In danbarch/dfba: What the Package Does (One Line, Title Case)

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(DFBA)

Theoretical Background

Many studies have two variates where each variate is a score on an ordinal scale (e.g., an integer on a $1,\ldots,M$ scale). Such data are typically organized into a rank-ordered matrix of frequency values where the element in the $[I, J]$ cell is the frequency of occasions where one variate has a rank value of $I$ while the corresponding rank for the other variate is $J$. For such matrices, Goodman and Kruskal (1954) provided a frequentist distribution-free concordance correlation statistic that has come to be called the Goodman and Kruskal's gamma or the $G$ statistic (Siegel \& Castellan, 1988). The dfba_gamma() function provides a corresponding Bayesian distribution-free analysis given the input of a rank-ordered matrix.

Chechile (2020) showed that the Goodman-Kruskal gamma is equivalent to the more general Kendall $\tau_A$ nonparametric correlation coefficient. Historically, gamma was considered a different metric from $\tau$ because, typically, the version of $\tau$ in standard use was $\tau_B$, which is a flawed metric because it does not properly correct for ties. It is important to point out that the commands cor(x, y, method = "kendall") and cor.test(x, y, method = "kendall") (from the stats package) return the $\tau_B$ correlation, which is incorrect when there are ties.

The correct $\tau_A$ is computed by the dfba_bivariate_concordance() function (see the vignette for the dfba_bivariate_concordance() function for more details and examples about the difference between $\tau_A$ and $\tau_B$). The dfba_gamma() function is similar to the dfba_bivariate_concordance() function; the main difference is that the dfba_gamma() function deals with data that are organized in advance into a rank-ordered table or matrix, whereas the input for the dfba_bivariate_concordance() function are two paired vectors x and y of continuous values.

The gamma statistic is equal to:

\begin{equation} G = \frac{n_c-n_d}{n_c+n_d}, (#eq:gammastat) \end{equation}

where $n_c$ is the number of occasions when the variates change in a concordant way, and $n_d$ is the number of occasions when the variates change in a discordant fashion. The value of $n_c$ for an order matrix is the sum of terms for each $[I, J]$ that are equal to $n_{ij}N^{+}{ij}$, where $n{ij}$ is the frequency for cell $[I, J]$ and $N^{+}{ij}$ is the sum of the frequencies in the matrix where the row value is greater than $I$ and where the column value is greater than $J$. The value $n_d$ is the sum of terms for each $[I, J]$ that are $n{ij}N^{-}{ij}$, where $N^{-}{ij}$ is the sum of the frequencies in the matrix where row value is greater than $I$ and the column value is less than $J$. The $n_c$ and $n_d$ values computed in this fashion are respectively equal to $n_c$ and $n_d$ values found when the bivariate measures are entered as paired vectors into the dfba_bivariate_concordance() function.

As with the dfba_bivariate_concordance() function, the Bayesian analysis focuses on the population concordance proportion parameter $\phi$, which is linked to the $G$ statistic because $G=2\phi-1$. The likelihood function is proportional to $\phi^{n_c}(1-\phi)^{n_d}$. Similar to the Bayesian analysis for the concordance parameter in the dfba_bivariate_concordance() function, the prior distribution is a beta distribution with shape parameters $a_0$ and $b_0$, and the posterior distribution is the conjugate beta distribution where shape parameters are $a = a_0 + n_c$ and $b = b_0 + n_d$.

Using the dfba_gamma() Function

The dfba_gamma() function has one required argument x that must be an object in the form of a matrix or a table.

Example

The following example demonstrates how to create a matrix of data and to analyze it using the dfba_gamma() function.

N <- matrix(c(38, 4, 5, 0, 6, 40, 1, 2, 4, 8, 20, 30),
            ncol = 4,
            byrow = TRUE)

colnames(N) <- c('C1', 'C2', 'C3', 'C4')

rownames(N) <- c('R1', 'R2', 'R3')

A <- dfba_gamma(N)

A

plot(A)

The dfba_gamma() function also has three optional arguments; listed with their respective default arguments, they are: a0 = 1, b0 = 1, and prob_interval = .95 The a0 and b0 arguments are the shape parameters for the prior beta distribution; the default value of $1$ for each corresponds to a uniform prior. The prob_interval argument specifies the probability value for the interval estimate of the $\phi$ concordance parameter.

References

Chechile, R.A. (2020). Bayesian Statistics for Experimental Scientists: A General Introduction Using Distribution-Free Methods. Cambridge: MIT Press.

Goodman, L. A., and Kruskal, W. H. (1954). Measures of association for cross classifications. Journal of the American Statistical Association, 49, 732-764.

Siegel, S., and Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. New York: McGraw-Hill.

danbarch/dfba documentation built on Jan. 30, 2024, 6:51 p.m.