February 15, 2017

tl;dr: The German Psychological Society developed and adopted new recommendations for data sharing that fully embrace openness, transparency and scientific integrity. Key message is that raw data are an essential part of an empirical publication and must be openly shared. The recommendations also give very practical advice on how to implement these values, such as “When should data providers be asked to be co-authors in a data reuse project?” and “How to deal with participant privacy?”.

In the last year, the discussion in our field moved from “Do we have a replication crisis?” towards “Yes, we have a problem, and what can and should we change? How can be implement it?”. I think that we need both top-down changes on an institutional level, combined with bottom-up approaches, such as local Open Science Initiatives. Here, I want to present one big institutional change concerning open data.

The German Research Foundation (DFG), the largest public funder of research in Germany, updated their policy on data sharing, which can be summarized in a single sentence: **Publicly funded research, including the raw data, belongs to the public**. Consequently, all research data from a DFG funded project should be made open immediately, or at least a couple of months after finalization of the research project (see [1] and [2]). Furthermore, the DFG asked all scientific disciplines to develop more specific guidelines which implement these principles in their respective discipline.

The German Psychological Society (Deutsche Gesellschaft für Psychologie, DGPs) installed a working group (Andrea Abele-Brehm, Mario Gollwitzer and me) who worked for one year on such recommendations for psychology.

In the development of the document, we tried to be very inclusive and to harvest the wisdom of the crowd. A first draft (Feb 2016) was discussed for 6 weeks in an internet forum where all DGPs members could comment. Based on this discussion (and many additional personal conversations), a revised version was circulated and discussed in person with a smaller group of interested members (July 2016) and a representative of the DFG. Furthermore, we had regular contact to the “Fachkollegium Psychologie” of the DFG (i.e., the group of people which decides about funding decisions in psychology; meanwhile, the members of the Fachkollegium have changed on a rotational basis). Finally, the chair persons of all sections of the DGPs and the speakers of the young members had another possibility to comment. On September 17, the recommendations were officially adopted by the society.

I think this thorough and iterative process was very important for two reasons: First, it definitely improved the quality of the document, because we got so many great ideas and comments from the members, ironing out some inconsistencies and covering some edge cases. Second, it was important in order to get people on board. As this new open data guideline of the DFG causes a major change in the way we do our everyday scientific work, we wanted to talk to and convince as many people as possible from the early steps on. Of course not every single of the >4,000 members is equally convinced, but the topic now has considerable attention in the society.

Hence, one focus was consensus and inclusivity. At the same time, we had the goal to develop bold and forward-looking guidelines that really address the current challenges of the field, and not to settle on the lowest common denominator. For this goal, we had to find a balance between several, sometimes conflicting, values.

*Research transparency ⬌ privacy rights.* A first specialty of psychology is that we do not investigate rocks or electrons, but human subjects who have privacy rights. In a nutshell, privacy rights have to be respected, and in case of doubt they win over openness. But if data can be properly anonymized, there’s no problem in open sharing; one possibility to share non-anonymous research data are “scientific use files”, where access is restricted to scientists. If data cannot be shared due to privacy (or other) reasons, this has to be made transparent in the paper. (Hence, the recommendations are PRO compatible). The recommendations give clear guidance on privacy issues and gives practical advice, for example, on how to write your informed consent that you actually are able to share the data afterwards.

*Data reuse ⬌ right of first usage.* A second balance concerns an optimal reuse of data on the one hand, and the right of first usage of the original authors. In the discussion phase during the development of the recommendations, several people expressed the fear of “research parasites”, who “steal” the data from hard-working scientists. A very common gut feeling is: “The data belong to me”. But, as we are publicly funded researchers with publicly funded research projects, the answer is quite clear: the data belong to the public. There is no copyright on raw data. On the other hand, we also need incentives for original researchers to generate data in the first place. Data generators of course have the right of first usage, and the recommendations allow to extend this right by an embargo of 5 more years (see below). But at the end of the day, publicly funded research data belongs to the public, and everybody can reuse it. If data are open by default, a guideline also must discuss and define how data reuse should be handled. Our recommendations make suggestions in which cases a co-authorship should be offered to the data providers and in which cases this is not necessary.

*Verification ⬌ fair treatment of original authors.* Finally, research should be verifiable, but with a fair treatment of the original authors. The guidelines say that whenever a reanalysis of a data set is going to be published (and that also includes blog posts or presentations), the original authors have to be informed about this. They cannot prevent the reanalysis, but they have the chance to react to it.

We distinguish two types of data sharing:

*Type 1 data sharing* means that all raw data should be openly shared that is necessary to reproduce the results reported in a paper. Hence, this can be only a subset of all available variables in the full data set: The subset which is needed to reproduce these specific results. **The primary data are an essential part of an empirical publication, and a paper without that simply is not complete.**

*Type 2 data sharing* refers to the release of the full data set of a funded research project. The DGPs recommendations claim that after the end of a DFG-funded

project all data – even data which has not yet been used for publications – should be made open. Unpublished null results, or additional, exploratory variables now have to chance to see the light and to be reused by other researchers. Experience tells that not all planned papers have been written after the official end date of a project. Therefore, the recommendations allow that the right of first usage can be extended with an embargo period of up to 5 years, where the (so far unpublished) data do not have to be made public. The embargo option only applies to data that has not yet been used for publications. Hence, typically an embargo cannot be applied to Type 1 data sharing.

To summarize, I think these recommendations are the most complete, practical, and specific guidelines for data sharing in psychology to date. (Of course much more details are in the recommendations themselves). They fully embrace openness, transparency and scientific integrity. Furthermore, they do not proclaim detached ethical principles, but give very practical guidance on how to actually implement data sharing in psychology.

What are the next steps? The president of the DGPs, Prof. Conny Antoni, and the secretary Prof. Mario Gollwitzer already contacted other psychological societies (APA, APS, EAPP, EASP, EFPA, SIPS, SESP, SPSP) and introduced our recommendations. The Board of Scientific Affairs of EFPA – the European Federation of Psychologists’ Associations – already expressed its appreciation of the recommendations and will post them on their website. Furthermore, it will discuss them in an invited symposium on the European Congress of Psychology in Amsterdam this year. A mid-term goal will also be to check compatibility with existing other guidelines and to think about a harmonization of several guidelines within psychology.

As other scientific disciplines in Germany also work on their specific implementations of the DFG guidelines, it will be interesting to see whether there are common lines (although there certainly will be persisting and necessary differences between the requirements of the fields). Finally, we are in contact with the new Fachkollegium at the DFG, with the goal to see how the recommendations can and should be used in the process of funding decisions.

If your field also implements such recommendations/guidelines, don’t hesitate to contact us.

Schönbrodt, F., Gollwitzer, M., & Abele-Brehm, A. (2017). Der Umgang mit Forschungsdaten im Fach Psychologie: Konkretisierung der DFG-Leitlinien. *Psychologische Rundschau, 68*, 20–35. doi:10.1026/0033-3042/a000341. [PDF German][PDF English]

*(English translation by Malte Elson, Johannes Breuer, and Zoe Magraw-Mickelson)*

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January 17, 2017

*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.

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 H_{Fisher }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.

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: H_{0}: µ ≤ 4.50 and H_{1}: 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”.

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).

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.

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.

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.)

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

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)

# 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)

January 10, 2017

*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).

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”):

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 **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: H_{Fisher}: 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: H_{Muriel}: SR = 0.8.

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 H_{Fisher}, but does not completely rule out that Mrs. Bristol could be right with her hypothesis. More formally, we could describe this position as: P(H_{Fisher}) = 0.6 and P(H_{Muriel}) = 0.4. This means that his prior odds are P(H_{Fisher})/P(H_{Muriel}) = 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(H_{Muriel}) = 0.9) than to the one of Mr. Fisher (P(H_{Fisher}) = 0.1). His prior odds are therefore P(H_{Fisher})/P(H_{Muriel}) = 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.

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:

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):

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):

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:

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:

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.

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.

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