Ordinal Predicted Variable

Transcription

1 Ordinal Predicted Variable Tim Frasier Copyright Tim Frasier This work is licensed under the Creative Commons Attribution 4.0 International license. Click here for more information.

2 Goals and General Idea

3 Goals When would we use this type of analysis? When the predicted variable is ordinal! Places in a race (1st, 2nd, 3rd, etc.) Surveys on a Likert scale (5 = strongly agree, 4 = agree, 3 = neutral 2 = disagree, 1 = strongly disagree) Scaled responses (good, mediocre, bad) etc.

4 Characteristics Ordinal data are kind of a pain to deal with Know order, but not necessarily equally spaced How much do you like fish (1-hate to 5-love)? May be harder to go from 1 2 than 4 5 As predictor variables increase, should sequentially step through predicted values How can we ensure this happens?

5 Characteristics Suppose ordinal data with 7 levels There will be cut-off points (thresholds) between levels, indicating where it switches from one to another (indicated here as θs) If there are k levels, there will be k-1 of these thresholds From Kruschke (2015) p. 673

6 Characteristics How do we get probabilities for each level? Cumulative distribution From Kruschke (2015) p. 673

7 Characteristics Now values range from 0 to 1 Probability for each level is the cumulative area up to the threshold just above that level minus the cumulative area up to the threshold just below that level Call each threshold point an α value cumulative proportion α1 α2 α3 α4 α5 α response

8 Characteristics For first category, probability is cumulative probability for that value, minus zero Considering the mean and sd of the underlying distribution Cumulative normal distribution in JAGS is pnorm cumulative proportion α1 α2 α3 α4 α5 α response

9 Characteristics For second category, probability is cumulative probability for that value, minus that for the first category cumulative proportion α1 α2 α3 α4 α5 α response

10 Characteristics For third category, probability is cumulative probability for that value, minus that for the second category cumulative proportion α1 α2 α3 α4 α5 α response

11 Characteristics For fourth category, probability is cumulative probability for that value, minus that for the third category cumulative proportion α1 α2 α3 α4 α5 α response

12 Characteristics For fifth category, probability is cumulative probability for that value, minus that for the fourth category cumulative proportion α1 α2 α3 α4 α5 α response

13 Characteristics For sixth category, probability is cumulative probability for that value, minus that for the fifth category cumulative proportion α1 α2 α3 α4 α5 α response

14 Characteristics For seventh category, probability is one, minus the cumulative probability for the 6th category cumulative proportion α1 α2 α3 α4 α5 α response

15 Characteristics One problem: Our α values are only relative to one another, and have no absolute position, as is Could add any constant to raw (i.e. non-cumulative) values, and recover the same estimates Like sliding our distribution up and down the x-axis, our α estimates would remain the same Real problems for MCMC process (any value is reasonable!)

16 Characteristics One solution: Pin down distribution by specifying the two extreme α values Estimate the rest, relative to these, which is all that matters Will specify this in the data list

17 Characteristics The mean (μ) of this distribution is the result of the additive effect of our predictor variables Our standard equation for the effects of the predictor variables goes into this μ

18 Characteristics Distribution does not have to look normal for the normal distribution to be appropriate All the following histograms were generated by a normal distribution From Kruschke (2015) p. 673

19 Characteristics What we are estimating: 1. The α values for all but the first and last thresholds 2. The mean (μ) of the underlying distribution (based on the additive effect of the predictor variables) 3. The standard deviation (σ) of the underlying distribution 4. Other appropriate distribution parameters if not using the normal distribution

20 The Data

21 Data Fake data generated from code in Kruschke (2011) ord <- read.table("ordinaldata.csv", header = TRUE, sep = ",")

22 Data Fake data generated from code in Kruschke (2011) Ordinal predicted variable ord <- read.table("ordinaldata.csv", header = TRUE, sep = ",")

23 Data Fake data generated from code in Kruschke (2011) Two metric predictor variables ord <- read.table("ordinaldata.csv", header = TRUE, sep = ",")

24 Data Can use the pairs function to plot the data, and get some idea of potential patterns (keeping in mind the issue of interactions) pairs(ord, pch = 16, col = rgb(0, 0, 1, 0.3))

25 Data Exploration Y X1 X

26 Data Exploration Looks like a negative relationship between X1 & Y, and a positive relationship between X2 & Y Y X1 X

27 Data Exploration Use the table function to get frequencies for each ordinal response ytable <- table(ord$y) ytable

28 Data Exploration Make as a data frame and format properly ytable.df <- as.data.frame(ytable) ytable.df[, 1] <- as.numeric(as.character(ytable.df[, 1]))

29 Data Exploration Plot the data plot(ytable.df[, 1], ytable.df[, 2], type = "h", ylab = "Frequency", xlab = "response", lwd = 4, col = rgb(0, 0, 1, 0.5)) Frequency response

30 Data Exploration Can also transpose this to the cumulative distribution of your data, if you want to Frequency response # Get proportions pr_y <- ytable / nrow(ord) # Get cumulative proportions cum_pr_y <- cumsum(pr_y)

31 Data Exploration Can also transpose this to the cumulative distribution of your data, if you want to An R function that calculates the cumulative sums of a vector Frequency response # Get proportions pr_y <- ytable / nrow(ord) # Get cumulative proportions cum_pr_y <- cumsum(pr_y)

32 Data Exploration # Plot plot(ytable.df[, 1], cum_pr_y, type = "b", lwd = 2, ylab = cumulative proportion", xlab = "response", ylim = c(0, 1), col = blue ) cumulative proportion response

33 Frequentist Approach

34 Frequentist Approch I don t know polr function from the MASS package seems to be an option, but I couldn t get it to work (with limited time)

35 Bayesian Approach

36 Load Libraries & Functions library(runjags) library(coda) source("plotpost.r")

37 Organize the Data y <- ord$y N <- length(y) nlevels <- length(unique(y)) x1 <- ord$x1 x2 <- ord$x2

38 Organize the Data y <- ord$y N <- length(y) nlevels <- length(unique(y)) x1 <- ord$x1 x2 <- ord$x2 Making a variable with the number of response levels will make your code more generic

39 Standardize the Metric Variables # x1 x1mean <- mean(x1) x1sd <- sd(x1) zx1 <- (x1 - x1mean) / x1sd # x2 x2mean <- mean(x2) x2sd <- sd(x2) zx2 <- (x2 - x2mean) / x2sd

40 Create a List For Alpha Values #--- Create a list for anchored alpha values ---# # with beginning and ending values, but the # # rest will be filled in by the MCMC process # # (have them as "NA" for now). # # # alpha <- rep(na, nlevels - 1) alpha[1] < # Set first value alpha[nlevels - 1] <- nlevels # Set last value

41 Make Data List For JAGS datalist = list( y = y, nlevels = nlevels, N = N, x1 = zx1, x2 = zx2, alpha = alpha )

42 Define the Model Doesn t lend itself well to a diagram We ll just walk through the code

43 modelstring = " model { #--- The likelihood ---# for (i in 1:N) { Note that I have rearranged things from how I did it before, to try to add clarity. We ll walk through it. # The standard part of our equation mu[i] <- b0 + (b1 * x1[i]) + (b2 * x2[i]) # log-odds for each value being in first category p[i, 1] <- pnorm(alpha[1], mu[i], 1 / sigma^2) # log-odds for each value being in categories between the highest and the lowest for (j in 2:(nLevels - 1)) { p[i, j] <- max(0, pnorm(alpha[j], mu[i], 1 / sigma^2) - pnorm(alpha[j - 1], mu[i], 1 / sigma^2)) # log-odds for each value being in the highest category p[i, nlevels] <- 1 - pnorm(alpha[nlevels - 1], mu[i], 1/sigma^2)... # Now, fit the y data to a categorical distribution # with the characteristics we just calculated y[i] ~ dcat(p[i, 1:nLevels])

44 modelstring = " model { #--- The likelihood ---# for (i in 1:N) { The black box into which we can put any equations that we have dealt with before (or more) # The standard part of our equation mu[i] <- b0 + (b1 * x1[i]) + (b2 * x2[i]) # log-odds for each value being in first category p[i, 1] <- pnorm(alpha[1], mu[i], 1 / sigma^2) # log-odds for each value being in categories between the highest and the lowest for (j in 2:(nLevels - 1)) { p[i, j] <- max(0, pnorm(alpha[j], mu[i], 1 / sigma^2) - pnorm(alpha[j - 1], mu[i], 1 / sigma^2)) # log-odds for each value being in the highest category p[i, nlevels] <- 1 - pnorm(alpha[nlevels - 1], mu[i], 1/sigma^2)... # Now, fit the y data to a categorical distribution # with the characteristics we just calculated y[i] ~ dcat(p[i, 1:nLevels])

45 modelstring = " model { #--- The likelihood ---# for (i in 1:N) {...describe the mean of the normal distribution describing the data # The standard part of our equation mu[i] <- b0 + (b1 * x1[i]) + (b2 * x2[i]) # log-odds for each value being in first category p[i, 1] <- pnorm(alpha[1], mu[i], 1 / sigma^2) # log-odds for each value being in categories between the highest and the lowest for (j in 2:(nLevels - 1)) { p[i, j] <- max(0, pnorm(alpha[j], mu[i], 1 / sigma^2) - pnorm(alpha[j - 1], mu[i], 1 / sigma^2)) # log-odds for each value being in the highest category p[i, nlevels] <- 1 - pnorm(alpha[nlevels - 1], mu[i], 1/sigma^2)... # Now, fit the y data to a categorical distribution # with the characteristics we just calculated y[i] ~ dcat(p[i, 1:nLevels])

46 modelstring = " model { #--- The likelihood ---# for (i in 1:N) { # The standard part of our equation mu[i] <- b0 + (b1 * x1[i]) + (b2 * x2[i]) The probability of each data point being in the first category is the cumulative probability up to the first threshold, based on the mean and sd of the underlying distribution # log-odds for each value being in first category p[i, 1] <- pnorm(alpha[1], mu[i], 1 / sigma^2) # log-odds for each value being in categories between the highest and the lowest for (j in 2:(nLevels - 1)) { p[i, j] <- max(0, pnorm(alpha[j], mu[i], 1 / sigma^2) - pnorm(alpha[j - 1], mu[i], 1 / sigma^2)) # log-odds for each value being in the highest category p[i, nlevels] <- 1 - pnorm(alpha[nlevels - 1], mu[i], 1/sigma^2)... # Now, fit the y data to a categorical distribution # with the characteristics we just calculated y[i] ~ dcat(p[i, 1:nLevels])

47 modelstring = " model { #--- The likelihood ---# for (i in 1:N) { # The standard part of our equation mu[i] <- b0 + (b1 * x1[i]) + (b2 * x2[i]) # log-odds for each value being in first category p[i, 1] <- pnorm(alpha[1], mu[i], 1 / sigma^2) The probability of each data point being in each of the middle categories is the cumulative probability up to the top threshold for the given category, minus the cumulative probability up to the bottom threshold for the given category, based on the mean and sd of the underlying distribution # log-odds for each value being in categories between the highest and the lowest for (j in 2:(nLevels - 1)) { p[i, j] <- max(0, pnorm(alpha[j], mu[i], 1 / sigma^2) - pnorm(alpha[j - 1], mu[i], 1 / sigma^2)) # log-odds for each value being in the highest category p[i, nlevels] <- 1 - pnorm(alpha[nlevels - 1], mu[i], 1/sigma^2)... # Now, fit the y data to a categorical distribution # with the characteristics we just calculated y[i] ~ dcat(p[i, 1:nLevels])

48 modelstring = " model { #--- The likelihood ---# for (i in 1:N) { # The standard part of our equation mu[i] <- b0 + (b1 * x1[i]) + (b2 * x2[i]) This is just a safety net. If the difference calculated to the right is less than zero, zero will be used. Included because probabilities can t be less than zero. # log-odds for each value being in first category p[i, 1] <- pnorm(alpha[1], mu[i], 1 / sigma^2) # log-odds for each value being in categories between the highest and the lowest for (j in 2:(nLevels - 1)) { p[i, j] <- max(0, pnorm(alpha[j], mu[i], 1 / sigma^2) - pnorm(alpha[j - 1], mu[i], 1 / sigma^2)) # log-odds for each value being in the highest category p[i, nlevels] <- 1 - pnorm(alpha[nlevels - 1], mu[i], 1/sigma^2)... # Now, fit the y data to a categorical distribution # with the characteristics we just calculated y[i] ~ dcat(p[i, 1:nLevels])

49 modelstring = " model { #--- The likelihood ---# for (i in 1:N) { # The standard part of our equation mu[i] <- b0 + (b1 * x1[i]) + (b2 * x2[i]) The probability of each data point being in the last category is one minus the cumulative probability up to the lower threshold for that category, based on the mean and sd of the underlying distribution # log-odds for each value being in first category p[i, 1] <- pnorm(alpha[1], mu[i], 1 / sigma^2) # log-odds for each value being in categories between the highest and the lowest for (j in 2:(nLevels - 1)) { p[i, j] <- max(0, pnorm(alpha[j], mu[i], 1 / sigma^2) - pnorm(alpha[j - 1], mu[i], 1 / sigma^2)) # log-odds for each value being in the highest category p[i, nlevels] <- 1 - pnorm(alpha[nlevels - 1], mu[i], 1/sigma^2)... # Now, fit the y data to a categorical distribution # with the characteristics we just calculated y[i] ~ dcat(p[i, 1:nLevels])

50 modelstring = " model { #--- The likelihood ---# for (i in 1:N) { What we ve created in the last few lines is a probability matrix for each data point being in each of our response categories... # The standard part of our equation mu[i] <- b0 + (b1 * x1[i]) + (b2 * x2[i]) # log-odds for each value being in first category p[i, 1] <- pnorm(alpha[1], mu[i], 1 / sigma^2) # log-odds for each value being in categories between the highest and the lowest for (j in 2:(nLevels - 1)) { p[i, j] <- max(0, pnorm(alpha[j], mu[i], 1 / sigma^2) - pnorm(alpha[j - 1], mu[i], 1 / sigma^2)) # log-odds for each value being in the highest category p[i, nlevels] <- 1 - pnorm(alpha[nlevels - 1], mu[i], 1/sigma^2)... # Now, fit the y data to a categorical distribution # with the characteristics we just calculated y[i] ~ dcat(p[i, 1:nLevels])

51 modelstring = " model { #--- The likelihood ---# for (i in 1:N) {...these are used to describe the categorical distribution that is ultimately fit to the observed response variables # The standard part of our equation mu[i] <- b0 + (b1 * x1[i]) + (b2 * x2[i]) # log-odds for each value being in first category p[i, 1] <- pnorm(alpha[1], mu[i], 1 / sigma^2) # log-odds for each value being in categories between the highest and the lowest for (j in 2:(nLevels - 1)) { p[i, j] <- max(0, pnorm(alpha[j], mu[i], 1 / sigma^2) - pnorm(alpha[j - 1], mu[i], 1 / sigma^2)) # log-odds for each value being in the highest category p[i, nlevels] <- 1 - pnorm(alpha[nlevels - 1], mu[i], 1/sigma^2)... # Now, fit the y data to a categorical distribution # with the characteristics we just calculated y[i] ~ dcat(p[i, 1:nLevels])

52 Define the Model... #--- The Priors ---# # intercept and effect coefficients b0 ~ dnorm((1 + nlevels) / 2, 1 / nlevels^2) b1 ~ dnorm(0, 1 / nlevels^2) b2 ~ dnorm(0, 1 / nlevels^2) # Sigma sigma ~ dunif(nlevels / 1000, nlevels * 10) # Intermediate alpha values (we set the min and max # values in our initial data list) for (j in 2:(nLevels - 2)) { alpha[j] ~ dnorm(j + 0.5, 1 / 2^2) " writelines(modelstring, con = "model.txt")

53 Define the Model... #--- The Priors ---# # intercept and effect coefficients b0 ~ dnorm((1 + nlevels) / 2, 1 / nlevels^2) b1 ~ dnorm(0, 1 / nlevels^2) b2 ~ dnorm(0, 1 / nlevels^2) # Sigma sigma ~ dunif(nlevels / 1000, nlevels * 10) # Intermediate alpha values (we set the min and max # values in our initial data list) for (j in 2:(nLevels - 2)) { alpha[j] ~ dnorm(j + 0.5, 1 / 2^2) " writelines(modelstring, con = "model.txt") Our mean value (b0) should be about in the centre of our categories, and the sd is the number of categories (can t be beyond this)

54 Define the Model... #--- The Priors ---# # intercept and effect coefficients b0 ~ dnorm((1 + nlevels) / 2, 1 / nlevels^2) b1 ~ dnorm(0, 1 / nlevels^2) b2 ~ dnorm(0, 1 / nlevels^2) # Sigma sigma ~ dunif(nlevels / 1000, nlevels * 10) # Intermediate alpha values (we set the min and max # values in our initial data list) for (j in 2:(nLevels - 2)) { alpha[j] ~ dnorm(j + 0.5, 1 / 2^2) " writelines(modelstring, con = "model.txt") Same logic for sd here

55 Define the Model... #--- The Priors ---# # intercept and effect coefficients b0 ~ dnorm((1 + nlevels) / 2, 1 / nlevels^2) b1 ~ dnorm(0, 1 / nlevels^2) b2 ~ dnorm(0, 1 / nlevels^2) # Sigma sigma ~ dunif(nlevels / 1000, nlevels * 10) # Intermediate alpha values (we set the min and max # values in our initial data list) for (j in 2:(nLevels - 2)) { alpha[j] ~ dnorm(j + 0.5, 1 / 2^2) " writelines(modelstring, con = "model.txt") Our prior for sigma comes from a uniform distribution with a minimum value of our number of levels divided by 1000, and a maximum value of the number of levels times 10

56 Define the Model... #--- The Priors ---# # intercept and effect coefficients b0 ~ dnorm((1 + nlevels) / 2, 1 / nlevels^2) b1 ~ dnorm(0, 1 / nlevels^2) b2 ~ dnorm(0, 1 / nlevels^2) # Sigma sigma ~ dunif(nlevels / 1000, nlevels * 10) # Intermediate alpha values (we set the min and max # values in our initial data list) for (j in 2:(nLevels - 2)) { alpha[j] ~ dnorm(j + 0.5, 1 / 2^2) " writelines(modelstring, con = "model.txt") Only need alpha priors for the middle values because outer ones were specified. These will come from a normal distribution with mean of that level value plus 0.5, and a sd of 2. Note that what you choose here should be based on what values you used to specify the outer alphas.

57 Specify Initial Values initslist <- function() { list( b0 = rnorm(n = 1, mean = (1 + nlevels) / 2, sd = nlevels), b1 = rnorm(n = 1, mean = 0, sd = nlevels), b2 = rnorm(n = 1, mean = 0, sd = nlevels), sigma = runif(n = 1, min = nlevels / 1000, max = nlevels * 10) )

58 Specify MCMC Parameters and Run runjagsout <- run.jags( method = "simple", model = "model.txt", monitor = c("b0", "b1", "b2", "sigma", "alpha"), data = datalist, inits = initslist, n.chains = 3, adapt = 500, burnin = 1000, sample = 20000, thin = 1, summarise = TRUE, plots = FALSE)

59 Specify MCMC Parameters and Run runjagsout <- run.jags( method = "simple", model = "model.txt", monitor = c("b0", "b1", "b2", "sigma", "alpha"), data = datalist, inits = initslist, n.chains = 3, Note adapt that = there 500, is a lot going on in this burnin = 1000, model. As a result, it takes substantially sample = 20000, longer thin than = 1, our other ones. This one takes about summarise 10 min = on TRUE, my computer, and my plots = FALSE) computer is fairly fast.

60 Next Steps (On Your Own) Retrieve the data and take a peek at the structure Test model performance Extract & parse results Convert back to original scale

61 View Posteriors

62 Plotting Posterior Distributions β 0 par(mfrow = c(1, 1)) histinfo = plotpost(b0, xlab = bquote(beta[0])) mean = % HDI β 0

63 Plotting Posterior Distributions β 1 & β 2 par(mfrow = c(1, 2)) histinfo = plotpost(b1, xlab = bquote(beta[1]), main = "x1") histinfo = plotpost(b2, xlab = bquote(beta[2]), main = "x2") x1 x2 mean = mean = % HDI % HDI β β 2

64 Plotting Posterior Distributions β 1 & β 2 x1 has a credible negative effect, and x2 has par(mfrow = c(1, 2)) a credible positive effect. These are on a histinfo = plotpost(b1, xlab = bquote(beta[1]), main = "x1") histinfo = plotpost(b2, real xlab scale. = No bquote(beta[2]), need to transform main them = for "x2") interpretation. x1 mean = x2 mean = % HDI % HDI β β 2

65 Posterior Predictive Check

66 Posterior Predictive Check Code is clunky and slow, but should make sense (and work!) Takes about 10 minutes on my computer...so be patient! Code predicts for the entire data set (N = 200), but could use a subset For each step in the chain, count the number of individuals assigned to each level (given the predictor variables) Compare the mean and HDIs of these predictions relative to true values

67 Posterior Predictive Check source("hdiofmcmc.r") # Create a matrix to hold results ypostpred <- matrix(0, nrow = chainlength, ncol = nlevels) # For each step in the chain... for (i in 1:chainLength) { # Initialize holders (counters for each level) counter1 <- 0 counter2 <- 0 counter3 <- 0 counter4 <- 0 counter5 <- 0 counter6 <- 0 counter7 <- 0...

68 Posterior Predictive Check source("hdiofmcmc.r") Starting a loop that will go through every step in the chain # Create a matrix to hold results ypostpred <- matrix(0, nrow = chainlength, ncol = nlevels) # For each step in the chain... for (i in 1:chainLength) { # Initialize holders (counters for each level) counter1 <- 0 counter2 <- 0 counter3 <- 0 counter4 <- 0 counter5 <- 0 counter6 <- 0 counter7 <- 0...

69 Posterior Predictive Check source("hdiofmcmc.r") # Create a matrix to hold results ypostpred <- matrix(0, nrow = chainlength, ncol = nlevels) # For each step in the chain... for (i in 1:chainLength) { # Initialize holders (counters for each level) counter1 <- 0 counter2 <- 0 counter3 <- 0 counter4 <- 0 counter5 <- 0 counter6 <- 0 counter7 < For each response level, initialize a counter that will keep track of how many results were assigned to each level (re-zeroed for each step in the chain)

70 Posterior Predictive Check... # For each individual... for (j in 1:N) { Then (for each step in the chain), for each individual (data point) calculate the mean, and then the probability of them being in each response level, using equations we have seen before. # Calculate mean mu = b0[i] + (b1[i] * x1[j]) + (b2[i] * x2[j]) # Calculate their probability of being in each level levelprobs <- rep(0, times = nlevels) levelprobs[1] <- pnorm(alpha[i, 1], mu, sigma[i]) levelprobs[2] <- pnorm(alpha[i, 2], mu, sigma[i]) - pnorm(alpha[i, 1], mu, sigma[i]) levelprobs[3] <- pnorm(alpha[i, 3], mu, sigma[i]) - pnorm(alpha[i, 2], mu, sigma[i]) levelprobs[4] <- pnorm(alpha[i, 4], mu, sigma[i]) - pnorm(alpha[i, 3], mu, sigma[i]) levelprobs[5] <- pnorm(alpha[i, 5], mu, sigma[i]) - pnorm(alpha[i, 4], mu, sigma[i]) levelprobs[6] <- pnorm(alpha[i, 6], mu, sigma[i]) - pnorm(alpha[i, 5], mu, sigma[i]) levelprobs[7] <- 1 - pnorm(alpha[i, 6], mu, sigma[i])...

71 # Find item number for highest value levelid <- which.max(levelprobs) # Increase counter for appropriate group if(levelid == 1) { counter1 <- counter1 + 1 else { if (levelid == 2) { counter2 <- counter2 + 1 else { if (levelid == 3) { counter3 <- counter3 + 1 else { if (levelid == 4) { counter4 <- counter4 + 1 else { if (levelid == 5) { counter5 <- counter5 + 1 else { if (levelid == 6 ) { counter6 <- counter6 + 1 else { counter7 <- counter7 + 1

72 # Find item number for highest value levelid <- which.max(levelprobs) # Increase counter for appropriate group if(levelid == 1) { counter1 <- counter1 + 1 else { if (levelid == 2) { counter2 <- counter2 + 1 else { if (levelid == 3) { counter3 <- counter3 + 1 else { if (levelid == 4) { counter4 <- counter4 + 1 else { if (levelid == 5) { counter5 <- counter5 + 1 else { if (levelid == 6 ) { counter6 <- counter6 + 1 else { counter7 <- counter7 + 1 Identify for which response level the individual has the highest probability

73 # Find item number for highest value levelid <- which.max(levelprobs) # Increase counter for appropriate group if(levelid == 1) { counter1 <- counter1 + 1 else { if (levelid == 2) { counter2 <- counter2 + 1 else { if (levelid == 3) { counter3 <- counter3 + 1 else { if (levelid == 4) { counter4 <- counter4 + 1 else { if (levelid == 5) { counter5 <- counter5 + 1 else { if (levelid == 6 ) { counter6 <- counter6 + 1 else { counter7 <- counter7 + 1 Increment the appropriate counter. For each step in the chain, these counters will be the number of individuals predicted to be in each response level

74 Posterior Predictive Check... # Place results in results matrix ypostpred[i, 1] <- counter1 ypostpred[i, 2] <- counter2 ypostpred[i, 3] <- counter3 ypostpred[i, 4] <- counter4 ypostpred[i, 5] <- counter5 ypostpred[i, 6] <- counter6 ypostpred[i, 7] <- counter7 ypredmeans = apply(ypostpred, 2, median, na.rm = TRUE) ypredhdi = apply(ypostpred, 2, HDIofMCMC)

75 Posterior Predictive Check Fill the appropriate row (step in chain) of the ypostpred matrix with the counts for each response level.... # Place results in results matrix ypostpred[i, 1] <- counter1 ypostpred[i, 2] <- counter2 ypostpred[i, 3] <- counter3 ypostpred[i, 4] <- counter4 ypostpred[i, 5] <- counter5 ypostpred[i, 6] <- counter6 ypostpred[i, 7] <- counter7 ypostpred will have one row for each step in the chain, indicating how many individuals were assigned to each response level (columns) ypredmeans = apply(ypostpred, 2, median, na.rm = TRUE) ypredhdi = apply(ypostpred, 2, HDIofMCMC)

76 Posterior Predictive Check... # Place results in results matrix ypostpred[i, 1] <- counter1 ypostpred[i, 2] <- counter2 ypostpred[i, 3] <- counter3 ypostpred[i, 4] <- counter4 ypostpred[i, 5] <- counter5 ypostpred[i, 6] <- counter6 ypostpred[i, 7] <- counter7 Calculate the mean and HDI for each response level, across all steps in the chain ypredmeans = apply(ypostpred, 2, median, na.rm = TRUE) ypredhdi = apply(ypostpred, 2, HDIofMCMC)

77 Posterior Predictive Check # Plot original data hist(y, breaks = c(0.5, (1:nLevels + 0.5)), main = "", col = "skyblue", border = "white") # Add predicted means points(x = 1:nLevels, y = ypredmeans, pch = 16) # Add HDI bars segments(x0 = 1:nLevels, y0 = ypredhdi[1, ], x1 = 1:nLevels, y1 = ypredhdi[2, ], lwd = 2) Frequency y

78 Questions?

79 Creative Commons License Anyone is allowed to distribute, remix, tweak, and build upon this work, even commercially, as long as they credit me for the original creation. See the Creative Commons website for more information. Click here to go back to beginning