Bayes factor functions for reporting outcomes of hypothesis tests Valen E. Johnson1Sandipan Pramanik1Rachael Shudde1

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Bayes factor functions for reporting outcomes

of hypothesis tests

Valen E. Johnson,1∗Sandipan Pramanik,1Rachael Shudde1

1Department of Statistics, Texas A&M University,

3143 TAMU, College Station, TX 77843-3143, USA

∗To whom correspondence should be addressed; E-mail: vejohnson@tamu.edu

Bayes factors represent the ratio of probabilities assigned to data

by competing scientiﬁc hypotheses. Drawbacks of Bayes factors

are their dependence on prior speciﬁcations that deﬁne null and

alternative hypotheses and difﬁculties encountered in their com-

putation. To address these problems, we deﬁne Bayes factor func-

tions (BFF) directly from common test statistics. BFFs depend on

a single non-centrality parameter that can be expressed as a func-

tion of standardized effect sizes, and plots of BFFs versus effect

size provide informative summaries of hypothesis tests that can be

easily aggregated across studies. Such summaries eliminate the

need for arbitrary P-value thresholds to deﬁne “statistical signif-

icance.” BFFs are available in closed form and can be computed

easily from z,t,χ2, and Fstatistics.

Two approaches are commonly used to summarize evidence from statistical

hypothesis tests: P-values and Bayes factors. P-values are more frequently re-

ported. As noted in the American Statistical Association Statement on Statistical

Signiﬁcance and P-values, the signiﬁcance of many published scientiﬁc ﬁndings

are based on P-values, even though this index “is commonly misused and mis-

interpreted. This has led to some scientiﬁc journals discouraging the use of P-

values, and some scientists and statisticians recommending their abandonment ...

arXiv:2210.00049v1 [math.ST] 30 Sep 2022

Informally, a P-value is the probability under a speciﬁed statistical model that a

statistical summary of the data would be equal to or more extreme than its ob-

served value.” (1). P-values do not provide a direct measure of support for either

the null or alternative hypotheses, and their use in deﬁning arbitrary thresholds

for deﬁning statistical signiﬁcance has long been a subject of intense debate; see,

for example, (2–8). Interpreting evidence provided by P-values across replicated

studies can also be challenging.

Bayes factors represent the ratio of the marginal probability assigned to data

by competing hypotheses and, when combined with prior odds assigned between

hypotheses, yield an estimate of the posterior odds that each hypothesis is true.

That is,

posterior odds =Bayes factor ×prior odds,(1)

or, more precisely,

P(H1|x)

P(H0|x)=m1(x)

m0(x)×P(H1)

P(H0).(2)

Here, P(Hi|x)denotes the posterior probability of hypothesis Higiven data

x;P(Hi)denotes the prior probability assigned to Hi; and mi(x)denotes the

marginal probability (or probability density function) assigned to the data under

hypothesis Hi, for i= 0 (null) or i= 1 (alternative).

The marginal density of the data under the alternative hypothesis is given by

m1(x) = ZΘ

f(x|θ)π1(θ)d θ. (3)

In null hypothesis signiﬁcance tests (NHSTs), the marginal density of the data un-

der the null hypothesis, m0(x), is simply the sampling density of the data assumed

under the null hypothesis. The function π1(θ)represents the prior density for the

parameter of interest θunder the alternative hypothesis, i.e., the alternative prior

density.

The speciﬁcation of π1(θ)is problematic and, as a consequence, numerous

Bayes factors based on “default” alternative prior densities have been proposed.

These include (9–16). Nonetheless, the value of a Bayes factor depends on the

alternative prior density used in its deﬁnition, and it is generally difﬁcult to justify

or interpret any single default choice. In addition, the numerical calculation of

many Bayes factors is difﬁcult, often requiring specialized software, and each of

these problems is exacerbated in high-dimensional settings (17). A more detailed

description of the Bayesian hypothesis testing framework and controversies sur-

rounding the deﬁnition of default alternative prior densities may be found in, for

example, (18) or (19).

We propose several modiﬁcations to existing Bayes factor methodology de-

signed to enhance the report of scientiﬁc ﬁndings. First, we deﬁne Bayes factors

directly from standard z,t,χ2, and Ftest statistics (20). Under the null hypothe-

ses, the distribution of these test statistics is known. Under alternative hypotheses,

the asymptotic distributions of these test statistics often depend only on scalar-

valued non-centrality parameters. Thus, the speciﬁcation of the prior density that

deﬁnes the alternative hypothesis is simpliﬁed, and no prior densities need to be

speciﬁed under the null hypothesis.

Second, we express prior densities on non-centrality parameters as functions

of standardized effect sizes and construct Bayes Factor Functions (BFFs) that vary

with these standardized effects. BFFs thus make the connection between Bayes

factors and prior assumptions more apparent. BFFs also facilitate the integration

of evidence across multiple studies of the same phenomenon.

Third, the prior densities we propose for non-centrality parameters are special

cases of non-local alternative prior (NAP) densities. These densities are iden-

tically zero when the non-centrality parameter is zero. This property makes it

possible to more quickly accumulate evidence in favor of both true null and true

alternative hypotheses (21–23); this particular feature of NAP densities is dis-

cussed in detail in (24).

Finally, we provide closed-form expressions for Bayes factors and BFFs. These

expressions eliminate computational difﬁculties sometimes encountered when cal-

culating other Bayes factors.

1 Mathematical framework

Theorems describing default Bayes factors based on z,t,F, and χ2statistics are

provided below. In each case, the Bayes factors depend on a hyperparameter τ2.

Procedures for setting τ2as a function of standardized effect size are described in

Section 1.1. Proofs of the theorems appear in the Supplemental Material.

Notationally, we write a|b∼D(b)to indicate that a random variable ahas

distribution Dthat depends on a parameter vector b.N(a, b)denotes the normal

distribution with mean aand variance b;T(ν, λ)denotes a tdistribution with

νdegrees-of-freedom and non-centrality parameter λ;F(k, m, λ)denotes an F

distribution on k, m degrees-of-freedom and non-centrality parameter λ;G(α, λ)

denotes a gamma random variable with shape parameter αand rate parameter

λ;H(ν, λ)denotes a χ2distribution on νdegrees of freedom and non-centrality

parameter λ; and J(µ0, τ2)denotes a normal moment distribution with mean µ0

and rate parameter τ2(21). We use lower case letters to denote corresponding

densities; e.g., a gamma density evaluated at xis written g(x|α, λ).

The density of a J(µ0, τ2)random variable can be written as

j(x|µ0, τ2) = (x−µ0)2

√2πτ3exp −(x−µ0)2

2τ2.(4)

The modes of this distribution occur at x=µ0±√2τ. A plot of a normal moment

density appears in the left panel of Figure 1. If a∼J(0, τ2), then a2has a

G[3/2,1/(2τ2)] distribution, as displayed in the right panel of Figure 1.

−4 −2 0 2 4

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Normal Moment Density

0 2 4 6 8

0.05 0.10 0.15 0.20 0.25

Scaled Gamma Density

Figure 1: Normal moment prior density, J(0,1) (left panel), and scaled Gamma

density, G3

2,1

2(right panel).

The ﬁrst two theorems extend results in (24) for parametric hypothesis tests

for normally distributed data to more general cases of zand ttests. Theorems 3

and 4 provide new results for χ2and Ftests. Throughout, BF10(x|ψ)denotes

the Bayes factor in favor of the alternative hypothesis based on a test statistic x

for a hyperparameter value ψ.

Theorem 1: z test.Assume the distributions of a random variable zunder the

null and alternative hypotheses are described by

H0:z∼N(0,1),(5)

H1:z|λ∼N(λ, 1), λ |τ2∼J(0, τ2).(6)

Then the Bayes factor in favor of the alternative hypothesis is

BF10(z|τ2)=(τ2+ 1)−3/21 + τ2z2

τ2+ 1exp τ2z2

2(τ2+ 1).(7)

Theorem 2: t test.Assume the distributions of a random variable tunder the null

and alternative hypotheses are described by

H0:t∼T(ν, 0),(8)

H1:t|λ∼T(ν, λ), λ |τ2∼J(0, τ2).(9)

Then the Bayes factor in favor of the alternative hypothesis is

BF10(t|τ2)=(τ2+ 1)−3/2r

sν+1

21 + qt2

s,(10)

where

r= 1 + t2

ν, s = 1 + t2

ν(1 + τ2),and q=τ2(ν+ 1)

ν(1 + τ2).

Theorem 3: χ2test.Assume the distributions of a random variable hunder the

null and alternative hypotheses are described by

H0:h∼H(k, 0),(11)

H1:h|λ∼H(k, λ), λ |τ2∼Gk

2+ 1,1

2τ2.(12)

Then the Bayes factor in favor of the alternative hypothesis is

BF10(h|τ2)=(τ2+ 1)−k/2−11 + τ2

k(τ2+ 1)hexp τ2h

2(τ2+ 1).(13)

For k= 1 and z2=h, the Bayes factors in equation (13) has the same value

as the Bayes factor speciﬁed in equation (7). The choice of the shape parameter

as k/2 + 1 for the gamma density (a scaled χ2

k+2 random variable) in equation

(12) was based on the fact that χ2

νdistributions are not 0 at the origin for integer

degrees of freedom ν < 3. Thus, they are not NAP densities for ν < 3and

typically are not able to provide strong evidence in favor of true null hypotheses

without large sample sizes (21).

Theorem 4: F test.Assume the distributions of a random variable funder the

null and alternative hypotheses are described by

H0:f∼F(k, m, 0),(14)

H1:f|λ∼F(k, m, λ), λ |τ2∼Gk

2+ 1,1

2τ2.(15)

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BayesfactorfunctionsforreportingoutcomesofhypothesistestsValenE.Johnson,1SandipanPramanik,1RachaelShudde11DepartmentofStatistics,TexasA&MUniversity,3143TAMU,CollegeStation,TX77843-3143,USATowhomcorrespondenceshouldbeaddressed;E-mail:vejohnson@tamu.eduBayesfactorsrepresenttheratioofprobabilitiesass...

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Bayes factor functions for reporting outcomes of hypothesis tests Valen E. Johnson1Sandipan Pramanik1Rachael Shudde1

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