mcreg {mcrPioda}

Comparison of Two Measurement Methods Using Regression Analysis

Description

mcreg is used to compare two measurement methods by means of regression analysis. Available methods comprise ordinary and weighted linear regression, Deming and weighted Deming regression and Passing-Bablok regression. Point estimates of regression parameters are computed together with their standard errors and confidence intervals.

Usage

mcreg(
  x,
  y = NULL,
  error.ratio = 1,
  alpha = 0.05,
  mref.name = NULL,
  mtest.name = NULL,
  sample.names = NULL,
  method.reg = c("PaBa", "LinReg", "WLinReg", "Deming", "WDeming", "PaBaLarge", "PBequi",
    "TS", "MDeming", "MMDeming", "NgMMDeming", "PiMMDeming"),
  method.ci = c("bootstrap", "jackknife", "analytical", "nestedbootstrap"),
  method.bootstrap.ci = c("quantile", "Student", "BCa", "tBoot"),
  nsamples = 999,
  nnested = 25,
  rng.seed = NULL,
  rng.kind = "Mersenne-Twister",
  iter.max = 30,
  threshold = 1e-06,
  na.rm = FALSE,
  NBins = 1e+06,
  kM = 1.345,
  tauMM = 4.685,
  priorSlope = 1,
  priorIntercept = 0,
  bdPoint = 0.5,
  slope.measure = c("radian", "tangent"),
  methodlarge = TRUE
)

Arguments

x

measurement values of reference method, or two column matrix.

y

measurement values of test method.

error.ratio

ratio between squared measurement errors of reference and test method, necessary for Deming regression (Default 1).

alpha

value specifying the 100(1-alpha)% confidence level for confidence intervals (Default is 0.05).

mref.name

name of reference method (Default "Method1").

mtest.name

name of test Method (Default "Method2").

sample.names

names of cases (Default "S##").

method.reg

regression method. It is possible to choose between five regression methods: "LinReg" - ordinary least square regression.
"WLinReg" - weighted ordinary least square regression.
"Deming" - Deming regression.
"WDeming" - weighted Deming regression.
"MDeming" - Huber M-Deming regression.
"MMDeming" - Huber MM-Deming regression.
"NgMMDeming" - new generation MM-Deming regression.
"PiMMDeming" - Prior informed MM-Deming regression.
"TS" - Theil-Sen regression.
"PBequi" - equivariant Passing-Bablok regression.
"PaBa" - Passing-Bablok regression.
"PaBaLarge" - approximative Passing-Bablok regression for large datasets, operating on NBins classes of constant slope angle which each slope is classified to instead of building the complete triangular matrix of all N*N/2 slopes.

method.ci

method of confidence interval calculation. The function contains four basic methods for calculation of confidence intervals for regression coefficients. "analytical" - with parametric method.
"jackknife" - with leave one out resampling.
"bootstrap" - with ordinary non-parametric bootstrap resampling.
"nested bootstrap" - with ordinary non-parametric bootstrap resampling.

method.bootstrap.ci

bootstrap based confidence interval estimation method.

nsamples

number of bootstrap samples.

nnested

number of nested bootstrap samples.

rng.seed

integer number that sets the random number generator seed for bootstrap sampling. If set to NULL currently in the R session used RNG setting will be used.

rng.kind

type of random number generator for bootstrap sampling. Only used when rng.seed is specified, see set.seed for details.

iter.max

maximum number of iterations for weighted Deming iterative algorithm.

threshold

numerical tolerance for weighted Deming iterative algorithm convergence.

na.rm

remove measurement pairs that contain missing values (Default is FALSE).

NBins

number of bins used when 'reg.method="PaBaLarge"' to classify each slope in one of 'NBins' bins covering the range of all slopes

kM

Huber's k for the M weighting, default kM = 1.345

tauMM

Tukey's tau for bisquare redescending weighting function, default tauMM = 4,685

priorSlope

starting slope value for PiMMDeming, default priorSlope = 1

priorIntercept

starting intercept value for PiMMDeming, default priorIntercept = 0

bdPoint

Proportion of data points selected for the highly robust M regression used for the determination of the starting parameters. Default 0.5

slope.measure

angular measure of pairwise slopes used for exact PaBa regression (see below for details).
"radian" - for data sets with even sample numbers median slope is calculated as average of two central slope angles.
"tangent" - for data sets with even sample numbers median slope is calculated as average of two central slopes (tan(angle)).

methodlarge

Boolean. This parameter applies only to regmethod="PBequi" and "TS". If TRUE, a quasilinear algorithm is used. If FALSE, a quadratic algorithm is used which is faster for less than several hundred data pairs.

Details

The regression analysis yields regression coefficients 'Intercept' and 'Slope' of the regression Testmethod = Intercept + Slope * Referencemethod. There are methods for computing the systematical bias between reference and test method at a decision point Xc, Bias(Xc) = Intercept + (Slope-1) * Xc, accompanied by its corresponding standard error and confidence interval. One can use plotting method plotBias for a comprehensive view of the systematical bias.

Weighted regression for heteroscedastic data is available for linear and Deming regression and implemented as a data point weighting with the inverted squared value of the reference method. Therefore calculation of weighted regression (linear and Deming) is available only for positive values (>0). Passing-Bablok regression is only available for non-negative values (>=0).

Confidence intervals for regression parameters and bias estimates are calculated either by using analytical methods or by means of resampling methods ("jackknife", "bootstrap", "nested bootstrap"). An analytical method is available for all types of regression except for weighted Deming. For Passing-Bablok regression the option "analytical" calculates confidence intervals for the regression parameters according to the non-parametric approach given in the original reference.

The "jackknife" (or leave one out resampling) method was suggested by Linnet for calculating confidence intervals of regression parameters of Deming and weighted Deming regression. It is possible to calculate jackknife confidence intervals for all types of regression. Note that we do not recommend this method for Passing-Bablok since it has a tendency of underestimating the variability (jackknife is known to yield incorrect estimates for errors of quantiles).

The bootstrap method requires additionally choosing a value for method.bootstrap.ci. If bootstrap is the method of choice, "BCa", t-bootstrap ("tBoot") and simple "quantile" confidence intervals are recommended (See Efron B. and Tibshirani R.J.(1993),Carpenter J., Bithell J. (2000)). The "nestedbootstrap" method can be very time-consuming but is necessary for calculating t-bootstrap confidence intervals for weighted Deming or Passing-Bablok regression. For these regression methods there are no analytical solutions for computing standard errors, which therefore have to be obtained by nested bootstrapping.

Note that estimating resampling based confidence intervals for Passing–Bablok regressions can take very long time for larger data sets due to the high computational complexity of the algorithm. To mitigate this drawback an adaption of the Passing-Bablok algorithm has been implemented ("PaBaLarge"), which yields approximative results. This approach does not build the complete upper triangular matrix of all 'n*(n-1)/2' slopes. It subdivides the range of slopes into 'NBins' classes, and sorts each slope into one of these bins. The remaining steps are the same as for the exact "PaBa" algorithm, except that these are performed on the binned slopes instead of operating on the matrix of slopes.

Our implementation of exact Passing-Bablok regression ("PaBa") provides two alternative metrics for regression slopes which can result in different regression estimates. As a robust regression method PaBa is essentially invariant to the parameterization of regression slopes, however in the case of an even number of all pairwise slopes the two central slopes are averaged to estimate the final regression slope. In this situation using an angle based metric (slope.measure="radian") will result in a regression estimate that is geometrically centered between the two central slopes, whereas the tangent measure (slope.measure="tangent") proposed in Passing and Bablok (1983) will be geometrically biased towards a higher slope. See below for a pathological example. Note that the difference between the two measures is negligible for data sets with reasonable sample size (N>20) and correlation.

Equivariant Passing-Bablok regression as proposed by Bablok et al. (1988) (see also Dufey 2020) is not bound to slopes near 1 and therefore not only applicable for method comparison but also for method transformation, i.e., when two methods yield results on a different scale. Like ordinary Passing-Bablok regression, the method is robust. This method should be preferred over the older "PaBa" and "PaBalarge" algorithms. Both slope measures "radian" and "tangent" are available as are methods for the determination of confidence intervals -analytical and bootstrap. By default (methodlarge=TRUE), a modified algorithm (Dillencourt et al., 1992) is used which scales quasilinearly and requires little memory. Alternatively (methodlarge=F), a simpler implementation which scales quadratically and is more memory intensive may be called. While point estimates coincide for both implementations, analytic confidence intervals differ slightly. Same holds true for the Theil-Sen estimator, which is a robust alternative to linear regression. Like linear regression, it assumes that x-values are error free.

Value

"MCResult" object containing regression results. The function getCoefficients or printSummary can be used to obtain or print a summary of the results. The function getData allows to see the original data. An S4 object of class "MCResult" containing at least the following slots:

data

measurement data in wide format, one pair of observations per sample. Includes samples ID, reference measurement, test measurement.

para

numeric matrix with estimates for slope and intercept, corresponding standard deviations and confidence intervals.

mnames

character vector of length two containing names of analytical methods.

regmeth

type of regression type used for parameter estimation.

cimeth

method used for calculation of confidence intervals.

error.ratio

ratio between squared measurement errors of reference and test method, necessary for Deming regression.

alpha

confidence level using for calculation of confidence intervals.

Author(s)

Ekaterina Manuilova [e-mail], Andre Schuetzenmeister [e-mail], Fabian Model [e-mail], Sergej Potapov [e-mail], Florian Dufey [e-mail], Jakob Raymaekers [e-mail]

References

Bland, J. M., Altman, D. G. (1986) Statistical methods for assessing agreement between two methods of clinical measurement. Lancet, i: 307–310.

Linnet, K. (1993) Evaluation of Regression Procedures for Methods Comparison Studies. CLIN. CHEM. 39/3, 424–432.

Linnet, K. (1990) Estimation of the Linear Relationship between the Measurements of two Methods with Proportional Errors. Statistics in Medicine, Vol. 9, 1463–1473.

Neter, J., Wassermann, W., Kunter, M. (1985) Applied Statistical Models. Richard D. Irwing, INC.

Looney, S. W. (2010) Statistical Methods for Assessing Biomarkers. Methods in Molecular Biology, vol. 184: Biostatistical Methods. Human Press INC.

Passing, H., Bablok, W. (1983) A new biometrical procedure for testing the equality of measurements from two different analytical methods. Application of linear regression procedures for method comparison studies in clinical chemistry, Part I. J Clin Chem Clin Biochem. Nov; 21(11):709–20.

Bablok, W., Passing, H., Bender, R., & Schneider, B. (1988) A general regression procedure for method transformation. Application of linear regression procedures for method comparison studies in clinical chemistry, Part III. Clinical Chemistry and Laboratory Medicine, 26(11): 783–790.

Dillencourt, M. B., Mount, D. M., & Netanyahu, N. S. (1992) A randomized algorithm for slope selection. International Journal of Computational Geometry & Applications, 2(01): 1–27.

Dufey, F. (2020) Derivation of Passing-Bablok regression from Kendall's tau. The International Journal of Biostatistics, 16(2): 20190157. https://doi.org/10.1515/ijb-2019-0157

Raymaekers, J., Dufey, F. (2022) Equivariant Passing-Bablok regression in quasilinear time. arXiv preprint arXiv:2202.08060. https://doi.org/10.48550/arXiv.2202.08060

Efron, B., Tibshirani, R.J. (1993) An Introduction to the Bootstrap. Chapman and Hall.

Carpenter, J., Bithell, J. (2000) Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians. Stat Med, 19 (9), 1141–1164.

CLSI EP9-A2. Method Comparison and Bias Estimation Using Patient Samples; Approved Guideline.

See Also

plotDifference, plot.mcr, getResiduals, plotResiduals, calcResponse, calcBias, plotBias, compareFit

Examples

library("mcrPioda")
data(creatinine,package="mcrPioda")
x <- creatinine$serum.crea
y <- creatinine$plasma.crea
# Deming regression fit.
# The confidence intercals for regression coefficients
# are calculated with analytical method
model1<- mcreg(x,y,error.ratio=1,method.reg="Deming", method.ci="analytical",
               mref.name = "serum.crea", mtest.name = "plasma.crea", na.rm=TRUE)
# Results
printSummary(model1)
getCoefficients(model1)
plot(model1)
# Deming regression fit.
# The confidence intervals for regression coefficients
# are calculated with bootstrap (BCa) method
model2<- mcreg(x,y,error.ratio=1,method.reg="Deming",
               method.ci="bootstrap", method.bootstrap.ci = "BCa",
               mref.name = "serum.crea", mtest.name = "plasma.crea", na.rm=TRUE)
compareFit(model1, model2) 

## Pathological example of Passing-Bablok regression where measure for slope angle matters  
x1 <- 1:10; y1 <- 0.5*x1; x <- c(x1,y1); y <- c(y1,x1) 
m1 <- mcreg(x,y,method.reg="PaBa",method.ci="analytical",slope.measure="radian",
            mref.name="X",mtest.name="Y")
m2 <- mcreg(x,y,method.reg="PaBa",method.ci="analytical",slope.measure="tangent",
            mref.name="X",mtest.name="Y")
plot(m1, add.legend=FALSE,identity=FALSE,
     main="Radian vs. tangent slope measures in Passing-Bablok regression\n(pathological example)",
     ci.area=FALSE,add.cor=FALSE)
plot(m2, ci.area=FALSE,reg.col="darkgreen",reg.lty=2,identity=FALSE,add.legend=FALSE,
     draw.points=FALSE,add=TRUE,add.cor=FALSE)
includeLegend(place="topleft",models=list(m1,m2),model.names=c("PaBa Radian","PaBa Tangent"),
              colors=c("darkblue","darkgreen"),lty=c(1,2),design="1",digits=2)