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**
POLS 6386 MEASUREMENT THEORY
Seventh Assignment
Due 18 March 2003**

- The aim of this problem is to study the relationship between correlated
dimensions and their associated singular values (eigenvalues). This is a
continuation of problem 1 of Homework 6. Download
the following
**R**program:

R Program to Perform SVD on Bivariate Normal and Display Ratio of Singular Values

Below is a listing of the program and the graph that it creates:**# # Program to Illustrate Singular Value Decomposition # # Draws n points from bivariate normal distribution # and draws graph of ratio of 1st over 2nd # Singular Values # nrow <- 1000 ncol <- 2 # # Number of Iterations # nitr <- 39 # # These two commands create a matrix that is 39 by 4 and filled # with zeroes # w <- rep(0,nitr*4) dim(w) <- c(nitr,4) # # xinc is the increment -- that is, the value that is subtracted # from the previous value of the correlation after every pass # through the loop below. Note that it equals .05 # xinc <- 2/(nitr+1) # # Initial Value for Correlation # rcor <- .95 # # Note that at the first pass through the loop r=.95, the 2nd pass # r=.90, third pass r=.85, etc. At the 39th pass, r=-.95. # iii <- 1 while (iii <= nitr) { # # Create Variance-Covariance Matrix # # 1.0 0.9 # 0.9 1.0 # Sigma <- matrix(c(1,rcor,rcor,1),2,2) # # Call Bivariate Normal with zero mean and # Sigma Var-Cov Matrix, Place in X # X <- mvrnorm(n=nrow,rep(0,2),Sigma) # # Perform Singular Value Decomposition # xsvd <- svd(X) # # The Two Lines Below Put the Singular Values in a # Diagonal Matrix -- The first one creates an # identity matrix and the second command puts # the singular values on the diagonal # Lambda <- diag(ncol) diag(Lambda) <- xsvd$d # # Compute U*LAMBDA*V' for check below # XX <- xsvd$u %*% Lambda %*% t(xsvd$v) # # Compute Fit of SVD -- This is just the sum of squared # error -- Note that ssesvd should be zero! # # i <- 0 j <- 0 ssesvd <- 0 while (i < nrow) { i <- i + 1 j <- 0 while (j < ncol) { j <- j + 1 ssesvd <- ssesvd + (X[i,j] - XX[i,j])**2 } } # # Store the correlation, fit, and two singular values for plotting # purposes # w[iii,1] <- rcor w[iii,2] <- ssesvd w[iii,3] <- xsvd$d[1] w[iii,4] <- xsvd$d[2] # # Increment correlation value # rcor <- rcor - xinc # # Increment Loop counter # iii <- iii+1 # # End of Big Loop # } # # Plot Commands # plot(w[,1],w[,3]/w[,4],xlab="Correlation Between Dimensions",ylab="Ratio of 1st to 2nd Singular Value",pch=16,col="blue") mtext(side=3,line=1.5,"Bivariate Normal Random Draws \nRatio of Singular Values",font=2)**

Note that the figure above is symmetric! Consequently, we only need to be concerned with studying correlation values between zero and 1.0.

- Modify the
**R**program above so that it runs from .98 to .00 in steps of .02. Turn in your program listing and turn in the resulting graph. It should look something like this:

You can insert text inside the graph with the command:

**text(.3,7.0,"1000 by 2 Matrix \n.02 Size Steps")**

- Run your
**R**program from .99 to .00 in steps of .01. Turn in the graph.

- Run your
**R**program from .98 to .00 in steps of .02 with 2000 by 2 matrices. Turn in the graph.

- Run your
**R**program from .98 to .00 in steps of .02 with 5000 by 2 matrices. Turn in the graph.

- Run your
**R**program from .99 to .00 in steps of .01 with 100 by 2 matrices. Turn in the graph.

- Modify the
- This problem combines parts of
problem 2 of homework 3 with parts of
problem 2 of homework 6. We are going
to study
the recent vote to invoke cloture on the Miguel A. Estrada Nomination. We are
going to create coordinates for the 107
^{th}Senate using the double-centered squared distance matrix computed from the agreement scores and then produce smoothed histograms of those Senators who served in both the 107^{th}and the 108^{th}who voted for or against cloture. Download the new agreement score program**HOUSYM3.EXE**, the 107^{th}roll call data, and the control card file (the latter two files were used in problem 2 of homework 3).

New Agreement Score Program

107^{th}Senate Roll Call Matrix

Control Card File for Agreement Score Program

The first few lines of the agreement score file**SEN107.DAT**should look like this:

Use**1079991099 0USA 200 BUSH 100 92 92 99 97 96 etc etc 1074970041 0ALABAMA 20001SESSIONS 92 100 92 89 86 86 etc etc 1079465941 0ALABAMA 20001SHELBY 92 92 100 90 87 83 etc etc etc etc etc****Epsilon**to strip off the header of each row so that you only have the agreement score matrix itself. Download the double-centering program below:

Double-Centering Program For Symmetric Agreement Score Matrix

Below is a listing of the program:**# # double_center_6.r -- Double-Center Program # # The Data Are assumed to be Squared Distances -- Data Must # Be Transformed to Squared Distances Below # T <- matrix(scan("D:/R_Files/sen107_ascore.dat",0),ncol=103,byrow=TRUE) sennames <- read.csv("D:/R_Files/sen_107.txt",header=F) attach(sennames) # # Catch The Missing Data # TT <- ifelse(T <= 0,50,T) T <- TT # # nrow <- length(T[,1]) ncol <- length(T[1,]) xrow <- NULL xcol <- NULL matrixmean <- 0 matrixmean2 <- 0 # # Transform the Matrix # i <- 0 while (i < nrow) { i <- i + 1 j <- 0 while (j < ncol) { j <- j + 1 # # This is the Normal Transformation # TT[i,j] <- ((100 - T[i,j])/50)**2 # } } T <- TT # # # 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 # 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) } } ev <- eigen(TT) DIM.1 <- ev$vec[,1]*sqrt(ev$val[1]) DIM.2 <- ev$vec[,2]*sqrt(ev$val[2]) #****SEN107_ASCORE.DAT**is the agreement score matrix and**SEN_107.TXT**has the roll call vote. For example, here is the first few lines of my**SEN_107.TXT**file:

Note that I use "1" for Yea, "6" for Nay, and "0" for no longer in the Senate. I also treated President Bush as if he voted for Cloture.**99,USA , 200,1,Bush 41,ALABAMA, 200,1,Sessions 41,ALABAMA, 200,1,Shelby 81,ALASKA , 200,1,Murkowski 81,ALASKA , 200,1,Stevens 61,ARIZONA, 200,1,Kyl 61,ARIZONA, 200,1,McCain 42,ARKANSA, 200,0,Hutchinson 42,ARKANSA, 100,6,Lincoln 71,CALIFOR, 100,6,Boxer etc etc etc**

- Construct the
**SEN_107.TXT**file and turn in a listing of it.

- Produce a Density plot of the Yea and Nay voters on the Cloture vote
for the Estrada nomination. If you do it correctly, it should look something
like the following:

Turn in your graph*and*the**R**code that you used to create the graph.

- Produce a Density plot of the
*Democrat*Yea and Nay voters on the Cloture vote for the Estrada nomination. If you do it correctly, it should look something like the following:

Turn in your graph*and*the**R**code that you used to create the graph.

- Construct the
- The aim of this problem is to show you how to use
**metric unfolding**to analyze thermometer scores. To do this you need to run a program that unfolds the thermometer scores. We are going to analyze the 1968 feeling thermometers. Download the the program, control card file, and data file and place them in the same directory.

Metric Unfolding Program (MLSMU6)

Control Card File for Metric Unfolding Program

1968 Election Data

The 1968 Election Data file contains the same variables that we have used in the past plus the thermometer scores and voting information for the respondents. The variables are:

The control card file for the metric unfolding procedure is shown below. The first line has the name of the data file. The first number in the second line is the number of stimuli, the next two numbers are the minimum and maximum number of dimensions to estimate, and the "10" is the number of iterations.**idno respondent id number partyid strength of party id -- 0 to 6 income raw income category incomeq income quintile -- 1 to 5 race 0 = white, 1 = black sex 0 = man, 1 = woman south 0 = north, 1 = south education 1=HS, 2=SC, 3=College age age in years uulbj lbj position urban unrest uuhhh humphrey pos urban unrest uunixon nixon position urban unrest uuwallace wallace pos urban unrest uuself self placement urban unrest vnmlbj lbj pos vietnam vnmhhh hhh pos vietnam vnmnixon nixon pos vietnam vnmwallace wallace pos vietnam vnmself self placement vietnam voted 1=voted, 5=did not vote votedfor who voted for -- 1 = humphrey, 2= nixon, 3=wallace wallace wallace therm humphrey humphrey thermometer nixon nixon thermometer mccarthy mccarthy thermometer reagan reagan thermometer rockefeller rockefeller thermometer lbj lbj thermometer romney romney thermometer kennedy robert kennedy thermometer muskie muskie thermometer agnew agnew thermometer lemay "bombs away with Curtis LeMay" thermometer**

The third line contains some "antique" options we will never use. The only numbers that matter on this line are the "4" which indicates the number of identifying characters to read off each line of the data file (e.g., the respondent id number), and the "2" at the end. This is the number of missing data codes which appear in the sixth line.

The first number in the fourth line is a tolerance value -- leave it as is. The next three numbers are parameters to transform the input data into squared distances. In this case, let amx=-.02, bmx=2.0, and cmx=2.0. The following equation transforms the thermometers into squared distances:

**d**^{2}= (amx*t+bmx)^{cmx}

where t = input data. This formula takes a linear transformation of the input data to the power cmx. With amx = -.02, bmx = 2.0, and cmx = 2.0, this is equivalent to subtracting the thermometer score from 100, dividing by 50, and then squaring. This converts t from a 0-100 scale to a 4-0 scale. If the data, t, are distances, set amx = 1.0, bmx = 0.0, and cmx = 2.0. If the data are correlations, set amx = -1.0, bmx = 1.0, and cmx=2.0 or 1.0 if the correlations are initially regarded as unsquared or squared distances respectively.

The next value, "1.5", is the maximum absolute expected coordinate value on any dimension. It is used for plotting purposes. If the squared distances are confined to a 4-0 scale, xmax=1.5 is usually sufficient. The last two numbers, "0.0" and "100.0", are the minimum and maximum expected values of the input data. These are used to catch coding errors in the input data. Anything out of range is treated as missing data.

The fifth line is the format of the data file and the sixth line contains the missing data codes.

Finally, the last 12 lines are labels for the stimuli.**OLS68B.DAT 12 2 2 10 0 0 1 1 0 4 2 .001 -0.02 2.0 2.0 1.5 0.0 100.0 (1X,4A1,60X,12F3.0) 98 99 WALLACE HUMPHREY NIXON MCCARTHY REAGAN ROCKEFELLER LBJ ROMNEY R.KENNEDY MUSKIE AGNEW LEMAY**- Run
**MLSMU6**. It will produce an output file called**FORT.22**. The first 20 lines look like this:

The first two columns after the names are the two dimensional coordinates. The first 12 lines are the coordinates for the political candidates and lines 13 onward are the coordinates for the respondents. Use**WALLACE 1.2646 0.5154 217.4823 0.5541 1242.0000 HUMPHREY -0.5559 0.3738 114.7892 0.6968 1252.0000 NIXON 0.1480 -0.5415 123.2209 0.5319 1250.0000 MCCARTHY -0.6251 -0.4938 151.8926 0.3854 1204.0000 REAGAN 0.3080 -0.8895 131.8091 0.4380 1212.0000 ROCKEFELLER -0.5579 -0.5995 148.1413 0.3724 1229.0000 LBJ -0.5223 0.4905 147.0334 0.5573 1253.0000 ROMNEY -0.4736 -0.7866 111.3147 0.3434 1167.0000 R.KENNEDY -0.4245 0.2351 148.8571 0.5418 1242.0000 MUSKIE -0.6611 0.1660 126.0836 0.4862 1177.0000 AGNEW 0.2341 -0.8706 114.1418 0.4675 1180.0000 LEMAY 1.1901 0.4267 174.3242 0.4601 1188.0000 1681 -0.0285 0.2555 0.7918 0.6824 12.0000 1124 -0.1768 0.2692 1.4788 0.6992 12.0000 78 0.5707 -0.1514 3.5611 0.2141 12.0000 553 0.1376 0.1064 0.1597 0.7047 9.0000 7 0.2542 0.1235 1.2634 0.0116 12.0000 412 0.2781 0.0867 0.1024 0.6197 12.0000 631 0.5017 0.1088 1.1196 0.0742 12.0000 1316 0.2175 -0.5842 1.1568 0.8577 12.0000 etc etc etc etc etc etc****R**to plot the 12 candidates in two dimensions. This plot should be very similar to the one you did for question 2 of homework 5.

- Use
**Epsilon**to insert the**voted**and**voted for**variables into**FORT.22**(strip off the candidate coordinates first). Turn in the**Epsilon**macro you used to do the insertion and the first 20 lines of the file.

- Use
**R**to make two-dimensional plots of the Voters, Non-Voters, Humphrey Voters only, Nixon Voters only, and Wallace Voters only. For example, your Humphrey Voter plot should look something like this:

Label each plot appropriately and use solid dots to plot the respondents. Turn in all these plots.

- Run