# bayesfactor_parameters: Bayes Factors (BF) for a Single Parameter In DominiqueMakowski/bayestestR: Understand and Describe Bayesian Models and Posterior Distributions

## Description

This method computes Bayes factors against the null (either a point or an interval), based on prior and posterior samples of a single parameter. This Bayes factor indicates the degree by which the mass of the posterior distribution has shifted further away from or closer to the null value(s) (relative to the prior distribution), thus indicating if the null value has become less or more likely given the observed data.

When the null is an interval, the Bayes factor is computed by comparing the prior and posterior odds of the parameter falling within or outside the null interval (Morey & Rouder, 2011; Liao et al., 2020); When the null is a point, a Savage-Dickey density ratio is computed, which is also an approximation of a Bayes factor comparing the marginal likelihoods of the model against a model in which the tested parameter has been restricted to the point null (Wagenmakers et al., 2010; Heck, 2019).

Note that the `logspline` package is used for estimating densities and probabilities, and must be installed for the function to work.

`bayesfactor_pointnull()` and `bayesfactor_rope()` are wrappers around `bayesfactor_parameters` with different defaults for the null to be tested against (a point and a range, respectively). Aliases of the main functions are prefixed with `bf_*`, like `bf_parameters()` or `bf_pointnull()`.

For more info, in particular on specifying correct priors for factors with more than 2 levels, see the Bayes factors vignette.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111``` ```bayesfactor_parameters( posterior, prior = NULL, direction = "two-sided", null = 0, verbose = TRUE, ... ) bayesfactor_pointnull( posterior, prior = NULL, direction = "two-sided", null = 0, verbose = TRUE, ... ) bayesfactor_rope( posterior, prior = NULL, direction = "two-sided", null = rope_range(posterior), verbose = TRUE, ... ) bf_parameters( posterior, prior = NULL, direction = "two-sided", null = 0, verbose = TRUE, ... ) bf_pointnull( posterior, prior = NULL, direction = "two-sided", null = 0, verbose = TRUE, ... ) bf_rope( posterior, prior = NULL, direction = "two-sided", null = rope_range(posterior), verbose = TRUE, ... ) ## S3 method for class 'numeric' bayesfactor_parameters( posterior, prior = NULL, direction = "two-sided", null = 0, verbose = TRUE, ... ) ## S3 method for class 'stanreg' bayesfactor_parameters( posterior, prior = NULL, direction = "two-sided", null = 0, verbose = TRUE, effects = c("fixed", "random", "all"), component = c("conditional", "location", "smooth_terms", "sigma", "zi", "zero_inflated", "all"), parameters = NULL, ... ) ## S3 method for class 'brmsfit' bayesfactor_parameters( posterior, prior = NULL, direction = "two-sided", null = 0, verbose = TRUE, effects = c("fixed", "random", "all"), component = c("conditional", "location", "smooth_terms", "sigma", "zi", "zero_inflated", "all"), parameters = NULL, ... ) ## S3 method for class 'blavaan' bayesfactor_parameters( posterior, prior = NULL, direction = "two-sided", null = 0, verbose = TRUE, ... ) ## S3 method for class 'data.frame' bayesfactor_parameters( posterior, prior = NULL, direction = "two-sided", null = 0, verbose = TRUE, ... ) ```

## Arguments

 `posterior` A numerical vector, `stanreg` / `brmsfit` object, `emmGrid` or a data frame - representing a posterior distribution(s) from (see 'Details'). `prior` An object representing a prior distribution (see 'Details'). `direction` Test type (see 'Details'). One of `0`, `"two-sided"` (default, two tailed), `-1`, `"left"` (left tailed) or `1`, `"right"` (right tailed). `null` Value of the null, either a scalar (for point-null) or a range (for a interval-null). `verbose` Toggle off warnings. `...` Arguments passed to and from other methods. (Can be used to pass arguments to internal `logspline::logspline()`.) `effects` Should results for fixed effects, random effects or both be returned? Only applies to mixed models. May be abbreviated. `component` Should results for all parameters, parameters for the conditional model or the zero-inflated part of the model be returned? May be abbreviated. Only applies to brms-models. `parameters` Regular expression pattern that describes the parameters that should be returned. Meta-parameters (like `lp__` or `prior_`) are filtered by default, so only parameters that typically appear in the `summary()` are returned. Use `parameters` to select specific parameters for the output.

## Details

This method is used to compute Bayes factors based on prior and posterior distributions.

#### One-sided & Dividing Tests (setting an order restriction)

One sided tests (controlled by `direction`) are conducted by restricting the prior and posterior of the non-null values (the "alternative") to one side of the null only (Morey & Wagenmakers, 2014). For example, if we have a prior hypothesis that the parameter should be positive, the alternative will be restricted to the region to the right of the null (point or interval). For example, for a Bayes factor comparing the "null" of `0-0.1` to the alternative `>0.1`, we would set `bayesfactor_parameters(null = c(0, 0.1), direction = ">")`.

It is also possible to compute a Bayes factor for dividing hypotheses - that is, for a null and alternative that are complementary, opposing one-sided hypotheses (Morey & Wagenmakers, 2014). For example, for a Bayes factor comparing the "null" of `<0` to the alternative `>0`, we would set `bayesfactor_parameters(null = c(-Inf, 0))`.

## Value

A data frame containing the (log) Bayes factor representing evidence against the null.

## Setting the correct `prior`

For the computation of Bayes factors, the model priors must be proper priors (at the very least they should be not flat, and it is preferable that they be informative); As the priors for the alternative get wider, the likelihood of the null value(s) increases, to the extreme that for completely flat priors the null is infinitely more favorable than the alternative (this is called the Jeffreys-Lindley-Bartlett paradox). Thus, you should only ever try (or want) to compute a Bayes factor when you have an informed prior.

(Note that by default, `brms::brm()` uses flat priors for fixed-effects; See example below.)

It is important to provide the correct `prior` for meaningful results.

• When `posterior` is a numerical vector, `prior` should also be a numerical vector.

• When `posterior` is a `data.frame`, `prior` should also be a `data.frame`, with matching column order.

• When `posterior` is a `stanreg` or `brmsfit` model:

• `prior` can be set to `NULL`, in which case prior samples are drawn internally.

• `prior` can also be a model equivalent to `posterior` but with samples from the priors only. See `unupdate()`.

• Note: When `posterior` is a `brmsfit_multiple` model, `prior` must be provided.

• When `posterior` is an `emmGrid` object:

• `prior` should be the `stanreg` or `brmsfit` model used to create the `emmGrid` objects.

• `prior` can also be an `emmGrid` object equivalent to `posterior` but created with a model of priors samples only.

• Note: When the `emmGrid` has undergone any transformations (`"log"`, `"response"`, etc.), or `regrid`ing, then `prior` must be an `emmGrid` object, as stated above.

## Interpreting Bayes Factors

A Bayes factor greater than 1 can be interpreted as evidence against the null, at which one convention is that a Bayes factor greater than 3 can be considered as "substantial" evidence against the null (and vice versa, a Bayes factor smaller than 1/3 indicates substantial evidence in favor of the null-model) (Wetzels et al. 2011).

## Note

There is also a `plot()`-method implemented in the see-package.

## Author(s)

Mattan S. Ben-Shachar

## References

• Wagenmakers, E. J., Lodewyckx, T., Kuriyal, H., and Grasman, R. (2010). Bayesian hypothesis testing for psychologists: A tutorial on the Savage-Dickey method. Cognitive psychology, 60(3), 158-189.

• Heck, D. W. (2019). A caveat on the Savage–Dickey density ratio: The case of computing Bayes factors for regression parameters. British Journal of Mathematical and Statistical Psychology, 72(2), 316-333.

• Morey, R. D., & Wagenmakers, E. J. (2014). Simple relation between Bayesian order-restricted and point-null hypothesis tests. Statistics & Probability Letters, 92, 121-124.

• Morey, R. D., & Rouder, J. N. (2011). Bayes factor approaches for testing interval null hypotheses. Psychological methods, 16(4), 406.

• Liao, J. G., Midya, V., & Berg, A. (2020). Connecting and contrasting the Bayes factor and a modified ROPE procedure for testing interval null hypotheses. The American Statistician, 1-19.

• Wetzels, R., Matzke, D., Lee, M. D., Rouder, J. N., Iverson, G. J., and Wagenmakers, E.-J. (2011). Statistical Evidence in Experimental Psychology: An Empirical Comparison Using 855 t Tests. Perspectives on Psychological Science, 6(3), 291–298. doi: 10.1177/1745691611406923

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37``` ```library(bayestestR) if (require("logspline")) { prior <- distribution_normal(1000, mean = 0, sd = 1) posterior <- distribution_normal(1000, mean = .5, sd = .3) bayesfactor_parameters(posterior, prior) } ## Not run: # rstanarm models # --------------- if (require("rstanarm") && require("emmeans") && require("logspline")) { contrasts(sleep\$group) <- contr.orthonorm # see vingette stan_model <- stan_lmer(extra ~ group + (1 | ID), data = sleep) bayesfactor_parameters(stan_model) bayesfactor_parameters(stan_model, null = rope_range(stan_model)) # emmGrid objects # --------------- group_diff <- pairs(emmeans(stan_model, ~group)) bayesfactor_parameters(group_diff, prior = stan_model) } # brms models # ----------- if (require("brms")) { contrasts(sleep\$group) <- contr.orthonorm # see vingette my_custom_priors <- set_prior("student_t(3, 0, 1)", class = "b") + set_prior("student_t(3, 0, 1)", class = "sd", group = "ID") brms_model <- brm(extra ~ group + (1 | ID), data = sleep, prior = my_custom_priors ) bayesfactor_parameters(brms_model) } ## End(Not run) ```

DominiqueMakowski/bayestestR documentation built on July 27, 2021, 4:12 p.m.