Statistics

Assessing the evidential value of journals with p-curve, R-index, TIVA, etc: A comment on Motyl et al. (2017) with new data

Recently, Matt Motyl et al. (2017) posted a pre-print paper in which they contrasted the evidential value of several journals in two time periods (2003-2004 vs. 2013-2014). The paper sparked a lot of discussion in Facebook groups [1][2], blog posts commenting on the paper (DataColada, Dr. R), and a reply from the authors. Some of the discussion was about which effect sizes should have been coded for their analyses. As it turns out, several working groups had similar ideas to check the evidential value of journals (e.g., Leif Nelson pre-registered a very similar project here). In an act of precognition, we did a time-reversed conceptual replication of of Motyl et al.’s study already in 2015:

Anna Bittner, a former student of LMU Munich, wrote her bachelor thesis on a systematic comparison of two journals in social psychology – one with a very high impact factor (Journal of Personality and Social Psychology (JPSP), IF = 5.0 at that time), another with a very low IF (Journal of Applied Social Psychology (JASP), IF = 0.8 at that time – not to be confused with the great statistical software JASP). By coincidence (or precognition), we also chose the year 2013 for our analyses. Read her full blog post about the study below.

In a Facebook discussion about the Motyl et al. paper, Michael Inzlicht wrote: “But, is there some way to systematically test a small sample of your p-values? […] Can you randomly sample 1-2% of your p-values, checking for errors and then calculating your error rate? I have no doubt there will be some errors, but the question is what the rate is.“. As it turns out, we also did our analysis on the 2013 volume of JPSP. Anna followed the guidelines by Simonsohn et al. That means, she only selected one effect size per sample, selected the focal effect size, and wrote a detailed disclosure table. Hence, we do not think that the critique in the DataColada blog post applies to our coding scheme. The disclosure table includes quotes from the verbal hypotheses, quotes from results section, and which test statistic was selected. Hence, you can directly validate the choice of effect sizes. Furthermore, all test statistics can be entered into the p-checker app, where you get TIVA, R-Index, p-curve and more diagnostic tests (grab the JPSP test statistics & JASP test statistics).

Is our field “Rotten to the core”? (CC-BY by Andy Hay; https://www.flickr.com/photos/andyhay/8196333166)

Now we can compare two completely independently coded p-curve disclosure tables about a large set of articles. Any disagreement of course does not mean that one party is right and the other is wrong. But it will be interesting to see the amount of agreement.

Here comes Anna’s blog post about her own study. Anna Bittner is now doing her Master of Finance at the University of Melbourne.


Don’t judge a journal by its cover (or its impact factor)

by Anna Bittner and Felix Schönbrodt

The recent discoveries on staggeringly low replicability in psychology have come as a shock to many and led to a discussion on how to ensure better research practices are employed in the future. To this end, it is necessary to find ways to efficiently distinguish good research from bad and research that contains evidential value from such that does not.

In the past the impact factor (IF) has often been the favored indicator of a journal’s quality. To check whether a journal with a higher IF does indeed publish the “better” research in terms of evidential value, we compared two academic journals from the domain of social psychology: The Journal of Personality and Social Psychology (JPSP, Impact Factor = 5.031) and the Journal of Applied Social Psychology (JASP, Impact Factor = 0.79).

For this comparison, Anna has analysed and carefully hand-coded all studies with hypothesis tests starting in January 2013 and progressing chronologically until about 110 independent test statistics for each journal were acquired. See the full report (in German) in Anna’s bachelor thesis. These test statistics were fed into the p-checker app (Version 0.4; Schönbrodt, 2015) that analyzed them with the tools p-curve, TIVA und R-Index.

All material and raw data is available on OSF: https://osf.io/pgc86/

P-Curve

P-curve (Simonsohn, Nelson, & Simmons, 2014) takes a closer look at all significant p-values and plots them against their relative frequency. This results in a curve that will ideally have a lot of very small p-values (<.025) and much fewer larger p–values (>.025). Another possible shape is a flat curve, which will occur when researchers only investigate null effects and selectively publish those studies that obtained a p-value < .05 by chance. under the null hypothesis each individual p-value is equally likely and the distribution is even. P-curve allows to test whether the empirical curve is flatter than the p-curve that would be expected at any chosen power.

Please note that p-curve assumes homogeneity (McShane, Böckenholt, & Hansen, 2016). Lumping together diverse studies from a whole journal, in contrast, guarantees heterogeneity. Hence, trying to recover the true underlying effect size/power is of limited usefulness here.

The p-curves of both JPSP and JASP were significantly right-skewed, which suggests that both journal’s output cannot be explained by pure selective publication of null effects that got significant by chance. JASP’s curve, however, had a much stronger right-skew, indicating stronger evidential value:

TIVA

TIVA (Schimmack, 2014a) tests for an insufficient variance of p-values: If no p-hacking and no publication bias was present, the variance of p-values should be at least 1. An value below 1 is seen as indicator of publication bias and/or p-hacking. However, variance can and will be much larger than 1 when studies of different sample and effect sizes are included in the analysis (which was inevitably the case here). Hence, TIVA is a rather weak test of publication bias when heterogeneous studies are combined: A p-hacked set of heterogeneous effect sizes can still result in a high variance in TIVA. Publication bias and p-hacking reduce the variance, but heterogeneity and different sample sizes can increase the variance in a way that the TIVA is clearly above 1, even if all studies in the set are severely p-hacked.

As expected, neither JPSP nor JASP attained a significant TIVA result, which means the variance of p-values was not significantly smaller than 1 for either journal. Descriptively, JASP had a higher variance of 6.03 (chi2(112)=674, p=1), compared to 1.09 (chi2(111)=121, p=.759) for the JPSP. Given the huge heterogeneity of the underlying studies, a TIVA variance in JPSP close to 1 signals a strong bias. This, again, is not very surprising. We already knew before with certainty that our literature is plagued by huge publication bias.

The descriptive difference in TIVA variances can be due to less p-hacking, less publication bias, or more heterogeneity of effect sizes and sample sizes in JASP compared to JPSP. Hence, drawing firm conclusions from this numerical difference is difficult; but the much larger value in JASP can be seen as an indicator that the studies published there paint a more realistic picture.

(Note: The results here differ from the results reported in Anna’s bachelor thesis, as the underlying computation has been improved. p-checker now uses logged p-values, which allows more precision with very small p-values. Early versions of p-checker underestimated the variance when extremely low p-values were present).

Unfortunately, Motyl et al. do not report the actual variances from their TIVA test (only the test statistics), so a direct comparison of our results is not possible.

R-Index

The R-Index (Schimmack, 2014b) is a tool that aims to quantify the replicability of a set of studies. It calculates the difference between median estimated power and success rate, which results in the so called inflation rate. This inflation rate is then subtracted from the median estimated power, resulting in the R-Index. Here is Uli Schimmack’s interpretation of certain R-Index values: “The R-Index is not a replicability estimate […] I consider an R-Index below 50 an F (fail).  An R-Index in the 50s is a D, and an R-Index in the 60s is a C.  An R-Index greater than 80 is considered an A”.

Here, again, the JASP was ahead. It obtained an R-Index of .60, whereas the JPSP landed at .49.

Both journals had success rates of around 80%, which is much higher than what would be expected with the average power and effect sizes found in psychology (Bakker, van Dijk, & Wicherts, 2012). It is known and widely accepted that journals tend to publish significant results over non-significant ones.

Motyl et al. report an R-Index of .52 for 2013-2014 for high impact journals, which is very close to our value of .49.


Summary

The comparison between JPSP and JASP revealed a better R-Index, a more realistic TIVA variance and a more right-skewed p-curve for the journal with the much lower IF. As the studies had roughly comparable sample sizes (JPSP: Md = 86, IQR: 54 – 124; JASP: Md = 114, IQR: 65 – 184), I would bet some money that more studies from JASP replicate then from JPSP.

A journal’s prestige does not protect it from research submissions that contain QRPs – contrarily it might lead to higher competition between reseachers and more pressure to submit a significant result by all means. Furthermore, higher rejection rates of a journal also leave more room for “selecting for significance”. In contrast, a journal that must publish more or less every submission it gets to fill up its issues simply does not have much room for this filter. With the currently applied tools, however, it is not possible to make a distinction between p-hacking and publication bias: they only detect patterns in test statistics that can be the result of both practices.

References

Bakker, M., van Dijk, A., & Wicherts, J. M. (2012). The rules of the game called psychological science. Perspectives on Psychological Science, 7(6), 543-554.

McShane, B. B., Böckenholt, U., & Hansen, K. T. (2016). Adjusting for publication bias in meta-analysis: An evaluation of selection methods and some cautionary notes. Perspectives on Psychological Science, 11, 730–749. doi:10.1177/1745691616662243

Schimmack, U. (2014b). Quantifying Statistical Research Integrity: The Replicabilty-Index.

Schimmack, U. (2014a, December 30). The Test of Insufficient Variance (TIVA): A New Tool for the Detection of Questionable Research Practices. Retrieved from https://replicationindex.wordpress.com/2014/12/30/the-test-of-insufficient-variance-tiva-a-new-tool-for-the-detection-of-questionable-research-practices/

Schimmack, U. (2015, September 15). Replicability-Ranking of 100 Social Psychology Departments [Web log post]. Retrieved from https://replicationindex.wordpress.com/2015/09/15/replicability-ranking-of-100-social-psychology-departments/

Schönbrodt, F. (2015). p-checker [Online application]. Retrieved from http://shinyapps.org/apps/p-checker/

Simonsohn, U., Nelson, L. D., & Simmons, J. P. (2014). P-curve: A key to the file-drawer. Journal of Experimental Psychology: General, 143(2), 534.

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Two meanings of priors, part II: Quantifying uncertainty about model parameters

by Angelika Stefan & Felix Schönbrodt

This is the second part of “Two meanings of priors”. The first part explained a first meaning – “priors as subjective probabilities of models”. While the first meaning of priors refers to a global appraisal of existing hypotheses, the second meaning of priors refers to specific assumptions which are needed in the process of hypothesis building. The two kinds of priors have in common that they are both specified before concrete data are available. However, as it will hopefully become evident from the following blog post, they differ significantly from each other and should be distinguished clearly during data analysis.

The second meaning of priors: Prior distribution for parameters

In order to know how well evidence supports a hypothesis compared to another hypothesis, one must know the concrete specifications of each hypothesis. For example, in the tea tasting experiment, each hypothesis was characterized by a specific probability (e.g., the success rate of exactly 0.5 in HFisher of the previous blog post). What might sound trivial at first – deciding on the concrete specifications of a hypothesis – is in fact one of the major challenges when doing Bayesian statistics. Scientific theories are often imprecise, resulting in more than one plausible way to derive a hypothesis. With deciding upon one specific hypothesis, often new auxiliary assumptions are made. These assumptions, which are needed in order to specify a hypothesis adequately, are called “priors” as well. They influence the formulation and interpretation of the likelihood (which gives you the plausibility of data under a specific hypothesis). We will illustrate this in an example.

How much money do we spend for lunch?

Fig. 2: A typical lunch in Germany: Schweinshaxe, Brezn and 0.5 liter of beer.

A food company conducts market research in a large German city. They know from a recent representative enquiry by the German Federal Statistical Office that Germans spend on average 4.50 € for their lunch (standard deviation: 0.60 €). Now they want to know if the inhabitants of one specific city spend more money for their lunch compared to the German average. They expect lunch expenses to be especially high in this city because of the generally high living costs. In a traditional testing procedure in inferential statistics the food company would formulate two hypotheses to test their assumption: a null and an alternative hypothesis: H0: µ ≤ 4.50 and H1: µ > 4.50.

In Bayesian hypothesis testing, the formulation of the hypotheses has to be more precise than this. We need precise hypotheses as a basis for the likelihood functions which assign probability values to possible states of reality. The traditional formulation, µ > 4.50, is too vague for that purpose: Is any lunch cost above 4.50€ a priori equally likely? Is it plausible that a lunch costs 1,000,000€ on average? Probably not. Not every state of reality is, a priori, equally plausible. “Models connect theory to data“ (Rouder, Morey, & Wagenmakers, 2016), and a model that predicts everything predicts nothing.

As Bayesian statisticians we therefore must ask ourselves: Which values are more plausible given that our hypotheses are true? Of course, our knowledge differs from case to case in this point. Sometimes, we may be able to commit to a very small range of plausible values or even to a single value (in this case, we would call the respecting hypothesis a “point hypothesis”). Theories in physics sometimes predict a single state of reality: “If this theory is true, then the mass of a Chicks boson is exactly 1.52324E-16 gr”.

 

Point hypotheses in the food company example

More often, however, our knowledge about plausible values under a certain theory might be less precise, leading to a wider range of plausible values. Hence, the prior in the second sense defines the probability of a parameter value given a hypothesis, p(θ | H1).

Let us come back to the food company example. Their null hypothesis might be that there is no difference between the city in the focus of their research project and the German average. Hence, the null hypothesis predicts an average lunch cost of 4.50€. With the alternative hypothesis, it becomes slightly more complex. They assume that average lunch expenses in the city should be higher than the German average, so the most plausible value under the alternative hypothesis should be higher than 4.5. However, they may deem it very improbable that the mean lunch expenses are more than two standard deviations higher than the German average (so, for example, it should be very improbable that someone spends more than, say, 10 EUR for lunch even in the expensive city). With this knowledge, they can put most plausibility on values in a range from 4.5 to 5.7 (4.5 + 2 standard deviations). They could further specify their hypothesis by claiming that the most plausible value should be 5.1, i.e., one standard deviation higher than the German average. The elements of these verbal descriptions of the alternative hypothesis can be summarized in a truncated normal distribution that is centered over 5.1 and truncated at 4.5 (as the directional hypothesis does not predict values in the opposite direction).

 

Truncated normal distribution centered over 5.1 and truncated at 4.5

With this model specification, the researchers would place 13% of the probability mass on values larger than 2SD of the general population (i.e., > 5.7).

Making it even more complex, they could quantify their uncertainty about the most plausible value (i.e., the maximum of the density distribution) by assigning another distribution to it. For example, they could build a normal distribution around it, with a mean of 5.1 and a standard deviation of 0.3. This would imply that in their opinion, 5.1 is the “most plausible most plausible value” but that values between 4.8 and 5.4 are also potential candidates for the most plausible value.

Why making it complicated if it’s so simple in traditional hypothesis testing?

What you can notice in the example about the development of hypotheses is that the market researchers have to make auxiliary assumptions on top of their original hypothesis (which was H1: µ > 4.5). If possible, these prior plausibilities should be informed by theory or by previous empirical data. Specifying alternative hypothesis in this way may seem to be an unnecessary burden compared to traditional hypothesis testing where these extra assumptions seemingly are not necessary. Except that they are necessary. Without going into detail in this blog post, we recommend to read Rouder et al.’s (2016a) “Is there a free lunch in inference?“, with the bottom line that principled and rational inference needs specified alternative hypotheses. (For example, in Neyman-Pearson testing, you also need to specify a precise alternative hypothesis that refers to the “smallest effect size of interest”)

Furthermore, readers might object: “Researchers rarely have enough background knowledge to specify models that predict data“. Rouder et al. (2016b) argue that this critique is overstated, as (1) with proper elicitation, researchers often know much more than they initially think, (2) default models can be a starting point if really no information is available, and (3) several models can be explored without penalty.

Do the two kinds of priors depend on each other in some way?

A question that may come to your mind soon after you understood the difference between the two kinds of priors is: If they both are called “priors”, do they depend on each other in some way? Does the formulation of your “personal prior plausibility of a hypothesis” (like the skeptical observer’s prior on Mrs. Bristol’s tea tasting abilities) influence the specification of your model (like the hypothesis specification in the second example) or vice versa?

The straightforward answer to this question is “no, they don’t”. This can be easily illustrated in a case where the prior conviction of a researcher runs against the hypothesis he or she wants to test. The food company in the second example has sophisticatedly determined the likelihood of the two hypotheses (H0 and H1), which they want to pit against each other. They are probably considerably convinced that the specification of the alternative hypothesis describes reality better than the specification of the null hypothesis. In a simplified form, their prior odds (i.e., priors in the first sense) can be described as a ratio like 10:1. This would mean that they deem the alternative hypothesis ten times as likely as the null hypothesis. However, another food company, may have prior odds of 3:5 while conducting the same test (i.e., using the same prior plausibilities of model parameters). This shows that priors in the first sense are independent of priors in the second sense. Priors in the first sense change with different personal convictions while priors in the second sense remain constant. Similarly, prior beliefs can change after seeing the data – the formulation of the model (i.e., what a theory predicts) stays the same. (As long as the theory, from which the model specification is derived, does not change. In an estimation context, the model parameters are updated by the data.)

Summary of both blog posts

The term “prior” has two meanings in the context of Bayesian hypothesis testing. The first one, usually applied in Bayes factor analysis, is equivalent to a prior subjective probability of a hypothesis (“how plausible do you deem a hypothesis compared to another hypothesis before seeing the data”). The second meaning refers to the assumptions made in the specification of the model of the hypotheses which are needed to derive the likelihood function. These two meanings of the term “prior” have to be distinguished clearly during data analysis, especially as they do not depend on each other in any way. Some researchers (e.g., Dienes, 2016) therefore suggest to call only priors in the first sense “priors” and speak about “specification of the model” when referring to the second meaning.

Read the first part of this blog post: Priors as the plausibility of models

References

Dienes, Z. (2011). Bayesian versus orthodox statistics: Which side are you on?. Perspectives On Psychological Science, 6(3), 274-290. http://doi:10.1177/1745691611406920

Dienes, Z. (2016). How Bayes factors change scientific practice. Journal Of Mathematical Psychology, 7278-89. http://doi:10.1016/j.jmp.2015.10.003

Lindley, D. V. (1993). The analysis of experimental data: The appreciation of tea and wine. Teaching Statistics, 15(1), 22-25. http://dx.doi.org/10.1111/j.1467-9639.1993.tb00252.x

Rouder, J. N., Morey, R. D., Verhagen, J., Province, J. M., & Wagenmakers, E. J. (2016a). Is there a free lunch in inference? Topics in Cognitive Science, 8, 520–547. http://doi.org/10.1111/tops.12214

Rouder, J. N., Morey, R. D., & Wagenmakers, E. J. (2016b). The Interplay between Subjectivity, Statistical Practice, and Psychological Science. Collabra, 2(1), 6–12. http://doi.org/10.1525/collabra.28

library(truncnorm)

# parameters for H1 model
M <- 5.1
SD <- 0.5

range <- seq(4.5, 7, by=.01)
plausibility <- dtruncnorm(range, a=4.5, b=Inf, mean=M, sd=SD)
plot(range, plausibility, type="l", xlim=c(4, 7), axes=FALSE, ylab="Plausibility", xlab="Lunch cost in €", mgp = c(2.5, 1, 0))
axis(1)

# Get the axis ranges, draw arrow
u <- par("usr")
points(u[1], u[4], pch=17, xpd = TRUE)
lines(c(u[1], u[1]), c(u[3], u[4]), xpd = TRUE)

abline(v=4.5, lty="dotted")

# What is the probability of values > 5.7?
1-ptruncnorm(5.7, a=4.5, b=Inf, mean=M, sd=SD)
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Two meanings of priors, part I: The plausibility of models

by Angelika Stefan & Felix Schönbrodt

When reading about Bayesian statistics, you regularly come across terms like “objective priors“, “prior odds”, “prior distribution”, and “normal prior”. However, it may not be intuitively clear that the meaning of “prior” differs in these terms. In fact, there are two meanings of “prior” in the context of Bayesian statistics: (a) prior plausibilities of models, and (b) the quantification of uncertainty about model parameters. As this often leads to confusion for novices in Bayesian statistics, we want to explain these two meanings of priors in the next two blog posts*. The current blog post covers the the first meaning of priors (link to part II).

The first meaning of “prior”: Prior plausibility for models

In this context, the term “prior” incorporates the personal assumptions of a researcher on the probability of a hypothesis (p(H1)) relative to a competing hypothesis, which has the probability p(H2). Hence, the meaning of this prior is “how plausible do you deem a model relative to another model before looking at your data”. The ratio of the two priors of the models, that is “how probable do you consider H1 compared to H2”, is called “prior odds”: p(H1) / p(H2).

The first meaning of priors is used in the context of Bayes factor analysis, where you compare two different hypotheses. In Bayes factor analysis, prior odds are updated by the likelihood ratio of the two hypotheses, which contains the information from the data, and result in the posterior odds (“what you believe after looking at your data”):

\large \frac{Posterior_{H1}}{Posterior_{H2}} =\frac{Likelihood_{H1}}{Likelihood_{H2}}\cdot\frac{Prior_{H1}}{Prior_{H2}}

The prior belief is called “subjective”, but this label does not imply that it is “arbitrary”, “unprincipled”, or “irrational”. In contrast, the prior belief can (and preferably should) be informed by previous data or experiences. For example, it can be a belief that started with an equipoise (50/50) position, but has been repeatedly updated by data. But within the bounds of rationality and consistency, people still can come to considerably different prior beliefs, and might have good arguments for their choice – that’s why it is called “subjective”. But initially differing degrees of belief will converge as more and more evidence comes in. We will observe this in the following example.

The incredible tea tasting abilities of Lady Muriel Bristol

Fig. 1: Muriel Bristol surprising her husband (William) by making the correct guess for the fifth time in a row (reenacted scene)

The classical experiment of tasting tea has already been described in the context of Bayesian hypothesis testing by Lindley (1993). We will present a simpler form here. Dr. Muriel Bristol, a scientist working in the field of alga biology who was acquainted to the statistician R. A. Fisher, claimed that she could discriminate whether milk is put in a cup before or after the tea infusion during the process of preparing tea with milk. However, Mr. Fisher considered this very unlikely.

So they decided to run an experiment: Muriel Bristol tasted several cups of tea in a row, making a guess on the preparation procedure for each cup. Unlike in the original story, where inferential statistics were consulted to solve the disagreement, we will employ Bayesian statistics to track how prior convictions in this example change. If Muriel Bristol makes her guesses only based on chance as Mr. Fisher supposes, she has a probability of success of 50% in each trial. Before observing her performance, Mr. Fisher should therefore consider it very likely that she is right about the procedure in about 50% of the trials across all trials. We can therefore assume a point hypothesis: HFisher: Success rate (SR) = 0.5. Muriel Bristol, on the other hand, is very confident in her divination skills. She claims to get 80% of trials correct, which can be equally captured in a point hypothesis: HMuriel: SR = 0.8.

Two observers with different prior beliefs

To introduce prior beliefs about hypotheses and show how they change with upcoming evidence, we want to introduce two additional persons. The first one is a slightly skeptical observer who tends to favor HFisher, but does not completely rule out that Mrs. Bristol could be right with her hypothesis. More formally, we could describe this position as: P(HFisher) = 0.6 and P(HMuriel) = 0.4. This means that his prior odds are P(HFisher)/P(HMuriel) = 3:2. Fisher’s hypothesis is 1.5 times more likely to him than Muriel Bristol’s hypothesis.

The second additional person we would like to introduce is William, Muriel Bristol’s loving husband who fervently advocates her position. He knows his wife would never (or at least very rarely) make wrong claims, concerning tea preparation and all others issues of their marriage. He therefore assigns a much higher subjective probability to her hypothesis (P(HMuriel) = 0.9) than to the one of Mr. Fisher (P(HFisher) = 0.1). His prior odds are therefore P(HFisher)/P(HMuriel) = 1:9. Please note that the content of the hypotheses (the proposed success rates 0.5 and 0.8, which are the parameters of the model) is logically independent of the probability of the hypotheses (priors) that our two observers have.

How to update prior beliefs about hypotheses

During the process of hypothesis testing, these two priors are updated with the existing evidence. It is reported that Muriel Bristol’s performance at the experiment was extraordinarily good. We therefore assume that out of the first 6 trials of the experiment she got 5 correct.

With this information, we can now compute the likelihood of the data under each of the hypotheses (for more information on the computation of likelihoods, see Alexander Etz’s blog:

L_{Fisher} = 0.5^5 \cdot 0.5^1 =  0.016

L_{Muriel} = 0.8^5 \cdot 0.2^1 = 0.066

The computation of the likelihoods does not involve the prior model probabilities of our observers. What can be seen is that the data are more likely under Muriel Bristol’s hypothesis than under Mr. Fisher’s. This should not come as a surprise as Muriel Bristol claimed that she could make a very high percentage of right guesses and the data show a very high percentage of right guesses whereas Mr. Fisher assumed a much lower percentage of right guesses. To emphasize this difference in likelihoods and to assign it a numerical value, we can compute the likelihood ratio (Bayes factor):

BF_{FM} = LR_{FM} = \frac{L_{Fisher}}{L_{Muriel}} = 0.016 / 0.066 = 0.238

BF_{MF} = LR_{MF} = \frac{L_{Muriel}}{L_{Fisher}} = 0.066 / 0.016 = 4.19

This ratio means that the data are 4.19 times as likely under Mrs. Bristol’s hypothesis as under Mr. Fisher’s hypothesis. It does not matter how you order the likelihoods in the fraction, the meaning remains constant (see this blog post).

How does this likelihood change the prior odds of both our slightly skeptical observer and William Bristol? Bayes theorem shows that prior odds can be updated by multiplying them with the likelihood ratio (Bayes factor):

\frac{ Posterior(H_{Fisher}) } {Posterior(H_{Muriel})} =  \frac{Prior(H_{Fisher})}{Prior(H_{Muriel})} \cdot \text {Bayes Factor}

First, we will focus on the posterior odds of the slightly skeptical observer. To remember, the slightly skeptical observer had assigned a probability of 0.6 to Mr. Fisher’s hypothesis and a probability of 0.4 to Muriel Bristol’s hypothesis before seeing the data, which resulted in prior odds of 3:2 for Mr. Fisher’s hypothesis. How do these convictions change now when the experiment has conducted? To examine this, we simply have to insert all known values in the equation:

\frac {3}{2} \cdot 0.238 = 0.357 = \frac {1}{2.8}

This shows that the prior odds of the slightly skeptical observer changed from 3:2 to posterior odds of 1:2.8. This means that whereas before the experiment the slightly skeptical observer had deemed Mr. Fisher’s hypothesis more plausible than Mrs. Bristol’s hypothesis, he changed his opinion after seeing the data, now preferring Mrs. Bristol’s hypothesis over Mr. Fisher’s.

The same equation can be applied to William Bristol’s prior odds:

\frac {1}{9} \cdot 0.238 = 0.0264 = \frac {1}{37.9}

What we can notice is that after taking the data into consideration both prior odds display a higher amount of agreement with Muriel Bristol and reduced confidence in Mr. Fisher’s hypothesis. Whereas the convictions of the slightly skeptical observer were changed in favor of Muriel Bristol’s hypothesis after the experiment, William Bristol’s prior convictions were strengthened.

Something else you can notice is that compared to William Bristol the slightly skeptical observer still assigns a higher plausibility to Mr. Fisher’s hypothesis. This rank order between the two priors will remain no matter what the data look like. Even if Muriel Bristol made, say, 100/100 correct guesses, the slightly skeptical observer would trust less in her hypothesis than her husband. However, with increasing evidence the absolute difference between both observers will decrease more and more.

Summary

This blog post explained the first meaning of “prior” in the context of Bayesian statistics. It can be defined as the subjective plausibility a researcher assigns to a hypothesis compared to another hypothesis before seeing the data. As illustrated in the tea-tasting example, these prior beliefs are updated with upcoming evidence in the research process. In the next blog post, we will explain a second meaning of “priors”: The quantification of uncertainty about model parameters.

Continue reading part II: Quantifying uncertainty about model parameters

We want to thank Eric-Jan Wagenmakers for helpful comments on a previous version of the post.

*As a note: Both meanings in fact can be unified, but for didactic purpose we think it makes sense to keep them distinct as a start.

References

Dienes, Z. (2011). Bayesian versus orthodox statistics: Which side are you on?. Perspectives On Psychological Science, 6(3), 274-290. http://doi:10.1177/1745691611406920

Dienes, Z. (2016). How Bayes factors change scientific practice. Journal Of Mathematical Psychology, 7278-89. http://doi:10.1016/j.jmp.2015.10.003

Lindley, D. V. (1993). The analysis of experimental data: The appreciation of tea and wine. Teaching Statistics, 15(1), 22-25. http://dx.doi.org/10.1111/j.1467-9639.1993.tb00252.x

Rouder, J. N., Morey, R. D., Verhagen, J., Province, J. M., & Wagenmakers, E. J. (2016a). Is there a free lunch in inference? Topics in Cognitive Science, 8, 520–547. http://doi.org/10.1111/tops.12214

Rouder, J. N., Morey, R. D., & Wagenmakers, E. J. (2016b). The Interplay between Subjectivity, Statistical Practice, and Psychological Science. Collabra, 2(1), 6–12. http://doi.org/10.1525/collabra.28

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