cb.pd: Compute accuracy estimates and maximum likelihood estimates... In merror: Accuracy and Precision of Measurements

Description

Compute accuracy estimates and maximum likelihood estimates of precision for the constant bias measurement error model using paired data.

Usage

 `1` ```cb.pd(x, conf.level = 0.95, M = 40) ```

Arguments

 `x` n (no. of items) x N (no. of methods) matrix or data.frame containing the measurements. N must be >= 3 and n > N. `conf.level` Chosen onfidence level. `M` Maximum no.of iterations to reach convergence.

Details

Measurement Error Model:

x[i,k] = alpha[i] + beta[i]*mu[k] + epsilon[i,k]

where x[i,k] is the measurement by the ith method for the kth item, i = 1 to N, k = 1 to n, mu[k] is the true value for the kth item, epsilon[i,k] is the Normally distributed random error with variance sigma[i] squared for the ith method and the kth item, and alpha[i] and beta[i] are the accuracy parameters for the ith method.

The imprecision for the ith method is sigma[i]. If all alphas are zeroes and all betas are ones, there is no bias. If all betas equal 1, then there is a constant bias. Otherwise there is a nonconstant bias.

ME (method of moments estimator) and MLE are the same for N=3 instruments except for a factor of (n-1)/n: MLE = (n-1)/n * ME

Using paired differences forces Constant Bias model (beta[1] = beta[2] = ... = beta[N]). Also, the process variance CANNOT be estimated.

Value

 `conf.level ` Confidence level used. `sigma.table ` Table of accuracy and precision estimates and confidence intervals. `n.items ` No. of items. `N.methods ` No. of methods `Grubbs.initial.sigma2 ` N vector of initial imprecision estimates using Grubbs' method `sigma2 ` N vector of variances that measure the method imprecision. `sigma2.se2 ` N vector of squared standard errors of the estimated imprecisions (variances). `alpha.cb ` N vector of estimated alphas for constant bias model. `alpha.ncb ` N vector of estimated alphas for nonconstant bias model `beta ` N vector of hypothesized betas for the constant bias model - all ones. `df ` N vector of estimated degrees of freedom. `chisq.low ` N vector of chi-square values for the lower tail (used to compute the ci upper bound). `chisq.low ` N vector of chi-square values for the upper tail (used to compute the ci lower bound). `lb ` N vector of lower bounds for confidence intervals `ub ` N vector of upper bounds for confidence intervals

Author(s)

Richard A. Bilonick

References

Jaech, J. L. (1985) Statistical Analysis of Measurement Errors. New York: Wiley.

`ncb.od`, `lrt`
 ```1 2``` ```data(pm2.5) cb.pd(pm2.5) ```