t-test of judge bias

Transcription

1 t-test of judge bias Xiaoqian Yang Goal: conduct a linear regression to fit the fodel: with constraints for each j, and s j,i = λ i µ j β j,c(i) ɛ j,i (1) µ j = 0 (2) j β j,c = 0 (3) c β j,c = 0 (4) j for each c. In the code below, a model matrix is constructed. The related coefficients and p-values are obtained with the function lsfit(). (Only those of the interactions are taken because that is what we are interested in.) Then the coefficients and p-values are put in two matrices where the rows stand for judges and the columns stand for countries. > ### function to get the model matrix > getcontrasts <- function(x, quality = FALSE, jtend = FALSE, r = 7) { n <- nrow(x) var1 <- names(x)[1] # how to use these variables? var2 <- names(x)[2] levels1 <- levels(x[,1]) levels2 <- levels(x[,2]) n1 <- length(levels1) # number of judges n2 <- length(levels2) # number of countries xmis <- table(x[,1], x[,2]) == 0 # subset is/not null xcon <- xmis # whether accessible by contrast for (i in 1:nrow(xmis)) xcon[i, max(which(!xmis[i,]))] <- "r" for (i in 1:ncol(xmis)) xcon[max(which(!xmis[,i])), i] <- "c" xcon[xcon!="c" & xcon!="r"] <- "" p <- sum(!xmis & xcon == "") # number of parameters to estimate 1

2 betapos <- matrix(0, n1, n2) # what are the parameters num <- table(x[,1], x[,2]) # number of data for each group k <- 1 for (i in 1:n1) for (j in 1:n2) { if (!xmis[i,j] & xcon[i,j] == "") { betapos[i,j] <- k k <- k1 rx <- array(0, dim = c(n1, n2, p)) # rows in the model matrix that correspond to a specific judge and country ans <- matrix(0, n, p) # model matrix colnames(ans) <- rep('', p) for (i in 1:n1) { for (j in 1:n2) { if (!xmis[i,j]) { if (xcon[i,j] == "") { rx[i,j,][betapos[i,j]] <- 1 ans[x[,1] == levels1[i] & x[,2] == levels2[j], ] <- matrix(rep(rx[i,j,], num[i,j]), ncol = p, byrow = T) else { if (xcon[i,j] == "r") { rx[i,j,] <- -colsums(rx[i,1:(j-1),]) ans[x[,1] == levels1[i] & x[,2] == levels2[j], ] <- matrix(rep(rx[i,j,], num[i,j]), ncol = p, byrow = T) else { rx[i,j,] <- -colsums(rx[1:(i-1),j,]) ans[x[,1] == levels1[i] & x[,2] == levels2[j], ] <- matrix(rep(rx[i,j,], num[i,j]), ncol = p, byrow = T) colnames(ans)[betapos[i,j]] <- paste(levels1[i], levels2[j], sep = '/') if (quality) { ans1 <- matrix(0, n, round(n/r)) # add the quality for each dive colnames(ans1) <- rep('', round(n/r)) for (i in 1:round(n/r)) { # do this without loop? ans1[(r*i-r1):(r*i),i] <- 1 colnames(ans1)[i] <- paste('d', i, sep = '') 2

3 ans <- cbind(ans, ans1) if (jtend) { ans1 <- matrix(0, n, n1-1) # add each judge's score tendency colnames(ans1) <- levels1[1:(n1-1)] for (i in 1:(n1-1)) { ans1[,i] <- (x$judge == levels1[i])-(x$judge == levels1[n1]) ans <- cbind(ans, ans1) pval <- matrix(na, n1, n2) # store the pvalues in Judge*Country ma bias <- matrix(na, n1, n2) lsbase <- ls.print(lsfit(ans, y, intercept = FALSE), print.it = F) k <- 1 for (i in 1:n1) for (j in 1:n2) { if (!xmis[i,j] & xcon[i,j] == "") { bias[i,j] <- lsbase$coef[[1]][k,1] pval[i,j] <- lsbase$coef[[1]][k,4] k <- k1 rownames(pval) <- levels1 colnames(pval) <- levels2 rownames(bias) <- levels1 colnames(bias) <- levels2 return(list("mm" = ans, "p-value" = pval, 'bias' = bias)) Next apply the function above to the diving data. Eight countries had to be removed from the dataset to eliminate colinearity. Here is results of the linear regression: > x <- read.csv(" as.is=true) > x <- x[x$country!= 'ARG',] > x <- x[x$country!= 'GEO',] > x <- x[x$country!= 'ARM',] > x <- x[x$country!= 'AZE',] > x <- x[x$country!= 'CZE',] > x <- x[x$country!= 'HKG',] > x <- x[x$country!= 'ZIM',] 3

4 > x <- x[x$country!= 'FIN',] > y <- x$jscore > theinput <- data.frame(judge=factor(x$judge), Country=factor(x$Country)) > res <- getcontrasts(theinput, quality = T, jtend = T) > J <- unique(x$judge) > JC <- rep(na, length(j)) > pown <- rep(na, length(j)) > for (i in 1:length(J)) { if (sum(x$judge == J[i] & x$jcountry == x$country)>0) { JC[i] <- unique(x[x$judge == J[i], ]$JCountry) pown[i] <- res$p[j[i], JC[i]] > pres <- data.frame(judge = J, Jcountry = JC, P_value = pown) > pres <- pres[!is.na(pres$p_value),] > pres <- pres[order(pres$jc),] > pres Judge Jcountry P_value 14 BARNETT Madeleine AUS e HUBER Peter AUT e-03 3 BOYS Beverley CAN e SEAMAN Kathy CAN e-01 8 WANG Facheng CHN e XU Yiming CHN e-02 1 RUIZ-PEDREGUERA Rolando CUB e-02 7 CRUZ Julia ESP e-03 5 BOUSSARD Michel FRA e BOOTHROYD Sydney GBR e-03 9 ALT Walter GER e BURK Hans-Peter GER e MENA Jesus MEX e-04 6 CALDERON Felix PUR e GEISSBUHLER Michael SUI e McFARLAND Steve USA e-03 Output is the p-values of the 17 interactions that stand for the bias of judges toward divers from their countries. These results coincide with results of the paper Assessing Judging Bias: An Example From the 2000 Olympic Games. Next try to replicate Figure 1 from the paper. Successfully replicated the figure. The correctness of the model matrix is verified. Note: Also tried implementing Yu s Lagrange optimization algorithm to check that the two algorithms are equivalent. If qualities of each dive is added to the regression as variables, this algorithm cannot be applied to the big dataset (at least not for my computer) since it 4

5 1 AUS AUT BLR BRA CAN CHN COL CUB ESP FRA GBR GER GRE HUN INA JPN ITA KAZ KOR MAS MEX PER PHI PRK PUR STEWART Anthea McFARLAND Steve LINDBERG Mathz GEISSBUHLER Michael ZAITSEV Oleg CALDERON Felix HOOD Robin GEAR Dennis JOHNSON Bente MENA Jesus KELEMEN Ildiko BURK Hans Peter ALT Walter BOOTHROYD Sydney BOUSSARD Michel CRUZ Julia HOOD Robin CERMAKOVA Maria RUIZ PEDREGUERA Rolando XU Yiming WANG Facheng SEAMAN Kathy BOYS Beverley HUBER Peter BARNETT Madeleine 1 5

6 requires huge space allocation for the matrix operations. (How do the functions lm/lsfit avoid that matrix multiplication?) So only the toy data is used. Here is the implementation: > getbeta <- function(x) { n <- nrow(x) var1 <- names(x)[1] var2 <- names(x)[2] levels1 <- levels(x[,1]) levels2 <- levels(x[,2]) n1 <- length(levels1) n2 <- length(levels2) ans <- matrix(0, n, n1) # model matrix for (i in 1:n1) { ans[x[,1] == levels1[i], i] <- 1 ans1 <- matrix(0, n, n1*n2) for (i in 1:n1) { for (j in 1:n2) { ans1[x[,1] == levels1[i] & x[,2] == levels2[j], (i-1)*n2j] <- 1 ans <- cbind(ans, ans1) ans1 <- matrix(0, n, n/3) for (i in 1:(n/3)) { ans1[(3*i-2):(3*i),i] <- 1 ans <- cbind(ans, ans1) xmis <- table(x[,1], x[,2]) == 0 # construct matrix H H <- matrix(0, n1n2sum(xmis), n1n1*n2n/3) H[1, 1:n1] <- 1 for (i in 1:n1) { H[i1, (n1(i-1)*n21):(n1(i-1)*n2n2)] <- 1 for (i in 1:(n2-1)) { H[1n1i, seq(n11i-1, n1n1*n2i-n2, by = n2)] <- 1 for (i in 1:length(which(t(xmis)))) { H[1n1(n2-1)i, n1which(t(xmis))[i]] <- 1 X <- ans H <- t(h) 6

7 A1 <- cbind(t(x)%*%x, H) A2 <- cbind(t(h), matrix(0,ncol(h),ncol(h))) A <- rbind(a1,a2) b <- rbind(t(x)%*%y, matrix(0, ncol(h), 1)) beta <- solve(a, b) bias <- matrix(beta[(n11):(n1n1*n2)], nrow=n1, byrow=true) colnames(bias) <- levels1 rownames(bias) <- levels2 return(bias) > x <- read.csv(" as.is=true) > theinput <- data.frame(judge=factor(x$judge), Country=factor(x$Country)) > y <- x$jscore > bias1 <- getbeta(theinput) > bias1 BOUSSARD Michel CALDERON Felix McFARLAND Steve MENA Jesus CHN e e MEX e e PUR e e USA e e > bias2 <- getcontrasts(theinput, quality = T, jtend = T, r = 3)$bias > bias2 CHN MEX PUR USA BOUSSARD Michel NA NA CALDERON Felix NA McFARLAND Steve NA NA NA MENA Jesus NA NA NA NA The results are identical, the algorithms are both correct approaches to fit our linear model. One problem with Lagrange optimization approach is that it is not as simple as in contrast sum linear model to work out the p-values (need to deduct distribution of ˆβ theoretically). 7