MMSE2: Minimum mean squared error estimation with additional...

In starima74/AddInf: The package functions implement statistical estimation with the use of additional information

Minimum mean squared error estimation with additional information.

Description

This function implements statistical estimation with the use of additional information. This additional information can come from multiple indepedent data sources. Each source is reported by a mean estimate and its variance. In contrast with the MMSE function where the dataset is formally used, the MMSE2 uses only a variance-covariance matrix K, and it also assumes a that the vector of biases c is known.

Usage

  MMSE(K, c, Add.Info, thetahat, thetahat.length, eig.cutoff = 0.9)

Arguments

K	a variance covariance matrix. See the example below
c	a vector of biases
Add.Inf	a data frame with additional information, which consists of a columm of means (Means) and a columm of variances (Vars). The vector of indicators of possible biases Biases uses 0 to define unbiased additional information and 1 for possible bias.
thetahat	an estimated value for a parameter of interest
betahat	an estimated value for an auxiliary parameter
eig.cutoff	a percent cutoff for eigen values to approximate a matrix inverse

Value

Theta.Est	An estimate of the parameter of interest without the use of addtional information
Theta.Est.Var	A variance covariance matrix of Theta.Est
Theta.Hat	A new estimate of the parameter of interest with the use of addtional information
Theta.Hat.Var	A variance covariance matrix of Theta.Hat

Author(s)

Sergey Tarima ([e-mail]) and Roman Vygon ([e-mail])

References

Tarima S.S. and Dmitriev Y.G. Statistical estimation with possibly incorrect model assumptions. The Bulletin of Tomsk State University: control, computing, informatics 2009, 3(8):87-99

Tarima S.S., Vexler A., Singh S., Robust mean estimation under a possibly incorrect log-normality assumption. Communications in Statistics - Simulation and Computation, 42: 316-326, 2013

Sergey Tarima and Kadam Patel. The Association between Gestational Age and 3rd Grade Standardized Reading Score. MCW Biostatistics technical report #68. https://www.mcw.edu/-/media/MCW/Departments/Biostatistics

Examples

### A variance covariance matrix of adjuated and unadjusted
### estimates of differences in reading scores between children 
### born at the gestational age of 37 and 39:41 weeks

K <- matrix(c(0.0467408718, 0.0499675797, 0.0499675797, 0.0768166235),2,2)

### an estimate of the parameter of interest (the adjusted 
### estimate of differences (in percents) in reading scores
### between children born at the gestational age of 37 and 39:41 weeks
thetahat <- 0.173

### an estimate of the auxiliary parameter (the unadjusted 
### estimate of differences in reading scores between children 
### born at the gestational age of 37 and 39:41 weeks
betahat <- -0.952

### (Empty) Additional Information (on the auxiliary parameter)
Add.Info.Means <- list()
Add.Info.Vars <- list()
Add.Info.Biases <- list()

### the additional data source says the unadjuted difference 
### is -0.74 with a standard error = 0.098 (variance = 0.009604)
### this information may be biased as it came from 
### a different population

Add.Info.Means[[1]] <- -0.74
Add.Info.Vars[[1]] <- 0.009604
Add.Info.Biases[[1]] <- 1 # biased additional information

### create the data frame where the additional information will be saved
Add.Info <- data.frame(Means = rep(NA,1), 
                       Vars = rep(NA,1), 
                       Biases = rep(NA,1))

### Adding the additional information to the data frame
Add.Info$Means = Add.Info.Means
Add.Info$Vars = Add.Info.Vars
Add.Info$Biases = Add.Info.Biases

### estimation based on both data sources 
res <- MMSE2(K, -0.05, Add.Info, thetahat, betahat,
            eig.cutoff = 1)

### the new estimate and its confidence interval
lo  <- res$Theta.Est - 1.96*sqrt(res$Theta.Est.MSE)
est <- res$Theta.Est
hi  <- res$Theta.Est + 1.96*sqrt(res$Theta.Est.MSE)
c(lo, est, hi)

### the empirical (no additional information) estimate 
### and its confidence interval
theta_lo  <- res$Theta.Hat - 1.96*sqrt(res$Theta.Hat.Var)
theta_est <- res$Theta.Hat
theta_hi  <- res$Theta.Hat + 1.96*sqrt(res$Theta.Hat.Var)
c(theta_lo, theta_est, theta_hi)

### a sensitivity analysis for a range of bais values 

ra <- -10:10/11
n <- length(ra) 
CIs <- data.frame(LO=rep(NA,n),EST=rep(NA,n),HI=rep(NA,n))
s=0
for(delta in ra) { 
 s=s+1
 res <- MMSE2(K, delta, Add.Info, thetahat, betahat,
            eig.cutoff = 1)
 lo  <- res$Theta.Est - 1.96*sqrt(res$Theta.Est.MSE)
 est <- res$Theta.Est
 hi  <- res$Theta.Est + 1.96*sqrt(res$Theta.Est.MSE)
 CIs[s,] <- c(lo, est, hi)
}

plot(ra,CIs$EST,ylim=c(-0.5,1),type="l",lwd=3,
    ylab="Estimated difference and 95% CI",xlab="bias")
lines(ra,CIs$LO)
lines(ra,CIs$HI)
lines(ra,rep(theta_lo,n),lty=2)
lines(ra,rep(theta_hi,n),lty=2)
lines(ra,rep(theta_est,n),lty=2,lwd=3)
lines(ra,rep(0,n),lty=3,lwd=1)

starima74/AddInf documentation built on June 13, 2022, 12:08 p.m.