This example illustrates how to use the MCMC procedure to fit a Bayesian multinomial model for categorical response data measured on an ordinal scale. The example also demonstrates how use the MCMC procedure to compute posterior means, credible intervals, and posterior distributions of the parameters and odds ratios for the multinomial model.
Researchers study the results of a taste test on three different brands of ice cream. They want to assess the testers’ preference of the three brands. The taste of each brand is rated on a five-point scale from very good to very bad. the five points correspond to response variables Y1 through Y5, where Y1 represents very good and Y5 represents very bad. Response variables contain the number of taste testers who rate each brand in each category. The very bad taste level (Y5) is used as the reference response level. Two dummy variables, BRAND1 and BRAND2, are created to indicate Brands 1 and 2, respectively. Brand 3 is set as the reference level in this example and is represented in the data set when both of the dummy variables equal zero.
The following statements create the ICECREAM data set:
data icecream; input y1-y5 brand; if brand = 1 then brand1 = 1; else brand1 = 0; if brand = 2 then brand2 = 1; else brand2 = 0; keep y1-y5 brand1 brand2; datalines; 70 71 151 30 46 1 20 36 130 74 70 2 50 55 140 52 50 3 ;
Multinomial ordinal models occur frequently in applications such as food testing, survey response, or anywhere order matters in the categorical response. Categorical data with an ordinal response correspond to multinomial models based on cumulative response probabilities (. In this data set, the ordered response variable is the taste tester’s rating for a brand of ice cream.
Let the random variable for brand , and let response level , be from an multinomial ordinal model with mutually exclusive, discrete response levels and probability mass function
where represents the number of people from th brand in the th response level. For the grouped data, let denote the number of testers who taste the th brand of ice cream and let . Let denote the probability that the response of brand falls into the th response level, and let . Let denote the corresponding cumulative probability that the response falls in the th level or below, so . The transformed cumulative probabilities are linear functions of the covariates written as , where refers to the logit link function, represents the effects for the covariates, and . Let represent the baseline value of the transformed cumulative probability for category such that the constraint holds for all (. Then
and the group probabilities for the th levels are as follows:
The likelihood function for the counts and corresponding covariates is
There are six parameters in the likelihood: the intercepts through and the regression parameters and that correspond to the relative Brand 1 and 2 effects, respectively.
Suppose the following prior distributions are placed on the six parameters, where indicates a prior distribution and indicates a conditional prior distribution:
The joint prior distribution of through is the product of Equation 6 through 9. The prior distributions in Equation 7 through 9 represent truncated normal distributions with mean 0, variance 100, and the designated lower bound. The lower bound ensures that the order restriction on is sustained.
Using Bayes’ theorem, the likelihood function and prior distributions determine the posterior distribution of the parameters as follows:
PROC MCMC obtains samples from the desired posterior distribution. You do not need to specify the exact form of the posterior distribution.
The odds ratio for comparing one brand to another can be written as
for . The odds ratio is useful for interpreting how the taste preference for the different brands of ice cream cpmpares. For this example, Brand 3 is set as the reference level, which implies that .
The following SAS statements fit the Bayesian multinomial ordinal model. The PROC MCMC statement invokes the procedure and specifies the input data set. The NBI= option specifies the number of burn-in iterations. The NMC= option specifies the number of posterior simulation iterations. The THIN=10 option specifies that one of every 10 samples is kept. The SEED= option specifies a seed for the random number generator (the seed guarantees the reproducibility of the random stream). The PROPCOV=QUANEW option uses the estimated inverse Hessian matrix as the initial proposal covariance matrix. The MONITOR= option outputs analysis on selected symbols of interest in the program.
ods graphics on; proc mcmc data=icecream nbi=10000 nmc=25000 thin=10 seed=1181 propcov=quanew monitor=(beta1 beta2 or12 or13 or23); array theta; array gamma; parms theta1-theta4 beta1 beta2; prior beta: ~ normal(0,var=1000); prior theta1 ~ normal(0,var=100); prior theta2 ~ normal(0,var=100,lower=theta1); prior theta3 ~ normal(0,var=100,lower=theta2); prior theta4 ~ normal(0,var=100,lower=theta3); mu = beta1*brand1 + beta2*brand2; do j = 1 to 4; gamma[j] = logistic(theta[j] + mu); end; pi1 = gamma1; pi2 = gamma2 - gamma1; pi3 = gamma3 - gamma2; pi4 = gamma4 - gamma3; pi5 = 1 - sum(of pi1-pi4); llike = logmpdfmultinom(of y1-y5, of pi1-pi5); model dgeneral(llike); beginnodata;; or12 = exp(beta1-beta2); or13 = exp(beta1); or23 = exp(beta2); endnodata;; run; ods graphics off;
Each of the two ARRAY statements associates a name with a list of variables and constants. The first ARRAY statement specifies names for the intercept parameters. The second ARRAY statement contains the parameters.
The MU assignment statement calculates . The DO loop and coinciding GAMMA assignment statements calculate for as in Equation 1. The five ETA assignment statements calculate the individual probabilities that an observation falls into the th response level as in Equation 2 to Equation 4.
The LLIKE statement uses the multivariate LOGMPDFMULTIM function to calculate the value of the log likelihood function as in Equation 5. The multinomial density is not listed in the section "Standard Distributions" in the PROC MCMC documentation, so you use the DGENERAL function in the MODEL statement. The letter "D" stands for discrete. The log likelihood is the input parameter because the DGENERAL function must be specified on the logarithm scale. Since the multivariate log likelihood function takes the dependent variable into account, you do not need to explicitly state the dependent variable in the MODEL statement.
The statements within the BEGINNODATA and ENDNODATA statements calculate the three odds ratios for pairwise comparisons of ice cream brands according to Equation 11. The statements are enclosed within the BEGINNODATA and ENDNODATA block to reduce unnecessary observation-level computations.
Figure 1 displays diagnostic plots to assess whether the Markov chains have converged.
The trace plot in Figure 1 indicates that the chain appears to have reached a stationary distribution. It also has good mixing and is dense. The autocorrelation plot indicates low autocorrelation and efficient sampling. Finally, the kernel density plot shows the smooth, unimodal shape of posterior marginal distribution for . The remaining diagnostic plots (not shown here) similarly indicate good convergence in the other parameters.
Figure 2 displays a number of convergence diagnostics, including Monte Carlo standard errors, autocorrelations at selected lags, Geweke diagnostics, and the effective sample sizes.
|Monte Carlo Standard Errors|
|Parameter||Lag 1||Lag 5||Lag 10||Lag 50|
|Parameter||z||Pr > |z||
|Effective Sample Sizes|
Figure 3 reports summary and interval statistics for the regression parameters and odds ratios. The odds ratios provide the relative difference in one brand with respect to another and indicate whether there is a significant brand effect. The odds ratio for Brand 1 and Brand 2 is the multiplicative change in the odds of a taste tester preferring Brand 1 compared to the odds of the tester preferring Brand 2. The estimated odds ratio () value is 2.8366 with a corresponding 95% equal-tail credible interval of . Similarly, the odds ratio for Brand 1 and Brand 3 is 1.4787 with a 95% equal-tail credible interval of . Finally, the odds ratio for Brand 2 compared to Brand 3 is 0.5271 with a 95% equal-tail credible interval of . The lower categories indicate the favorable taste results; so Brand 1 scored significantly better when compared to Brand 2 or 3. Brand 2 scored less favorably when compared to Brand 3.
|Parameter||Alpha||Equal-Tail Interval||HPD Interval|