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**
POLI 279 MEASUREMENT THEORY
Third Assignment
Due 2 May 2007**

- The aim of this problem is to show you how to use
**R**to simulate a complex model of stimulus evaluation. The idea is that people use a simple mental model to compare two stimuli. The distinguishing features of the stimuli are assumed to be represented by dimensions in a simple geometric model. The stimuli are positioned on the dimensions according to the levels of the attributes represented by the dimensions. People are assumed to perceive the stimuli correctly with some random error. When asked to perform a stimulus comparison, people draw a momentary psychological value from a very tight error distribution around the locations of each of the stimuli. Their judgement of similarity is assumed to be an exponential function of the psychological distance between the two stimuli --**e**-- where^{-kd}**d**is the distance between the two momentary psychological values expressed as points in psychological space, and**k**is a scaling constant (see the figure below).

What we are going to do is use a program to simulate this process and numerically compute theof**expected value****e**for 40 values of the true distance (from 0 to 2) between the true locations of the stimuli.^{-kd}

Use the following**R**program below to do this experiment:

Shepard-Ennis-Nofosky_1.r -- Program to test the Shepard-Ennis_Nofosky Model of Stimulus Comparison

Here is what the program looks like:

The program will produce a graph that looks similar to this:**# # # normal_2.r -- Test of Shepard_Ennis_Nofosky model of Stimulus # Comparison # # Exemplars have bivariate normal distributions. One point is # drawn from each distribution and the distance computed. # This distance is then exponentiated -- exp(-d) -- and # this process is repeated m times to get a E[exp(-d)]. # # Note that when STDDEVX is set very low then this is equivalent # to a legislator's ideal point so that the stimulus comparison # is between the ideal point and the momentary draw from the # Exemplar distribution. # # Set Means and Standard Deviations of Two Bivariate Normal Distributions # # Remove all objects just to be safe # rm(list=ls(all=TRUE)) # STDDEVX <- .001 Initialize Means and Standard STDDEVY <- .4 Deviations of the Two Normal Distributions XMEAN <- 0 YMEAN <- 0 # # Set Number of Draws to Get Expected Value # m <- 10000 # # Set Number of Points # n <- 40 # # Set Increment for Distance Between Two Bivariate Normal Distributions # XINC <- 0.05 Note that n*XINC = 2.0 # x <- NULL Initialize vectors y <- NULL z <- NULL xdotval <- NULL ydotval <- NULL zdotval <- NULL iii <- 0 while (iii <= n) { ****Start of Outer Loop -- it Executes Until y has n=40 Entries x <- 0 i <- 0 iii <- iii + 1 while (i <= m) { ****Start of Inner Loop -- it Executes m=10000 Times # # Draw points from the two bivariate normal distributions # Draw two numbers randomly from Normal Distribution with Mean XMEAN and stnd. dev. of STDDEVX and put them in the vector xdotval xdotval <- c(rnorm(2,XMEAN,STDDEVX)) ydotval <- c(rnorm(2,YMEAN,STDDEVY)) # Draw two numbers randomly from Normal Distribution with Mean YMEAN and stnd. dev. of STDDEVY and put them in the vector ydotval # Compute distance Between two randomly drawn points # Calculate Euclidean Distance Between xdotval and ydotval zdotval <- sqrt((xdotval[1]-ydotval[1])**2 + (xdotval[2]-ydotval[2])**2) # # Exponentiate the distance # zdotval <- exp(-zdotval) # # Compute Expected Value # x <- x + zdotval/m Store exp(-d) in x -- note the division by m so that x will contain i <- i + 1 the mean after m=10000 trials } ****End of Inner Loop # # Store Expected Value and Distance Between Means of Two Distributions # y[iii] <- x z[iii] <- sqrt(2*YMEAN^2) YMEAN <- YMEAN + XINC Increment YMEAN by .05 } ****End of Outer Loop # # The type="n" tells R to not display anything # plot(z,y,xlim=c(0,3),ylim=c(0,.7),type="n", main="", xlab="", ylab="",font=2) # # Another Way of Doing the Title # title(main="Test of Shepard-Ennis-Nofosky, .001, .4\nTest of Ideal Point vs. Outcome Model") # Main title mtext("Test of Shepard-Ennis-Nofosky, .001, .4\nTest of Ideal Point vs. Outcome Model",side=3,line=1.00,cex=1.2,font=2) # x-axis title mtext("True Distance Between Stimuli",side=1,line=2.75,font=2,cex=1.2) # y-axis title mtext("Expected Value of Observed Similarity",side=2,line=2.75,font=2,cex=1.2) ## # An Alternative Way to Plot the Points and the Lines # points(z,y,pch=16,col="blue",font=2) lines(z,y,lty=1,lwd=2,font=2,col="red") #**

- Run the Program and turn in the resulting plot.

- Use
**Epsilon**to change**STDDEVX**and**STDDEVY**in**Shepard-Ennis-Nofosky_1.R**, re-load it into**R**, and produce a plot like the above. For example, change**STDDEVX**to .1 and**STDDEVY**to .4 and re-run the program.*Change the "title" each time to reflect the different values*! Pick 3 pairs of reasonable values of**STDDEVX**and**STDDEVY**and produce plots for each experiment.

- Run the Program and turn in the resulting plot.
- The aim of this problem is to show you how to double-center a
matrix in
**R**and extract the eigenvalues and eigenvectors of the matrix. Use the**R**program below to do this exercise:

Double_Center_Drive_Data.r -- Program to Double-Center a Matrix of Squared Distances-
Drive2.txt -- 11 City Driving Distances Data

The program will produce a graph that looks similar to this:**# # double_center_drive_data.r -- Double-Center Program # # Data Must Be Transformed to Squared Distances Below # # ATLANTA 0000 2340 1084 715 481 826 1519 2252 662 641 2450 # BOISE 2340 0000 2797 1789 2018 1661 891 908 2974 2480 680 # BOSTON 1084 2797 0000 976 853 1868 2008 3130 1547 443 3160 # CHICAGO 715 1789 976 0000 301 936 1017 2189 1386 696 2200 # CINCINNATI 481 2018 853 301 0000 988 1245 2292 1143 498 2330 # DALLAS 826 1661 1868 936 988 0000 797 1431 1394 1414 1720 # DENVER 1519 891 2008 1017 1245 797 0000 1189 2126 1707 1290 # LOS ANGELES 2252 908 3130 2189 2292 1431 1189 0000 2885 2754 370 # MIAMI 662 2974 1547 1386 1143 1394 2126 2885 0000 1096 3110 # WASHINGTON 641 2480 443 696 498 1414 1707 2754 1096 0000 2870 # CASBS 2450 680 3160 2200 2330 1720 1290 370 3110 2870 0000 # # Remove all objects just to be safe # rm(list=ls(all=TRUE)) # library(MASS) library(stats) # # T <- matrix(scan("C:/ucsd_course/drive2.txt",0),ncol=11,byrow=TRUE) # nrow <- length(T[,1]) ncol <- length(T[1,]) # Another way of entering names -- Note the space after the name names <- c("Atlanta ","Boise ","Boston ","Chicago ","Cincinnati ","Dallas ","Denver ", "Los Angeles ","Miami ","Washington ","CASBS ") # # pos -- a position specifier for the text. Values of 1, 2, 3 and 4, # respectively indicate positions below, to the left of, above and # to the right of the specified coordinates # namepos <- rep(2,nrow) # TT <- rep(0,nrow*ncol) dim(TT) <- c(nrow,ncol) TTT <- rep(0,nrow*ncol) dim(TTT) <- c(nrow,ncol) # xrow <- NULL xcol <- NULL xcount <- NULL matrixmean <- 0 matrixmean2 <- 0 # # Transform the Matrix # i <- 0 while (i < nrow) { i <- i + 1 xcount[i] <- i Ignore -- Just a diagnostic j <- 0 while (j < ncol) { j <- j + 1 # # Square the Driving Distances # TT[i,j] <- (T[i,j]/1000)**2 Note the division by 1000 for scale purposes # } } # # Put it Back in T # T <- TT TTT <- sqrt(TT) This creates***distances*-- see below # # # Call Double Center Routine From R Program # cmdscale(....) in stats library # The Input data are DISTANCES!!! Not Squared Distances!! # Note that the R routine does not divide # by -2.0 # ndim <- 2 # dcenter <- cmdscale(TTT,ndim, eig=FALSE,add=FALSE,x.ret=TRUE) # # returns double-center matrix as dcenter$x if x.ret=TRUE # # Do the Division by -2.0 Here # TTT <- (dcenter$x)/(-2.0) Here is the Double-Centered Matrix # # # Below is the old Long Method of Double-Centering # # Compute Row and Column Means # i <- 0 while (i < nrow) { i <- i + 1 xrow[i] <- mean(T[i,]) } i <- 0 while (i < ncol) { i <- i + 1 xcol[i] <- mean(T[,i]) } matrixmean <- mean(xcol) matrixmean2 <- mean(xrow) # # Double-Center the Matrix Using old Long Method # Compute comparison as safety check # i <- 0 while (i < nrow) { i <- i + 1 j <- 0 while (j < ncol) { j <- j + 1 TT[i,j] <- (T[i,j]-xrow[i]-xcol[j]+matrixmean)/(-2) This is the Double-Center Operation } } # # Run some checks to make sure everything is correct # xcheck <- sum(abs(TT-TTT)) This checks to see if both methods are the same numerically # # # Perform Eigenvalue-Eigenvector Decomposition of Double-Centered Matrix # ev <- eigen(TT) # # Find Point furthest from Center of Space # aaa <- sqrt(max((abs(ev$vec[,1]))**2 + (abs(ev$vec[,2]))**2)) These two commands do the same thing bbb <- sqrt(max(((ev$vec[,1]))**2 + ((ev$vec[,2]))**2)) # # Weight the Eigenvectors to Scale Space to Unit Circle # torgerson1 <- ev$vec[,1]*(1/aaa)*sqrt(ev$val[1]) torgerson2 <- -ev$vec[,2]*(1/aaa)*sqrt(ev$val[2]) Note the Sign Flip # plot(torgerson1,torgerson2,type="n",asp=1, main="", xlab="", ylab="", xlim=c(-3.0,3.0),ylim=c(-3.0,3.0),font=2) points(torgerson1,torgerson2,pch=16,col="red",font=2) text(torgerson1,torgerson2,names,pos=namepos,offset=0.2,col="blue",font=2) Experiment with the offset value # # # Main title mtext("Double-Centered Driving Distance Matrix \n Torgerson Coordinates",side=3,line=1.25,cex=1.5,font=2) # x-axis title mtext("West --- East",side=1,line=2.75,cex=1.2,font=2) # y-axis title mtext("South --- North",side=2,line=2.5,cex=1.2,font=2) #

- Run the Program and turn in the resulting plot.

- Change the values in "namepos" until you have a nice looking plot
*with no overlapping names and no names cut off by the margins*and turn in the resulting plot.

- Use
**Epsilon**to add a 12^{th}city (you will have to get a driving atlas to do this) to**Drive2.txt**and call the matrix**Drive3.txt**. Modify the**R**program to read and plot the 12 by 12 driving distance matrix. Turn in**Drive3.txt**, the resulting plot, and the modified**R**program.

- Run the Program and turn in the resulting plot.
- In this problem we are going to analyze Eckman's famous
data. Use the*color circle***R**program below to do this exercise:

Double_Center_Color_Circle.r -- Program to Double-Center a Matrix of Squared Distances-
Color_Circle.txt -- 14 Color Similarities Data

Run the program and you should see something like this:**# # # double_center_color_circle.r -- Double-Center Program # # Data Must Be Transformed to Squared Distances Below # # **************** The Eckman Color Data ***************** # # 434 INDIGO 100 86 42 42 18 6 7 4 2 7 9 12 13 16 # 445 BLUE 86 100 50 44 22 9 7 7 2 4 7 11 13 14 # 465 42 50 100 81 47 17 10 8 2 1 2 1 5 3 # 472 BLUE-GREEN 42 44 81 100 54 25 10 9 2 1 0 1 2 4 # 490 18 22 47 54 100 61 31 26 7 2 2 1 2 0 # 504 GREEN 6 9 17 25 61 100 62 45 14 8 2 2 2 1 # 537 7 7 10 10 31 62 100 73 22 14 5 2 2 0 # 555 YELLOW-GREEN 4 7 8 9 26 45 73 100 33 19 4 3 2 2 # 584 2 2 2 2 7 14 22 33 100 58 37 27 20 23 # 600 YELLOW 7 4 1 1 2 8 14 19 58 100 74 50 41 28 # 610 9 7 2 0 2 2 5 4 37 74 100 76 62 55 # 628 ORANGE-YELLOW 12 11 1 1 1 2 2 3 27 50 76 100 85 68 # 651 ORANGE 13 13 5 2 2 2 2 2 20 41 62 85 100 76 # 674 RED 16 14 3 4 0 1 0 2 23 28 55 68 76 100 # # # Remove all objects just to be safe # rm(list=ls(all=TRUE)) # library(MASS) library(stats) # # rcx.file <- "c:/ucsd_course/color_circle.txt" # # Standard fields and their widths # rcx.fields <- c("colorname","x1","x2","x3","x4","x5", "x6","x7","x8","x9","x10","x11","x12","x13","x14") rcx.fieldWidths <- c(18,4,4,4,4,4,4,4,4,4,4,4,4,4,4) # # Read the vote data from fwf # U <- read.fwf(file=rcx.file,widths=rcx.fieldWidths,as.is=TRUE,col.names=rcx.fields) dim(U) ncolU <- length(U[1,]) Note the "trick" I used here T <- U[,2:ncolU] The 1st column of U has the color names # # nrow <- length(T[,1]) ncol <- length(T[1,]) # Even though I have the color names in U this is more convenient # because there are no leading or trailing blanks names <- c("434 Indigo","445 Blue","465","472 Blue-Green","490","504 Green","537", "555 Yellow-Green","584","600 Yellow","610","628 Orange-Yellow", "651 Orange","674 Red") # # pos -- a position specifier for the text. Values of 1, 2, 3 and 4, # respectively indicate positions below, to the left of, above and # to the right of the specified coordinates # namepos <- rep(2,nrow) # TT <- rep(0,nrow*ncol) dim(TT) <- c(nrow,ncol) TTT <- rep(0,nrow*ncol) dim(TTT) <- c(nrow,ncol) # xrow <- NULL xcol <- NULL xcount <- NULL matrixmean <- 0 matrixmean2 <- 0 # # Transform the Matrix # i <- 0 while (i < nrow) { i <- i + 1 xcount[i] <- i j <- 0 while (j < ncol) { j <- j + 1 # # Transform the Color "Agreement Scores" into Distances # TT[i,j] <- ((100-T[i,j])/50)**2 Note the Transformation from a 100 to 0 scale # to a 0 to 4 (squared distance) scale } } # # Put it Back in T # T <- TT TTT <- sqrt(TT) # # # Call Double Center Routine From R Program # cmdscale(....) in stats library # The Input data are DISTANCES!!! Not Squared Distances!! # Note that the R routine does not divide # by -2.0 # ndim <- 2 # dcenter <- cmdscale(TTT,ndim, eig=FALSE,add=FALSE,x.ret=TRUE) # # returns double-center matrix as dcenter$x if x.ret=TRUE # # Do the Division by -2.0 Here # TTT <- (dcenter$x)/(-2.0) # # # Below is the old Long Method of Double-Centering # # Compute Row and Column Means # i <- 0 while (i < nrow) { i <- i + 1 xrow[i] <- mean(T[i,]) } i <- 0 while (i < ncol) { i <- i + 1 xcol[i] <- mean(T[,i]) } matrixmean <- mean(xcol) matrixmean2 <- mean(xrow) # # Double-Center the Matrix Using old Long Method # Compute comparison as safety check # i <- 0 while (i < nrow) { i <- i + 1 j <- 0 while (j < ncol) { j <- j + 1 TT[i,j] <- (T[i,j]-xrow[i]-xcol[j]+matrixmean)/(-2) } } # # Run some checks to make sure everything is correct # xcheck <- sum(abs(TT-TTT)) # # # Perform Eigenvalue-Eigenvector Decomposition of Double-Centered Matrix # ev <- eigen(TT) # # Find Point furthest from Center of Space # aaa <- sqrt(max((abs(ev$vec[,1]))**2 + (abs(ev$vec[,2]))**2)) bbb <- sqrt(max(((ev$vec[,1]))**2 + ((ev$vec[,2]))**2)) # # Weight the Eigenvectors to Scale Space to Unit Circle # torgerson1 <- ev$vec[,1]*(1/aaa)*sqrt(ev$val[1]) torgerson2 <- -ev$vec[,2]*(1/aaa)*sqrt(ev$val[2]) # plot(torgerson1,torgerson2,type="n",asp=1, main="", xlab="", ylab="", xlim=c(-3.0,3.0),ylim=c(-3.0,3.0),font=2) points(torgerson1,torgerson2,pch=16,col="red",font=2) text(torgerson1,torgerson2,names,pos=namepos,offset=0.2,col="blue",font=2) # # # Main title mtext("The Color Circle \n Torgerson Coordinates",side=3,line=1.25,cex=1.5,font=2) # x-axis title mtext("",side=1,line=2.75,cex=1.2,font=2) # y-axis title mtext("",side=2,line=2.5,cex=1.2,font=2) #**

- Run the program and turn in this figure.

- Change the values in "namepos" until you have a nice looking plot
*with no overlapping names and no names cut off by the margins*and turn in the resulting plot. Use the**offset**parameter in the text command to space the names away from the points. Also*flip the dimensions as appropriate so that the configuration matches the one in Borg and Goenen*!

- Check out:

Color Circle in Color

Use the**colors()**command in**R**to find the corresponding colors and then use the text command to color the label for the corresponding point; e.g.,**text(torgerson1[10],torgerson2[10],"Yellow",pos=4,col="yellow",font=2)**

You will have to override the labeling command in the double-center program to do this. Turn in the plot (**hard copy in B/W**) and the listing of your*and*e-mail it to me so I can see it!**R**program.

- Plot the eigenvalues that are stored in ev$value. The simplest plot is to use the command at the
**R**prompt:

**plot(1:14,ev$value)**

and you will get the following really lame graph:

Turn this into a nicer graph. To do this, check out the**R**code for Figure 5.4 of my book which is posted on my website at:

Chapter 5 Page

Turn in a final graph and a listing of the**R**code you use to create it.

- Run the program and turn in this figure.
- Run Tables 1.1, 1.3, 1.4, and 4.1 from Borg and Groenen through
**KYST**. Note that all four tables are**similarities**so that you can use the Supreme Court example as a template. (Note that you will have to enter the diagonals for tables 1.3 and 1.4 -- 9 and 7, respectively.) For example, for Table 4.1, if you entered the data as follows:

then change the format statement to:**434_INDIGO 100 86 42 42 18 6 7 4 2 7 9 12 13 16 445_BLUE 86 100 50 44 22 9 7 7 2 4 7 11 13 14 465 42 50 100 81 47 17 10 8 2 1 2 1 5 3 472_BLUE-GREEN 42 44 81 100 54 25 10 9 2 1 0 1 2 4 490 18 22 47 54 100 61 31 26 7 2 2 1 2 0 504_GREEN 6 9 17 25 61 100 62 45 14 8 2 2 2 1 537 7 7 10 10 31 62 100 73 22 14 5 2 2 0 555_YELLOW-GREEN 4 7 8 9 26 45 73 100 33 19 4 3 2 2 584 2 2 2 2 7 14 22 33 100 58 37 27 20 23 600_YELLOW 7 4 1 1 2 8 14 19 58 100 74 50 41 28 610 9 7 2 0 2 2 5 4 37 74 100 76 62 55 628_ORANGE-YELLOW 12 11 1 1 1 2 2 3 27 50 76 100 85 68 651_ORANGE 13 13 5 2 2 2 2 2 20 41 62 85 100 76 674_RED 16 14 3 4 0 1 0 2 23 28 55 68 76 100**

**(18X,101F4.0)**

Be sure to change the missing values parameter to a negative number because there are some zeroes in the matrix. Your**KYST**file should look like the following:**TORSCA PRE-ITERATIONS=3 DIMMAX=3,DIMMIN=1 PRINT HISTORY,PRINT DISTANCES COORDINATES=ROTATE ITERATIONS=50 REGRESSION=DESCENDING DATA,LOWERHALFMATRIX,DIAGONAL=PRESENT,CUTOFF=-.01 EKMAN'S COLOR DATA EXAMPLE 14 1 1 (18X,101F4.0) ****the color data**** COMPUTE STOP**- Run all tables from 3 to 1 dimensions, report the Stress values, and
use
**R**to graph the results for each table in two dimensions.

- Produce Shepard Diagrams for each two dimensional solution in the same
format as Homework 2.

- Run all tables from 3 to 1 dimensions, report the Stress values, and
use