ggmr: Solves the Generalized Gauss Markov Regression model

Description

Fits the linear model using covariance matrices on the predictor, the response and covariance matrix between predictor and response, according to ISO/TS 28037 (2010).

Usage

ggmr(x, y, Ux = diag(0, length(x)), 
			Uy = diag(1, length(x)), 
			Uxy = diag(0, length(x)),
			subset = rep(TRUE, length(x)), 
			tol = sqrt(.Machine$double.eps), max.iter = 100, alpha = 0.05,
			coef.H0 = c(0, 1))

Arguments

numeric vector, the predictor values

numeric vector, the response values

numeric matrix, the variance matrix of the predictor

numeric matrix, the variance matrix of the response

Uxy

numeric matrix, the covariance matrix between predictor and the response

subset

a logical vector or a numeric vector with the position to be considered

tol

numeric, the maximum allowed error tolerance, tolerance is relative

max.iter

integer, the maximum number of allowed iterations

alpha

numeric, the significance level used on testing H0

coef.H0

the coeffients for hypothesis testing purposes

Value

a list with the following elements

coefficients

estimated coefficients

cov

covariance matrix of the estimated coefficients

estimated latent unobservable variables

chisq.validation

chi-squared statistic for model validation

chisq.ht

chi-squared statistic of the observed values for the hypothesis testing

chisq.cri

chi-squared critical value

p.value

probability of observing a validation statistic equal or larger then the sampled just by chance

curr.iter

current number of iterations used

curr.tol

current relative tolerance

References

ISO/TS 28037 (2010). Determination and Use of straight-line calibration functions https://www.iso.org/standard/44473.html

Examples

Run this code

# NOT RUN {
require(MASS)

# Example ISO 28037 (2010) Section 6. table 6
d<- data.frame(
 x=c(1.0, 2.0, 3.0, 4.0, 5.0, 6.0),
 y=c(3.2, 4.3, 7.6, 8.6, 11.7, 12.8),
 uy=c(0.5, 0.5, 0.5, 1.0, 1.0, 1.0)
)
# estimates
ggmr.res <- ggmr(d$x, d$y, Uy=diag(d$uy^2), coef.H0=c(0, 2), tol = 1e-10)
ggmr.res$coefficients
sqrt(diag(ggmr.res$cov))
ggmr.res$cov[1, 2]
ggmr.res$chisq.validation
ggmr.res$chisq.cri
# reference values
# coefficients = c(0.885, 2.057)
# se = c(0.530, 0.178)
# cov = -0.082
# validation.stat = 4.131
# critical.value = 9.488

# lm() estimates the coefficients correctly but 
# fails to reproduce the standard errors
summary(lm(y~x, data=d, weights=1/d$uy^2)) 
# coefficients = c(0.8852, 2.0570)
# se = c(0.5383, 0.1808)

# Example ISO 28037 (2010) Section 7. table 10
d <- data.frame(
 x = c(1.2, 1.9, 2.9, 4.0, 4.7, 5.9),
 y = c(3.4, 4.4, 7.2, 8.5, 10.8, 13.5)
)
Ux = diag(c(0.2, 0.2, 0.2, 0.2, 0.2, 0.2))^2
Uy = diag(c(0.2, 0.2, 0.2, 0.4, 0.4, 0.4))^2

# estimates
ggmr.res <- ggmr(d$x, d$y, Ux, Uy, coef.H0=c(0, 2), tol = 1e-10)
ggmr.res$coefficients
sqrt(diag(ggmr.res$cov))
ggmr.res$cov[1, 2]
ggmr.res$chisq.validation
ggmr.res$chisq.cri
# reference values
# coefficients = c(0.5788, 2.1597)
# se = c(0.4764, 0.1355)
# cov = -0.0577
# validation.stat = 2.743
# critical.value = 9.488

# Example ISO 28037 (2010) Section 10. table 25
d<- data.frame(
 x=c(50.4, 99.0, 149.9, 200.4, 248.5, 299.7, 349.1),
 y=c(52.3, 97.8, 149.7, 200.1, 250.4, 300.9, 349.2)
)

Ux<- matrix(c(
 0.50, 0.00, 0.25, 0.00, 0.25, 0.00, 0.25,
 0.00, 1.25, 1.00, 0.00, 0.00, 1.00, 1.00,
 0.25, 1.00, 1.50, 0.00, 0.25, 1.00, 1.25,
 0.00, 0.00, 0.00, 1.25, 1.00, 1.00, 1.00,
 0.25, 0.00, 0.25, 1.00, 1.50, 1.00, 1.25,
 0.00, 1.00, 1.00, 1.00, 1.00, 2.25, 2.00,
 0.25, 1.00, 1.25, 1.00, 1.25, 2.00, 2.50
), 7, 7)

Uy<- matrix(1.00, 7, 7) + diag(4.00, 7)

Uxy<- matrix(0, 7, 7)

# estimates
ggmr.res<- ggmr(d$x, d$y, Ux, Uy, Uxy)
ggmr.res$coefficients
sqrt(diag(ggmr.res$cov))
ggmr.res$cov[1, 2]
ggmr.res$chisq.validation
ggmr.res$chisq.cri
# reference values
# coefficients = c(0.3424, 1.0012)
# se = c(2.0569, 0.0090)
# cov = -0.0129
# validation.stat = 1.772
# critical.value = 11.070

# }