errorsINseveral {DAAG}

Simulation of classical errors in x model, with multiple explanatory variables.

Description

Simulates $y-$ and $x-$values for a classical “errors in $x$” linear regression model. One or more $x-$values are subject to random measurement error, independently of the corresponding covariate values that are measured without error.

Usage

errorsINseveral(n = 1000, a0 = 2.5, beta = c(1.5, 0), mu = 12.5, SDyerr = 0.5,
default.Vpar = list(SDx = 2, rho = -0.5, timesSDx = 1.5),
V = with(default.Vpar, matrix(c(1, rho, rho, 1), ncol = 2) * SDx^2),
xerrV = with(default.Vpar, matrix(c(1, 0, 0, 0), ncol = 2) * (SDx * timesSDx)^2),
parset = NULL, print.summary = TRUE, plotit = TRUE)

Arguments

n

Number of observations

a0

Intercept in linear regression model

beta

Regression coefficients. If one coefficient only is given, this will be repeated as many times as necessary

mu

Vector of covariate means.

SDyerr

SD of $y$, conditional on the covariates measured without error

default.Vpar

Parameters for the default model with two explanatory variables,

V

Variance-covariance matrix for the z's, measured without error. (These are generated from a multivariate normal distribution, mainly as a matter of convenience)

xerrV

Variance-covariance matrix for the added “errors in x”

parset

Parameter list (theme) in a form suitable for supplying to trellis.par.set().

print.summary

If TRUE, print summary details of the regression results from the simulation.

plotit

If TRUE, plot the fitted values for the model with covariates with error, against the fitted values for covariates without error.

Details

With default arguments, simulates a model in which two covariates are in contention, the first measured without error, and the second with coefficient 0 in the model that includes both covariates measured without error.

Value

ERRfree

Data frame holding covariates without error, plus $y$

addedERR

Data frame holding covariates with error, plus $y$

Author(s)

John Maindonald

References

Data Analysis and Graphics Using R, 3rd edn, Section 6.8.1

See Also

errorsINx

Examples

library(lattice)
function(n=1000, a0=2.5, beta=c(1.5,0), mu=12.5, SDyerr=0.5, 
           default.Vpar=list(SDx=2, rho=-0.5, timesSDx=1.5),
           V=with(default.Vpar, matrix(c(1,rho,rho,1), ncol=2)*SDx^2),
           xerrV=with(default.Vpar, matrix(c(1,0,0,0), ncol=2)*(SDx*timesSDx)^2),
           parset=NULL, print.summary=TRUE, plotit=TRUE){
    m <- dim(V)[1]
    if(length(mu)==1)mu <- rep(mu,m)
    ow <- options(warn=-1)
    xxmat <- sweep(matrix(rnorm(m*n, 0, 1), ncol=m) %*% chol(V), 2, mu, "+")
    errxx <- matrix(rnorm(m*n, 0, 1), ncol=m) %*% chol(xerrV, pivot=TRUE)
    options(ow)
    dimnames(xxmat)[[2]] <- paste("z", 1:m, sep="")
    xxWITHerr <- xxmat+errxx
    xxWITHerr <- data.frame(xxWITHerr)
    names(xxWITHerr) <- paste("xWITHerr", 1:m, sep="")
    xxWITHerr[, "y"] <- a0 + xxmat %*% matrix(beta,ncol=1) + rnorm(n, sd=SDyerr)
    err.lm <- lm(y ~ ., data=xxWITHerr)
    xx <- data.frame(xxmat)
    names(xx) <- paste("z", 1:m, sep="")
    xx$y <- xxWITHerr$y
    xx.lm <- lm(y ~ ., data=xx)
    B <- coef(err.lm)
    b <- coef(xx.lm)
    SE <- summary(err.lm)$coef[,2]
    se <- summary(xx.lm)$coef[,2]
    if(print.summary){
      beta0 <- c(mean(xx$y)-sum(beta*apply(xx[,1:m],2,mean)), beta)
      tab <- rbind(beta0, b, B)
      dimnames(tab) <- list(c("Values for simulation",
                              "Estimates: no error in x1",
                              "LS Estimates: error in x1"),
                            c("Intercept", paste("b", 1:m, sep="")))
      tabSE <- rbind(rep(NA,m+1),se,SE)
      rownames(tabSE) <- rownames(tab)
      colnames(tabSE) <- c("SE(Int)", paste("SE(", colnames(tab)[-1],")", sep=""))
      tab <- cbind(tab,tabSE)
      print(round(tab,3))
    }
    if(m==2 & print.summary){
      tau <- default.Vpar$timesSDx
      s1 <- sqrt(V[1,1])
      s2 <- sqrt(V[2,2])
      rho <- default.Vpar$rho
      s12 <- s1*sqrt(1-rho^2)
      lambda <- (1-rho^2)/(1-rho^2+tau^2)
      gam12 <- rho*sqrt(V[1,1]/V[2,2])
      expB2 <- beta[2]+beta[1]*(1-lambda)*gam12
      print(c("Theoretical attenuation of b1" = lambda, "Theoretical b2" = expB2))
    }
    if(is.null(parset))parset <- simpleTheme(col=c("gray40","gray40"),
                                             col.line=c("black","black"))
    if(plotit){
      library(lattice)
      zhat <- fitted(xx.lm)
      xhat <- fitted(err.lm)
      plt <- xyplot(xhat ~ zhat, aspect=1, scales=list(tck=0.5),
                    panel=function(x,y,...){
                      panel.xyplot(x,y,type="p",...)
                      panel.abline(lm(y ~ x), lty=2)
                      panel.abline(0,1)
                    },
                    xlab="Fitted values; regress on exact z",
                    ylab="Fitted values; regress on x = xWITHerr",
                    key=list(space="top", columns=2,
                      text=list(lab=c("Line y=x", "Regression fit to points")),
                      lines=list(lty=1:2)),
                    par.settings=parset
                    )
      print(plt)}
    invisible(list(ERRfree=xx, addedERR=xxWITHerr))
  }