In this vignette, we will explain how to compute a Bayes factor for informed hypotheses on multiple binomial parameters.

## Model and Data

As example for an inequality-constrained hypothesis on multiple independent binomials, we will use the dataset, journals, which is included in the package multibridge. The dataset is based on a study by Nuijten et al. (2016) which evaluated the prevalence of statistical reporting errors (i.e., inconsistencies between the reported test statistic and degrees of freedom, and the reported p-value) in the field of psychology. In total, the dataset contains a summary of statistical reporting errors of 16,695 research articles reporting results from null hypothesis significance testing (NHST). The selected articles were published in eight major journals in psychology between 1985 to 2013: Developmental Psychology (DP), Frontiers in Psychology (FP), the Journal of Applied Psychology (JAP), the Journal of Consulting and Clinical Psychology (JCCP), Journal of Experimental Psychology: General (JEPG), the Journal of Personality and Social Psychology (JPSP), the Public Library of Science (PLoS), and Psychological Science (PS).

# load the package and data
library(multibridge)
data(journals)
journals
##   journal articles_downloaded articles_with_NHST perc_articles_with_NHST
## 1      DP                3379               2607                    77.2
## 2      FP                2139                702                    32.8
## 3     JAP                2782               1638                    58.9
## 4    JCCP                3519               2413                    68.6
## 5    JEPG                1184                821                    69.3
## 6    JPSP                5108               4346                    85.1
## 7    PLOS               10299               2487                    24.1
## 8      PS                2307               1681                    72.9
##   nr_NHST mean_nr_NHST_per_article_with_NHST
## 1   37658                              14.44
## 2   10149                              14.46
## 3   15134                               9.24
## 4   27429                              11.37
## 5   18921                              23.05
## 6  101621                              23.38
## 7   31539                              12.68
## 8   15654                               9.31
##   mean_nr_NHST_per_article_all_included errors dec_errors perc_errors
## 1                                 11.14   3821        654       10.15
## 2                                  4.74   1045        134       10.30
## 3                                  5.44   1146        302        7.57
## 4                                  7.79   3396        463       12.38
## 5                                 15.98   1495        210        7.90
## 6                                 19.89   8484       1139        8.35
## 7                                  3.06   4151        488       13.16
## 8                                  6.79   1423        191        9.09
##   perc_dec_errors perc_articles_with_errors perc_articles_with_dec_errors
## 1            1.74                     50.90                         15.15
## 2            1.32                     50.85                         11.97
## 3            2.00                     33.64                         12.45
## 4            1.69                     48.90                         11.56
## 5            1.11                     54.81                         12.91
## 6            1.12                     57.62                         15.81
## 7            1.55                     49.70                         11.50
## 8            1.22                     39.74                          6.48
##   APAfactor
## 1 0.7827710
## 2 0.7408296
## 3 0.7321825
## 4 0.7710022
## 5 0.8031367
## 6 0.8106632
## 7 0.6701959
## 8 0.7795777

The model that we will use assumes that the elements in the vector of successes $$x_1, \cdots, x_K$$ and the elements in the vector of total number of observations $$n_1, \cdots, n_K$$ in the $$K$$ categories follow independent binomial distributions. The parameter vector of the binomial success probabilities, $$\theta_1, \cdots, \theta_K$$, contains the probabilities of observing a value in a particular category; here, it reflects the probabilities of a statistical reporting error in one of the 8 journals. The parameter vector $$\theta_1, \cdots, \theta_K$$ are drawn from independent beta distributions with parameters $$\alpha_1, \cdots, \alpha_K$$ and $$\beta_1, \cdots, \beta_K$$. The model can be described as follows:

\begin{align} x_1 \cdots x_K & \sim \prod_{k = 1}^K \text{Binomial}(\theta_k, n_k) \\ \theta_1 \cdots \theta_K &\sim \prod_{k = 1}^K \text{Beta}(\alpha_k, \beta_k) \\ \end{align}

Here, we test the inequality-constrained hypothesis $$\mathcal{H}_r$$ formulated by Nuijten et al. (2016) that the prevalence for statistical reporting errors for articles published in social psychology journals (i.e., JPSP) is higher than for articles published in other journals. We will test this hypothesis against the encompassing hypothesis $$\mathcal{H}_e$$ without any constraints. In addition, this hypothesis will be tested against the null hypothesis $$\mathcal{H}_0$$ that all journals have the same prevalence to include an article with a statistical reporting error:

\begin{align*} \mathcal{H}_r &: (\theta_{\text{DP}}, \theta_{\text{FP}}, \theta_{\text{JAP}} , \theta_{\text{JCCP}} , \theta_{\text{JEPG}} , \theta_{\text{PLoS}}, \theta_{\text{PS}}) < \theta_{\text{JPSP}} \\ \mathcal{H}_e &: \theta_{\text{DP}} , \theta_{\text{FP}} , \cdots , \theta_{\text{JPSP}}\\ \mathcal{H}_0 &: \theta_{\text{DP}} = \theta_{\text{FP}} = \cdots = \theta_{\text{JPSP}}. \end{align*}

To evaluate the inequality-constrained hypothesis, we need to specify (1) a vector with observed successes, and (2) a vector containing the total number of observations, (3) the informed hypothesis, (4) a vector with prior parameters alpha for each binomial proportion, (5) a vector with prior parameters beta for each binomial proportion, and (6) the labels of the categories of interest (i.e., journal names):

# since percentages are rounded to two decimal values, we round the articles
# with an error to receive integer values (step 1)
x <- round(journals$articles_with_NHST * (journals$perc_articles_with_errors/100))
# total number of articles (step 2)
n <- journals$articles_with_NHST # Specifying the informed Hypothesis (step 3) Hr <- c('JAP , PS , JCCP , PLOS , DP , FP , JEPG < JPSP') # Prior specification (step 4 and 5) # We assign a uniform beta distribution to each binomial propotion a <- rep(1, 8) b <- rep(1, 8) # categories of interest (step 6) journal_names <- journals$journal

## Analysis of BFre

With this information, we can now conduct the analysis with the function binom_bf_informed(). Since we are interested in quantifying evidence in favor of the restricted hypothesis, we set the Bayes factor type to BFre. For reproducibility, we are also setting a seed with the argument seed:

ineq_results <- multibridge::binom_bf_informed(x=x, n=n, Hr=Hr, a=a, b=b,
factor_levels=journal_names,
bf_type = 'BFre',
seed = 2020)
m1 <- summary(ineq_results)
m1
## Bayes factor analysis
##
##  Hypothesis H_e:
##  All parameters are free to vary.
##
##  Hypothesis H_r:
##  JAP , PS , JCCP , PLOS , DP , FP , JEPG < JPSP
##
## Bayes factor estimate BFre:
##
## 7.4289
##
## Based on 1 independent inequality-constrained hypothesis.
##
## Relative Mean-Square Error:
##
## 0.066
##
## Posterior Median and Credible Intervals Of Marginal Beta Distributions:
##           alpha     beta  2.5%   50% 97.5%
## 1  JAP 1 + 551  1 + 1087 0.314 0.337 0.360
## 2   PS 1 + 668  1 + 1013 0.374 0.397 0.421
## 3 JCCP 1 + 1180 1 + 1233 0.469 0.489 0.509
## 4 PLOS 1 + 1236 1 + 1251 0.477 0.497 0.517
## 5   DP 1 + 1327 1 + 1280 0.490 0.509 0.528
## 6   FP 1 + 357  1 + 345  0.472 0.509 0.545
## 7 JEPG 1 + 450  1 + 371  0.514 0.548 0.582
## 8 JPSP 1 + 2504 1 + 1842 0.561 0.576 0.591

From the summary of the results we can see that the overall relative mean-square error of $$0.066$$ is quite high, which might suggest unstable results. This becomes apparent if we look at the result as percentage error which can be extracted from the output object:

ineq_results$bf_list$error_measures
##          re2        cv percentage
## 1 0.06597936 0.2568645   25.6864%

Here the percentage error is at almost 25.6864%. For that reason, we will rerun the binom_bf_informed() with more samples.

ineq_results <- multibridge::binom_bf_informed(x=x, n=n, Hr=Hr, a=a, b=b,
factor_levels=journal_names,
bf_type = 'BFre',
seed = 2020,
niter = 2e4)
m2 <- summary(ineq_results)

With more samples, the percentage error is considerably smaller:

ineq_results$bf_list$error_measures
##           re2         cv percentage
## 1 0.003796574 0.06161634    6.1616%

Now, the overall relative mean-square error is $$0.0038$$, which translates to a percentage error of 6.1616%.

The data are more likely under the informed hypothesis than under the alternative hypothesis; in fact we collected moderate evidence for the informed hypothesis. The results suggest that the data are 7.29 more likely under the informed hypothesis than under the hypothesis that all parameters are free to vary.

## Analysis of BFr0

If we would like to compare the inequality-constrained hypothesis $$\mathcal{H}_r$$ against the null hypothesis $$\mathcal{H}_0$$ which states that the probability for a statistical reporting error is equal across all journals, we can set the Bayes factor type in binom_bf_equality() to BFr0.

eq_results      <- multibridge::binom_bf_informed(x=x, n=n, Hr=Hr, a=a, b=b,
factor_levels=journal_names,
bf_type = 'BFr0',
seed = 2020,
niter = 2e4)
m3 <- summary(eq_results)
m3
## Bayes factor analysis
##
##  Hypothesis H_0:
##  All parameters are exactly equal.
##
##  Hypothesis H_r:
##  JAP , PS , JCCP , PLOS , DP , FP , JEPG < JPSP
##
## Bayes factor estimate BFr0:
##
## 5.3802e+68
##
## Based on 1 independent inequality-constrained hypothesis.
##
## Relative Mean-Square Error:
##
## 0.0038
##
## Posterior Median and Credible Intervals Of Marginal Beta Distributions:
##           alpha     beta  2.5%   50% 97.5%
## 1  JAP 1 + 551  1 + 1087 0.314 0.337 0.360
## 2   PS 1 + 668  1 + 1013 0.374 0.397 0.421
## 3 JCCP 1 + 1180 1 + 1233 0.469 0.489 0.509
## 4 PLOS 1 + 1236 1 + 1251 0.477 0.497 0.517
## 5   DP 1 + 1327 1 + 1280 0.490 0.509 0.528
## 6   FP 1 + 357  1 + 345  0.472 0.509 0.545
## 7 JEPG 1 + 450  1 + 371  0.514 0.548 0.582
## 8 JPSP 1 + 2504 1 + 1842 0.561 0.576 0.591

The resulting Bayes factor that compares the informed to the null hypothesis provides extreme evidence for the informed hypothesis; the data are 5.3810^{68} more likely under the informed hypothesis than under the null hypothesis. But since the data provided only moderate evidence against the encompassing hypotheses (i.e., BFre=7.29), it would be more sensible to say that this result suggests a misspecification of the null model rather than a well specified informed hypothesis.

# References

Nuijten, Michèle B, Chris HJ Hartgerink, Marcel ALM van Assen, Sacha Epskamp, and Jelte M Wicherts. 2016. “The Prevalence of Statistical Reporting Errors in Psychology (1985–2013).” Behavior Research Methods 48: 1205–26.