# Markov Chain Monte Carlo for the Two-Dimensional Pairwise Comparisons Model with Dirichlet Process Prior in Yu and Quinn (2021) This function generates a sample from the posterior distribution of a model for pairwise comparisons data with a probit link. Unlike standard models for pairwise comparisons data, in this model the latent attribute of each item being compared is a vector in two-dimensional Euclidean space. ```r MCMCpaircompare2dDP( pwc.data, theta.constraints = list(), burnin = 1000, mcmc = 20000, thin = 1, verbose = 0, seed = NA, gamma.start = NA, theta.start = NA, store.theta = TRUE, store.gamma = FALSE, tune = 0.3, procrustes = FALSE, alpha.start = 1, cluster.max = 100, cluster.mcmc = 500, alpha.fixed = TRUE, a0 = 1, b0 = 1, ... ) ``` ## Arguments - `pwc.data`: A data.frame containing the pairwise comparisons data. Each row of `pwc.data` corresponds to a single pairwise comparison. `pwc.data` needs to have exactly four columns. The first column contains a unique identifier for the rater. Column two contains the unique identifier for the first item being compared. Column three contains the unique identifier for the second item being compared. Column four contains the unique identifier of the item selected from the two items being compared. If a tie occurred, the entry in the fourth column should be NA. The identifiers in columns 2 through 4 must start with a letter.Examples are provided below. - `theta.constraints`: A list specifying possible simple equality or inequality constraints on the item parameters. A typical entry in the list has one of three forms: `itemname=list(d,c)` which will constrain the dth dimension of theta for the item named `itemname` to be equal to c, `itemname=list(d,"+")` which will constrain the dth dimension of theta for the item named `itemname` to be positive, and `itemname=list(d, "-")` which will constrain the dth dimension of theta for the item named `itemname` to be negative. - `burnin`: The number of burn-in iterations for the sampler. - `mcmc`: The number of Gibbs iterations for the sampler. - `thin`: The thinning interval used in the simulation. The number of Gibbs iterations must be divisible by this value. - `verbose`: A switch which determines whether or not the progress of the sampler is printed to the screen. If `verbose` is greater than 0 output is printed to the screen every `verbose`th iteration. - `seed`: The seed for the random number generator. If NA, the Mersenne Twister generator is used with default seed 12345; if an integer is passed it is used to seed the Mersenne twister. The user can also pass a list of length two to use the L'Ecuyer random number generator, which is suitable for parallel computation. The first element of the list is the L'Ecuyer seed, which is a vector of length six or NA (if NA a default seed of `rep(12345,6)` is used). The second element of list is a positive substream number. See the MCMCpack specification for more details. - `gamma.start`: The starting value for the gamma vector. This can either be a scalar or a column vector with dimension equal to the number of raters. If this takes a scalar value, then that value will serve as the starting value for all of the gammas. The default value of NA will set the starting value of each gamma parameter to $\pi/4$. - `theta.start`: Starting values for the theta. Can be either a numeric scalar, a J by 2 matrix (where J is the number of items compared), or NA. If a scalar, all theta values are set to that value (except elements already specified via theta.contraints. If NA, then non constrained elements of theta are set equal to 0, elements constrained to be positive are set equal to 0.5, elements constrained to be negative are set equal to -0.5 and elements with equality constraints are set to satisfy those constraints. - `store.theta`: Should the theta draws be returned? Default is TRUE. - `store.gamma`: Should the gamma draws be returned? Default is TRUE. - `tune`: Tuning parameter for the random walk Metropolis proposal for each gamma_i. `tune` is the width of the uniform proposal centered at the current value of gamma_i. Must be a positive scalar. - `procrustes`: Should the theta and gamma draws be post-processed with a Procrustes transformation? Default is FALSE. The Procrustes target matrix is derived from the constrained elements of theta. Each row of theta that has both theta values constrained is part of the of the target matrix. Elements with equality constraints are set to those values. Elements constrained to be positive are set to 1. Elements constrained to be negative are set to -1. If `procrustes` is set to `TRUE` theta.constraints must be set so that there are at least three rows of theta that have both elements of theta constrained. - `alpha.start`: The starting value for the DP concentration parameter alpha. Must be a positive scalar. Defaults to 1. If `alpha.fixed` is set equal to TRUE, then alpha is held fixed at alpha.start. - `cluster.max`: The maximum number of clusters allowed in the approximation to the DP prior for gamma. Defaults to 100. Must be a positive integer. - `cluster.mcmc`: The number of additional MCMC iterations that are done to sample each cluster-specific gamma value within one main MCMC iteration. Must be a positive integer. Defaults to 500. Setting this to a lower value speeds runtime at the cost of (possibly) worse mixing. - `alpha.fixed`: Logical value indicating whether the DP concentration parameter alpha be held fixed (`TRUE`) or estimated (`FALSE`). - `a0`: The shape parameter of the gamma prior for alpha. This is the same parameterization of the gamma distribution as R's internal `rgamma()` function. Only relevant if `alpha.fixed` is set equal to `FALSE`. Defaults to 1. - `b0`: The rate parameter of the gamma prior for alpha. This is the same parameterization of the gamma distribution as R's internal `rgamma()` function. Only relevant if `alpha.fixed` is set equal to `FALSE`. Defaults to 1. - `...`: further arguments to be passed ## Returns An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package. Most of the column names of the mcmc object are self explanatory. Note however that the columns with names of the form "cluster.[raterID]" give the cluster membership of each rater at each stored MCMC iteration. Because of the possibility of label switching, the particular values of these cluster membership variables are not meaningful. What is meaningful is whether two raters share the same cluster membership value at a particular MCMC iteration. This indicates that those two raters were clustered together during that iteration. Finally, note that the "n.clusters" column gives the number of distinct gamma values at each iteration, i.e. the number of clusters at that iteration. ## Details `MCMCpaircompare2d` uses the data augmentation approach of Albert and Chib (1993) in conjunction with Gibbs and Metropolis-within-Gibbs steps to fit the model. The user supplies data and a sample from the posterior is returned as an `mcmc` object, which can be subsequently analyzed in the `coda` package. The simulation is done in compiled C++ code to maximize efficiency. Please consult the `coda` package documentation for a comprehensive list of functions that can be used to analyze the posterior sample. The model takes the following form: $$ i = 1,...,I \ \ \ \ (raters) $$ $$ j = 1,...,J \ \ \ \ (items) $$ $$ Y_{ijj'} = 1 \ \ if \ \ i \ \ chooses \ \ j \ \ over \ \ j' $$ $$ Y_{ijj'} = 0 \ \ if \ \ i \ \ chooses \ \ j' \ \ over \ \ j $$ $$ Y_{ijj'} = NA \ \ if \ \ i \ \ chooses \ \ neither $$ $$ \Pr(Y_{ijj'} = 1) = \Phi( \mathbf{z}_{i}' [\boldsymbol{\theta}_{j} -\boldsymbol{\theta}_{ j'} ]) $$ $$ \mathbf{z}_{i}=[\cos(\gamma_{i}), \ \sin(\gamma_{i})]' $$ The following priors are assumed: $$ \gamma_i \sim G $$ $$ G \sim \mathcal{DP}(\alpha G_0) $$ $$ G_0 = \mathcal{U}nif(0, \pi/2) $$ $$ \alpha \sim \mathcal{G}amma(a_0, b_0) $$ $$ \boldsymbol{\theta}_j \sim\mathcal{N}_{2}(\mathbf{0}, \mathbf{I}_{2}) $$ For identification, some $\boldsymbol{\theta}_j$s are truncated above or below 0, or fixed to constants. ## Examples ```r ## Not run: ## a synthetic data example set.seed(123) I <- 65 ## number of raters J <- 50 ## number of items to be compared ## 3 clusters: ## raters 1 to 5 put most weight on dimension 1 ## raters 6 to 10 put most weight on dimension 2 ## raters 11 to I put substantial weight on both dimensions gamma.true <- c(rep(0.05, 5), rep(1.50, 5), rep(0.7, I-10) ) theta1.true <- rnorm(J, m=0, s=1) theta2.true <- rnorm(J, m=0, s=1) theta1.true[1] <- 2 theta2.true[1] <- 2 theta1.true[2] <- -2 theta2.true[2] <- -2 theta1.true[3] <- 2 theta2.true[3] <- -2 n.comparisons <- 125 ## number of pairwise comparisons for each rater ## generate synthetic data according to the assumed model rater.id <- NULL item.1.id <- NULL item.2.id <- NULL choice.id <- NULL for (i in 1:I){ for (c in 1:n.comparisons){ rater.id <- c(rater.id, i+100) item.numbers <- sample(1:J, size=2, replace=FALSE) item.1 <- item.numbers[1] item.2 <- item.numbers[2] item.1.id <- c(item.1.id, item.1) item.2.id <- c(item.2.id, item.2) z <- c(cos(gamma.true[i]), sin(gamma.true[i])) eta <- z[1] * (theta1.true[item.1] - theta1.true[item.2]) + z[2] * (theta2.true[item.1] - theta2.true[item.2]) prob.item.1.chosen <- pnorm(eta) u <- runif(1) if (u <= prob.item.1.chosen){ choice.id <- c(choice.id, item.1) } else{ choice.id <- c(choice.id, item.2) } } } item.1.id <- paste("item", item.1.id+100, sep=".") item.2.id <- paste("item", item.2.id+100, sep=".") choice.id <- paste("item", choice.id+100, sep=".") sim.data <- data.frame(rater.id, item.1.id, item.2.id, choice.id) ## fit the model (should be run for more than 10500 iterations) posterior <- MCMCpaircompare2dDP(pwc.data=sim.data, theta.constraints=list(item.101=list(1,2), item.101=list(2,2), item.102=list(1,-2), item.102=list(2,-2), item.103=list(1,"+"), item.103=list(2,"-")), verbose=100, burnin=500, mcmc=10000, thin=5, cluster.mcmc=10, store.theta=TRUE, store.gamma=TRUE, tune=0.1) theta1.draws <- posterior[, grep("theta1", colnames(posterior))] theta2.draws <- posterior[, grep("theta2", colnames(posterior))] gamma.draws <- posterior[, grep("gamma", colnames(posterior))] theta1.post.med <- apply(theta1.draws, 2, median) theta2.post.med <- apply(theta2.draws, 2, median) gamma.post.med <- apply(gamma.draws, 2, median) theta1.post.025 <- apply(theta1.draws, 2, quantile, prob=0.025) theta1.post.975 <- apply(theta1.draws, 2, quantile, prob=0.975) theta2.post.025 <- apply(theta2.draws, 2, quantile, prob=0.025) theta2.post.975 <- apply(theta2.draws, 2, quantile, prob=0.975) gamma.post.025 <- apply(gamma.draws, 2, quantile, prob=0.025) gamma.post.975 <- apply(gamma.draws, 2, quantile, prob=0.975) ## compare estimates to truth par(mfrow=c(2,2)) plot(theta1.true, theta1.post.med, xlim=c(-2.5, 2.5), ylim=c(-2.5, 2.5), col=rgb(0,0,0,0.3)) segments(x0=theta1.true, x1=theta1.true, y0=theta1.post.025, y1=theta1.post.975, col=rgb(0,0,0,0.3)) abline(0, 1, col=rgb(1,0,0,0.5)) plot(theta2.true, theta2.post.med, xlim=c(-2.5, 2.5), ylim=c(-2.5, 2.5), col=rgb(0,0,0,0.3)) segments(x0=theta2.true, x1=theta2.true, y0=theta2.post.025, y1=theta2.post.975, col=rgb(0,0,0,0.3)) abline(0, 1, col=rgb(1,0,0,0.5)) plot(gamma.true, gamma.post.med, xlim=c(0, 1.6), ylim=c(0, 1.6), col=rgb(0,0,0,0.3)) segments(x0=gamma.true, x1=gamma.true, y0=gamma.post.025, y1=gamma.post.975, col=rgb(0,0,0,0.3)) abline(0, 1, col=rgb(1,0,0,0.5)) ## plot point estimates plot(theta1.post.med, theta2.post.med, xlim=c(-2.5, 2.5), ylim=c(-2.5, 2.5), col=rgb(0,0,0,0.3)) for (i in 1:length(gamma.post.med)){ arrows(x0=0, y0=0, x1=cos(gamma.post.med[i]), y1=sin(gamma.post.med[i]), col=rgb(1,0,0,0.2), len=0.05, lwd=0.5) } ## End(Not run) ``` ## References Albert, J. H. and S. Chib. 1993. ``Bayesian Analysis of Binary and Polychotomous Response Data.'' **J. Amer. Statist. Assoc.** 88, 669-679 Yu, Qiushi and Kevin M. Quinn. 2021. ``A Multidimensional Pairwise Comparison Model for Heterogeneous Perceptions with an Application to Modeling the Perceived Truthfulness of Public Statements on COVID-19.'' University of Michigan Working Paper. Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. ``MCMCpack: Markov Chain Monte Carlo in R.'', **Journal of Statistical Software**. 42(9): 1-21. tools:::Rd_expr_doi("10.18637/jss.v042.i09") . Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. **Scythe Statistical Library 1.0.** [http://scythe.wustl.edu.s3-website-us-east-1.amazonaws.com/](http://scythe.wustl.edu.s3-website-us-east-1.amazonaws.com/). Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2006. ``Output Analysis and Diagnostics for MCMC (CODA)'', **R News**. 6(1): 7-11. [https://CRAN.R-project.org/doc/Rnews/Rnews_2006-1.pdf](https://CRAN.R-project.org/doc/Rnews/Rnews_2006-1.pdf). ## See Also `plot.mcmc`,`summary.mcmc`, `MCMCpaircompare`, `MCMCpaircompare2dDP` ## Author(s) Qiushi Yu and Kevin M. Quinn

MCMCpaircompare2dDP function