Fit a generalized Deming regression

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Description

Find the MLE line relating x and y when both are measured with error. When the variances of x and y are constant and equal, this is the special case of Deming regression.

Usage

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deming(formula, data, subset, weights, na.action, cv=FALSE,
       xstd, ystd, stdpat, conf=.95, jackknife=TRUE, dfbeta=FALSE,
       x=FALSE, y=FALSE, model=TRUE)

Arguments

formula

a model formula with a single continuous response on the left and a single continuous predictor on the right.

data

an optional data frame, list or environment containing the variables in the model.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

weights

an optional vector of weights to be used in the fitting process. Should be NULL or a numeric vector.

I

na.action

a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options. The 'factory fresh' default is na.omit, the na.exclude option is often useful.

xstd

optional, the variable name of a vector that contains explicit error values for each of the predictor values. This data overrides the cv option if both are present.

ystd

optional, the variable name of a vector that contains explicit error values for each of the response values. This data overrides the cv option if both are present.

cv

constant coefficient of variation? The default of false corresponds to ordinary Deming regression, i.e., an assumption of constant error. A value of cv=TRUE corresponds to the assumption of constant coefficient of variation.

stdpat

pattern for the standard deviation, see comments below. If this is missing the default is based on the cv option.

conf

confidence level for the confidence interval

jackknife

compute a jackknife estimate of variance.

dfbeta

return the dfbeta matrix from the jackknife computation.

x,y, model

logicals. If TRUE the corresponding components of the fit (the model frame, the model matrix, or the response) is returned.

Details

Ordinary least squares regression minimizes the sum of distances between the y values and the regression line, Deming regression minimizes the sum of distances in both the x and y direction. As such it is often appropriate when both x and y are measured with error. A common use is in comparing two assays, each of which is designed to quantify the same compound.

The standard deviation of the x variate variate will often be of the form s(c+dx) for c and d some constants, where s is the overal scale factor; similarly for y with constants e and f. Ordinary Deming regression corresponds to c=1 and d=0, i.e., constant variation over the range of the data. A more realistic assumption for many laboratory measurments is c=0 and d=1, i.e., constant coefficient of variation. Laboratory tests are often assumed to have constant coefficient of variation rather than constant variance.

There are 3 ways to specify the variation. The first is to directly set the pattern of (c,d,e,f) for the $x$ and $y$ standard deviations. If this is omitted, a default of (0,1,0,1) or (1,0,1,0) is chosen, based on whether the codecv option is TRUE or FALSE, respectively. As a third option, the user can specifiy xstd and ystd directly as vectors of data values. In this last case any values for the stdpat or ccs options are ignored. Note that the two calls deming(y ~ x, cv=TRUE) and deming(y ~ x, xstd=x, ystd=y) are subtly different. In the first the standard deviation values are based on the data, and in the second they will be based on the fitted values. The two outcomes will often be nearly identical.

Although a cv option of TRUE is often much better justified constant variance, the assumption of perfectly constant CV is questionable. Most actual biologic assays will have both a constant and a proportional component of error, with the former becoming dominant for values near zero and the latter dominant elsewhere. If all of the results are far from zero, however, the constant part may be ignored.

Value

a object of class 'deming' containing the components:

coefficient

the coefficient vector, containing the intercept and slope.

variance

The jackknife or bootstrap estimate of variance

ci

bootstrap confidence intervals, if nboot >0

dfbeta

pptionally, the dfbeta residuals. A 2 column matrix, each row is the change in the coefficient vector if that observation is removed from the data.

Author(s)

Terry Therneau

References

BD Ripley and M Thompson, Regression techniques for the detection of analytical bias, Analyst 112:377-383, 1987.

K Linnet, Estimation of the linear relationship between the measurements of two methods with proportional errors. Statistics in Medicine 9:1463-1473, 1990.

Examples

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# Data from Ripley and Thompson
fit <- deming(aes ~ aas, data=arsenate, xstd=se.aas, ystd=se.aes)
print(fit)
## Not run: 
            Coef se(coef) lower 0.95 upper 0.95
Intercept 0.1064   0.2477    -0.3790     0.5919
Slope     0.9730   0.1430     0.6928     1.2532

   Scale= 1.358 

## End(Not run)
plot(1:30, fit$dfbeta[,2]) #subject 22 has a large effect on the slope

# Constant proportional error fit (constant CV)
fit2 <- deming(new.lot ~ old.lot, ferritin, cv=TRUE,
                subset=(period==3))