3 Using the dfba_beta_contrast Function
The dfba_beta_contrast function has seven arguments where the first three are required and the last four have defaults, so those four arguments are optional. The three required arguments are: n1_vec, n2_vec and contrast_vec.
Each of these arguments are vectors of \(K\) elements. The n1_vec argument is a vector of the \(n1\) values for the \(K\) separate binomial groups. The n2_vec argument is a vector of the corresponding \(n2\) values for the \(K\). The \(n_1\) and \(n_2\) values for each binomial are defined in the same way as for the dfba_binomial function.
In addition to the three required arguments, the dfba_beta_contrast function has the following four optional arguments:
a0_vecb0_vecprob_intervalsamples
The a0_vec and b0_vec arguments are vectors of, respectively, the \(a_0\) and \(b_0\) shape parameters of the \(K\) prior distributions for the separate beta distributions. The default for both of these inputs is a vector of \(1\)’s for both prior shape parameters, which corresponds to a uniform prior for each condition. The prob_interval argument is the value for the interval estimate; the default value for prob_interval is \(.95\). Finally, the samples argument is the number of Monte Carlo sampled values for the contrast. The default for this input is set to \(10000\). Please note that is not recommended to use fewer than \(10000\) samples and thus argument values less than \(10000\) are not allowed.
3.1 Example
As an example, for four separate groups of data where the binomial frequencies \((n_1,~n_2)\) are: \(G_1:(22,~18)\), \(G_2:(15,~25)\), \(G_3:(13,~27)\), and \(G_4:(21,~19)\), respectively, the corresponding contrast vectors arguments would be:
A contrast vector argument (contrast_vec) to compare groups \(G1\) and \(G3\) versus groups \(G2\) and \(G4\) can be defined as:
Using the defaults for the optional arguments (a0_vec, b0_vec, prob_interval, and samples), the Bayesian analysis of the contrast is given by:
contrast_example <- dfba_beta_contrast(n1_vec = n1_responses,
n2_vec = n2_responses,
contrast_vec = G13_vs_G24)
contrast_example
#> Bayesian Contrasts
#> ========================
#> Contrast Weights
#> 0.5 -0.5 0.5 -0.5
#> Exact posterior contrast mean
#> -0.01190476
#> Monte Carlo sampling results
#> Number of Monte Carlo Samples
#> 10000
#> Equal-tail 95% Probability Interval
#> Lower Limit Upper Limit
#> -0.158797 0.1364627
#> Posterior Probability that Contrast is Positive
#> 0.4299
#> Prior Probability that Contrast is Positive
#> 0.5016
#> Bayes Factor Estimate that Contrast is Positive
#> 0.7492675The plot() method produces a visualization of the prior and posterior distributions for the \(\phi\) parameter. Note: a plot of the posterior distribution without the prior distribution is given by including the argument plot.prior = FALSE (the default is plot.prior = TRUE).
Because the estimates for the probability that \(\Delta>0\) and the Bayes factor are based on the random set of \(10,000\) vectors drawn for \(\Delta\), those values can vary when another random set of vectors are drawn. To decrease the variability between any set of random value, the user can increase the value for samples argument from the default value.
Although Monte Carlo sampling is somewhat variable on different implementations of the dfba_beta_contrast function, there is one result that does not vary. The output value exact posterior contrast mean (mean) has an analytic value that does not depend on Monte Carlo sampling. From elementary probability theory, the expected value \(E(\Delta)\) or the mean of a linear combination of \(K\) independent random variables is
\[\begin{equation} \begin{aligned} E(\Delta) & = \psi_1 E(\phi_1) +\ldots+ \psi_K E(\phi_K) \\ & = \sum_{i=1}^{K} \psi_i \frac{a_i}{a_i+b_i} \end{aligned} \tag{3.1} \end{equation}\]
where \(a_i\) and \(b_i\), for \(i=1,\ldots,K\) are the shape parameters for the \(K\) separate posterior beta distributions.
Finally, it should be stressed that a contrast is a specific comparison among the \(K\) binomial conditions. Presumably, if a scientist observed \(K\) groups, there was a reason for the \(K\) conditions in the first place. In the above example, the contrast examined is only one possible way to compare the four groups. Contrasts are widely used in parametric analysis of variance (ANOVA) models (Kirk, 2013). From the ANOVA literature, it is useful to mention the idea of orthogonal contrasts (non-correlated comparisons). Contrast \(\Psi_A=(\psi_{1A},\ldots,\psi_{KA})\) and contrast \(\Psi_B=(\psi_{1B},\cdots,\psi_{KB})\) are orthogonal if \(\sum_{i=1}^{K} \psi_{iA} \psi_{iB}=0\). For \(K\) groups, there are \(K-1\) orthogonal contrasts possible. For the example above, the contrast coefficients are \(\Psi_A=(.5,-.5,.5,-.5)\). Two other orthogonal contrasts to this vector might be \(\Psi_B=(.5,.5,-.5,-.5)\) and \(\Psi_C=(.5,-.5,-.5,.5)\). Note that each of these three contrasts are mutually orthogonal to the others. However, that set of three orthogonal contrasts is not unique. For example, an alternative set of three orthogonal contrasts might be \(\Psi_{A'}=(1,-1,0,0)\), \(\Psi_{B'}=(0,0,1,-1)\) and \(\Psi_{C'}=(.5,.5,-.5,-.5)\). While the dfba_beta_contrast function can produce any linear contrast among the \(K\) independent beta distributed variates, the user needs to keep in mind the specific contrasts that make sense given the purposes of the research study.