 times more likely to be occured under H_0 than under H_1“, while the same situation would be a BF_{10} of 0.5 (i.e., the reciprocal value 1/2). Intuitively, I prefer larger values to be “better”, and as I usually would like to have evidence for H_1 instead of H_0, I usually prefer the BF_{10}. However, if your goal is to show evidence for the H0, then BF_{01} is easier to communicate (compare: “Data occured 0.1 more likely under the alternative” vs. “Data show 10 times more evidence for the null than for the alternative”).

The second distinction is, whether one reports the raw BF, or the natural logarithm of the BF (The log(BF) has also been called “weight of evidence“; Good, 1985). The logarithm has the advantage that the scale above 1 (evidence for H_1) is identical to the scale below 1 (evidence for H_0). In the previous example, a BF_{10} of 2 is equivalent to a BF_{01} of 0.5. Taking the log of both values leads to log(BF_{10}) = 0.69 and log(BF_{01}) = -0.69: Same value, reversed sign. This makes the log(BF) ideal for visualization, as the scale is linear in both directions. Following graphic shows the relationship between raw/log BF:

Bayesfactor_Overview1

Figure 1

 

As you can see in the Table of Figure 1, different authors use different flavors. This often makes sense, as we sometimes want to communicate evidence for the H1, and sometimes for the H0. However, for the uninitiated it can be sometimes confusing.

Usually, tables in publication report the raw BF (raw- or raw+). Plots, in contrast, typically use the log scale, for example:

Bildschirmfoto 2014-10-15 um 10.48.25

 

Figure 2 shows conversion paths of the different BF flavors:

Bayesfactor_Overview2

 

 

The user interface functions of the BayesFactor package always print the raw BF_{10}. Internally, however, the BF is stored as log(BF_{10}).

Hence, you have to be careful when you directly use the backend utility functions, such as ttest.tstat. These functions return the log(BF_{10}). As the conversion table shows, you have to exp() that number to get the raw BF. Check the documentation of the functions if you are unsure which flavor is returned.

Related posts: Exploring the robustness of Bayes Factors: A convenient plotting function

References

Berger, J. O. (2006). Bayes factors. In S. Kotz, N. Balakrishnan, C. Read, B. Vidakovic, & N. L. Johnson (Eds.), Encyclopedia of Statistical Sciences, vol. 1 (2nd ed.) (pp. 378–386). Hoboken, NJ: Wiley.
Good, I. J. (1985). Weight of evidence: A brief survey. In J. M. Bernardo, M. H. DeGroot, D. V. Lindley, & A. F. M. Smith (Eds.), Bayesian Statistics 2 (pp. 249–270). Elsevier.

Morey, R. D. & Rouder, J. N. (2011). Bayes factor approaches for testing interval null hypotheses. Psychological Methods, 16(4), 406–419. PMID: 21787084. doi:10.1037/a0024377

Morey, R. D. & Rouder, J. N. (2013). {BAYESFACTOR}: computation of bayes factors for common designs. R package version 0.9.4. Retrieved from http://CRAN.R- project.org/package=BayesFactor

Rouder, J. N., Morey, R. D., Speckman, P. L., & Province, J. M. (2012). Default bayes factors for ANOVA designs. Journal of Mathematical Psychology, 56(5), 356–374. doi:10.1016/j.jmp.2012.08.001

Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin & Review, 16(2), 225–237.

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Exploring the robustness of Bayes Factors: A convenient plotting function

One critique frequently heard about Bayesian statistics is the subjectivity of the assumed prior distribution. If one is cherry-picking a prior, of course the posterior can be tweaked, especially when only few data points are at hand. For example, see the Scholarpedia article on Bayesian statistics:

In the uncommon situation that the data are extensive and of simple structure, the prior assumptions will be unimportant and the assumed sampling model will be uncontroversial. More generally we would like to report that any conclusions are robust to reasonable changes in both prior and assumed model: this has been termed inference robustness

(David Spiegelhalter and Kenneth Rice (2009) Bayesian statistics. Scholarpedia, 4(8):5230.)

Therefore, it is suggested that …

In particular, audiences should ideally fully understand the contribution of the prior distribution to the conclusions. (ibid)

In the example of Bayes factors for t tests (Rouder, Speckman, Sun, Morey, & Iverson, 2009), the assumption that has to be defined a priori is the effect size δ expected under the H1. In the BayesFactor package for R, this can be adjusted via the r parameter. By default, it is set to 0.5, but it can be made wider (larger r’s, which means one expects larger effects) or narrower (r’s close to zero, which means one expects smaller effects in the population).

In their reanalysis of Bem’s ESP data, Wagenmakers, Wetzels, Borsboom, Kievit, and van der Maas (2011, PDF) proposed a robustness analysis for Bayes factors (BF), which simply shows the BF for a range of priors. If the conclusion is the same for a large range of priors, it could be judged to be robust (this is also called a “sensitivity analysis”).

I wrote an R function that can generate plots like this. Here’s a reproduction of Wagenmakers’ et al (2011) analysis of Bem’s data – it looks pretty identical:

## Bem data, two sided
# provide t values, sample sizes, and the location(s) of the red dot(s)
# set forH1 to FALSE in order to vertically flip the plot.
# Usually I prefer higher BF to be in favor of H1, but I flipped it in order to match Wagenmakers et al (2011)
BFrobustplot(
     ts=c(2.51, 2.55, 2.23, 1.74, 1.92, 2.39, 2.03, 1.8, 1.31, 2.96),
     ns=c(100, 97, 100, 150, 100, 150, 99, 150, 200, 50),
     dots=1, forH1 = FALSE)

 

Bayes factor robustness analysis, two-sided (cf. Wagenmakers et al., 2011)

Bayes factor robustness analysis, two-sided (cf. Wagenmakers et al., 2011)

 

You can throw in as many t values and corresponding sample sizes as you want. Furthermore, the function can compute one-sided Bayes factors as described in Wagenmakers and Morey (2013). If this approach is applied to the Bem data, the plot looks as following – everything is shifted a bit into the H1 direction:

# Bem data one-sided
BFrobustplot(
     ts=c(2.51, 2.55, 2.23, 1.74, 1.92, 2.39, 2.03, 1.8, 1.31, 2.96),
     ns=c(100, 97, 100, 150, 100, 150, 99, 150, 200, 50),
     dots=1, sides="one", forH1 = FALSE)

 

Bayes factor robustness analysis, one-sided (cf. Wagenmakers et al., 2011; Wagenmakers and Morey, 2013)

Bayes factor robustness analysis, one-sided (cf. Wagenmakers et al., 2011; Wagenmakers and Morey, 2013)

Finally, here’s the function:

## This source code is licensed under the FreeBSD license
## (c) 2013 Felix Schönbrodt

#' @title Plots a comparison of a sequence of priors for t test Bayes factors
#'
#' @details
#'
#'
#' @param ts A vector of t values
#' @param ns A vector of corresponding sample sizes
#' @param rs The sequence of rs that should be tested. r should run up to 2 (higher values are implausible; E.-J. Wagenmakers, personal communication, Aug 22, 2013)
#' @param labels Names for the studies (displayed in the facet headings)
#' @param dots Values of r's which should be marked with a red dot
#' @param plot If TRUE, a ggplot is returned. If false, a data frame with the computed Bayes factors is returned
#' @param sides If set to "two" (default), a two-sided Bayes factor is computed. If set to "one", a one-sided Bayes factor is computed. In this case, it is assumed that positive t values correspond to results in the predicted direction and negative t values to results in the unpredicted direction. For details, see Wagenmakers, E. J., & Morey, R. D. (2013). Simple relation between one-sided and two-sided Bayesian point-null hypothesis tests.
#' @param nrow Number of rows of the faceted plot.
#' @param forH1 Defines the direction of the BF. If forH1 is TRUE, BF > 1 speak in favor of H1 (i.e., the quotient is defined as H1/H0). If forH1 is FALSE, it's the reverse direction.
#'
#' @references
#'
#' Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t-tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin and Review, 16, 225-237.
#' Wagenmakers, E.-J., & Morey, R. D. (2013). Simple relation between one-sided and two-sided Bayesian point-null hypothesis tests. Manuscript submitted for publication
#' Wagenmakers, E.-J., Wetzels, R., Borsboom, D., Kievit, R. & van der Maas, H. L. J. (2011). Yes, psychologists must change the way they analyze their data: Clarifications for Bem, Utts, & Johnson (2011)

BFrobustplot <- function(
    ts, ns, rs=seq(0, 2, length.out=200), dots=1, plot=TRUE,
    labels=c(), sides="two", nrow=2, xticks=3, forH1=TRUE)
{
    library(BayesFactor)
   
    # compute one-sided p-values from ts and ns
    ps <- pt(ts, df=ns-1, lower.tail = FALSE)   # one-sided test
   
    # add the dots location to the sequences of r's
    rs <- c(rs, dots)
       
    res <- data.frame()
    for (r in rs) {
       
        # first: calculate two-sided BF
        B_e0 <- c()
        for (i in 1:length(ts))
            B_e0 <- c(B_e0, exp(ttest.tstat(t = ts[i], n1 = ns[i], rscale=r)$bf))
   
        # second: calculate one-sided BF
        B_r0 <- c()
        for (i in 1:length(ts)) {
            if (ts[i] > 0) {
                # correct direction
                B_r0 <- c(B_r0, (2 - 2*ps[i])*B_e0[i])
            } else {
                # wrong direction
                B_r0 <- c(B_r0, (1 - ps[i])*2*B_e0[i])
            }
        }
   
        res0 <- data.frame(t=ts, n=ns, BF_two=B_e0, BF_one=B_r0, r=r)
        if (length(labels) > 0) {
            res0$labels <- labels
            res0$heading <- factor(1:length(labels), labels=paste0(labels, "\n(t = ", ts, ", df = ", ns-1, ")"), ordered=TRUE)
        } else {
            res0$heading <- factor(1:length(ts), labels=paste0("t = ", ts, ", df = ", ns-1), ordered=TRUE)
        }
        res <- rbind(res, res0)
    }
   
    # define the measure to be plotted: one- or two-sided?
    res$BF <- res[, paste0("BF_", sides)]

   
    # Flip BF if requested
    if (forH1 == FALSE) {
        res$BF <- 1/res$BF
    }
       
   
    if (plot==TRUE) {
        library(ggplot2)
        p1 <- ggplot(res, aes(x=r, y=log(BF))) + geom_line() + facet_wrap(~heading, nrow=nrow) + theme_bw() + ylab("log(BF)")
        p1 <- p1 + geom_hline(yintercept=c(c(-log(c(30, 10, 3)), log(c(3, 10, 30)))), linetype="dotted", color="darkgrey")
        p1 <- p1 + geom_hline(yintercept=log(1), linetype="dashed", color="darkgreen")
   
        # add the dots
        p1 <- p1 + geom_point(data=res[res$r %in% dots,], aes(x=r, y=log(BF)), color="red", size=2)
   
        # add annotation
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=-2.85, label=paste0("Strong~H[", ifelse(forH1==TRUE,0,1), "]"), hjust=1, vjust=.5, size=3, color="black", parse=TRUE)
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=-1.7 , label=paste0("Moderate~H[", ifelse(forH1==TRUE,0,1), "]"), hjust=1, vjust=.5, size=3, color="black", parse=TRUE)
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=-.55 , label=paste0("Anectodal~H[", ifelse(forH1==TRUE,0,1), "]"), hjust=1, vjust=.5, size=3, color="black", parse=TRUE)
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=2.86 , label=paste0("Strong~H[", ifelse(forH1==TRUE,1,0), "]"), hjust=1, vjust=.5, size=3, color="black", parse=TRUE)
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=1.7  , label=paste0("Moderate~H[", ifelse(forH1==TRUE,1,0), "]"), hjust=1, vjust=.5, size=3, color="black", parse=TRUE)
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=.55  , label=paste0("Anectodal~H[", ifelse(forH1==TRUE,1,0), "]"), hjust=1, vjust=.5, vjust=.5, size=3, color="black", parse=TRUE)
       
        # set scale ticks
        p1 <- p1 + scale_y_continuous(breaks=c(c(-log(c(30, 10, 3)), 0, log(c(3, 10, 30)))), labels=c("-log(30)", "-log(10)", "-log(3)", "log(1)", "log(3)", "log(10)", "log(30)"))
        p1 <- p1 + scale_x_continuous(breaks=seq(min(rs), max(rs), length.out=xticks))

        return(p1)
    } else {
        return(res)
    }
}

 


References

Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t-tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin and Review, 16, 225-237. [for a PDF, see bottom of this page]
Wagenmakers, E.-J., & Morey, R. D. (2013). Simple relation between one-sided and two-sided Bayesian point-null hypothesis tests. Manuscript submitted for publication (website)
Wagenmakers, E.-J., Wetzels, R., Borsboom, D., Kievit, R. & van der Maas, H. L. J. (2011). Yes, psychologists must change the way they analyze their data: Clarifications for Bem, Utts, & Johnson (2011) [PDF]

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