The Bayesian counterpart of the paired t test, the analysis of
bayes_one_sample_t applied to the within-pair differences.
It reports the posterior of the standardized difference
\(\delta = \mu_D / \sigma_D\) under the default
Jeffreys-Zellner-Siow (JZS) prior, a Cauchy prior on \(\delta\) with
Jeffreys priors on the nuisance parameters (the variance), summarized by
its median, mean, a credible interval, and the probability statement
\(P(\delta > 0 \mid \mathrm{data})\). The JZS default Bayes factor
(Rouder, Speckman, Sun, Morey, & Iverson, 2009) is also reported. See
bayes_one_sample_t for the model, the package's
interpretive stance, and the computational details (exact quadrature,
no Monte Carlo error).
Usage
bayes_paired_t(
x = NULL,
y = NULL,
mean_diff = NULL,
sd_diff = NULL,
n = NULL,
prior_location = 0,
prior_scale = sqrt(2)/2,
prior_mean = NULL,
prior_sd = NULL,
conf_level = 0.95
)Arguments
- x, y
Numeric vectors of paired observations, the same length, in matching order. The analysis is of
x - y. Omit both to supply summary statistics of the differences instead.- mean_diff, sd_diff, n
Summary statistics of the paired differences: their mean, their standard deviation, and the number of pairs. These are the quantities a paper's paired t test reports. Supply either raw data or all three summary values, never both.
- prior_location
Location of the Cauchy prior on \(\delta\). Defaults to 0, the JZS prior; a nonzero value centers the prior on an expected effect (Gronau, Ly, & Wagenmakers, 2020).
- prior_scale
The way to adjust the prior. It is the scale (width) of the Cauchy prior on the standardized effect \(\delta\) (the JZS prior), so a user who wants a more or less informative prior sets
prior_scale: larger values say larger effects are plausible a priori, smaller values concentrate the prior near zero. The default \(\sqrt{2}/2 \approx 0.707\) is the JZS “medium” prior. Fully custom or subjective priors beyond the Cauchy family are not supported by the BayesFactor engine.- prior_mean, prior_sd
Mean and standard deviation of a normal prior on \(\delta\), for prior beliefs stated as moments. Supplying them selects the normal prior; they cannot be combined with the Cauchy arguments. See the prior section of Details.
- conf_level
Probability mass of the credible interval.
Value
A data.frame (class dmar_tbl) with the same
rows as bayes_one_sample_t, where \(\delta\) is the
standardized within-pair difference and the raw rows are on the
difference scale; n is the number of pairs.
Specifying the prior. The default prior on the standardized
effect \(\delta\) is the JZS Cauchy centered at zero. Its
prior_scale \(r\) is not a standard deviation: a Cauchy has
no mean and no variance (those integrals diverge), so beliefs stated
as prior moments cannot be expressed through it. What the scale does
fix is the quartiles: half the prior mass lies within
\(\pm r\) of the location, so the default \(r = \sqrt{2}/2\)
says a 50 percent prior bet that \(|\delta| < 0.71\). A directional
prior keeps the Cauchy and moves prior_location (Gronau, Ly, &
Wagenmakers, 2020). A researcher who thinks in prior moments instead
sets prior_mean and prior_sd, which use a normal prior
with exactly those moments; the two families are exclusive.
The families are also linked by an exact identity: a Cauchy with
location \(\mu\) and scale \(r\) is a normal prior
\(N(\mu, r^2/z^2)\) whose \(z\) is standard normal, that is, a
normal prior whose variance you are not sure of. Choosing the Cauchy
is therefore choosing a normal prior with built-in doubt about its
own width, which is why its tails are heavier and its Bayes factors
more conservative. A normal matched to the Cauchy's interquartile
range has prior_sd = 1.4826 * prior_scale. The full posterior
of \(\delta\) is returned in the "posterior" attribute as a
data frame of delta and density, so any posterior
probability, not only the reported ones, can be computed from it.
References
Gronau, Q. F., Ly, A., & Wagenmakers, E.-J. (2020). Informed Bayesian t-tests. The American Statistician, 74(2), 137–143. doi:10.1080/00031305.2018.1562983
Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin & Review, 16(2), 225–237. doi:10.3758/PBR.16.2.225
Jeffreys, H. (1961). Theory of probability (3rd ed.). Oxford University Press.
Zellner, A., & Siow, A. (1980). Posterior odds ratios for selected regression hypotheses. In J. M. Bernardo, M. H. DeGroot, D. V. Lindley, & A. F. M. Smith (Eds.), Bayesian statistics: Proceedings of the First International Meeting (pp. 585–603). University of Valencia Press.
See also
bayes_one_sample_t for the model and stance;
bayes_independent_t for unpaired groups;
probability_of_superiority_paired and
randomization_test_paired for other paired analyses.
Other Bayesian t analyses:
bayes_independent_t(),
bayes_one_sample_t()
Author
Ken Kelley kkelley@nd.edu
Examples
set.seed(113)
before <- rnorm(30, 100, 12)
after <- before + rnorm(30, 3, 6)
bayes_paired_t(after, before)
#> term value
#> delta_posterior_median 0.624
#> delta_posterior_mean 0.626
#> delta_lower 0.237
#> delta_upper 1.02
#> p_delta_positive 0.999
#> raw_posterior_median 3.85
#> raw_lower 1.47
#> raw_upper 6.31
#> bf_10 35.8
#> bf_01 0.0279
#> t 3.69
#> df 29
#> prior_location 0
#> prior_scale 0.707
#> n 30
#>
#> Confidence level: 95%