Regression of Differences against Sums

Method Comparison Studies

DragonflyStats.github.io

Blackwood & Bradley (1988, 1991) and Bartko 1994

  • The Bradley Blackwood procedure is a simultaneous test for equality of the variance and mean of a pair of data sets.

  • A linear regression is fitted according to the sums and differences.

  • The Bradley-Blackwood procedure tests whether the regression coefficients in the regression of the pair wise case-wise difference (X1-X2) versus the case-wise averages ((X1+X2)/2) are significantly different from zero.

  • The null hypothesis is that mean and variance of both methods equal.

  • The test statistic is provided in , and the critical value is an F value at % significance, with degrees of freedom 2 and n-2, where n is the number of paired values.

  • This approach was amended by Bartko 1994 to be uses in the Method comparison study.

  • It is noticeably a simultaneous test for bias and precision.

Grubbs Procedure

Fuse Data Set

This Data set is the running times of 20 mechanical fuses measured by operators stopping two independent clocks.

Fuse <- 1:20
###Measuerement by first Instrument
Y1 <- c(4.85,4.93,4.75,4.77,4.67,4.87,4.67,4.94,4.85,4.75,
        4.83,4.92,4.74,4.99,4.88,4.95,4.95,4.93,4.92,4.89)
###Measuerement by second Instrument
Y2 <- c(5.09,5.04,4.95,5.02,4.90,5.05,4.90,5.15,5.08,4.98,
        5.04,5.12,4.95,5.23,5.07,5.23,5.16,5.11,5.11,5.08)

Identity Plot

par(font.lab=2,font.axis=2,cex=1.7)
plot(Y1,Y2,pch=16,col="red",ylim=c(4.5,5.5),xlim=c(4.5,5.5))
title("Identity Plot - Grubbs's Fuse Data")
abline(a=0,b=1,col="blue",lwd=2)

Preliminary Calculations / Bland-Altman Analysis

Ds = Y2-Y1
Ss = Y2+Y1

According to conventionally used procedure, there is a significant inter-method bias between both methods.

## Mean of Case-wise Differences
mean(Ds)
## [1] 0.2105
## Limits of Agreement
c(mean(Ds)-1.96*sd(Ds),mean(Ds)+1.96*sd(Ds))
## [1] 0.1421658 0.2788342
## Paired t-test
t.test(Y2,Y1,paired=T)$p.value
## [1] 1.281159e-16

Using the BB procedure

Case-wise difference regressed on case-wise sums.

FitBB <- lm(Ds ~ Ss)

summary(FitBB)$coefficients
##                 Estimate Std. Error    t value  Pr(>|t|)
## (Intercept)  0.304367826  0.4330005  0.7029272 0.4910933
## Ss          -0.009466777  0.0436616 -0.2168216 0.8307859
anova(FitBB)
## Analysis of Variance Table
## 
## Response: Ds
##           Df    Sum Sq    Mean Sq F value Pr(>F)
## Ss         1 0.0000602 0.00006016   0.047 0.8308
## Residuals 18 0.0230348 0.00127971

Scatterplot of BB regression

par(font.lab=2,font.axis=2,cex=1.7)
plot(Ss,Ds,pch=16,col="red",ylab="Case-wise Differences",xlab="Case-wise Sums",ylim=c(-0.05,0.35))
title("Grubbs's Fuse Data")
abline(coef(FitBB),col="blue",lwd=2)
abline(h=0,col="green",lwd=1.5)