A complete guide to the bayes factor test

The Bayes factor test is an interesting thing. Some Bayesians advocate it unequivalently, whereas others reject the notion of testing altogether, Bayesian or otherwise. This post takes a critical look at the Bayes factor, attempting to tease apart the ideas to get to the core of what it’s really doing. If you're used to frequentist tests and want to understand Bayes factors, this post is for you.

The Bayes factor test goes all the way back to Jeffreys’ early book on the Bayesian approach to statistics [Jeffreys1939]. Evidence for an alternative hypothesis H1 against that of the null hypothesis H0 is summarized by a quantity known as the Bayes factor. The Bayes factor is just the ratio of the data likelihoods, under both hypotheses and integrating out any nuisance parameters:

B10 : = p(D|H1) p(D|H0).

If this ratio is large, we can conclude that there is strong evidence for the alternative hypothesis. In contrast, if the inverse of this ratio is large, we have evidence supporting the null hypothesis. The definition of what is strong evidence is subjective, but usually a Bayes factor of 10 or more is considered sufficient.

The standard Bayes factor test against the point-null

Although Bayes factors are sometimes used for testing simple linear regression models against more complex ones, by far the most common test in practice is the analogue to the frequentist t-test, the Bayes factor t-test. Under the assumption of normality with unknown variance, it tests a null hypothesis of zero mean against non-zero mean.

This test is implemented in the BayesFactor R package with the ttestBF method. For the pairwise case it can be invoked with bf <- ttestBF(x = xdata, y=ydata). For a non-pairwise test just x can be passed in.

Statistical details of the test To apply the test we must formally define the two hypotheses in the form of statistical models. Jeffreys’ book is very pragmatic, and the modelling choices for this test reflect that. Since this is a t-test, we of course assume a normal distribution for the data for both hypothesis, so it just remains to just define the priors on it’s parameters in both cases.

We have essentially 3 parameters we have to place priors on: the mean μ of the alternative, the variance under the alternative σ12 and the variance under the null σ02. The μ under the null is assumed to be 0. We have to be extremely careful about the priors we place on these parameters. Reasonable seeming choices can lead to divergent integrals or non-sensical results very easily. We discuss this further below.

Variance Because the variance appears in both the numerator and denominator of the Bayes factor (the null and the alternative), we can use a “non-informative” improper prior of the form p(σ2) = 1σ2 for both. This prior causes the marginal likelihood integrals to go to infinity, but this cancels out between the numerator and denominator giving a finite value.

Mean The choice of a prior for the mean is a little more subjective. We can’t use an improper prior here, as the mean prior appears only in the numerator, and without the convenient cancelation with factors in the denominator, the Bayes factor won’t be well defined. Jeffreys argues for a Cauchy prior, which is just about the most vague prior you can use that still gives a convergent integral. The scale parameter of the Cauchy is just fixed to the standard deviation in his formulation. The BayesFactor package uses 0.707 times the standard deviation instead as the default, but this can be overridden.

What are the downsides compared to a frequentist t-test? The main downside is that the BF test is not as good by some frequentist measures than the tests that are designed to be as good as possible with respect to those measures. In particular, consider the standard measure of power at γ: the probability of picking up an effect of size γ under repeated replications of the experiment. Depending on the effect size and the number of samples n, a Bayesian t-test often requires 2-4 times as much data to match the power of a frequentist t-test.

Implementation details The Bayes factor requires computing marginal likelihoods, which is a quite distinct problem from the usual posterior expectations we compute when performing Bayesian estimation instead of hypothesis testing. The marginal likelihood is just an integral over the parameter space, which can be computed using numerical integration when the parameter space is small. For this t-test example, the BayesFactor package uses Gaussian quadrature for the alternative hypothesis. The null hypothesis doesn’t require any integration since it consists of a single point.

Jefferys lived in the statistical era of tables and hand computation, so he derived several formulas that can be used to approximate the Bayes factor. The following formula in terms of the t-statistic t = x̄(n 1)s2 works well for n in the hundreds or thousands:

B10 2 πnexp t22.

The sensitivity to priors

There are two types of priors that appear in the application of Bayes factors; The priors on the hypothesis p(H0), p(H1), and the priors on parameters that appears in marginal likelihoods, the p(0|H0), p(1|H1). Your prior belief ratio p(H1)p(H0) is updated to form your posterior beliefs using the simple equation:

p(H1|D) p(H0|D) = p(D|1)p(1|H1)d1 p(D|0)p(0|H0)d0 p(H1) p(H0).

The ratio p(H1)p(H0) is not formally considered part of the Bayes factor, but it plays an important role. It encodes your prior beliefs about the hypotheses. If you are using Bayes factors in a subjective fashion, to update your beliefs in a hypothesis after seeing data, then it is crucial to properly encode your prior belief in which hypothesis is more likely to be true. However, in practice everybody always takes this ratio to be 1, following the ideals of objective Bayesian reasoning, where one strives to introduce as little prior knowledge as possible into the problem. If your writing a scientific paper, it’s reasonable to use 1 as anybody reading the paper can apply there own ratio as a correction to your Bayes factor, reflecting there own prior beliefs.

The prior ratio of the hypotheses plays an interesting part in the calibration of the Bayes factor test. One way of testing Bayes factors is to simulate data from the priors, and compare the empirical Bayes factors over many simulations with the true ratios. This involving sampling a hypothesis, H0 or H1, then sampling then D. If the probability of sampling H0 and H1 is not equal at that first stage, then the observed Bayes factor will be off by the corresponding p(H1)p(H0). This is somewhat of a tautology, as we are basically just numerically verifying the Bayes factor equation as written. Nevertheless, it shows that if the priors on the hypotheses don’t reflect reality, then the Bayes factor won’t either. Blind application of a Bayes factor test won’t automatically give you interpretable “beliefs”.

Parameter priors

The dependence of Bayes factors on the parameter priors p(1|H1) and p(0|H0) is really at the core of understanding Bayes factors. It is a far more concrete problem than the hypothesis likelihoods, as there is no clear “objective” choices we can make. Additionally, intuitions from Bayesian estimation problems do not carry over to hypothesis testing.

Consider the simple BF t-test given above. We described the test using the Cauchy prior on the location parameter of the Gaussian. Often in practice people use large vague Gaussian priors instead when doing Bayesian estimation (as opposed to hypothesis testing), as they tend to be more numerically stable than the Cauchy. However, it has been noted by several authors [Berger and Pericchi2001] that simply replacing the Cauchy with a wide Gaussian in the BF t-test causes serious problems. Suppose we use a standard deviation of ν in this wide prior (ν large). It turns out that Bayes factor scales asymptotically like:

1 νnexp t22.

where t = x̄(n 1)s2 is the standard t-statistic of the data. The large ν that was chosen with the intent that it would be ’non-informative’ (i.e. minimally effect the result) turns out to directly divide the BF!

This is the opposite situation from estimation, where as ν increases is behaves more and more like the improper flat prior p() 1. We can’t use the flat prior in the BF t-test, but if we assume that the sample standard deviation s matches the true standard deviation, we can perform a BF z-test instead. Using a flat prior on , we get a BF of:

s2π n exp z22,

Because the flat prior appears in the numerator only there is no clear scaling choice for it, so this expression is not in any sense calibrated (Several approaches to calibration appear in the literature, see Robert [1993]).

It is interesting to see what happens when we apply the Cauchy prior, as used in the standard form of the test. In that case we see approximate scaling like [Robert et al.2009]:

2 πnexp t22.

A similar formula appears when the data standard deviation is assumed to be known also:

2 πn 1 + t2 n exp t22.

The 1 + t2 n part can also be written 1 + x̄ s2 in terms of the known standard deviation and the empirical mean, and clearly it is constant asymptotically; the expression on the whole scales similarly as in the unknown variance case.

Lindley’s paradox

Lindley’s paradox refers to a remarkable difference in the behavior of Bayes factor tests compared to frequentist tests as the sample size increases. Under a frequentist t-test, we compute the t statistic and compare it against a threshold that depends on n, and that decreases as n increases, (Recall the definition of the t statistic: t = x̄(n 1)s2). In contrast the effective threshold used by the BF t-test actually increases as n increases, although slowly, following roughly log n. This difference is illustrated in the following plot lifted from Rouder et al. [2009]:

PIC

The effect of this difference is that for large sample sizes, a frequentist test can reject the null with small p (say p = 0.05), while a BF t-test run on the same data can strongly support the null hypothesis! As a concrete example, if we generate n = 50,000 synthetic data points with mean 250000, variance 1, then the BF t-test yields 0.028 for the alternative hypothesis (i.e. 36:1 odds for the null being true), whereas a frequentist t-test rejects the null with p = 0.0478 (two-sided test).

It’s all well and good to argue the philosophical differences between Bayesian and frequentist approaches, but when they disagree so strongly, it becomes a practical issue. The key issue is the handling of small-effects. As n increases and s2 stays fixed, a fixed t value of say 2 represents a smaller and smaller measured effect, namely:

x̄ 2 n.

Detecting small effects requires a large sample size, and the Bayesian t-test greatly prefers the null hypothesis over that of a small effect. This is usually described positively as a Occam’s razor effect, but it does have real consequences when we are attempting to find true small effects. Essentially, the BF test will require much more data pick up a small effect then a corresponding frequentist test. Often, this is viewed from the opposite side; that frequentist tests too easily reject the null hypothesis for large sample sizes. This is more of an issue when testing more complex models such as linear regressions with multiple coefficients. Unless the data was actually generated from a linear equation with Gaussian noise, large samples will inevitably reject simple null models. This is in contrast on simple point null hypothesis testing, where we can appeal to asymptotic normality.

Possible changes from the default Bayes factor test

Here we attempt to answer the following question: are the properties of the default Bayes factor test indicative of BF tests in general? As we see it, there are two main ways to modify the test: We can use a non-point hypothesis as the null, or we can change the alternative hypothesis in some fashion. We discuss these possibilities and their combination separately.

Changing the alternative hypothesis

The default Bayes factor t-test is formulated from a objectivist point of view: it’s designed so that the data can “speak for its self”, the priors used are essentially as vague as possible while still yielding a usable test. What about if we choose the prior for the alternative that gives the strongest evidence for the alternative instead? This obviously violates the likelihood principle as we choose the prior after seeing the data, but it is nevertheless instructive in that it indicates how large the difference between point NHST tests and BF tests are.

A reasonable class of priors to consider is those that are non-increasing in terms of || , as they have a single mode at = 0. We discuss other priors below. Table 1 shows a comparison of p-values, likelihood ratios from the Neyman-Pearson likelihood ratio test, and a bound on BFs from any possible prior of this form. It is immediately obvious that according to this bound, any reasonable choice of prior yields much less evidence for the alternative then that suggested by the p-value. For example, the difference at p=0.001 is 18 fold, and the difference is larger for smaller p.







Method










t-statistic 1.645 1.960 2.576 3.291





p-value 0.1 0.05 0.01 0.001





LR 0.258:10.146:10.036:10.0044:1





BF bound0.644:10.409:10.123:1 0.018:1





Table 1: Comparison of methods for point-null normal data tests, reproduced from Berger and Wolpert [1988].

Although we disregarded multi-modal alternative priors above, they actually form an interesting class of tests. In terms of conservativeness, their BF is easily seen to be bounded by the likelihood ratio when using a prior concentrated on the maximum likelihood solution ̂:

p(D| = 0) p(D| = ̂).

This ratio is simplify the Neyman-Pearson likelihood ratio for composite hypothesis, and it’s values are also shown in Table 1. This values are still more conservative than p-values by a factor of 2-5. Note that in a Neyman-Pearson test such as the frequentist t-test, this ratio is not read-off directly as an odds ratio as we are doing here; that’s why this NP odds ratio doesn’t agree directly with the p-value.

The interesting aspect of multi-modal priors is that they allow you to formulate a test that is more powerful for finding evidence of the null hypothesis. It can be shown that under the default BF t-test, evidence for the null accumulates very slowly when the null is true, where as evidence for the alternative accumulates exponentially fast when the alternative is true. Multi-modal priors can be formulated that fix this imbalance [Johnson and Rossell2010]. One simple case is a normal or Cauchy prior with a notch around the null = 0 set to have low probability.

If we go a step further and consider the limits of sequences of improper priors, the Bayes factor can take essentially any value. Some particular “reasonable” sequences actually yield values very similar to t-test’s p-values [Robert1993] as their posterior probabilities, at least between 0.1 and 0.01. Their construction also places 0 probability mass in the neighborhood of the null point in the limit, so it is similar to the multi-modal priors discussed in the previous paragraph. Although this improper prior does actually avoid Lindley’s paradox, it is more of a theoretical construction; it is not recommend in practice, even by it’s discoverer [Robert1993].

Consequences for Lindley’s paradox

As discussed above, when restricting ourselves to well behaved proper priors, “Lindley’s paradox” as such can not be rectified by a careful choice of a proper prior on the alternative hypothesis. When one takes a step back and considers the consequences of this, Lindley’s paradox becomes much clearer, as a statement about objective priors rather than BF tests in general.

The central idea is this: It requires a lot of evidence to detect small effects when you use vague priors. The idea of running a long experiment when you use a vague prior is nonsensical from an experimental design point of view: large amounts of data are useful for picking up small effects, and if you expect such a small effect you should encode this sensibly in your prior. The default Cauchy prior used in the BF t-test essentially says that you expect a 50% chance that the absolute effect size (Cohen’s d) is larger than 1, which is a completely implausible in many settings. For instance, if you are trying to optimize conversion rates on a website with say a 10% base conversion rate, an effect size of “1” is a 4x increase in sales! Some software defaults to 0.707 instead of 1 for the Cauchy scale parameter, but the effect is largely the same

Experimental design considerations are important in the choice of a prior, and the Bayesian justification of the Bayes factor depends on our priors being reasonable. The choice of using a particular BF test is not a full experimental design, just as for a frequentist it is not sufficient to just decide on particular test without regard to the eventual sample size. It is necessary to examine other properties of the test, so determine if it will behave reasonably in the experimental setting in which we will apply it. So in effect, Lindley’s paradox is just the statement that a poorly designed Bayesian experiment won’t agree with a less poorly designed frequentist experiment.

Put another way, if the Bayes factor is comparing two hypothesis, both of which are unlikely under the data, you won’t get reasonable results. You can’t identify this from examining the Bayes factor, you have to look at the experiment holistically. The extreme case of this is when neither hypothesis includes the true value of . In such cases, the BF will generally not converge to any particular value, and may show strong evidence in either direction.

An example

It is illustrative to discuss an example in the literature of this problem: a high profile case where the Bayesian and frequentist results are very different under a default prior, and more reasonable under a subjective prior. One such case is the study by Bem [2011] on precognition. The subject matter is controversial; Bem gives results for 9 experiments, in which 8 show evidence for the existence of precognition at the p = 0.05 level! This obviously extraordinary claim lead to extensive examination of the results from the Bayesian view-point.

In Wagenmakers et al. [2011a], the raw data of Bem [2011] is analyzed using the default Bayesian t-test. They report Bayes factors towards the alternative (in decreasing order) of 5.88, 1.81, 1.63, 1.05, 0.88, 0.58, 0.47, 0.32, 0.29, 0.13 . So only one experiment showed reasonable evidence of precognition (5.88:1 for), and if you combine the results you get 19:1 odds against an effect.

In contrast, Bem et al. [2011] performed a Bayes factor analysis using informative priors on effect sizes, uses values considered reasonable under non-precognitive circumstances. The 3 largest Bayes factors for the existence of precognition they got were 10.1, 5.3 and 4.9, indicating substantial evidence!

The interesting thing here is that the vague prior used by Wagenmakers et al. [2011a] placed a large prior probability on implausibly large effect sizes. Even strong proponents of ESP don’t believe in such large effects, as they would have been easily detected in other past similar experiments. Somewhat counter-intuitively, this belief that ESP effects are strong if they exist at all yields a weaker posterior belief in the existence of ESP!

It should be noted that Wagenmakers et al. [2011b] responded to this criticism, but somewhat unsatisfactory in my opinion. Of course, even with the use of correct subjective Bayesian priors on the alternative hypothesis, these experiments are not sufficient to convince a skeptic (such as my self) of the existence of ESP! If one factors in the prior p(H1)p(H0) and updates their beliefs correctly, you may go from a 1:1-million belief in ESP existing to a 1:76 belief (if you use the naive combined BF of 13,669 from Bem et al. [2011]).

Most reasonable people would consider the chance that a systematic unintentional error was made during the experiments to be greater than 1 in a thousand, and so not update their beliefs quite as severely. Regardless, several pre-registered replications of Bem’s experiments have failed to show any evidence of precognition [Ritchie et al.2012].

Recommendations when using point-nulls

The general consensus in the literature is that single Bayes factors should not be reported. Instead the BFs corresponding to several different possible priors for the alternative hypothesis should be reported. Only if the conclusions are stable for a wide range of priors can they be reported as definitive. If the results are not stable, then it must be argued why a particular prior, chosen subjectively, is the best choice.

Bounds giving the largest or smallest possible Bayes factor over reasonable classes of priors, such as the bounds given in Berger and Jefferys [1991], can also be reported. The consideration of multiple possible priors is generally called “robust” Bayesian analysis.

Non-point nulls

As we showed above, under essentially any point-null Bayes factor t-test, the results are much more conservative than frequentist t-tests on the same data. But is this indicative of BF tests in general? It turns out Bayes factor tests that don’t use point-nulls behave much more like their frequentists counterparts [Casella and Berger1987].

But first, lets just point out that point-null hypotheses are fundamentally weird things. When we start assigning positive probability mass to single values, we move from the comfortable realm of simple distribution functions we can plot on a whiteboard, into the Terra pericolosa that is measure theory. As anybody who has done a measure theory course knows, common sense reasoning can easily go astray even on simple sounding problems. The “surprising” Occam’s razor effect seen in point-null BF tests is largely attributable to this effect. It can be argued that any prior that concentrates mass on a single point is not “impartial”, effectively favoring that point disproportionately [Casella and Berger1987].

One-sided tests

The simplest alternative to a point null test is a postive versus negative effect test, where the null is taken to be the entire negative (or positive) portion of the real line. This test is very similar in effect to a frequentist one sided test. Unlike the point-null case, both the alternative and null hypotheses spaces are the same dimension; in practice this means we can use the same unnormalized priors on both without running into issues. This test can be preformed using the BayesFactor R package with the ttestBF method, althought somewhat indirectly, by taking the ratio of two seperate point null tests:

bfInterval < ttestBF(x = ldata, y=rdata, nullInterval=c(Inf,0)) 
bfNonPoint < bfInterval[2] / bfInterval[1] 
print(bfNonPoint)

Applying one-sided tests in the frequentist setting is routine (in fact often advocated [Cho and Abe2013]), it is surprising then that their theoretical properties are very different to two-sided tests. If we look at both Bayesian and frequentist approaches from the outside, using decision theory (with quadratic loss), we see a fundamental difference. For the one sided case, the Fisher p-value approach and obeys a weak kind of optimality known as admissibility [Hwang et al.1992]: no approach strictly dominates it. In contrast, in the two-sided case neither the frequentist or the standard Bayesian approach is admissible, and neither dominates the other!

What does this tell us about sequential testing settings? The use of Bayes factor tests is sometimes advocated in the sequential testing setting, as their interpretation is not affected by early stopping. “Sequential testing” is when a test of some form of test is applied multiple times during the experiment, with the experiment potentially stopping early if the test indicates strong evidence of an effect. Using frequentist tests sequentially is problematic; the false-positive rate is amplified and special considerations must be made to account for this. See my previous post for details on how to perform the frequentist tests correctly.

The interpretation of the Bayes factor in contrast is unaffected by early stopping. Naive application of a point-null BF test does seem to perform reasonable in a sequential setting, as it’s naturally conservative nature results in few false positives being detected. However, as we pointed out above, one-sided non-point null tests are equally valid, and they are not particularly conservative. Just applying one sequentially gives you high false-positive rates comparable to the frequentist test. The false-positive rate is a frequentist property, and as such it is quite unrelated to the Bayesian interpretation of the posterior distribution.

Indeed, in most sequential A/B testing situations, such as website design tests, the one-sided non-point test seems the most reasonable, as the two hypothesis correspond more directly to the two courses of action that will be taken after the test concludes: Switch all customers to design A or design B.

So, should I use a point null hypothesis?

Here we start to move outside the realm of mathematics and start to meander into the realm of philosophy. You have to consider the purpose of your hypothesis test. Point-nulls enforce a strong Occam’s razor effect, and depending on the purpose of your test, this may or may not be appropriate.

The examples and motivation used for null hypothesis testing in the original formulation of the problem by Jeffreys came from physics. In physics problems testing for absolute truth is a much clearer proposition. Either an exotic particle exists or it doesn’t, there is no in-between. This is far cry from the psychology, marketing and economics problems that are more commonly dealt with today, where we always expect some sort of effect, although potentially small and possibly in the opposite direction from what we guessed apriori.

You need a real belief in the possibility of zero effect in order to justify a point null test. This is a real problem in practice. In contrast, by using non-point null tests you will be in closer agreement with frequentist results, and you will need less data to make conclusions, so there is a lot of advantages to avoiding point null tests.

It is interesting to contrast modern statistical practice in high energy physics (HEP) with that of other areas where statistics is applied. It is common practice to seek “five sigma” of evidence before making a claim of a discovery in HEP fields. In contrast to the usual p 0.05 requirements, the difference is vast. Five sigma corresponds to about a 1-in-1million p value. However, if you view the five sigma approach as a heuristic approximation to the point-null BF test, the results are much more comparable. The equivalent t statistic needed for rejection of the point null under the Bayesian test scales with the amount of data. HEP experiments often average over millions of data points, and it is in that multiple-million region where the rejection threshold, when rephrased in terms of the t statistic, is roughly five.

Concluding guidelines for applying Bayes factor tests

  • Don’t use point-null Bayesian tests unless the point null hypothesis is physically plausible.
  • Always perform a proper experimental design, including the consideration of frequentist implications of your Bayesian test.
  • Try to use an informative prior if you are using a point null test; non-informative priors are both controversial and extremely conservative. Ideally show the results under multiple reasonable priors.
  • Don’t blindly apply the Bayes factor test in sequential testing situations. You need to use decision theory.

References

   Daryl J. Bem. Feeling the future: Experimental evidence for anomalous retroactive influences on cognition and affect. Journal of Personality and Social Psychology, 2011.

   Daryl J. Bem, Jessica Utts, and Wesley O. Johnson. Must psychologists change the way they analyze their data? a response to wagenmakers, wetzels, borsboom, & van der maas. 2011.

   James O. Berger and William H. Jefferys. The application of robust bayesian analysis to hypothesis testing and occam’s razor. Technical report, Purdue University, 1991.

   James O. Berger and Luis R. Pericchi. Objective bayesian methods for model selection: Introduction and comparison. IMS Lecture Notes - Monograph Series, 2001.

   J.O. Berger and R.L. Wolpert. The Likelihood Principle. Institute of Mathematical Statistics. Lecture notes : monographs series. Institute of Mathematical Statistics, 1988. ISBN 9780940600133. URL https://books.google.com.au/books?id=7fz8JGLmWbgC.

   George Casella and Roger L. Berger. Reconciling bayesian and frequentist evidence in the one-sided testing problem. Journal of the American Statistical Association, 82(397):106–111, 1987. ISSN 0162-1459. doi: 10.1080/01621459.1987.10478396.

   Hyun-Chul Cho and Shuzo Abe. Is two-tailed testing for directional research hypotheses tests legitimate? Journal of Business Research, 66(9):1261 – 1266, 2013. ISSN 0148-2963. doi: http://dx.doi.org/10.1016/j.jbusres.2012.02. 023. URL http://www.sciencedirect.com/science/article/pii/S0148296312000550. Advancing Research Methods in Marketing.

   Jiunn Tzon Hwang, George Casella, Christian Robert, Martin T. Wells, and Roger H. Farrell. Estimation of accuracy in testing. Ann. Statist., 20(1):490–509, 03 1992. doi: 10.1214/aos/1176348534. URL http://dx.doi.org/10.1214/aos/1176348534.

   Harold Jeffreys. Theory of Probability. 1939.

   V. E. Johnson and D. Rossell. On the use of non-local prior desities in bayesian hypothesis tests. Journal of the Royal Statistical Society, 2010.

   Stuart J. Ritchie, Richard Wiseman, and Christopher C. French. Failing the future: Three unsuccessful attempts to replicate bem’s ‘retroactive facilitation of recall’ effect. PLoS ONE, 2012.

   Christian P. Robert. A note on jeffreys-lindley paradox. Statistica Sinica, 1993.

   Christian P. Robert, Nicolas Chopin, and Judith Rousseau. Harold jeffreys’s theory of probability revisited. Statistical Science, 2009.

   Jeffrey N. Rouder, Paul l. Speckman, Dongchu Sun, Richard D. Morey, and Geoffrey Iverson. Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin & Review, 2009.

   Eric Jan Wagenmakers, Ruud Wetzels, Denny Borsboom, and Han van der Maas. Why psychologists must change the way they analyze their data: The case of psi. Journal of Personality and Social Psychology, 2011a.

   Eric Jan Wagenmakers, Ruud Wetzels, Denny Borsboom, and Han van der Maas. Why psychologists must change the way they analyze their data: The case of psi: Clarifications for bem, utts, and johnson (2011). Journal of Personality and Social Psychology, 2011b.

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