The program automatically simulates the values of the Rasch model parameters, and
# randomly generates a data set from those parameters, before computing all the 36
# person fit statistics. However, simple modifications of the commands in the "Initiate()"
# function allow the user to input his/her own data set, and/or input his/her parameter
# estimates (of any IRT model), before computing the 36 fit statistics.
#
#
#
# THREE STEPS TO CALCULATE 36 PERSON FIT STATISTICS
#
# (1) ENTER APPROPRIATE SPECIFICATIONS WITHIN THE
#
"Initiate" FUNCTION BELOW, INCLUDING DATA,
#
RASCH PARAMETER ESTIMATES, NUMBER OF PERSONS, NUMBER OF ITEMS,
#
ETC.
#
# (2) SAVE THIS PROGRAM, AND THEN,
#
WHILE STILL IN THIS COMMAND WINDOW, HIT "F10"
#
TO READ YOUR UPDATES
#
# (3) WITH YOUR CURSOR, HIGHLIGHT THIS COMMAND: RUNIT()
#
AND THEN HIT "F10" TO GENERATE THE RESULTS
#
OF THE 36 PERSON FIT STATISTICS IN THE BOTTOM OF THIS WINDOW
#
THE RESULTS WILL ASLO APPEAR IN A MATRIX NAMED "RESULTS",
#
IN THE "Object Explorer"WINDOW OF S-PLUS
INITIATE <<- function(){
options(object.size=8000000)
# PROGRAM SETUP--------------------------------------------------------------------------------#-----------------------------------------------------------------------------------------------------------------------# IN THE NEXT SIX LINES ENTER THE NUMBER (N) OF PERSONS,
# THE NUMBER (L) OF TEST ITEMS, YOU WISH TO ANALYZE;
# ALSO ENTER THE NUMBER OF ITEMS IN EACH OF THE THREE
# TEST SUBSETS (FOR PERSON FIT METHODS BASED ON GROUPING TEST ITEMS)
#-----------------------------------------------------------------------------------------------------------------------N <<- 100
# ENTER Number of persons
L <<- 40
# ENTER Number of test items
S <<- 3
# ENTER Number of item subsets
s.1 <<- 13
# ENTER Number of item in subset 1
s.2 <<- 14
# ENTER Number of item in subset 2
s.3 <<- 13
# ENTER Number of items in subset 3
#
#
#======================================================================
# Simulate parameters of a Rasch model
#======================================================================
theta <<- sort(runif(N,-3,3)) ##THIS COMMAND SIMULATES L PERSON ABILITY PARAMETERS
FROM A UNIFORM U(-3,3) DISTRIBUTION
#BUT YOU MAY YOU MAY INSTEAD ENTER ANY SET OF N
ESTIMATES OF THE ABILITY PARAMETER
delta <<- sort(runif(L,-3,3)) ##THIS COMMAND SIMULATES L ITEM DIFFICULTY PARAMETER
VALUES FROM A UNIFORM U(-3,3) DISTRIBUTION
#BUT HERE YOU MAY INSTEAD ENTER ANY SET OF L ESTIMATES
OF THE ITEM DIFFICULTY PARAMETER
logs <<- (matrix(theta,N,1) %*% matrix(1,1,L)) - (matrix(1,N,1) %*% matrix(delta,1,L))
probs <<- (1+exp(-logs))^-1
delta.matrix <<- matrix(1,N,1) %*% matrix(delta,1,L)
#
#
#======================================================================#
# Generate a Simulated Data Set (or enter your own data set)
#======================================================================
# NOTE: YOU MAY INSTEAD WISH TO ENTER YOUR OWN (REAL) DATA MATRIX
# INTO AN OBJECT CALLED "Data"
# IF YOU WISH TO USE REAL DATA,
# THEN YOU DON'T NEED TO USE THE NEXT 5 LINES OF CODE
Data <<- ifelse(matrix(runif(N*L),N,L)>probs,0,1)
# Generate data
Data <<- t(rbind(Data,(N-colSums(Data))))
Data <<- t(Data[order(Data[,(N+1)]),-(N+1)])
Data <<- cbind(Data,(rowSums(Data)))
Data <<- Data[order(Data[,(L+1)]),-(L+1)]
#-------------------------------------------------------------------------------------------------------- ---------------return()}
# ******YOU NEED NOT MODIFY ANY OF THE REMAINING CODE BELOW
# ******YOU NEED NOT MODIFY ANY OF THE REMAINING CODE BELOW
# ******YOU NEED NOT MODIFY ANY OF THE REMAINING CODE BELOW
FITANALYSIS1 <<- function(){
# elements of Van der Flier statistics
#-------------------------------------------------------------------------------------------------------# Person/item scores & proportions
#
r <<- rowSums(Data)
# Person score r
i.sc <<- colSums(Data)
# Item score s
r.matrix <<- matrix(r,N,1) %*% matrix(1,1,L)
s.matrix <<- matrix(1,N,1) %*% matrix(i.sc,1,L)
p.cor.n <<- (r/L)
# Proportion correct by person n
p.cor.i <<- (i.sc/N)
# Proportion correct by item i
p.cor.i.matrix <<- t(matrix(p.cor.i,L,N))
#
# Create Guttman & Anti-Guttman matrices
#
gutt <- r.matrix - t(matrix(1:L,L,N))
gutt <<- ifelse(gutt<0,0,1)
antigutt <- (L-r.matrix) - t(matrix(1:L,L,N))
antigutt <<- ifelse(antigutt<0,1,0)
log.p.i <<- log(p.cor.i.matrix / (1-p.cor.i.matrix))
U.a <<- rowSums(gutt * log.p.i,na.rm=T)
U.b <<- rowSums(Data * log.p.i,na.rm=T)
U.c <<- rowSums(antigutt * log.p.i,na.rm=T)
a.1 <<- rowSums(p.cor.i.matrix * log.p.i,na.rm=T)
a.2 <<- (rowSums(p.cor.i.matrix * (1-p.cor.i.matrix) * log.p.i,na.rm=T)) / (rowSums(p.cor.i.matrix * (1p.cor.i.matrix),na.rm=T))
a.3 <<- r - rowSums(p.cor.i.matrix,na.rm=T)
a <<- a.1 + (a.2*a.3)
E.U3 <<- (U.a-a) / (U.a-U.c)
b.1 <<- rowSums(p.cor.i.matrix * (1-p.cor.i.matrix) * (log.p.i^2),na.rm=T)
b.2 <<- (rowSums(p.cor.i.matrix * (1-p.cor.i.matrix) * log.p.i,na.rm=T))^2
b.3 <<- rowSums(p.cor.i.matrix * (1-p.cor.i.matrix),na.rm=T)
b <<- (b.1 - (b.2/b.3))
sd.U3 <<- (sqrt(b)) / (abs(U.a - U.c))
#
# elements of Sijitsma H statistic
#
sig.ij <<- var(t(Data),t(Data),unbiased=F)
diag(sig.ij) <<- rep(0,N)
H.num <<- apply(sig.ij,1,sum)
H.denom1 <<- matrix(p.cor.n,N,1) %*% matrix(1-p.cor.n,1,N) #Bi(1-Bj)
H.denom2 <<- matrix(1-p.cor.n,N,1) %*% matrix(p.cor.n,1,N) #(1-Bi)Bj
H.denom <<- ifelse(H.denom1>H.denom2,H.denom2,H.denom1)
diag(H.denom) <<- rep(0,N)
H.denom <<- apply(H.denom,1,sum)
#
# IRT probability elements
#
IRT.var <<- probs*(1-probs)
IRT.resid <<- (Data-probs)
IRT.sq.resid <<- (Data-probs)^2
IRT.sqstan.resid <<- IRT.sq.resid / IRT.var
inv.IRT.var <<- 1/IRT.var
U.sd <<- sqrt((rowSums(inv.IRT.var) - (4*L)) / (L^2))
W.sd <<- (sqrt((rowSums(IRT.var,na.rm=T)) - (4*(rowSums(IRT.var^2,na.rm=T))))) /
(rowSums(IRT.var,na.rm=T))
UW1.sd <<- sqrt((rowSums(inv.IRT.var[,1:s.1]) - (4*s.1))/(s.1^2))
UW2.sd <<- sqrt((rowSums(inv.IRT.var[,(s.1+1):(s.1+s.2)]) - (4*s.2))/(s.2^2))
UW3.sd <<- sqrt((rowSums(inv.IRT.var[,(s.1+s.2+1):L]) - (4*s.3))/(s.3^2))
UB.sd <<- (2 / (S-1))^.5
UBa <<- rowSums(IRT.sq.resid[,1:s.1],na.rm=T)/rowSums(IRT.var[,1:s.1],na.rm=T)
UBb <<rowSums(IRT.sq.resid[,(s.1+1):(s.1+s.2)],na.rm=T)/rowSums(IRT.var[,(s.1+1):(s.1+s.2)],na.rm=T)
UBc <<- rowSums(IRT.sq.resid[,(s.1+s.2+1):L],na.rm=T)/rowSums(IRT.var[,(s.1+s.2+1):L],na.rm=T)
#
# Elements of ECI statistics
group.probs <<- matrix(1,N,1) %*% matrix(colMeans(probs,na.rm=T),1,L)
IRT.probs.mean <<- matrix(rowMeans(probs,na.rm=T),N,1) %*% matrix(1,1,L)
PY <<- (probs - Data)
#
# Elements of Likelihood statistics
l.matrix <<- (Data*log(probs)) + ((1-Data)*log(1-probs))
l <<- rowSums(l.matrix,na.rm=T)
E.l.matrix <<- (probs*log(probs)) + ((1-probs)*log(1-probs))
E <<- rowSums(E.l.matrix,na.rm=T)
var.l.matrix <<- probs*(1-probs)*((log(probs/(1-probs)))^2)
var.l <<- rowSums(var.l.matrix,na.rm=T)
l.m.num <<- ((rowSums(l.matrix[,1:s.1],na.rm=T)) - (rowSums(E.l.matrix[,1:s.1],na.rm=T))) +
((rowSums(l.matrix[,(s.1+1):(s.1+s.2)],na.rm=T)) (rowSums(E.l.matrix[,(s.1+1):(s.1+s.2)],na.rm=T))) + ((rowSums(l.matrix[,(s.1+s.2+1):L],na.rm=T)) (rowSums(E.l.matrix[,(s.1+s.2+1):L],na.rm=T)))
l.m.den <<- (rowSums(var.l.matrix[,1:s.1],,na.rm=T)) +
((rowSums(var.l.matrix[,(s.1+1):(s.1+s.2)],,na.rm=T))) +
((rowSums(var.l.matrix[,(s.1+s.2+1):L],,na.rm=T)))
#
## elements of p value for Molenaar's M statistic
m.M1 <<- rowSums((-delta.matrix)*probs,na.rm=T)
sd.M.11 <<- rowSums((-delta.matrix)*(-delta.matrix)*probs*(1-probs),na.rm=T)
sd.M.12 <<- rowSums((-delta.matrix)*probs*(1-probs),na.rm=T)
sd.M.22 <<- rowSums(probs*(1-probs),na.rm=T)
kappa03 <<- rowSums(((-delta.matrix)^0)*(1^3)*probs*(1-probs)*(1-(2*probs)),na.rm=T)
kappa12 <<- rowSums(((-delta.matrix)^1)*(1^2)*probs*(1-probs)*(1-(2*probs)),na.rm=T)
kappa21 <<- rowSums(((-delta.matrix)^2)*(1^1)*probs*(1-probs)*(1-(2*probs)),na.rm=T)
kappa30 <<- rowSums(((-delta.matrix)^3)*(1^0)*probs*(1-probs)*(1-(2*probs)),na.rm=T)
beta.M <<- sd.M.12/sd.M.22
m.M <<- m.M1 - ((kappa12 - (beta.M*kappa03)) / (2*sd.M.22))
var.M <<- sd.M.11 - ((sd.M.12^2)/(sd.M.22))
gam <<- ((kappa30) - (3*beta.M*kappa21) + (3*(beta.M^2)*kappa12) - ((beta.M^3)*kappa03)) /
(sqrt(var.M)^3)
#
return()}
FITANALYSIS <<- function(){
#
#*************************************************************************************
**
# 1. Guttman statistic (Guttman, 1944)
#
g.data <- Data[,-1]
h.data <- Data[,-L]
Gni <- g.data * (1-h.data)
G <<- rowSums(Gni)
#*************************************************************************************
**
# 2. Normed Guttman statistic (Van der Flier, 1977)
#
Gstar <<- G / (r*(L-r))
#
#*************************************************************************************
**
# 3. Personal biserial correlation (Donlan & F. Fischer, 1968), non-parametric
#
pbs <<- diag(cor(t(Data),t(s.matrix),unbiased=F))
#
#*************************************************************************************
**
# 4. Caution index (Sato, 1975)
#
Caution <<- 1 - ((diag(var(t(Data),t(s.matrix),unbiased=F))) / (diag(var(t(gutt),t(s.matrix),unbiased=F))))
#
#*************************************************************************************
**
# 5. Modified Caution index (Harnisch & Linn, 1981)
#
MCI <<- (diag(var(t(gutt),t(p.cor.i.matrix),unbiased=F)) - diag(var(t(Data),t(p.cor.i.matrix),unbiased=F))) /
(diag(var(t(gutt),t(p.cor.i.matrix),unbiased=F)) - diag(var(t(antigutt),t(p.cor.i.matrix),unbiased=F)))
#
#*************************************************************************************
**
# 6. U3 (Van der Flier, 1982)
#
U3 <<- (U.a-U.b) / (U.a-U.c)
#
#*************************************************************************************
**
# 7. ZU3 (Van der Flier, 1982)
#
ZU3 <<- (U3 - E.U3) / (sd.U3)
#
#*************************************************************************************
**
#8. H statistic (Sijtsma, 1986)
#
H <<- H.num / H.denom
#
#*************************************************************************************
**
# 9-11. A,D,E (Kane & Brennan, 1980)
#
A <<- rowSums(Data*p.cor.i.matrix,na.rm=T)
D <<- rowSums(p.cor.i.matrix*gutt,na.rm=T) - A
Ei <<- A / rowSums(p.cor.i.matrix*gutt,na.rm=T)
#
#*************************************************************************************
**
# 12-17. Mean squares & standardized mean squares
#
U <<- rowMeans(IRT.sqstan.resid,na.rm=T)
ZU <<- ((U^(1/3))-1)*(3/(U.sd)) + (U.sd/3)
# Smith, 1986
W <<- rowSums(IRT.sq.resid,na.rm=T) / rowSums(IRT.var,na.rm=T)
ZW <<- ((W^(1/3))-1)*(3/(W.sd)) + (W.sd/3)
# Smith, 1986
ln.U <<- (log(U) + U - 1) * (sqrt((L-1)/8))
ln.W <<- (log(W) + W - 1) * (sqrt((L-1)/8))
#
#*************************************************************************************
**
# 18-27. ECIs
#
ECI1 <<- 1 - ((diag(var(t(Data),t(s.matrix),unbiased=F,na.method="include"))) /
(diag(var(t(probs),t(s.matrix),unbiased=F,na.method="include"))))
ECI2 <<- 1 - ((diag(var(t(Data),t(group.probs),unbiased=F,na.method="include"))) /
(diag(var(t(probs),t(group.probs),unbiased=F,na.method="include"))))
ECI3 <<- 1 - ((diag(cor(t(Data),t(group.probs),unbiased=F,na.method="include"))) /
(diag(cor(t(probs),t(group.probs),unbiased=F,na.method="include"))))
ECI4 <<- 1 - ((diag(var(t(Data),t(probs),unbiased=F,na.method="include"))) /
(diag(var(t(probs),t(group.probs),unbiased=F,na.method="include"))))
ECI5 <<- 1 - ((diag(cor(t(Data),t(probs),unbiased=F,na.method="include"))) /
(diag(cor(t(probs),t(group.probs),unbiased=F,na.method="include"))))
ECI6 <<- 1 - ((diag(var(t(Data),t(probs),unbiased=F,na.method="include"))) /
(rowVars(probs,unbiased=F,na.rm=T)))
ECI1z <<- (diag(var(t(PY),t(s.matrix),unbiased=F,na.method="include"))*L) / sqrt(rowSums((((s.matrix matrix(mean(i.sc),N,L))^2)*IRT.var),na.rm=T))
ECI2z <<- (diag(var(t(PY),t(group.probs),unbiased=F,na.method="include"))*L) /
sqrt(rowSums((((group.probs - matrix(mean(group.probs[1,]),N,L))^2)*IRT.var),na.rm=T))
ECI4z <<- (diag(var(t(PY),t(probs),unbiased=F,na.method="include"))*L) / sqrt(rowSums((((probs IRT.probs.mean)^2)*IRT.var),na.rm=T))
ECI6z <<- ECI4z
#
#*************************************************************************************
**
# 28-29. Likelihood l (Levine & Rubin, 1979) and standardized l (Drasgow et al., 1985)
#
l <<- l
lz <<- (l - E)/(sqrt(var.l))
#
#*************************************************************************************
**
# 30-31. M statisticS (Molenaar & Hoijtink, 1990)
#
M <<- (rowSums(Data*delta.matrix,na.rm=T)) * -1
ZM <<- (M - m.M) / sqrt(var.M)
M.pvalue <<- pnorm(ZM) - (((dnorm(ZM))*((ZM^2)-1)*gam)/6)
#Bedrick
#
#*************************************************************************************
**
# 32-36. Item group person fit statistics
#
D.theta.W1 <<- rowMeans(IRT.sq.resid[,1:s.1],na.rm=T)
D.theta.W2 <<- rowMeans(IRT.sq.resid[,(s.1+1):(s.1+s.2)],na.rm=T)
D.theta.W3 <<- rowMeans(IRT.sq.resid[,(s.1+s.2+1):L],na.rm=T)
D.theta <<- D.theta.W1 + D.theta.W2 + D.theta.W3
lzm <<- l.m.num / l.m.den
UB <<- (1/(S-1)) * (UBa+UBb+UBc)
ZUB <<- ((UB^(1/3))-1)*(3/(UB.sd)) + (UB.sd/3)
# Smith, 1986
ln.UB <<- (log(UB) + UB - 1) * (sqrt((S-1)/8))
UW1 <<- rowMeans(IRT.sqstan.resid[,1:s.1],na.rm=T)
ZUW1 <<- ((UW1^(1/3))-1)*(3/(UW1.sd)) + (UW1.sd/3)
# Smith, 1986
ln.UW1 <<- (log(UW1) + UW1 - 1) * (sqrt((s.1-1)/8))
UW2 <<- rowMeans(IRT.sqstan.resid[,(s.1+1):(s.1+s.2)],na.rm=T)
ZUW2 <<- ((UW2^(1/3))-1)*(3/(UW2.sd)) + (UW2.sd/3)
# Smith, 1986
ln.UW2 <<- (log(UW2) + UW2 - 1) * (sqrt((s.2-1)/8))
UW3 <<- rowMeans(IRT.sqstan.resid[,(s.1+s.2+1):L],na.rm=T)
ZUW3 <<- ((UW3^(1/3))-1)*(3/(UW3.sd)) + (UW3.sd/3)
# Smith, 1986
ln.UW3 <<- (log(UW3) + UW3 - 1) * (sqrt((s.3-1)/8))
ln.UW1 <<- ifelse((ln.UW1==-Inf) | (ln.UW1==+Inf),NA,ln.UW1)
ln.UW2 <<- ifelse((ln.UW2==-Inf) | (ln.UW2==+Inf),NA,ln.UW2)
ln.UW3 <<- ifelse((ln.UW3==-Inf) | (ln.UW3==+Inf),NA,ln.UW3)
#
return()}
RUNIT <<- function(){
INITIATE()
FITANALYSIS1()
FITANALYSIS()
RESULTS <<- cbind(G,Gstar)
RESULTS <<- cbind(RESULTS,pbs)
RESULTS <<- cbind(RESULTS,Caution)
RESULTS <<- cbind(RESULTS,MCI)
RESULTS <<- cbind(RESULTS,U3)
RESULTS <<- cbind(RESULTS,ZU3)
RESULTS <<- cbind(RESULTS,H)
RESULTS <<- cbind(RESULTS,A)
RESULTS <<- cbind(RESULTS,D)
RESULTS <<- cbind(RESULTS,Ei)
RESULTS <<- cbind(RESULTS,U)
RESULTS <<- cbind(RESULTS,ZU)
RESULTS <<- cbind(RESULTS,W)
RESULTS <<- cbind(RESULTS,ZW)
RESULTS <<- cbind(RESULTS,ln.U)
RESULTS <<- cbind(RESULTS,ln.W)
# (Drasgow et al. 199
RESULTS <<- cbind(RESULTS,ECI1)
RESULTS <<- cbind(RESULTS,ECI2)
RESULTS <<- cbind(RESULTS,ECI3)
RESULTS <<- cbind(RESULTS,ECI4)
RESULTS <<- cbind(RESULTS,ECI5)
RESULTS <<- cbind(RESULTS,ECI6)
RESULTS <<- cbind(RESULTS,ECI1z)
RESULTS <<- cbind(RESULTS,ECI2z)
RESULTS <<- cbind(RESULTS,ECI4z)
RESULTS <<- cbind(RESULTS,ECI6z)
RESULTS <<- cbind(RESULTS,l)
RESULTS <<- cbind(RESULTS,lz)
RESULTS <<- cbind(RESULTS,M)
RESULTS <<- cbind(RESULTS,M.pvalue)
RESULTS <<- cbind(RESULTS,D.theta)
RESULTS <<- cbind(RESULTS,lzm)
RESULTS <<- cbind(RESULTS,UB)
RESULTS <<- cbind(RESULTS,ZUB)
RESULTS <<- cbind(RESULTS,ln.UB)
RESULTS <<- round(RESULTS,3)
N <<- L <<- S <<- s.1 <<- s.2 <<- s.3 <<- logs <<- probs <<- delta.matrix <<- r <<- i.sc <<- r.matrix <<c()
s.matrix <<- p.cor.n <<- p.cor.i <<- p.cor.i.matrix <<- gutt <<- antigutt <<- antigutt <<- log.p.i <<- c()
U.a <<- U.b <<- U.c <<- a.1 <<- a.2 <<- a.3 <<- a <<- E.U3 <<- b.1 <<- b.2 <<- b.3 <<- b <<- sd.U3 <<c()
sig.ij <<- H.num <<- H.denom1 <<- H.denom2 <<- H.denom <<- var.l <<- c()
H.denom <<- IRT.var <<- IRT.resid <<- IRT.sq.resid <<- IRT.sqstan.resid <<- inv.IRT.var <<- c()
U.sd <<- W.sd <<- UW1.sd <<- UW2.sd <<- UW3.sd <<- UB.sd <<- UBa <<- UBb <<- UBc <<- c()
group.probs <<- IRT.probs.mean <<- PY <<- l.matrix <<- l <<- E.l.matrix <<- E <<- var.l.matrix <<- c()
l.m.num <<- l.m.den <<- m.M1 <<- sd.M.11 <<- sd.M.12 <<- sd.M.22 <<- kappa03 <<- kappa12 <<- c()
kappa21 <<- kappa30 <<- beta.M <<- m.M <<- var.M <<- gam <<- g.data <<- h.data <<- Gni <<- G <<c()
Gstar <<- pbs <<- Caution <<- MCI <<- U3 <<- ZU3 <<- H <<- A <<- D <<- Ei <<- U <<- ZU <<- W <<c()
ZW <<- ln.U <<- ln.W <<- ECI1 <<- ECI2 <<- ECI3 <<- ECI4 <<- ECI5 <<- ECI6 <<- ECI1z <<ECI2z <<- c()
ECI4z <<- ECI6z <<- ECI4z <<- l <<- lz <<- M <<- ZM <<- M.pvalue <<- D.theta.W1 <<- D.theta.W2
<<- c()
D.theta.W3 <<- D.theta <<- lzm <<- UB <<- ZUB <<- ln.UB <<- UW1 <<- ZUW1 <<- ln.UW1 <<- UW2
<<- c()
random <<- ZUW2 <<- ln.UW2 <<- UW3 <<- ZUW3 <<- ln.UW3 <<- ln.UW1 <<- ln.UW2 <<- ln.UW3
<<- c()
return(RESULTS)}