R: Non-Linear Hypothesis Testing

hypothesis.brmsfit {brms} | R Documentation

## Non-Linear Hypothesis Testing

### Description

Perform non-linear hypothesis testing for all model parameters.

### Usage

    ## S3 method for class 'brmsfit'
    hypothesis(
      x,
      hypothesis,
      class = "b",
      group = "",
      scope = c("standard", "ranef", "coef"),
      alpha = 0.05,
      robust = FALSE,
      seed = NULL,
      ...
    )

    hypothesis(x, ...)

    ## Default S3 method:
    hypothesis(x, hypothesis, alpha = 0.05, robust = FALSE, ...)

### Arguments

`x` | An `R` object. If it is no `brmsfit` object, it must be coercible to a `data.frame`. In the latter case, the variables used in the `hypothesis` argument need to correspond to column names of `x`, while the rows are treated as representing posterior draws of the variables.
`hypothesis` | A character vector specifying one or more non-linear hypothesis concerning parameters of the model.
`class` | A string specifying the class of parameters being tested. Default is "b" for population-level effects. Other typical options are "sd" or "cor". If `class = NULL`, all parameters can be tested against each other, but have to be specified with their full name (see also `variables`)
`group` | Name of a grouping factor to evaluate only group-level effects parameters related to this grouping factor.
`scope` | Indicates where to look for the variables specified in `hypothesis`. If `"standard"`, use the full parameter names (subject to the restriction given by `class` and `group`). If `"coef"` or `"ranef"`, compute the hypothesis for all levels of the grouping factor given in `"group"`, based on the output of `coef.brmsfit` and `ranef.brmsfit`, respectively.
`alpha` | The alpha-level of the tests (default is 0.05; see 'Details' for more information).
`robust` | If `FALSE` (the default) the mean is used as the measure of central tendency and the standard deviation as the measure of variability. If `TRUE`, the median and the median absolute deviation (MAD) are applied instead.
`seed` | A single numeric value passed to `set.seed` to make results reproducible.
`...` | Currently ignored.

### Details

Among others, `hypothesis` computes an evidence ratio (`Evid.Ratio`) for each hypothesis. For a one-sided hypothesis, this is just the posterior probability (`Post.Prob`) under the hypothesis against its alternative. That is, when the hypothesis is of the form `a > b`, the evidence ratio is the ratio of the posterior probability of `a > b` and the posterior probability of `a < b`. In this example, values greater than one indicate that the evidence in favor of `a > b` is larger than evidence in favor of `a < b`. For an two-sided (point) hypothesis, the evidence ratio is a Bayes factor between the hypothesis and its alternative computed via the Savage-Dickey density ratio method. That is the posterior density at the point of interest divided by the prior density at that point. Values greater than one indicate that evidence in favor of the point hypothesis has increased after seeing the data. In order to calculate this Bayes factor, all parameters related to the hypothesis must have proper priors and argument `sample_prior` of function `brm` must be set to `"yes"`. Otherwise `Evid.Ratio` (and `Post.Prob`) will be `NA`. Please note that, for technical reasons, we cannot sample from priors of certain parameters classes. Most notably, these include overall intercept parameters (prior class `"Intercept"`) as well as group-level coefficients. When interpreting Bayes factors, make sure that your priors are reasonable and carefully chosen, as the result will depend heavily on the priors. In particular, avoid using default priors.

The `Evid.Ratio` may sometimes be `0` or `Inf` implying very small or large evidence, respectively, in favor of the tested hypothesis. For one-sided hypotheses pairs, this basically means that all posterior draws are on the same side of the value dividing the two hypotheses. In that sense, instead of `0` or `Inf,` you may rather read it as `Evid.Ratio` smaller `1 / S` or greater `S`, respectively, where `S` denotes the number of posterior draws used in the computations.

The argument `alpha` specifies the size of the credible interval (i.e., Bayesian confidence interval). For instance, if we tested a two-sided hypothesis and set `alpha = 0.05` (5%) an, the credible interval will contain `1 - alpha = 0.95` (95%) of the posterior values. Hence, `alpha * 100`% of the posterior values will lie outside of the credible interval. Although this allows testing of hypotheses in a similar manner as in the frequentist null-hypothesis testing framework, we strongly argue against using arbitrary cutoffs (e.g., `p < .05`) to determine the 'existence' of an effect.

### Value

A `brmshypothesis` object.

### Author(s)

Paul-Christian Buerkner paul.buerkner@gmail.com

### See Also

`brmshypothesis`

### Examples

    ## Not run:
    ## define priors
    prior <- c(set_prior("normal(0,2)", class = "b"),
               set_prior("student_t(10,0,1)", class = "sigma"),
               set_prior("student_t(10,0,1)", class = "sd"))

    ## fit a linear mixed effects models
    fit <- brm(time ~ age + sex + disease + (1 + age|patient),
               data = kidney, family = lognormal(),
               prior = prior, sample_prior = "yes",
               control = list(adapt_delta = 0.95))

    ## perform two-sided hypothesis testing
    (hyp1 <- hypothesis(fit, "sexfemale = age + diseasePKD"))
    plot(hyp1)
    hypothesis(fit, "exp(age) - 3 = 0", alpha = 0.01)

    ## perform one-sided hypothesis testing
    hypothesis(fit, "diseasePKD + diseaseGN - 3 < 0")

    hypothesis(fit, "age < Intercept",
               class = "sd", group  = "patient")

    ## test the amount of random intercept variance on all variance
    h <- paste("sd_patient__Intercept^2 / (sd_patient__Intercept^2 +",
               "sd_patient__age^2 + sigma^2) = 0")
    (hyp2 <- hypothesis(fit, h, class = NULL))
    plot(hyp2)

    ## test more than one hypothesis at once
    h <- c("diseaseGN = diseaseAN", "2 * diseaseGN - diseasePKD = 0")
    (hyp3 <- hypothesis(fit, h))
    plot(hyp3, ignore_prior = TRUE)

    ## compute hypotheses for all levels of a grouping factor
    hypothesis(fit, "age = 0", scope = "coef", group = "patient")

    ## use the default method
    dat <- as.data.frame(fit)
    str(dat)
    hypothesis(dat, "b_age > 0")

    ## End(Not run)

[Package brms version 2.17.0 Index]