## Introduction

The starting point is a pilot study. The variability in this dataset (specifically the variance components, subsequently converted to mean squares), obtained by running the significance testing function StSignificanceTesting(), is used to extrapolate to the necessary numbers of readers and cases, in the pivotal study, to achieve the desired power. In this example, the observed effect size in the pilot study is used as the anticipated effect size for the pivotal study – this is generally not a good idea as discussed in Chapter 11 under “Cautionary notes”. Shown below, and the reader should confirm, is a first principles implementation of the relevant formulae in Chapter 11.

## Sample size estimation using the DBMH method

The Van Dyke dataset in file VanDyke.lrc, in "MRMC" format, is regarded as a pilot study. The command rocData <- DfReadDataFile(fileName, format = "MRMC") reads the data and saves it to a dataset object rocData. For more on data formats click here. The next line uses the function StSignificanceTesting() to apply method = "DBMH" analysis, the default, using the FOM = "Wilcoxon" figure of merit. The next line extracts the variance components varYTR, varYTC and varYEps (the Y’s denote pseudovalue based values). The next line extracts the effect size.

alpha <- 0.05
rocData <- dataset02 ##"VanDyke.lrc"
#fileName <- dataset03 ## "Franken1.lrc"
retDbm <- StSignificanceTesting(dataset = rocData, FOM = "Wilcoxon", method = "DBMH")
varYTR <- retDbm$varComp$varComp[3];varYTC <- retDbm$varComp$varComp[4];varYEps <- retDbm$varComp$varComp[6]
effectSize <- retDbm$ciDiffTrtRRRC$Estimate

The observed effect size is effectSize = -0.0438003, which, in this example, is used as the anticipated effect size, generally not a good idea. See Chapter 11 for nuances regarding the choice of this all important value. The following code snippet reveals the names and array indexing of the pseudovalue variance components.

retDbm$varComp #> varR varC varTR varTC varRC varErr #> 1 0.001534999 0.02724923 0.0002004025 0.0119753 0.01226473 0.0399716 For example, the treatment-reader pseudovalue variance component is the third element of retDbm$varComp.

### Random reader random case (RRRC)

This illustrates random reader random case sample size estimation. Assumed are 10 readers and 163 cases in the pivotal study. The non-centrality parameter is defined by:

$\Delta =\frac{JK\sigma _{Y;\tau }^{2}}{\left( \sigma _{Y;\varepsilon }^{2}+\sigma _{Y;\tau RC}^{2} \right)+K\sigma _{Y;\tau R}^{2}+J\max \left( \sigma _{Y;\tau C}^{2},0 \right)}$

The sampling distribution of the F-statistic under the AH is:

${{F}_{\left. AH \right|R}}\equiv \frac{MST}{MSTC}\tilde{\ }{{F}_{I-1,\left( I-1 \right)\left( K-1 \right),\Delta }}$ Also,

$\sigma _{Y;\tau }^{2}={{d}^{2}}/2$

where d is the observed effect size, i.e., effectSize. The formulae for calculating the mean-squares are in (Hillis and Berbaum 2004), implemented in UtilMeanSquares().

#RRRC
ncp <- (0.5*J*K*(effectSize)^2)/(K*varYTR+max(J*varYTC,0)+varYEps)
MS <- UtilMeanSquares(rocData, FOM = "Wilcoxon", method = "DBMH")
ddf <- (MS$msTR+max(MS$msTC-MS$msTRC,0))^2/(MS$msTR^2)*(J-1)
FCrit <- qf(1 - alpha, 1, ddf)
Power1 <- 1-pf(FCrit, 1, ddf, ncp = ncp)

The next line calculates the non centrality parameter, ncp = . Note that effectSize enters as the square. The UtilMeanSquares() function returns the mean-squares as a list (ignore the last two rows of output for now).

str(MS)
#> List of 9
#>  $msT : num 0.547 #>$ msR       : num 0.437
#>  $msC : num 0.397 #>$ msTR      : num 0.0628
#>  $msTC : num 0.0521 #>$ msRC      : num 0.0645
#>  $msTRC : num 0.04 #>$ msCSingleT: num [1:2] 0.336 0.16
#>  \$ msCSingleR: num [1:5] 0.1222 0.2127 0.1365 0.0173 0.1661

The next line calculates ddf = 12.822129. The remaining lines calculate the critical value of the F-distribution, FCrit = 4.680382 and statistical power = , which by design is close to 80%, i.e., the numbers of readers and cases were chosen to achieve this value.

### Fixed reader random case (FRRC)

This code illustrates fixed reader random case sample size estimation. Assumed are 10 readers and 133 cases in the pivotal study. The formulae are:

$\Delta =\frac{JK\sigma _{Y;\tau }^{2}}{\sigma _{Y;\varepsilon }^{2}+\sigma _{Y;\tau RC}^{2}+J\sigma _{Y;\tau C}^{2}}$

The sampling distribution of the F-statistic under the AH is:

${{F}_{\left. AH \right|R}}\equiv \frac{MST}{MSTC}\tilde{\ }{{F}_{I-1,\left( I-1 \right)\left( K-1 \right),\Delta }}$

#FRRC
ncp <- (0.5*J*K*(effectSize)^2)/(max(J*varYTC,0)+varYEps)
ddf <- (K-1)
FCrit <- qf(1 - alpha, 1, ddf)
Power2 <- 1-pf(FCrit, 1, ddf, ncp = ncp)

This time non centrality parameter, ncp = , ddf = 132, FCrit = 3.912875 and statistical power = . Again, be design, this is close to 80%. Note that when readers are regarded as a fixed effect, fewer cases are needed to achieve the desired power. Freezing out a source of variability results in a more stable measurement and hence fewer cases are needed to achieve the desired power.

### Random reader fixed case (RRFC)

This code illustrates random reader random case sample size estimation. Assumed are 10 readers and 53 cases in the pivotal study. The formulae are:

$\Delta =\frac{JK\sigma _{Y;\tau }^{2}}{\sigma _{Y;\varepsilon }^{2}+\sigma _{Y;\tau RC}^{2}+K\sigma _{Y;\tau R}^{2}}$

The sampling distribution of the F-statistic under the AH is:

${{F}_{\left. AH \right|C}}\equiv \frac{MST}{MSTR}\tilde{\ }{{F}_{I-1,\left( I-1 \right)\left( J-1 \right),\Delta }}$

#RRFC
ncp <- (0.5*J*K*(effectSize)^2)/(K*varYTR+varYEps)
ddf <- (J-1)
FCrit <- qf(1 - alpha, 1, ddf)
Power3 <- 1-pf(FCrit, 1, ddf, ncp = ncp)

This time non centrality parameter, ncp = , ddf = 9, FCrit = 5.117355 and statistical power = . Again, be design, this is close to 80%.

## Summary

For 10 readers, the numbers of cases needed for 80% power is largest (163) for RRRC, intermediate (133) for FRRC and least for RRFC (53). For all three analyses, the expectation of 80% power is met.

## References

Hillis, Stephen L., and K. S. Berbaum. 2004. “Power Estimation for the Dorfman-Berbaum-Metz Method.” Journal Article. Acad. Radiol. 11 (11): 1260–73.