• 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.
  • 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 piecewise linear nature of the plot, with sharp breaks, indicates that this is due to a finite dataset.
  • In contrast the following code shows a smooth plot, because it is a model predicted plot.

The distribution of number of lesions and weights

  • 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")



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