R wrapper that accesses C code to fit quantile value-added model

Description

qVA is the main function used to fit hierarchical model that produces quantile value-added estimates.

Usage

qVA(y,
    xmat,
    school,
    tau=0.5,
    draws=1100, burn=100, thin=1,
    priorVal=c(100^2, 1, 1, 1, 1, 0, 100^2),
    verbose=FALSE)

Arguments

Value

This function returns a list containing MCMC iterates that correspond to model parameters and school-specific quantile value-added estimates. In order to provide more detail, in what follows let

"T" - be the number of MCMC iterates collected (draws - burn)/thin,

"M" - be the number of schools,

"N" - be the total number of observations.

"p" - be the number of covariates

The output list contains the following

beta - an matrix of dimension (T, p) containing MCMC iterates associated with quantile regression covariate estimates.

alpha - an matrix of dimension (T, M) containing MCMC iterates assocated with school-specific random effects.

v - matrix of dimension (T, N) containing MCMC iterates of auxiliary variable.

a - matrix of dimension (T, 1) contaning MCMC iterates of mean of the school-specific random effects (alpha).

sig2a - a matrix of dimension (T, 1) containing MCMC iterates of variance of the school-specific random effects (alpha).

lambda - a matrix of dimension (T, M) containing MCMC interates associated with the lambda parameter of asymmetric laplace distribution.

cVA - a matrix of dimension (T, M) containing MCMC interates associated with conditional value-added for each school.

mVA - a matrix of dimension (T, M) containing MCMC iterates associated with the marginal value-added for each school.

qVA - a matrix of dimension (T, M) containing MCMC iterates associated with each schools quantile value-added.

Q - a matrix of dimension (T, N) containing MCMC iterates associated with the marginal quantile valued-added regression value for each student (i.e., averaging over school).

References

Page, Garritt L.; San Martín, Ernesto; Orellana, Javiera; Gonzalez, Jorge. (2017) “Exploring Complete School Effectiveness via Quantile Value-Added” Journal of the Royal Statistical Society: Series A 180(1) 315-340

Examples

# Example with synthetic data
tau <- 0.75

m <- 4 # number of schools
n <- 25 # number of students
N <- m*n
p <- 1 # number of covariates
betaT <- 0.5
alphaT <- seq(-10,10, length=m)

# Generate from the asymmetric Laplace
# using a mixture of a Normal and an Exponential
lambdaT <- 0.1;
xi <- rexp(N, 1/lambdaT)
epsilon <- (sqrt((lambdaT*2*xi)/(tau*(1-tau)))*rnorm(N,0,1) +
           (1-2*tau)/(tau*(1-tau))*xi)
epsilon <- rnorm(N,0,1)
alphavec <-  rep(alphaT, each=n)

x <- rnorm(N,250,1)
y <- x*betaT  + alphavec + epsilon

X <- cbind(x)


school <- rep(1:m, each=n)



fitQ3 <- qVA(y=y, xmat=X, school=school, tau=0.75, verbose=FALSE)

# quantile value-added estimates with 95% credible intervals for each school
qVA.est <- apply(fitQ3$qVA,2,mean)
qVA.int <- apply(fitQ3$qVA,2,function(x) quantile(x, c(0.025, 0.975)))

beta <- fitQ3$beta
alpha <- fitQ3$alpha

mVA <- fitQ3$mVA
cVA <- fitQ3$cVA
Q <- fitQ3$Q

# Plot results.
plot(x,y, col=rep(c("red","blue","green","orange"), each=n), pch=19)

# Plot Q3 quantile regression line for each school
lines(X[school==1,],
       (X[school==1,])*mean(beta) + apply(alpha,2,mean)[1], col='red', lwd=3)
lines(X[school==2,],
       (X[school==2,])*mean(beta) + apply(alpha,2,mean)[2], col='blue', lwd=3)
lines(X[school==3,],
       (X[school==3,])*mean(beta) + apply(alpha,2,mean)[3], col='green', lwd=3)
lines(X[school==4,],
       (X[school==4,])*mean(beta) + apply(alpha,2,mean)[4], col='orange', lwd=3)

# Plot the marginal VA for each school
points(tapply(X, school,mean), apply(mVA,2,mean),
       col=c("red","blue","green","orange"), pch=4, cex=2, lwd=2)

# Plot the conditional VA for each school
points(tapply(X, school,mean), apply(cVA,2,mean),
      col=c("red","blue","green","orange"),pch=10,cex=2, lwd=2)

# Plot the "global" Q3 quantile regression line.
points(X, apply(Q,2,mean), type='l', lwd=2)

R wrapper that accesses C code to fit temporal dependent value-added models for two cohorts

Description

tdVA is the main function used to fit the temporally dependent value-added model for two cohorts

Usage

tdVA(y1, xmat1,
     y2, xmat2,
     school1, school2,
     groupID=NULL,
     model=0,
     priors=c(0, 100^2, 1, 1, 1, 1, 0, 100^2, -1, 1, 0, 100^2, 0, 100^2),
     var.global=TRUE,
     MHsd=c(0.2),
     nchains=1,
     draws=50000, burn=40000, thin=10,
     verbose=FALSE)

Arguments

Value

This function returns a list that contains MCMC iterates for all the model parameters in addition to the value-added estimates and intervals for each of the two cohorts

Examples

# Generate data from model 1 of San Martin el al.
m <- 25 # number of schools
ni <- 20 # number of students per school
N <- m*ni

# specify parameter values to generate data
beta1 <- 0.6
beta2 <- 0.75;
sig21 <- 100;
sig22 <- 100;
tau2 <- 100
phi02 <- 0;
phi12 <- 0.75;
phi01 <- 0

X1 <- rnorm(N, 0, sqrt(200))
X2 <- rnorm(N, 0, sqrt(200))

alpha1 <- rnorm(m, phi01, sqrt(tau2))
alpha2 <- rnorm(m, phi02 + phi12*alpha1, sqrt(tau2*(1-phi12^2)))

Y1 <- rep(alpha1, each=ni) + X1*beta1 + rnorm(N, 0, sqrt(sig21))
Y2 <- rep(alpha2, each=ni) + X2*beta2 + rnorm(N, 0, sqrt(sig22))

# Create school vector indicating to which school each observation belongs
school1 <- rep(1:m, each=ni)
school2 <- rep(1:m, each=ni)


# design matrix only one covariate and no intercept
X1i <- cbind(X1)
X2i <- cbind(X2)


fit <- tdVA(y1=Y1,xmat1=X1i,y2=Y2,xmat2=X2i,
              school1=school1,school2=school2, groupID=NULL, model=2,
              var.global=TRUE, nchains=1)

# Value-added estimates of cohort 1 and 2 with 95% credible intervals.  See paper for details
cbind(fit$VA1.estimate,t(fit$VA1.intervals))
cbind(fit$VA2.estimate,t(fit$VA2.intervals))