logasympBF {EpiForsk}

Asymptotic Bayes factors

Description

The Bayesian equivalent of a significance test for H1: an unrestricted parameter value versus H0: of a specific parameter value based only on data D can be obtained from Bayes factor (BF). Then BF = Probability(H1|D) / Probability(H0|D) and is a Bayesian equivalent of a likelihood ratio. It is based on the same asymptotics as the ubiqutous chi-square tests. This BF only depends on the difference in deviance between the models corresponding to H0 and H1 (chisquare) and the dimension d of H1. This BF is monotone in chisquare (and hence the p-value p) for fixed d. It is thus a tool to turn p-values into evidence, also retrospectively. The expression for BF depends on a parameter lambda which expresses the ratio between the information in the prior and the data (likelihood). By default lambda = min(d / chisquare, lambdamax = 0.255). Thus, as chisquare goes to infinity we effectively maximize BF and hence the evidence favoring H1, and opposite for small chisquare has a well-defined watershed where we have equal preferences for H1 and H0. The value 0.255 corresponds to a watershed of 2, that is we prefer H1 when chisquare > d * 2 and prefer H0 when chisquare < d * 2, similar to having a BF that is a continuous version of the Akaike Information Criterion for model selection. For derivations and details, see Rostgaard (2023).

Usage

logasympBF(chisq = NA, p = NA, d = 1, lambda = NA, lambdamax = 0.255)

asympBF(chisq = NA, p = NA, d = 1, lambda = NA, lambdamax = 0.255)

invlogasympBF(logasympBF = NA, d = 1, lambda = NA, lambdamax = 0.255)

invasympBF(bf, d = 1, lambda = NA, lambdamax = 0.255)

watershed(chisq)

invwatershed(lambda)

Arguments

chisq

a non-negative number denoting the difference in deviance between the statistical models corresponding to H0 and H1

p

the p value corresponding to the test statistic chisq on d degrees of freedom

d

the dimension of H1, ⁠d => 1⁠

lambda

a strictly positive number corresponding to the ratio between the information in the prior and the data

lambdamax

an upper limit on lambda

logasympBF

log(bf)

bf

Bayes factor, a strictly positive number

Details

For fixed dimension d of the alternative hypothesis H1 asympBF(.) = exp(logasympBF(.)) maps a test statistic (chisquare) or a p-value p into a Bayes factor (the ratio between the probabilities of the models corresponding to each hypothesis). asympBF(.) is monotonely increasing in chisquare, attaining the value 1 when chisquare = d * watershed. The watershed is thus a device to specify what the user considers a practical null result by choosing lambdamax = watershed(watershed).

For sufficiently large values of chisquare, lambda is estimated as d/chisquare. This behavior can be overruled by specifying lambda. Using invasympBF(.) = exp(invlogasympBF(.)) we can map a Bayes factor, bf to a value of chisquare.

Likewise, we can obtain the watershed corresponding to a given lambdamax as invwatershed(lambdamax).

Generally in these functions we recode or ignore illegal values of parameters, rather than returning an error code. chisquare is always recoded as abs(chisquare). d is set to 1 as default, and missing or entered values of d < 1 are recoded as d = 1. Entered values of lambdamax <= 0 or missing are ignored. Entered values of lambda <= 0 or missing are ignored in invwatershed(.). we use abs(lambda) as argument, lambda = 0 results in an error.

Author(s)

KLP & KIJA

References

Klaus Rostgaard (2023): Simple nested Bayesian hypothesis testing for meta-analysis, Cox, Poisson and logistic regression models. Scientific Reports. https://doi.org/10.1038/s41598-023-31838-8

Examples

# example code

# 1. the example(s) from Rostgaard (2023)
asympBF(p = 0.19, d = 8) # 0.148411
asympBF(p = 0.19, d = 8, lambdamax = 100) # 0.7922743
asympBF(p = 0.19, d = 8, lambda = 100 / 4442) # 5.648856e-05
# a maximal value of a parameter considered practically null
deltalogHR <- -0.2 * log(0.80)
sigma <- (log(1.19) - log(0.89)) / 3.92
chisq <- (deltalogHR / sigma) ** 2
chisq # 0.3626996
watershed(chisq)
# leads nowhere useful chisq=0.36<2

# 2. tests for interaction/heterogeneity - real world examples
asympBF(p = 0.26, d = 24) # 0.00034645
asympBF(p = 0.06, d = 11) # 0.3101306
asympBF(p = 0.59, d = 7) # 0.034872

# 3. other examples
asympBF(p = 0.05) # 2.082664
asympBF(p = 0.05, d = 8) # 0.8217683
chisq <- invasympBF(19)
chisq # 9.102697
pchisq(chisq, df = 1, lower.tail = FALSE) # 0.002552328
chisq <- invasympBF(19, d = 8)
chisq # 23.39056
pchisq(chisq, df = 8, lower.tail = FALSE) # 0.002897385