Chapter 5 DBM method special cases
Special cases of DBM analysis are described here, namely fixed-reader random-case (FRRC), sub-special case of which is Single-reader multiple-treatment analysis, and random-reader fixed-case (RRFC).
5.2 Fixed-reader random-case (FRRC) analysis
The model is the same as in Eqn. (4.4) except one sets \(\sigma_{R}^{2}\) = \(\sigma_{\tau R}^{2}\) = 0 in Table 4.1. The appropriate test statistic is:
\[\begin{equation} \frac{E\left ( MST \right )}{E\left ( MSTC \right )} = \frac{\sigma_{\epsilon}^{2}+\sigma_{\tau RC}^{2}+J\sigma_{\tau C}^{2}+JK\sigma_{\tau}^{2}}{\sigma_{\epsilon}^{2}+\sigma_{\tau RC}^{2}+J\sigma_{\tau C}^{2}} \end{equation}\]
Under the null hypothesis \(\sigma_{\tau}^{2} = 0\):
\[\begin{equation} \frac{E\left ( MST \right )}{E\left ( MSTC \right )} = 1 \end{equation}\]
The F-statistic is (replacing expected with observed values):
\[\begin{equation} F_{DBM|R}=\frac{MST}{MSTC} \tag{5.1} \end{equation}\]
The observed value \(F_{DBM|R}\) (the Roe-Metz notation (Roe and Metz 1997b) is used which indicates that the factor appearing to the right of the vertical bar is regarded as fixed) is distributed as an F-statistic with \(\text{ndf}\) = \(I – 1\) and \(ddf = (I-1)(K-1)\); the degrees of freedom follow from the rows labeled \(T\) and \(TC\) in TBA Table Table 4.1. Therefore, the distribution of the observed value is (no Satterthwaite approximation needed this time as both numerator and denominator are simple mean-squares):
\[\begin{equation} F_{DBM|R} \sim F_{I-1,(I-1)(K-1)} \tag{5.2} \end{equation}\]
The null hypothesis is rejected if the observed value of the F- statistic exceeds the critical value:
\[\begin{equation} F_{DBM|R} > F_{1-\alpha,I-1,(I-1)(K-1)} \tag{5.3} \end{equation}\]
The p-value of the test is the probability that a random sample from the F-distribution TBA (4.1) Eqn. (9.39), exceeds the observed value:
\[\begin{equation} p=\Pr\left ( F> F_{DBM|R} \mid F \sim F_{I-1,(I-1)(K-1)} \right ) \tag{5.4} \end{equation}\]
The \((1-\alpha)\) confidence interval for the inter-treatment reader-averaged difference FOM is given by:
\[\begin{equation} CI_{1-\alpha}=\left ( \theta_{i \bullet} - \theta_{i' \bullet} \right ) \pm t_{\alpha/2,(I-1)(K-1)}\sqrt{2\frac{MST}{JK}} \tag{5.5} \end{equation}\]
5.2.1 Single-reader multiple-treatment analysis
With a single reader interpreting cases in two or more treatments, the reader factor must necessarily be regarded as fixed. The preceding analysis is applicable. One simply puts \(J = 1\) in the equations above.
5.2.1.1 Example 5: Code illustrating p-values for FRRC analysis, Van Dyke data
alpha <- 0.05
retMS <- UtilMeanSquares(dataset02)
I <- length(dataset02$ratings$NL[,1,1,1])
J <- length(dataset02$ratings$NL[1,,1,1])
K <- length(dataset02$ratings$NL[1,1,,1])
FDbmFR <- retMS$msT / retMS$msTC
ndf <- (I-1); ddf <- (I-1)*(K-1)
pValue <- 1 - pf(FDbmFR, ndf, ddf)
theta <- as.matrix(UtilFigureOfMerit(dataset02, FOM = "Wilcoxon"))
theta_i_dot <- array(dim = I)
for (i in 1:I) theta_i_dot[i] <- mean(theta[i,])
trtDiff <- array(dim = c(I,I))
for (i1 in 1:(I-1)) {
for (i2 in (i1+1):I) {
trtDiff[i1,i2] <- theta_i_dot[i1]- theta_i_dot[i2]
}
}
trtDiff <- trtDiff[!is.na(trtDiff)]
nDiffs <- I*(I-1)/2
std_DIFF_FOM_FRRC <- sqrt(2*retMS$msTC/J/K)
nDiffs <- I*(I-1)/2
CI_DIFF_FOM_FRRC <- array(dim = c(nDiffs, 3))
for (i in 1 : nDiffs) {
CI_DIFF_FOM_FRRC[i,1] <- qt(alpha/2,df = ddf)*std_DIFF_FOM_FRRC + trtDiff[i]
CI_DIFF_FOM_FRRC[i,2] <- trtDiff[i]
CI_DIFF_FOM_FRRC[i,3] <- qt(1-alpha/2,df = ddf)*std_DIFF_FOM_FRRC + trtDiff[i]
print(data.frame("pValue" = pValue,
"Lower" = CI_DIFF_FOM_FRRC[i,1],
"Mid" = CI_DIFF_FOM_FRRC[i,2],
"Upper" = CI_DIFF_FOM_FRRC[i,3]))
}
#> pValue Lower Mid Upper
#> 1 0.02103497 -0.08088303 -0.04380032 -0.006717613
retRJafroc <- StSignificanceTesting(dataset02, FOM = "Wilcoxon", method = "DBM")
data.frame("pValue" = retRJafroc$FRRC$FTests$p[1],
"Lower" = retRJafroc$FRRC$ciDiffTrt[1,"CILower"],
"Mid" = retRJafroc$FRRC$ciDiffTrt[1,"Estimate"],
"Upper" = retRJafroc$FRRC$ciDiffTrt[1,"CIUpper"])
#> pValue Lower Mid Upper
#> 1 0.021034969 -0.080883031 -0.043800322 -0.0067176131As one might expect, if one “freezes” reader variability, the FOM difference becomes significant, whether viewed from the point of view of the F-statistic exceeding the critical value, the observed p-value being smaller than alpha or the 95% CI for the difference FOM not including zero.
5.3 Random-reader fixed-case (RRFC) analysis
The model is the same as in TBA (4.1) Eqn. (9.4) except one puts \(\sigma_C^2 = \sigma_{\tau C}^2 =0\) in Table Table 4.1. It follows that:
\[\begin{equation} \frac{E(MST)}{E(MSTR)}=\frac{\sigma_\epsilon^2+\sigma_{\tau RC}^2+K\sigma_{\tau R}^2+JK\sigma_{\tau}^2}{\sigma_\epsilon^2+\sigma_{\tau RC}^2+K\sigma_{\tau R}^2} \end{equation}\]
Under the null hypothesis \(\sigma_\tau^2 = 0\):
\[\begin{equation} \frac{E(MST)}{E(MSTR)}=1 \end{equation}\]
Therefore, one defines the F-statistic (replacing expected values with observed values) by:
\[\begin{equation} F_{DBM|C} \sim \frac{MST}{MSTR} \tag{5.6} \end{equation}\]
The observed value \(F_{DBM|C}\) is distributed as an F-statistic with \(ndf = I – 1\) and \(ddf = (I-1)(J-1)\), see rows labeled \(T\) and \(TR\) in Table Table 4.1.
\[\begin{equation} F_{DBM|C} \sim F_{I-1,(I-1)(J-1))} \tag{5.7} \end{equation}\]
The null hypothesis is rejected if the observed value of the F statistic exceeds the critical value:
\[\begin{equation} F_{DBM|C} > F_{1-\alpha,I-1,(I-1)(J-1))} \tag{5.8} \end{equation}\]
The p-value of the test is the probability that a random sample from the distribution exceeds the observed value:
\[\begin{equation} p=\Pr\left ( F>F_{DBM|C} \mid F \sim F_{I-1,(I-1)(J-1)} \right ) \tag{5.9} \end{equation}\]
The confidence interval for inter-treatment differences is given by (TBA check this):
\[\begin{equation} CI_{1-\alpha}=\left ( \theta_{i \bullet} - \theta_{i' \bullet} \right ) \pm t_{\alpha/2,(I-1)(J-1)}\sqrt{2\frac{MSTR}{JK}} \tag{5.10} \end{equation}\]
5.3.0.1 Example 6: Code illustrating analysis for RRFC analysis, Van Dyke data
FDbmFC <- retMS$msT / retMS$msTR
ndf <- (I-1)
ddf <- (I-1)*(J-1)
pValue <- 1 - pf(FDbmFC, ndf, ddf)
nDiffs <- I*(I-1)/2
CI_DIFF_FOM_RRFC <- array(dim = c(nDiffs, 3))
for (i in 1 : nDiffs) {
CI_DIFF_FOM_RRFC[i,1] <- qt(alpha/2,df = ddf)*sqrt(2*retMS$msTR/J/K) + trtDiff[i]
CI_DIFF_FOM_RRFC[i,2] <- trtDiff[i]
CI_DIFF_FOM_RRFC[i,3] <- qt(1-alpha/2,df = ddf)*sqrt(2*retMS$msTR/J/K) + trtDiff[i]
print(data.frame("pValue" = pValue,
"Lower" = CI_DIFF_FOM_RRFC[i,1],
"Mid" = CI_DIFF_FOM_RRFC[i,2],
"Upper" = CI_DIFF_FOM_RRFC[i,3]))
}
#> pValue Lower Mid Upper
#> 1 0.041958752 -0.085020224 -0.043800322 -0.0025804202
data.frame("pValue" = retRJafroc$RRFC$FTests$p[1],
"Lower" = retRJafroc$RRFC$ciDiffTrt[1,"CILower"],
"Mid" = retRJafroc$RRFC$ciDiffTrt[1,"Estimate"],
"Upper" = retRJafroc$RRFC$ciDiffTrt[1,"CIUpper"])
#> pValue Lower Mid Upper
#> 1 0.041958752 -0.085020224 -0.043800322 -0.0025804202