Chapter 4 Data format and reading FROC data

4.1 TBA How much finished

90%

4.2 Introduction

In the Free-response Receiver Operating Characteristic (FROC) paradigm the observer searches each case for signs of localized disease and marks and rates localized regions that are sufficiently suspicious for presence of disease. FROC data consists of mark-rating pairs, where each mark is a localized-region that was considered sufficiently suspicious for presence of a localized lesion and the rating is it’s confidence level. As in the ROC paradigm, the rating can be an integer or quasi-continuous (e.g., 0 – 100), or a floating point value, as long as higher numbers represent greater confidence in presence of a lesion at the indicated region. This is termed a positive-directed confidence level scheme. By adopting a proximity criterion, the investigator classifies each mark as a lesion localization (LL) - if it is close to a real lesion - or a non-lesion localization (NL) otherwise.

The purpose of this chapter is to:

Explain the data format of the input Excel file for FROC datasets.
Explain the format of the FROC dataset.
Explain the lesion distribution array returned by UtilLesionDistr().
Explain the lesion weights array returned by UtilLesionWeightsDistr().
Details on the FROC paradigm are in my book (Chakraborty 2017).

The chapter is illustrated with a toy data file, R/quick-start/frocCr.xlsx in which readers ‘0’, ‘1’ and ‘2’ interpret 8 cases in two modalities, ‘0’ and ‘1’. The design is ‘factorial’, abbreviated to FCTRL in the software; this is also termed a ‘fully-crossed’ design. The Excel file has three worksheets named Truth, NL (or FP) and LL (or TP).

4.3 The `Truth` worksheet

The Truth worksheet contains 6 columns: CaseID, LesionID, Weight, ReaderID, ModalityID and Paradigm.
Since a diseased case may have more than one lesion, the first five columns contain at least as many rows as there are cases (images) in the dataset. There are 8 cases in the dataset and 12 rows of data, because some of the diseased cases contain more than one lesion.
CaseID: unique integers representing the cases in the dataset: ‘1’, ‘2’, ‘3’, the 3 non-diseased cases, and ‘70’, ‘71’, ‘72’, ‘73’, ‘74’, the 5 diseased cases. The ordering of the numbers is inconsequential.⁵
LesionID: integers 0, 1, 2, etc.,
- Each 0 represents a non-diseased case,
- Each 1 represents the first lesion on a diseased case, 2 the second lesion, if present, and so on.
- This field is zero for non-diseased cases ‘1’, ‘2’, ‘3’.
- For the first diseased case, i.e., ‘70’, it is 1 for the first lesion and 2 for the second lesion.
- For the second diseased case i.e., ‘71’, it is 1, as this case has only one lesion.
- For the third diseased case, i.e., ‘72’, it is 1 for the first lesion, 2 for the second lesion and 3 for the third lesion.
- For the fourth diseased case, i.e., ‘73’, it is 1 for the first lesion and 2 for the second lesion.
- For the fifth diseased case i.e., ‘74’, it is 1, as this case has only one lesion.
There are 3 non-diseased cases in the dataset (the number of 0’s in the LesionID column).
There are 5 diseased cases in the dataset (the number of 1’s in the LesionID column).
Weight or clinical importance - e.g., mortality associated with lesion:
- non-negative floating point values
- 0 for each non-diseased case
- For each diseased case values that sum to unity.
- A simple way to assign equal weights to all lesions in a case is to fill the Weight column with zeroes.
LesionID
- Diseased case 70 has two lesions, with LesionIDs ‘1’ and ‘2’, and weights 0.3 and 0.7.
- Diseased case 71 has one lesion, with LesionID = 1, and Weight = 1.
- Diseased case 72 has three lesions, with LesionIDs 1, 2 and 3 and weights 1/3 each.
- Diseased case 73 has two lesions, with LesionIDs 1, and 2 and weights 0.1 and 0.9.
- Diseased case 74 has one lesion, with LesionID = 1 and Weight = 1.
ReaderID: a comma-separated listing of readers, each represented by a unique text label, that have interpreted the case. In the example shown below each cell has the value ‘0, 1, 2’.
There are 3 readers in the dataset, as each cell in the ReaderID column contains ‘0, 1, 2’.
ModalityID: a comma-separated listing of modalities (or treatments), each represented by a unique integer, that apply to each case. In the example each cell has the value 0, 1. Each cell has to be text formatted.
There are 2 modalities in the dataset, as each cell in the ModalityID column contains ‘0, 1’.
Paradigm: The contents are FROC and FCTRL: this is an FROC dataset and the design is “factorial”.

4.4 Reading the FROC dataset

The example shown above corresponds to file R/quick-start/frocCr.xlsx in the project directory. The next code chunk reads this file into an R object x.

frocCr <- "R/quick-start/frocCr.xlsx"
x <- DfReadDataFile(frocCr, newExcelFileFormat = TRUE)
str(x)
#> List of 3
#>  $ ratings     :List of 3
#>   ..$ NL   : num [1:2, 1:3, 1:8, 1:2] 1.02 2.89 2.21 3.01 2.14 ...
#>   ..$ LL   : num [1:2, 1:3, 1:5, 1:3] 5.28 5.2 5.14 4.77 4.66 4.87 3.01 3.27 3.31 3.19 ...
#>   ..$ LL_IL: logi NA
#>  $ lesions     :List of 3
#>   ..$ perCase: int [1:5] 2 1 3 2 1
#>   ..$ IDs    : num [1:5, 1:3] 1 1 1 1 1 ...
#>   ..$ weights: num [1:5, 1:3] 0.3 1 0.333 0.1 1 ...
#>  $ descriptions:List of 7
#>   ..$ fileName     : chr "frocCr"
#>   ..$ type         : chr "FROC"
#>   ..$ name         : logi NA
#>   ..$ truthTableStr: num [1:2, 1:3, 1:8, 1:4] 1 1 1 1 1 1 1 1 1 1 ...
#>   ..$ design       : chr "FCTRL"
#>   ..$ modalityID   : Named chr [1:2] "0" "1"
#>   .. ..- attr(*, "names")= chr [1:2] "0" "1"
#>   ..$ readerID     : Named chr [1:3] "0" "1" "2"
#>   .. ..- attr(*, "names")= chr [1:3] "0" "1" "2"

This follows the general description in Chapter 2. The differences are described below.

The x$descriptions$type member indicates that this is an FROC dataset.
The x$lesions$perCase member is a vector whose contents reflect the number of lesions in each diseased case, i.e., 2, 1, 3, 2, 1 in the current example.
The x$lesions$IDs member indicates the labeling of the lesions in each diseased case.

x$lesions$IDs
#>      [,1] [,2] [,3]
#> [1,]    1    2 -Inf
#> [2,]    1 -Inf -Inf
#> [3,]    1    2    3
#> [4,]    1    2 -Inf
#> [5,]    1 -Inf -Inf

This shows that the lesions on the first diseased case are labeled ‘1’ and ‘2’. The -Inf is a filler used to denote a missing value. The second diseased case has one lesion labeled ‘1’. The third diseased case has three lesions labeled ‘1’, ‘2’ and ‘3’, etc.
The lesionWeight member is the clinical importance of each lesion. Lacking specific clinical reasons, the lesions should be equally weighted; this is not true for this toy dataset.

x$lesions$weights
#>           [,1]      [,2]      [,3]
#> [1,] 0.3000000 0.7000000      -Inf
#> [2,] 1.0000000      -Inf      -Inf
#> [3,] 0.3333333 0.3333333 0.3333333
#> [4,] 0.1000000 0.9000000      -Inf
#> [5,] 1.0000000      -Inf      -Inf

The first diseased case has two lesions, the first has weight 0.3 and the second has weight 0.7.
The second diseased case has one lesion with weight 1.
The third diseased case has three equally weighted lesions, each with weight 1/3. Etc.

4.5 The false positive (FP) ratings

These are found in the FP or NL worksheet.

It consists of 4 columns, of equal length. The common length is an integer random variable greater than or equal to zero. It could be zero if the dataset has no NL marks (a possibility if the lesions are very easy to find and the observer has perfect performance).
In the example dataset, the common length is 22.
ReaderID: the reader labels: these must be 0, 1, or 2, as declared in the Truth worksheet.
ModalityID: the modality labels: must be 0 or 1, as declared in the Truth worksheet.
CaseID: the labels of cases with NL marks. In the FROC paradigm NL events can occur on non-diseased and diseased cases.
FP_Rating: the floating point ratings of NL marks. Each row of this worksheet yields a rating corresponding to the values of ReaderID, ModalityID and CaseID for that row.
For ModalityID 0, ReaderID 0 and CaseID 1 (the first non-diseased case declared in the Truth worksheet), there is a single NL mark that was rated 1.02, corresponding to row 2 of the FP worksheet.
Diseased cases with NL marks are also recorded in the FP worksheet. Some examples are seen at rows 15, 16 and 21, 22, 23.
Rows 21 and 22 show that caseID = 71 got two NL marks, rated 2.24, 4.01.
Since this is the only case with two NL marks, it determines the length of the fourth dimension of the x$ratings$NL list member, 2. Absent this case, the length would have been one.
The case with the most NL marks determines the length of the fourth dimension of the x$ratings$NL list member.
The reader should confirm that the ratings in x$ratings$NL reflect the contents of the FP worksheet.

4.6 The true positive (TP) ratings

These are found in the TP or LL worksheet, see below.

This worksheet can only have diseased cases. The presence of a non-diseased case in this worksheet will generate an error.
The common vertical length, 31 in this example, is a-priori unpredictable. The maximum possible length, assuming every lesion is marked for each modality, reader and diseased case, is 9 X 2 X 3 = 54. The 9 comes from the total number of non-zero entries in the LesionID column of the Truth worksheet, the 2 from the number of modalities and 3 from the number of readers.
The fact that the actual length (31) is smaller than the maximum length (54) means that there are combinations of modality, reader and diseased cases on which some lesions were not marked.
As examples, line 2 in the worksheet, the first lesion in CaseID equal to 70 was marked (and rated 5.28) in ModalityID 0 and ReaderID 0. Line 3 in the worksheet, the second lesion in CaseID equal to 70 was also marked (and rated 4.65) in ModalityID 0 and ReaderID 0. However, lesions 2 and 3 in CaseID = 72 were not marked (line 5 in the worksheet indicates that for this modality-reader-case combination only the first lesion was marked).
The length of the fourth dimension of the x$ratings$LL list member, 3 in the present example, is determined by the diseased case (72) with the most lesions in the Truth worksheet.
The reader should confirm that the ratings in x$ratings$LL reflect the contents of the TP worksheet.

4.7 On the distribution of numbers of lesions in diseased cases

Consider a much larger dataset, dataset11, with structure as shown below (for descriptions of all embedded datasets the RJafroc documentation):

x <- dataset11
str(x)
#> List of 3
#>  $ ratings     :List of 3
#>   ..$ NL   : num [1:4, 1:5, 1:158, 1:4] -Inf -Inf -Inf -Inf -Inf ...
#>   ..$ LL   : num [1:4, 1:5, 1:115, 1:20] -Inf -Inf -Inf -Inf -Inf ...
#>   ..$ LL_IL: logi NA
#>  $ lesions     :List of 3
#>   ..$ perCase: int [1:115] 6 4 7 1 3 3 3 8 11 2 ...
#>   ..$ IDs    : num [1:115, 1:20] 1 1 1 1 1 1 1 1 1 1 ...
#>   ..$ weights: num [1:115, 1:20] 0.167 0.25 0.143 1 0.333 ...
#>  $ descriptions:List of 7
#>   ..$ fileName     : chr "dataset11"
#>   ..$ type         : chr "FROC"
#>   ..$ name         : chr "DOBBINS-1"
#>   ..$ truthTableStr: num [1:4, 1:5, 1:158, 1:21] 1 1 1 1 1 1 1 1 1 1 ...
#>   ..$ design       : chr "FCTRL"
#>   ..$ modalityID   : Named chr [1:4] "1" "2" "3" "4"
#>   .. ..- attr(*, "names")= chr [1:4] "1" "2" "3" "4"
#>   ..$ readerID     : Named chr [1:5] "1" "2" "3" "4" ...
#>   .. ..- attr(*, "names")= chr [1:5] "1" "2" "3" "4" ...

Focus for now in the 115 diseased cases.
The numbers of lesions in these cases is contained in x$lesions$perCase.

x$lesions$perCase
#>   [1]  6  4  7  1  3  3  3  8 11  2  4  6  2 16  5  2  8  3  4  7 11  1  4  3  4
#>  [26]  4  7  3  2  5  2  2  7  6  6  4 10 20 12  6  4  7 12  5  1  1  5  1  2  8
#>  [51]  3  1  2  2  3  2  8 16 10  1  2  2  6  3  2  2  4  6 10 11  1  2  6  2  4
#>  [76]  5  2  9  6  6  8  3  8  7  1  1  6  3  2  1  9  8  8  2  2 12  1  1  1  1
#> [101]  1  3  1  2  2  1  1  1  1  3  1  1  1  2  1

For example, the first diseased case contains 6 lesions, the second contains 4 lesions, the third contains 7 lesions, etc. and the last diseased case contains 1 lesion.
To get an idea of the distribution of the numbers of lesions per diseased cases, one could interrogate this vector as shown below using the which() function:

for (el in 1:max(x$lesions$perCase)) cat(
  "number of diseased cases with", el, "lesions = ", 
  length(which(x$lesions$perCase == el)), "\n")
#> number of diseased cases with 1 lesions =  25 
#> number of diseased cases with 2 lesions =  23 
#> number of diseased cases with 3 lesions =  13 
#> number of diseased cases with 4 lesions =  10 
#> number of diseased cases with 5 lesions =  5 
#> number of diseased cases with 6 lesions =  11 
#> number of diseased cases with 7 lesions =  6 
#> number of diseased cases with 8 lesions =  8 
#> number of diseased cases with 9 lesions =  2 
#> number of diseased cases with 10 lesions =  3 
#> number of diseased cases with 11 lesions =  3 
#> number of diseased cases with 12 lesions =  3 
#> number of diseased cases with 13 lesions =  0 
#> number of diseased cases with 14 lesions =  0 
#> number of diseased cases with 15 lesions =  0 
#> number of diseased cases with 16 lesions =  2 
#> number of diseased cases with 17 lesions =  0 
#> number of diseased cases with 18 lesions =  0 
#> number of diseased cases with 19 lesions =  0 
#> number of diseased cases with 20 lesions =  1

This tells us that 25 cases contain 1 lesion
Likewise, 23 cases contain 2 lesions
Etc.

4.7.1 Definition of `lesDistr` array

What is the fraction of (diseased) cases with 1 lesion, 2 lesions etc.

for (el in 1:max(x$lesions$perCase)) cat("fraction of diseased cases with", el, "lesions = ", 
                                              length(which(x$lesions$perCase == el))/length(x$ratings$LL[1,1,,1]), "\n")
#> fraction of diseased cases with 1 lesions =  0.2173913 
#> fraction of diseased cases with 2 lesions =  0.2 
#> fraction of diseased cases with 3 lesions =  0.1130435 
#> fraction of diseased cases with 4 lesions =  0.08695652 
#> fraction of diseased cases with 5 lesions =  0.04347826 
#> fraction of diseased cases with 6 lesions =  0.09565217 
#> fraction of diseased cases with 7 lesions =  0.05217391 
#> fraction of diseased cases with 8 lesions =  0.06956522 
#> fraction of diseased cases with 9 lesions =  0.0173913 
#> fraction of diseased cases with 10 lesions =  0.02608696 
#> fraction of diseased cases with 11 lesions =  0.02608696 
#> fraction of diseased cases with 12 lesions =  0.02608696 
#> fraction of diseased cases with 13 lesions =  0 
#> fraction of diseased cases with 14 lesions =  0 
#> fraction of diseased cases with 15 lesions =  0 
#> fraction of diseased cases with 16 lesions =  0.0173913 
#> fraction of diseased cases with 17 lesions =  0 
#> fraction of diseased cases with 18 lesions =  0 
#> fraction of diseased cases with 19 lesions =  0 
#> fraction of diseased cases with 20 lesions =  0.008695652

This tells us that fraction 0.217 of (diseased) cases contain 1 lesion
And fraction 0.2 of (diseased) cases contain 2 lesions
Etc.
This information is obtained using the function UtilLesionDistr()

lesDistr <- UtilLesionDistr(x)
lesDistr
#>       [,1]        [,2]
#>  [1,]    1 0.217391304
#>  [2,]    2 0.200000000
#>  [3,]    3 0.113043478
#>  [4,]    4 0.086956522
#>  [5,]    5 0.043478261
#>  [6,]    6 0.095652174
#>  [7,]    7 0.052173913
#>  [8,]    8 0.069565217
#>  [9,]    9 0.017391304
#> [10,]   10 0.026086957
#> [11,]   11 0.026086957
#> [12,]   12 0.026086957
#> [13,]   16 0.017391304
#> [14,]   20 0.008695652

The UtilLesionDistr() function returns an array with two columns and number of rows equal to the number of distinct non-zero values of lesions per case.
The first column contains the number of distinct non-zero values of lesions per case, 14 in the current example.
The second column contains the fraction of diseased cases with the number of lesions indicated in the first column.
The second column must sum to unity

sum(UtilLesionDistr(x)[,2])
#> [1] 1

The lesion distribution array will come in handy when it comes to predicting the operating characteristics from using the Radiological Search Model (RSM), as detailed in TBA Chapter 17.

4.8 Definition of `lesWghtDistr` array

This is returned by UtilLesionWeightsDistr().
This contains the same number of rows as lesDistr.
The number of columns is one plus the number of rows as lesDistr.
The first column contains the number of distinct non-zero values of lesions per case, 14 in the current example.
The second through the last columns contain the weights of cases with number of lesions per case corresponding to row 1.
Missing values are filled with -Inf.

lesWghtDistr <- UtilLesionWeightsDistr(x)
cat("dim(lesDistr) =", dim(lesDistr),"\n")
#> dim(lesDistr) = 14 2
cat("dim(lesWghtDistr) =", dim(lesWghtDistr),"\n")
#> dim(lesWghtDistr) = 14 21
cat("lesWghtDistr = \n\n")
#> lesWghtDistr =
lesWghtDistr
#>       [,1]       [,2]       [,3]       [,4]       [,5]       [,6]       [,7]
#>  [1,]    1 1.00000000       -Inf       -Inf       -Inf       -Inf       -Inf
#>  [2,]    2 0.50000000 0.50000000       -Inf       -Inf       -Inf       -Inf
#>  [3,]    3 0.33333333 0.33333333 0.33333333       -Inf       -Inf       -Inf
#>  [4,]    4 0.25000000 0.25000000 0.25000000 0.25000000       -Inf       -Inf
#>  [5,]    5 0.20000000 0.20000000 0.20000000 0.20000000 0.20000000       -Inf
#>  [6,]    6 0.16666667 0.16666667 0.16666667 0.16666667 0.16666667 0.16666667
#>  [7,]    7 0.14285714 0.14285714 0.14285714 0.14285714 0.14285714 0.14285714
#>  [8,]    8 0.12500000 0.12500000 0.12500000 0.12500000 0.12500000 0.12500000
#>  [9,]    9 0.11111111 0.11111111 0.11111111 0.11111111 0.11111111 0.11111111
#> [10,]   10 0.10000000 0.10000000 0.10000000 0.10000000 0.10000000 0.10000000
#> [11,]   11 0.09090909 0.09090909 0.09090909 0.09090909 0.09090909 0.09090909
#> [12,]   12 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333
#> [13,]   16 0.06250000 0.06250000 0.06250000 0.06250000 0.06250000 0.06250000
#> [14,]   20 0.05000000 0.05000000 0.05000000 0.05000000 0.05000000 0.05000000
#>             [,8]       [,9]      [,10]      [,11]      [,12]      [,13]  [,14]
#>  [1,]       -Inf       -Inf       -Inf       -Inf       -Inf       -Inf   -Inf
#>  [2,]       -Inf       -Inf       -Inf       -Inf       -Inf       -Inf   -Inf
#>  [3,]       -Inf       -Inf       -Inf       -Inf       -Inf       -Inf   -Inf
#>  [4,]       -Inf       -Inf       -Inf       -Inf       -Inf       -Inf   -Inf
#>  [5,]       -Inf       -Inf       -Inf       -Inf       -Inf       -Inf   -Inf
#>  [6,]       -Inf       -Inf       -Inf       -Inf       -Inf       -Inf   -Inf
#>  [7,] 0.14285714       -Inf       -Inf       -Inf       -Inf       -Inf   -Inf
#>  [8,] 0.12500000 0.12500000       -Inf       -Inf       -Inf       -Inf   -Inf
#>  [9,] 0.11111111 0.11111111 0.11111111       -Inf       -Inf       -Inf   -Inf
#> [10,] 0.10000000 0.10000000 0.10000000 0.10000000       -Inf       -Inf   -Inf
#> [11,] 0.09090909 0.09090909 0.09090909 0.09090909 0.09090909       -Inf   -Inf
#> [12,] 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333 0.08333333   -Inf
#> [13,] 0.06250000 0.06250000 0.06250000 0.06250000 0.06250000 0.06250000 0.0625
#> [14,] 0.05000000 0.05000000 0.05000000 0.05000000 0.05000000 0.05000000 0.0500
#>        [,15]  [,16]  [,17] [,18] [,19] [,20] [,21]
#>  [1,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#>  [2,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#>  [3,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#>  [4,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#>  [5,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#>  [6,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#>  [7,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#>  [8,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#>  [9,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#> [10,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#> [11,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#> [12,]   -Inf   -Inf   -Inf  -Inf  -Inf  -Inf  -Inf
#> [13,] 0.0625 0.0625 0.0625  -Inf  -Inf  -Inf  -Inf
#> [14,] 0.0500 0.0500 0.0500  0.05  0.05  0.05  0.05

Row 3 corresponds to 3 lesions per case and the weights are 1/3, 1/3 and 1/3.
Row 13 corresponds to 16 lesions per case and the weights are 0.06250000, 0.06250000, …, repeated 13 times.
Note that the number of rows is less than the maximum number of lesions per case (20).
This is because some configurations of lesions per case (e.g., cases with 13 lesions per case) do not occur in this dataset.

4.9 References

References

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

CaseID should not be so large that it cannot be represented in Excel by an integer; to be safe use unsigned short 8-bit integers. For example, 108057200 or 9971103254 are too large to be a valid caseID and may cause errors.↩︎