私は自分のために働いた解決策を投稿するのに非常に遅れています。改善の余地はあると思いますので、お気軽にコメントください！
この演習の目的は、提案の評価の線形変換が提案の選択にどの程度影響するかを理解することでした。アイデアは、「提案の質」を「評価者のバイアス」と「パネルのバイアス」から解きほぐすことでした。
これを行う1つの方法は、基本的にすべての評価をパネル平均に中央揃えし、次にすべての評価の全体平均とsdを使用してパネル中心の評価の平均/sd変換を実行することです。この手順は関数内にありますadj.scores。
プロポーザルは人によって評価されるため、これは重要です。プロポーザルの評価の成功には、多大な金銭的インセンティブ（助成金、契約など）がかかる可能性があります。
改善や競合する戦略についての考えは大歓迎です。
####################
########## proposal ratings project
########## 17 June 2011
########## original code by: jjb with help from es
##########------ functions to be read in and called when desired
########## applying this function to a single matrix will give detailed output
########## calculating generalizability theory components
########## not a very robust formulation, but a good place to start
########## for future, put panel facet on this design
g.pxr.long = function(x)
{
m.raters <<- colMeans(x)
n.raters <<- length(m.raters)
m.props <<- rowMeans(x)
n.props <<- length(m.props)
m.total <<- mean(x)
n.total <<- nrow(x)*ncol(x)
m.raters.2 <<- m.raters^2
m.props.2 <<- m.props^2
sum.m.raters.2 <<- sum(m.raters.2)
sum.m.props.2 <<- sum(m.props.2)
ss.props <<- n.raters*(sum.m.props.2) - n.total*(m.total^2)
ss.raters <<- n.props*(sum.m.raters.2) - n.total*(m.total^2)
ss.pr <<- sum(x^2) - n.raters*(sum.m.props.2) - n.props*(sum.m.raters.2) + n.total*(m.total^2)
df.props <- n.props - 1
df.raters <- n.raters - 1
df.pr <- df.props*df.raters
ms.props <- ss.props / df.props
ms.raters <- ss.raters / df.raters
ms.pr <- ss.pr / df.pr
var.p <- (ms.props - ms.pr) / n.raters
var.r <- (ms.raters - ms.pr) / n.props
var.pr <- ms.pr
out.1 <- as.data.frame( matrix(c( df.props, df.raters, df.pr,
ss.props, ss.raters, ss.pr,
ms.props, ms.raters, ms.pr,
var.p, var.r, var.pr), ncol = 4))
out.2 <- as.data.frame(matrix(c("p", "r", "pr" ), ncol = 1))
g.out.1 <- as.data.frame(cbind(out.2, out.1))
colnames(g.out.1) <- c(" source", " df ", " ss ", " ms ", " variance")
var.abs <- (var.r / n.raters) + (var.pr / n.raters)
var.rel <- (var.pr / n.raters)
var.xbar <- (var.p / n.props) + (var.r / n.raters) + (var.pr / (n.raters*n.props) )
gen.coef <- var.p / (var.p + var.rel)
dep.coef <- var.p / (var.p + var.abs)
out.3 <- as.data.frame(matrix(c(var.rel, var.abs, var.xbar, gen.coef, dep.coef), ncol=1))
out.3 <- round(out.3, digits = 4)
out.4 <- as.data.frame(matrix(c("relative error variance", "absolute error variance", "xbar error variance", "E(rho^2)", "Phi"), ncol=1))
g.out.2 <- as.data.frame(cbind(out.4,out.3))
colnames(g.out.2) <- c(" estimate ", " value")
outs <- list(g.out.1, g.out.2)
names(outs) <- c("generalizability theory: G-study components", "G-study Indices")
return(outs)
}
##########----- calculating confidence intervals using Chebychev, Cramer, and Normal
##########----- use this to find alpha for desired k
factor.choose = function(k)
{
alpha <- 1 / k^2
return(alpha)
}
conf.intervals = function(dat, alpha)
{
k <- sqrt( 1 / alpha ) # specifying what the k factor is...
x.bar <- mean(dat)
x.sd <- sd(dat)
n.obs <- length(dat)
sem <- x.sd / sqrt(n.obs)
ub.cheb <- x.bar + k*sem
lb.cheb <- x.bar - k*sem
ub.crem <- x.bar + (4/9)*k*sem
lb.crem <- x.bar - (4/9)*k*sem
ub.norm <- x.bar + qnorm(1-alpha/2)*sem
lb.norm <- x.bar - qnorm(1-alpha/2)*sem
out.lb <- matrix(c(lb.cheb, lb.crem, lb.norm), ncol=1)
out.ub <- matrix(c(ub.cheb, ub.crem, ub.norm), ncol=1)
dat <- as.data.frame(dat)
mean.raters <- as.matrix(rowMeans(dat))
dat$z.values <- matrix((mean.raters - x.bar) / x.sd)
select.cheb <- dat[ which(abs(dat$z.values) >= k ) , ]
select.crem <- dat[ which(abs(dat$z.values) >= (4/9)*k ) , ]
select.norm <- dat[ which(abs(dat$z.values) >= qnorm(1-alpha/2)) , ]
count.cheb <- nrow(select.cheb)
count.crem <- nrow(select.crem)
count.norm <- nrow(select.norm)
out.dat <- list()
out.dat <- list(select.cheb, select.crem, select.norm)
names(out.dat) <- c("Selected cases: Chebychev", "Selected cases: Cramer's", "Selected cases: Normal")
out.sum <- matrix(c(x.bar, x.sd, n.obs), ncol=3)
colnames(out.sum) <- c("mean", "sd", "n")
out.crit <- matrix(c(alpha, k, (4/9)*k, qnorm(1-alpha/2)) ,ncol=4 )
colnames(out.crit) <- c("alpha", "k", "(4/9)*k", "z" )
out.count <- matrix(c(count.cheb, count.crem, count.norm) ,ncol=3 )
colnames(out.count) <- c("Chebychev", "Cramer" , "Normal")
rownames(out.count) <- c("count")
outs <- list(out.sum, out.crit, out.count, out.dat)
names(outs) <- c("Summary of data", "Critical values", "Count of Selected Cases", "Selected Cases")
return(outs)
}
rm(list = ls())
set.seed(271828)
means <- matrix(c( rep(50,19), rep(70,1) ), ncol = 1) # matrix of true proposal values
bias.u <- matrix(c(0,2,4), nrow=1) # unidirectional bias
bias.b <- matrix(c(0,5, -5), nrow=1) # bidirectional bias
ones.u <- matrix(rep(1,ncol(bias.u)), nrow = 1) # number of raters is the number of columns (r)
ones.b <- matrix(rep(1,ncol(bias.b)), nrow = 1)
ones.2 <- matrix(rep(1,nrow(means)), ncol = 1) # number of proposals is the number of rows (p)
true.ratings <- means%*%ones.u # gives matrix of true proposal value for each rater (p*r)
uni.bias <- ones.2%*%bias.u
bid.bias <- ones.2%*%bias.b # gives matrix of true rater bias for each proposal (p*r)
pan.bias.pos <- matrix(20,nrow=nrow(true.ratings), ncol=ncol(true.ratings)) # gives a matrix of bias for every member in a panel (p*r)
n.val <- nrow(true.ratings)*ncol(true.ratings)
# true.ratings
# uni.bias
# bid.bias
# pan.bias.pos
library(MASS)
#####
##### generating replicate data...
#####
n.panels <- 10 # put in the number of replicate panels that should be produced
reps <- 2 # put in the number of times each bias condition should be included in a panel
t.reps <- list()
n.bias <- list()
u.bias <- list()
b.bias <- list()
pan.bias <- list()
n.bias.out <- list()
u.bias.out <- list()
b.bias.out <- list()
pan.bias.out <- list()
for (i in 1:n.panels)
{
{
for(j in 1:reps)
n.bias[[j]] <- true.ratings + matrix(round(rnorm(n.val,4,2), digits=0), nrow = nrow(means))
for(j in 1:reps)
u.bias[[j]] <- true.ratings + uni.bias + matrix(round(rnorm(n.val,4,2), digits=0), nrow = nrow(means))
for(j in 1:reps)
b.bias[[j]] <- true.ratings + bid.bias + matrix(round(rnorm(n.val,4,2), digits=0), nrow = nrow(means))
}
pan.bias[[i]] <- true.ratings + pan.bias.pos + matrix(round(rnorm(n.val,4,2), digits=0), nrow = nrow(means))
n.bias.out[[i]] <- n.bias
u.bias.out[[i]] <- u.bias
b.bias.out[[i]] <- b.bias
t.reps[[i]] <- c(n.bias, u.bias, b.bias, pan.bias[i])
}
# t.reps
# rm(list = ls())
##########----- this is the standardization formula to be applied to proposal ratings **WITHIN** a panel
adj.scores <- function(x, tot.dat)
{
t.sd <- sd(array(tot.dat))
t.mn <- mean(array(tot.dat))
ones.t.mn <- diag(1,ncol(x))
p <- nrow(x)
r <- ncol(x)
ones.total <- matrix(1,p,r)
r.sd <- diag(apply(x,2, sd))
r.mn <- diag(apply(x,2, mean))
den.r.sd <- ginv(r.sd)
b.shift <- x%*%den.r.sd
a <- t.mn*ones.t.mn - den.r.sd%*%r.mn
a.shift <- ones.total%*%a
l.x <- b.shift + a.shift
return(l.x)
}
##########----- applying standardization formula and getting
##########----- proposal means for adjusted and unadjusted ratings
adj.rep <- list()
m.adj <- list()
m.reg <- list()
for (i in 1:n.panels)
{
panel.data <- array(unlist(t.reps[[i]]))
adj.rep[[i]] <- lapply(t.reps[[i]], adj.scores, tot.dat = panel.data)
m.adj[[i]] <- lapply(adj.rep[[i]], rowMeans)
m.reg[[i]] <- lapply(t.reps[[i]], rowMeans)
}
##########-----
##########----- This function extracts the replicate proposal means for each set of reviews
##########----- across panels. So, if there are 8 sets of reviewers that are simulated 10 times,
##########----- this extracts the first set of means across the 10 replications and puts them together,
##########----- and then extracts the second set of means across replications and puts them together, etc..
##########----- The result will be 8 matrices made up of 10 columns with rows .
##########-----
##########----- first for unadjusted means
means.reg <- matrix(unlist(m.reg), nrow=nrow(means))
sets <- length(m.reg[[1]])
counter <- n.panels*length(m.reg[[1]])
m.reg.panels.set <- list()
for (j in 1:sets)
{
m.reg.panels.set[[j]] <- means.reg[ , c(seq( j, counter, sets))]
}
##########----- now for adjusted means
means.adj <- matrix(unlist(m.adj), nrow=nrow(means))
sets <- length(m.adj[[1]])
counter <- n.panels*length(m.adj[[1]])
m.adj.panels.set <- list()
for (j in 1:sets)
{
m.adj.panels.set[[j]] <- means.adj[ , c(seq( j, counter, sets))]
}
########## calculating msd as bias^2 and error^2
msd.calc <- function(x)
{
true.means <- means
calc.means <- as.matrix(rowMeans(x))
t.means.mat <- matrix(rep(true.means, n.panels), ncol=ncol(x))
c.means.mat <- matrix(rep(calc.means, n.panels), ncol=ncol(x))
msd <- matrix( rowSums( (x - t.means.mat )^2) / ncol(c.means.mat) )
bias.2 <- (calc.means - true.means)^2
e.var <- matrix( rowSums( (x - c.means.mat )^2) / ncol(c.means.mat ) )
msd <- matrix(c(msd, bias.2, e.var), ncol = 3)
colnames(msd) <- c("msd", "bias^2", "e.var")
return(msd)
}
########## checking work...
# msd.calc <- bias.2 + e.var
# all.equal(msd, msd.calc)
##########----- applying function to each set across panels
msd.reg.panels <- lapply(m.reg.panels.set, msd.calc)
msd.adj.panels <- lapply(m.adj.panels.set, msd.calc)
msd.reg.panels
msd.adj.panels
########## for the unadjusted scores, the msd is very large (as is expected)
########## for the adjusted scores, the msd is in line with the other panels
##########----- trying to evaluate impact of adjusting scores on proposal awards
reg.panel.1 <- matrix(unlist(m.reg[[1]]), nrow=nrow(means))
adj.panel.1 <- matrix(unlist(m.adj[[1]]), nrow=nrow(means))
reg <- matrix(array(reg.panel.1), ncol=1)
adj <- matrix(array(adj.panel.1), ncol=1)
panels.1 <- cbind(reg,adj)
colnames(panels.1) <- c("unadjusted", "adjusted")
cor(panels.1, method="spearman")
plot(panels.1)
######## identify(panels.1)
plot(panels.1, xlim = c(25,95), ylim = c(25,95) )
abline(0,1, col="red")
########## the adjustment greatly reduces variances of ratings
########## compare the projection of the panel means onto the respective margins
##########----- examining the selection of the top 10% of the proposals
pro.names <- matrix(seq(1,nrow(reg),1))
df.reg <- as.data.frame(cbind(pro.names, reg))
colnames(df.reg) <- c("pro", "rating")
df.reg$q.pro <- ecdf(df.reg$rating)(df.reg$rating)
df.adj <- as.data.frame(cbind(pro.names, adj))
colnames(df.adj) <- c("pro", "rating")
df.adj$q.pro <- ecdf(df.adj$rating)(df.adj$rating)
awards.reg <- df.reg[ which(df.reg$q.pro >= .90) , ]
awards.adj <- df.adj[ which(df.adj$q.pro >= .90) , ]
awards.reg[order(-awards.reg$q.pro) , ]
awards.adj[order(-awards.adj$q.pro) , ]
awards.reg[order(-awards.reg$pro) , ]
awards.adj[order(-awards.adj$pro) , ]
##### using unadjusted scores, the top 10% of proposals come mostly from one (biased) panel, and the rest are made up of the
##### highest scoring proposal (from the biased rater) from the remaining panels.
##### using the adjusted scores, the top 10% of proposals are made up of the highest scoring proposal (the biased rater) from the
##### several panels, and a few proposals from other panels. Doesn't seem to be a systematic explanation as to why...
#########----- treating rating data in an ANOVA model
ratings <- do.call("rbind", t.reps[[1]] )
panels <- factor(gl(7,20))
fit <- manova(ratings ~ panels)
summary(fit, test="Wilks")
adj.ratings <- do.call("rbind", adj.rep[[1]] )
adj.fit <- manova(adj.ratings ~ panels)
summary(adj.fit, test="Wilks")
##########----- thinking... extra ideas for later
##########----- calculating Mahalanobis distance to identify biased panels
mah.dist = function(dat)
{md.dat <- as.matrix(mahalanobis(dat, center = colMeans(dat) , cov = cov(dat) ))
md.pv <- as.matrix(pchisq(md.dat, df = ncol(dat), lower.tail = FALSE))
out <- cbind(md.dat, md.pv)
out.2 <- as.data.frame(out)
colnames(out.2) <- c("MD", "pMD")
return(out.2)
}
mah.dist(ratings)
mah.dist(adj.ratings)