Author: Maximilian Linde
Date: 2021-03-26

# Introduction to baymedr

#### Maximilian Linde

#### 2021-03-26

# baymedr: BAYesian inference for MEDical designs in R

`baymedr` is an R package with the goal of providing researchers with easy-to-use tools for the computation of Bayes factors for common biomedical research designs (see van Ravenzwaaij et al., 2019). Implemented are functions to test the equivalence (`equiv_bf()`), non-inferiority (`infer_bf()`), and superiority (`super_bf()`) of an experimental group (e.g., a new medication) compared to a control group (e.g., a placebo or an already existing medication) on a continuous dependent variable. A special focus of `baymedr` lies on a user-friendly interface, so that a wide variety or researchers (i.e., not only statisticians) can utilize `baymedr` for their analyses.

## General usage

All three functions for the three research designs (i.e., equivalence, non-inferiority, and superiority) allow the user to compute Bayes factors based on raw data (if arguments `x` and `y` are defined) or summary statistics (if arguments `n_x`, `n_y`, `mean_x`, `mean_y`, `sd_x`, and `sd_y` are defined). If summary statistics are used, the user has the option to specify `ci_margin` and `ci_level` instead of `sd_x` and `sd_y`. In general, arguments with ‘x’ as a name or suffix correspond to the control group and those with ‘y’ as a name or suffix refer to the experimental group. Importantly, the dependent variable must be continuous in order to obtain valid results.

Usage of the functions for equivalence (`equiv_bf()`), non-inferiority (`infer_bf()`), and superiority designs (`super_bf()`), results in S4 objects of classes `baymedrEquivalence`, `baymedrNonInferiority`, and `baymedrSuperiority`, respectively. Summary information are shown in the console by printing the created S4 object. To extract the Bayes factor from one of the three S4 objects, use the function `get_bf()`.

The Bayes factors resulting from `super_bf()` and `infer_bf()` quantify evidence in favor of the data under the alternative hypothesis (i.e., superiority and non-inferiority, respectively) relative to the data under the null hypothesis. In contrast, the Bayes factor resulting from `equiv_bf()` quantifies evidence in favor of the data under the null hypothesis (i.e., equivalence) relative to the data under the alternative hypothesis. In case the evidence for the data under the other hypothesis is desired, the user can take the reciprocal of the Bayes factor.

## The Cauchy prior distribution

Bayesian inference requires the specification of a prior distribution, which mirrors prior beliefs about the likelihood of parameter values. For the equivalence, non-inferiority, and superiority tests, the parameter of interest is the effect size between the experimental and control conditions (see, e.g., Rouder et al., 2009; van Ravenzwaaij et al., 2019). If relevant information is available, this knowledge could be expressed in an idiosyncratic prior distribution. Most of the time, however, relevant information is missing. In that case, it is reasonable to define a prior distribution that is as objective as possible. It has been argued that the Cauchy probability density function centered on 0 represents such a function (see, e.g., Rouder et al., 2009). The standard Cauchy distribution resembles a standard Normal distribution, except that the Cauchy distribution has less mass at the center but instead heavier tails. The center of the distribution is determined by the location parameter, while the width is specified by the scale parameter. By varying the scale of the Cauchy prior, the user can change the range of reasonable effect sizes. This is accomplished with the argument `prior_scale`.

## Random example data

In order to demonstrate the three functions within `baymedr`, we create an example dataset (data). There is a control group “con” and an experimental group “exp” (condition). Further, random numbers, sampled from the Normal distribution, within each group are created, serving as the dependent variable of interest (dv):

    set.seed(123456789)

    data <- data.frame(
      condition = rep(x = c("con", "exp"),
                      c(150, 180)),
      dv = c(rnorm(n = 150,
                   mean = 7.3,
                   sd = 3.4),
             rnorm(n = 180,
                   mean = 8.9,
                   sd = 3.1))
    )

    data[c(1:5, 151:155), ]
    #>     condition        dv
    #> 1         con  9.016566
    #> 2         con  8.645978
    #> 3         con 12.112828
    #> 4         con  4.844097
    #> 5         con  5.197586
    #> 151       exp  8.541017
    #> 152       exp 10.693952
    #> 153       exp 12.782147
    #> 154       exp  9.094879
    #> 155       exp  7.347376

## The superiority test (`super_bf()`)

With `super_bf()` we can test whether the experimental group is better than the control group. Importantly, sometimes low and sometimes high values on the measure of interest represent superiority, which can be specified with the argument `direction`. In the case where low values represent superiority we have BF-0, indicating that we quantify evidence for the data under the negative alternative hypothesis (i.e., H-) relative to the null hypothesis (i.e., H0). In the case where high values represent superiority we have BF+0, indicating that we quantify evidence for the data under the positive alternative hypothesis (i.e., H+) relative to the null hypothesis (i.e., H0). The default is that high values represent superiority.

We can use the raw data to compute a Bayes factor:

    mod_super_raw <- super_bf(
      x = data$dv[data$condition == "con"],
      y = data$dv[data$condition == "exp"]
    )

    mod_super_raw
    #> ******************************
    #> Superiority analysis
    #> --------------------
    #> Data:                         raw data
    #> H0 (non-superiority):         mu_y == mu_x
    #> H+ (superiority):             mu_y > mu_x
    #> Cauchy prior scale:           0.707
    #>
    #>     BF+0 (superiority) = 44.00
    #> ******************************

    get_bf(object = mod_super_raw)
    #> [1] 44.00176

Alternatively, if the raw data are not available, we can use summary statistics to compute a Bayes factor (cf. van Ravenzwaaij et al., 2019). The data were obtained from Skjerven et al. (2013):

    mod_super_sum <- super_bf(
      n_x = 201,
      n_y = 203,
      mean_x = 68.1,
      mean_y = 63.6,
      ci_margin = (15.5 - (-6.5)) / 2,
      ci_level = 0.95,
      direction = "low"
    )

    mod_super_sum
    #> ******************************
    #> Superiority analysis
    #> --------------------
    #> Data:                         summary data
    #> H0 (non-superiority):         mu_y == mu_x
    #> H- (superiority):             mu_y < mu_x
    #> Cauchy prior scale:           0.707
    #>
    #>     BF-0 (superiority) = 0.24
    #> ******************************

    get_bf(object = mod_super_sum)
    #> [1] 0.2364177

## The equivalence test (`equiv_bf()`)

With `equiv_bf()` we can test whether the experimental and the control groups are (practically) equivalent. With the argument `interval`, an equivalence interval can be specified. The argument `interval_std` can be used to specify whether the equivalence interval is given in standardized (TRUE; the default) or unstandardized (FALSE) units. However, in contrast to the frequentist equivalence test, `equiv_bf()` can also incorporate a point null hypothesis, which constitutes the default in `equiv_bf()` (i.e., `interval` = 0). The Bayes factor (i.e., BF01) resulting from `equiv_bf()` quantifies evidence for the data under the null hypothesis (i.e., H0) relative to the two-sided alternative hypothesis (i.e., H1).

We can use the raw data to compute a Bayes factor:

    mod_equiv_raw <- equiv_bf(
      x = data$dv[data$condition == "con"],
      y = data$dv[data$condition == "exp"],
      interval = 0.1,
    )

    mod_equiv_raw
    #> ******************************
    #> Equivalence analysis
    #> --------------------
    #> Data:                         raw data
    #> H0 (equivalence):             mu_y - mu_x > c_low AND mu_y - mu_x < c_high
    #> H1 (non-equivalence):         mu_y - mu_x < c_low OR mu_y - mu_x > c_high
    #> Equivalence interval:         Lower = -0.10; Upper = 0.10 (standardised)
    #>                               Lower = -0.33; Upper = 0.33 (unstandardised)
    #> Cauchy prior scale:           0.707
    #>
    #>     BF01 (equivalence) = 0.11
    #> ******************************

    get_bf(object = mod_equiv_raw)
    #> [1] 0.108721

Alternatively, if the raw data are not available, we can use summary statistics to compute a Bayes factor (cf. van Ravenzwaaij et al., 2019). The data were obtained from Steiner et al. (2015):

    mod_equiv_sum <- equiv_bf(
      n_x = 560,
      n_y = 538,
      mean_x = 8.683,
      mean_y = 8.516,
      sd_x = 3.6,
      sd_y = 3.6
    )

    mod_equiv_sum
    #> ******************************
    #> Equivalence analysis
    #> --------------------
    #> Data:                         summary data
    #> H0 (equivalence):             mu_y == mu_x
    #> H1 (non-equivalence):         mu_y != mu_x
    #> Equivalence interval:         Lower = -0.00; Upper = 0.00 (standardised)
    #>                               Lower = -0.00; Upper = 0.00 (unstandardised)
    #> Cauchy prior scale:           0.707
    #>
    #>     BF01 (equivalence) = 11.05
    #> ******************************

    get_bf(object = mod_equiv_sum)
    #> [1] 11.04945

## The non-inferiority test (`infer_bf()`)

With `infer_bf()` we can test whether the experimental group is not worse by a certain amount–which is given by the non-inferiority margin–than the control group. Importantly, sometimes low and sometimes high values on the dependent variable represent non-inferiority, which can be specified with the argument `direction`. In the case where low values represent non-inferiority we have BF-+, indicating that we quantify evidence for the data under the negative alternative hypothesis (i.e., H-) relative to the positive null hypothesis (i.e., H+). In the case where high values represent non-superiority we have BF+-, indicating that we quantify evidence for the data under the positive alternative hypothesis (i.e., H+) relative to the negative null hypothesis (i.e., H-). The default is that high values represent non-inferiority. The non-inferiority margin can be specified with the argument `ni_margin`. The argument `ni_margin_std` can be used to specify whether the non-inferiority margin is given in standardized (TRUE; the default) or unstandardized (FALSE) units.

We can use the raw data to compute a Bayes factor:

    mod_infer_raw <- infer_bf(
      x = data$dv[data$condition == "con"],
      y = data$dv[data$condition == "exp"],
      ni_margin = 1.5,
      ni_margin_std = FALSE
    )

    mod_infer_raw
    #> ******************************
    #> Non-inferiority analysis
    #> ------------------------
    #> Data:                         raw data
    #> H- (inferiority):             mu_y - mu_x < -ni_margin
    #> H+ (non-inferiority):         mu_y - mu_x > -ni_margin
    #> Non-inferiority margin:       0.45 (standardised)
    #>                               1.50 (unstandardised)
    #> Cauchy prior scale:           0.707
    #>
    #>     BF+- (non-inferiority) = 6.85e+10
    #> ******************************

    get_bf(object = mod_infer_raw)
    #> [1] 68546673627

Alternatively, if the raw data are not available, we can use summary statistics to compute a Bayes factor (cf. van Ravenzwaaij et al., 2019). The data were obtained from Andersson et al. (2013):

    mod_infer_sum <- infer_bf(
      n_x = 33,
      n_y = 32,
      mean_x = 17.1,
      mean_y = 13.6,
      sd_x = 8,
      sd_y = 9.8,
      ni_margin = 2,
      ni_margin_std = FALSE,
      direction = "low"
    )

    mod_infer_sum
    #> ******************************
    #> Non-inferiority analysis
    #> ------------------------
    #> Data:                         summary data
    #> H+ (inferiority):             mu_y - mu_x > ni_margin
    #> H- (non-inferiority):         mu_y - mu_x < ni_margin
    #> Non-inferiority margin:       0.22 (standardised)
    #>                               2.00 (unstandardised)
    #> Cauchy prior scale:           0.707
    #>
    #>     BF-+ (non-inferiority) = 79.59
    #> ******************************

    get_bf(object = mod_infer_sum)
    #> [1] 79.59441

## References

Gronau, Q. F., Ly, A., & Wagenmakers, E.-J. (2020). Informed Bayesian t-tests. The American Statistician, 74(2), 137-143.

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.

Schönbrodt, F. D., Wagenmakers, E.-J., Zehetleitner, M., & Perugini, M. (2017). Sequential hypothesis testing with Bayes factors: Efficiently testing mean differences. Psychological Methods, 22(2), 322-339.

van Ravenzwaaij, D., Monden, R., Tendeiro, J. N., & Ioannidis, J. P. A. (2019). Bayes factors for superiority, non-inferiority, and equivalence designs. BMC Medical Research Methodology, 19(1), 71.

Wagenmakers, E.-J., Marsman, M., Jamil, T., Ly, A., Verhagen, J., Love, J., … Morey, R. D. (2018). Bayesian inference for psychology. Part I: Theoretical advantages and practical ramifications. Psychonomic Bulletin & Review, 25(1), 35-57.