Effects of measurement error on the strength of concentration-response relationships in aquatic toxicology

Abstract

The effect that measurement error of predictor variables has on regression inference is well known in the statistical literature. However, the influence of measurement error on the ability to quantify relationships between chemical stressors and biological responses has received little attention in ecotoxicology. We present a common data-collection scenario and demonstrate that the relationship between explanatory and response variables is consistently underestimated when measurement error is ignored. A straightforward extension of the regression calibration method is to use a nonparametric method to smooth the predictor variable with respect to another covariate (e.g., time) and using the smoothed predictor to estimate the response variable. We conducted a simulation study to compare the effectiveness of the proposed method to the naive analysis that ignores measurement error. We conclude that the method satisfactorily addresses the problem when measurement error is moderate to large, and does not result in a noticeable loss of power in the case where measurement error is absent.

Appendix: R code for the bootstrap procedures used to calculate confidence intervals for the de-noising procedure.

# Assuming X,Y,time vectors are defined

library(‘SemiPar’) # for the smoothing function ‘spm’

n < - length(X) # number of observations

# Find the smoothed CCU

model.spm < - spm(X ~ f(time));

smoothed.X < - model.spm$fit$fitted;

# Fit the regression of Y onto smoothed X

model.dn < - lm(Y ~ smoothed.X);

Y.hat < - model.dn$fitted;

# Find the estimated values of epsilons and deltas

e < - Y - model.dn$fitted.values;

d < - X - smoothed.X;

# Bootstrap from e,d

bootstrap.slope < - rep(NA, 1000);

for(i in 1:10000){

index1 < - sample(1:n, n, replace = T);

index2 < - sample(1:n, n, replace = T);

X.boot < - smoothed.X + d[index1];

Y.boot < - Y.hat + e[index2];

model.spm < - spm(X.boot ~ f(time));

smoothed.X < - model.spm$fit$fitted;

model < - lm(Y ~ smoothed.X);

bootstrap.slope[i] < - model$coefficients;

}

quantile(bootstrap.slope, p=c(0.025, 0.975));