## Introduction

• The purpose of this vignette is to explain the operating characteristics predicted by the RSM. It relates to Chapter 17 in my book (Chakraborty 2017).
• This vignette is under development …
• Also to explain the difference between dataset members (lesionID, lesionWeight) and (lesDist, lesWghtDistr), which are RSM model parameters.

## The distinction between predicted curves and empirical curves

• Operating characteristics predicted by a model have zero sampling variability.
• Empirical operating characteristics, which apply to datasets, have non-zero sampling variability.
• If the model is corect, as the numbers of cases in the dataset increases, the empirical operating characteristic asymptotically approaches the predicted curve.

## The RSM model

• The 3 RSM parameters and two additional parameters characterizing the dataset determine the wAFROC curve.
• The 3 RSM parameters are $$\mu$$, $$\lambda$$ and $$\nu$$.
• The two dataset parameters are:
• The distribution of number of lesions per diseased case, lesDist.
• The distribution of lesion weights, lesWghtDistr.
• These parameters do not apply to individual cases; rather they refer to a large population (asymptotically infinite in size) of cases.
str(dataset04$lesionID) #> num [1:100, 1:3] 1 1 1 1 1 1 1 1 1 1 ... str(dataset04$lesionWeight)
#>  num [1:100, 1:3] 1 1 1 1 1 1 1 1 1 1 ...
• Note that the first index of both arrays is the case index for the 100 abnormal cases in this dataset.
• With finite number of cases the empirical operating characteristic (or for that matter any fitted operating characteristic) will have sampling variability as in the following example.

## The empirical wAFROC

p <- PlotEmpiricalOperatingCharacteristics(dataset04, opChType = "wAFROC")

## The distribution of number of lesions and weights

lesDistr
#>      [,1] [,2]
#> [1,]    1  0.2
#> [2,]    2  0.4
#> [3,]    3  0.1
#> [4,]    4  0.3
lesWghtDistr
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1  1.0 -Inf -Inf -Inf
#> [2,]    2  0.4  0.6 -Inf -Inf
#> [3,]    3  0.2  0.3  0.5 -Inf
#> [4,]    4  0.3  0.4  0.2  0.1
• The second column of lesDistr specifies the fraction of diseased cases with the number of lesions specified in the first column.
• The first column of lesWghtDistr is a copy of the first column of lesDistr. The remaining non--Inf entries are the weights.
• For cases with 1 lesion, the weight is 1.
• For cases with 2 lesions, the first lesion has weight 0.4 and the second lesion has weight 0.6, which sum to unity.
• For cases with 3 lesions, the respective weights are 0.2, 0.3 and 0.5, which sum to unity.
• For cases with 4 lesions, the respective weights are 0.3, 0.4, 0.2 and 0.1, which sum to unity.

## Other operating characteristics

• By changing OpChType one can generate other operating characteristics.
• Note that lesiion weights argument is not needed for ROC curves. It is only needed for wAFROC and wAFROC1 curves.
lesDistr <- rbind(c(1, 0.2), c(2, 0.4), c(3, 0.1), c(4, 0.3))
p <- PlotRsmOperatingCharacteristics(mu = 2, lambda = 1, nu = 0.6, OpChType = "ROC",
lesDistr = lesDistr,
legendPosition = "bottom")
p\$ROCPlot

## References

Chakraborty, Dev P. 2017. Observer Performance Methods for Diagnostic Imaging - Foundations, Modeling, and Applications with R-Based Examples. Book. Boca Raton, FL: CRC Press.