brian-j-smith@uiowa.edu

steve-hillis@uiowa.edu

**MRMCaov** is an R package for statistical comparison of diagnostic tests - such as those based on medical imaging - for which ratings have been obtained from multiple readers and on multiple cases. Features of the package include the following.

- Statistical comparisons of diagnostic tests with respect to reader performance metrics
- Comparisons based on the ANOVA model of Obuchowski-Rockette and the unified framework of Hillis
- Reader performance metrics for area under receiver operating characteristic curves (ROC AUCs), partial AUCs, expected utility of ROC curves, likelihood ratio of positive or negative tests, sensitivity, specificity, and user-defined metrics
- Parametric and nonparametric estimation and plotting of ROC curves
- Support for factorial, nested, and partially paired study designs
- Inference for random or fixed readers and cases
- Conversion of Obuchowski-Rockette to Roe, Metz & Hillis model parameters and vice versa
- DeLong, jackknife, and unbiased covariance estimation
- Compatibility with Microsoft Windows, MacOS, and Linux

Enter the following at the R console to install the package.

```
To cite MRMCaov in publications, please use the following two
references, including the R package URL.
Smith BJ, Hillis SL, Pesce LL (2023). _MCMCaov: Multi-Reader
Multi-Case Analysis of Variance_. R package version 0.3.0,
<https://github.com/brian-j-smith/MRMCaov>.
Smith BJ, Hillis SL (2020). "Multi-reader multi-case analysis of
variance software for diagnostic performance comparison of imaging
modalities." In Samuelson F, Taylor-Phillips S (eds.), _Proceedings
of SPIE 11316, Medical Imaging 2020: Image Perception, Observer
Performance, and Technology Assessment_, 113160K.
doi:10.1117/12.2549075 <https://doi.org/10.1117/12.2549075>,
<https://pubmed.ncbi.nlm.nih.gov/32351258>.
To see these entries in BibTeX format, use 'print(<citation>,
bibtex=TRUE)', 'toBibtex(.)', or set
'options(citation.bibtex.max=999)'.
```

```
@Manual{MRMCaov-package,
author = {Brian J Smith and Stephen L Hillis and Lorenzo L Pesce},
title = {{MCMCaov}: Multi-Reader Multi-Case Analysis of Variance},
year = {2023},
note = {R package version 0.3.0},
url = {https://github.com/brian-j-smith/MRMCaov},
}
@InProceedings{MRMCaov-SPIE2020,
author = {Brian J. Smith and Stephen L. Hillis},
title = {Multi-reader multi-case analysis of variance software for diagnostic performance comparison of imaging modalities},
booktitle = {Proceedings of SPIE 11316, Medical Imaging 2020: Image Perception, Observer Performance, and Technology Assessment},
editor = {Frank Samuelson and Sian Taylor-Phillips},
month = {16 March},
year = {2020},
pages = {113160K},
doi = {10.1117/12.2549075},
url = {https://pubmed.ncbi.nlm.nih.gov/32351258},
}
```

A common study design for comparing the diagnostic performance of imaging modalities, or diagnostic tests, is to obtain modality-specific ratings from multiple readers of multiple cases (MRMC) whose true statuses are known. In such a design, receiver operating characteristic (ROC) indices, such as area under the ROC curve (ROC AUC), can be used to quantify correspondence between reader ratings and case status. Indices can then be compared statistically to determine if there are differences between modalities. However, special statistical methods are needed when readers or cases represent a random sample from a larger population of interest and there is overlap between modalities, readers, and/or cases. An ANOVA model designed for these characteristics of MRMC studies was initially proposed by Dorfman et al. (Dorfman, Berbaum, and Metz 1992) and Obuchowski and Rockette (Obuchowski and Rockette 1995) and later unified and improved by Hillis and colleagues (Hillis et al. 2005; Hillis 2007, 2018; Hillis, Berbaum, and Metz 2008). Their models are implemented in the **MRMCaov** R package (Smith, Hillis, and Pesce 2022).

**MRMCaov** implements multi-reader multi-case analysis based on the Obuchowski and Rockette (1995) analysis of variance (ANOVA) model \[
\hat{\theta}_{ij} = \mu + \tau_i + R_j + (\tau R)_{ij} + \epsilon_{ij},
\] where \(i = 1,\ldots,t\) and \(j = 1,\ldots,r\) index diagnostic tests and readers; \(\hat{\theta}_{ij}\) is a reader performance metric, such as ROC AUC, estimated over multiple cases; \(\mu\) an overall study mean; \(\tau_i\) a fixed test effect; \(R_j\) a random reader effect; \((\tau R)_{ij}\) a random test \(\times\) reader interaction effect; and \(\epsilon_{ij}\) a random error term. The random terms \(R_j\), \((\tau R)_{ij}\), and \(\epsilon_{ij}\) are assumed to be mutually independent and normally distributed with 0 means and variances \(\sigma^2_R\), \(\sigma^2_{TR}\), and \(\sigma^2_\epsilon\).

The error covariances between tests and between readers are further assumed to be equal, resulting in the three covariances \[ \text{Cov}(\epsilon_{ij}, \epsilon_{i'j'}) = \left\{ \begin{array}{lll} \text{Cov}_1 & i \ne i', j = j' & \text{(different test, same reader)} \\ \text{Cov}_2 & i = i', j \ne j' & \text{(same test, same reader)} \\ \text{Cov}_3 & i \ne i', j \ne j' & \text{(different test, different reader)}. \end{array} \right. \] Obuchowski and Rockette (1995) suggest a covariance ordering of \(\text{Cov}_1 \ge \text{Cov}_2 \ge \text{Cov}_3 \ge 0\) based on clinical considerations. Hillis (2014) later showed that these can be replaced with the less restrictive orderings \(\text{Cov}_1 \ge \text{Cov}_3\), \(\text{Cov}_2 \ge \text{Cov}_3\), and \(\text{Cov}_3 \ge 0\). Alternatively, the covariance can be specified as the population correlations \(\rho_i = \text{Cov}_i / \sigma^2_\epsilon\).

In the Obuchowski-Rockette ANOVA model, \(\sigma^2_\epsilon\) can be interpreted as the performance metric variance for a single fixed reader and test; and \(\text{Cov}_1\), \(\text{Cov}_2\), and \(\text{Cov}_3\) as the performance metric covariances for the same reader of two different tests, two different readers of the same test, and two different readers of two different tests. These error variance and covariance parameters are estimated in the package by averaging the reader and test-specific estimates computed using jackknifing (Efron 1982) or, for empirical ROC AUC, an unbiased estimator (Gallas, Pennello, and Meyers 2007) or the method of DeLong (DeLong, DeLong, and Clarke-Pearson 1988).

Use of the **MRMCaov** package is illustrated with data from a study comparing the relative performance of cinematic presentation of MRI (CINE MRI) to single spin-echo magnetic resonance imaging (SE MRI) for the detection of thoracic aortic dissection (VanDyke et al. 1993). In the study, 45 patients with aortic dissection and 69 without dissection were imaged with both modalities. Based on the images, five radiologists rated patients disease statuses as 1 = definitely no aortic dissection, 2 = probably no aortic dissection, 3 = unsure about aortic dissection, 4 = probably aortic dissection, or 5 = definitely aortic dissection. Interest lies in estimating ROC curves for each combination of reader and modality and in comparing modalities with respect to summary statistics from the curves. The study data are included in the package as a data frame named `VanDyke`

.

```
## reader treatment case truth rating case2 case3
## 1 1 1 1 0 1 1.1 1.1
## 2 1 2 1 0 3 1.1 2.1
## 3 2 1 1 0 2 2.1 1.1
## 4 2 2 1 0 3 2.1 2.1
## 5 3 1 1 0 2 3.1 1.1
## 6 3 2 1 0 2 3.1 2.1
## 7 4 1 1 0 1 4.1 1.1
## 8 4 2 1 0 2 4.1 2.1
## 9 5 1 1 0 3 5.1 1.1
## 10 5 2 1 0 2 5.1 2.1
## 11 1 1 2 0 2 1.2 1.2
## 12 1 2 2 0 3 1.2 2.2
## 13 2 1 2 0 3 2.2 1.2
## 14 2 2 2 0 2 2.2 2.2
## 15 3 1 2 0 2 3.2 1.2
## 16 3 2 2 0 4 3.2 2.2
## 17 4 1 2 0 1 4.2 1.2
## 18 4 2 2 0 2 4.2 2.2
## 19 5 1 2 0 5 5.2 1.2
## 20 5 2 2 0 2 5.2 2.2
## ... with 1120 more rows
```

The study employed a factorial design in which each of the five radiologists read and rated both the CINE and SE MRI images from all 114 cases. The original study variables in the `VanDyke`

data frame are summarized below along with two additional `case2`

and `case3`

variables that represent hypothetical study designs in which cases are nested within readers (`reader`

) and within imaging modalities (`treatment`

), respectively.

Variable | Description |
---|---|

`reader` |
unique identifiers for the five radiologists |

`treatment` |
identifiers for the imaging modality (1 = CINE MRI, 2 = SE MRI) |

`case` |
identifiers for the 114 cases |

`truth` |
indicator for thoracic aortic dissection (1 = performed, 0 = not performed) |

`rating` |
five-point ratings given to case images by the readers |

`case2` |
example identifiers representing nesting of cases within readers |

`case3` |
example identifiers representing nesting of cases within treatments |

Data from other studies may be analyzed with the package and should follow the format of `VanDyke`

with columns for reader, treatment, and case identifiers as well as true event statuses and reader ratings. The variable names, however, may be different.

A multi-reader multi-case (MRMC) analysis, as the name suggests, involves multiple readers of multiple cases to compare reader performance metrics across two or more diagnostic tests. An MRMC analysis can be performed with a call to the `mrmc()`

function to specify a reader performance metric, study variables and observations, and covariance estimation method.

MRMC Function

`mrmc(response, test, reader, case, data, cov = jackknife)`

DescriptionReturns an

`mrmc`

class object of data that can be used to estimate and compare reader performance metrics in a multi-reader multi-case statistical analysis.

Arguments

`response`

: object defining true case statuses, corresponding reader ratings, and a reader performance metric to compute on them.`test`

,`reader`

,`case`

: variables containing the test, reader, and case identifiers for the`response`

observations.`data`

: data frame containing the response and identifier variables.`cov`

: function`jackknife`

,`unbiased`

, or`DeLong`

to estimate reader performance metric covariances.

The response variable in the `mrmc()`

specification is defined with one of the performance metrics described in the following sections. Results from `mrmc()`

can be displayed with `print()`

and passed to `summary()`

for statistical comparisons of the diagnostic tests. The summary call produces ANOVA results from a global test of equality of ROC AUC means across all tests and statistical tests of pairwise differences, along with confidence intervals for the differences and intervals for individual tests.

MRMC Summary Function

`summary(object, conf.level = 0.95)`

DescriptionReturns a

`summary.mrmc`

class object of statistical results from a multi-reader multi-case analysis.

Arguments

`object`

: results from`mrmc()`

.`conf.level`

: confidence level for confidence intervals.

Area under the ROC curve is a measure of concordance between numeric reader ratings and true binary case statuses. It provides an estimate of the probability that a randomly selected positive case will have a higher rating than a negative case. ROC AUC values range from 0 to 1, with 0.5 representing no concordance and 1 perfect concordance. AUC can be computed with the functions described below for binormal, binormal likelihood-ratio, and empirical ROC curves. Empirical curves are also referred to as trapezoidal. The functions also support calculation of partial AUC over a range of sensitivities or specificities.

ROC AUC Functions

`binormal_auc(truth, rating, partial = FALSE, min = 0, max = 1, normalize = FALSE)`

`binormalLR_auc(truth, rating, partial = FALSE, min = 0, max = 1, normalize = FALSE)`

`empirical_auc(truth, rating, partial = FALSE, min = 0, max = 1, normalize = FALSE)`

`trapezoidal_auc(truth, rating, partial = FALSE, min = 0, max = 1, normalize = FALSE)`

DescriptionReturns computed area under the receiver operating character curve estimated with a binormal model (

`binormal_auc`

), binormal likelihood-ratio model (`binormalLR_auc`

), or empirically (`empirical_auc`

or`trapezoidal_auc`

).

Arguments

`truth`

: vector of true binary case statuses, with positive status taken to be the highest level.`rating`

: numeric vector of case ratings.`partial`

: character string`"sensitivity"`

or`"specificity"`

for calculation of partial AUC, or`FALSE`

for full AUC. Partial matching of the character strings is allowed. A value of`"specificity"`

results in area under the ROC curve between the given`min`

and`max`

specificity values, whereas`"sensitivity"`

results in area to the right of the curve between the given sensitivity values.`min`

,`max`

: minimum and maximum sensitivity or specificity values over which to calculate partial AUC.`normalize`

: logical indicating whether partial AUC is divided by the interval width (`max - min`

) over which it is calculated.

In the example below, `mrmc()`

is called to compare CINE MRI and SE MRI treatments in an MRMC analysis of areas under binormal ROC curves computed for the readers of cases in the VanDyke study.

```
## Compare ROC AUC treatment means for the VanDyke example
est <- mrmc(
binormal_auc(truth, rating), treatment, reader, case, data = VanDyke
)
```

```
## Warning in binormal_params(data$truth, data$rating, ...):
## Binormal curve fit has AUC = 1 due to lack of interior points.
## Consider fitting an empirical curve instead.
```

The `print()`

function can be applied to `mrmc()`

output to display information about the reader performance metrics, including the

- value of variable
`truth`

(1) defining positive case status, - estimated performance metric values (
`data$binormal_auc`

) for each test (`$treatment`

) and reader (`$reader`

), - number of cases read at each level of the factors (
`N`

), and - error variance \(\sigma^2_\epsilon\) and covariances \(\text{Cov}_1\), \(\text{Cov}_2\), and \(\text{Cov}_3\).

```
## Call:
## mrmc(response = binormal_auc(truth, rating), test = treatment,
## reader = reader, case = case, data = VanDyke)
##
## Positive truth status: 1
##
## Response metric data:
##
## # A tibble: 10 × 2
## N data$binormal_auc $treatment $reader
## <dbl> <dbl> <fct> <fct>
## 1 114 0.933 1 1
## 2 114 0.890 1 2
## 3 114 0.929 1 3
## 4 114 0.970 1 4
## 5 114 0.833 1 5
## 6 114 0.951 2 1
## 7 114 0.935 2 2
## 8 114 0.928 2 3
## 9 114 1 2 4
## 10 114 0.945 2 5
##
## ANOVA Table:
##
## Df Sum Sq Mean Sq
## treatment 1 0.0041142 0.0041142
## reader 4 0.0104325 0.0026081
## treatment:reader 4 0.0037916 0.0009479
##
##
## Obuchowski-Rockette error variance and covariance estimates:
##
## Estimate Correlation
## Error 0.0010790500 NA
## Cov1 0.0003125013 0.2896078
## Cov2 0.0003115986 0.2887713
## Cov3 0.0001937688 0.1795735
```

MRMC statistical tests are performed with a call to `summary()`

. Results include a test of the global null hypothesis that performances are equal across all diagnostic tests, tests of their pairwise mean differences, and estimated mean performances for each one.

```
## Multi-Reader Multi-Case Analysis of Variance
## Data: VanDyke
## Factor types: Random Readers and Random Cases
## Covariance method: jackknife
##
## Experimental design: factorial
##
## Obuchowski-Rockette variance component and covariance estimates:
##
## Estimate Correlation
## reader 0.0007113799 NA
## treatment:reader 0.0002991713 NA
## Error 0.0010790500 NA
## Cov1 0.0003125013 0.2896078
## Cov2 0.0003115986 0.2887713
## Cov3 0.0001937688 0.1795735
##
##
## ANOVA global test of equal treatment binormal_auc:
## MS(T) MS(T:R) Cov2 Cov3 Denominator F df1
## 1 0.004114188 0.0009478901 0.0003115986 0.0001937688 0.001537039 2.676697 1
## df2 p-value
## 1 10.51753 0.1313668
##
##
## 95% CIs and tests for treatment binormal_auc pairwise differences:
## Comparison Estimate StdErr df CI.Lower CI.Upper t
## 1 1 - 2 -0.04056692 0.02479548 10.51753 -0.09544840 0.01431455 -1.636061
## p-value
## 1 0.1313668
##
##
## 95% treatment binormal_auc CIs (each analysis based only on data for the
## specified treatment):
## Estimate MS(R) Cov2 StdErr df CI.Lower CI.Upper
## 1 0.9109867 0.0027417526 0.0004612201 0.03177374 13.55866 0.8426301 0.9793433
## 2 0.9515536 0.0008142524 0.0001619772 0.01802298 15.91432 0.9133299 0.9897774
```

ROC curves estimated by `mrmc()`

can be displayed with `plot()`

and their parameters extracted with `parameters()`

.

```
## Warning in binormal_params(data$truth, data$rating, ...):
## Binormal curve fit has AUC = 1 due to lack of interior points.
## Consider fitting an empirical curve instead.
```

```
## Warning in binormal_params(data$truth, data$rating, ...):
## Binormal curve fit has AUC = 1 due to lack of interior points.
## Consider fitting an empirical curve instead.
```

```
## # A tibble: 10 × 3
## Group$reader $treatment a b
## <fct> <fct> <dbl> <dbl>
## 1 1 1 1.70 0.537
## 2 2 1 1.40 0.561
## 3 3 1 1.74 0.635
## 4 4 1 1.93 0.202
## 5 5 1 1.06 0.464
## 6 1 2 1.85 0.503
## 7 2 2 1.66 0.447
## 8 3 2 1.62 0.488
## 9 4 2 Inf 1
## 10 5 2 1.73 0.422
```

As an alternative to AUC as a summary of ROC curves, Abbey et al. (2013) propose an expected utility metric defined as \[ \text{EU} = \max_\text{FPR}(\text{TPR}(\text{FPR}) - \beta \times \text{FPR}), \] where \(\text{TPR}(\text{FPR})\) are true positive rates on the ROC curve, and FPR are false positive rates ranging from 0 to 1.

ROC Curve Expected Utility Functions

`binormal_eu(truth, rating, slope = 1)`

`binormalLR_eu(truth, rating, slope = 1)`

`empirical_eu(truth, rating, slope = 1)`

`trapezoidal_eu(truth, rating, slope = 1)`

DescriptionReturns expected utility of an ROC curve.

Arguments

`truth`

: vector of true binary case statuses, with positive status taken to be the highest level.`rating`

: numeric vector of case ratings.`slope`

: numeric slope (\(\beta\)) at which to compute expected utility.

Functions are provided to extract sensitivity from an ROC curve for a given specificity and vice versa.

ROC Curve Sensitivity and Specificity Functions

`binormal_sens(truth, rating, spec)`

`binormal_spec(truth, rating, sens)`

`binormalLR_sens(truth, rating, spec)`

`binormalLR_spec(truth, rating, sens)`

`empirical_sens(truth, rating, spec)`

`empirical_spec(truth, rating, sens)`

`trapezoidal_sens(truth, rating, spec)`

`trapezoidal_spec(truth, rating, sens)`

DescriptionReturns the sensitivity/specificity from an ROC curve at a specified specificity/sensitivity.

Arguments

`truth`

: vector of true binary case statuses, with positive status taken to be the highest level.`rating`

: numeric vector of case ratings.`spec`

,`sens`

: specificity/sensitivity on the ROC curve at which to return sensitivity/specificity.

Metrics for binary reader ratings are also available.

Sensitivity and Specificity Functions

`binary_sens(truth, rating)`

`binary_spec(truth, rating)`

DescriptionReturns the sensitivity or specificity.

Arguments

`truth`

: vector of true binary case statuses, with positive status taken to be the highest level.`rating`

: factor or numeric vector of 0-1 binary ratings.

```
## Compare sensitivity for binary classification
VanDyke$binary_rating <- VanDyke$rating >= 3
est <- mrmc(
binary_sens(truth, binary_rating), treatment, reader, case, data = VanDyke
)
```

```
## Call:
## mrmc(response = binary_sens(truth, binary_rating), test = treatment,
## reader = reader, case = case, data = VanDyke)
##
## Positive truth status: 1
##
## Response metric data:
##
## # A tibble: 10 × 2
## N data$binary_sens $treatment $reader
## <dbl> <dbl> <fct> <fct>
## 1 45 0.889 1 1
## 2 45 0.778 1 2
## 3 45 0.822 1 3
## 4 45 0.933 1 4
## 5 45 0.689 1 5
## 6 45 0.978 2 1
## 7 45 0.822 2 2
## 8 45 0.911 2 3
## 9 45 1 2 4
## 10 45 0.889 2 5
##
## ANOVA Table:
##
## Df Sum Sq Mean Sq
## treatment 1 0.023901 0.0239012
## reader 4 0.049679 0.0124198
## treatment:reader 4 0.007210 0.0018025
##
##
## Obuchowski-Rockette error variance and covariance estimates:
##
## Estimate Correlation
## Error 0.0023681257 NA
## Cov1 0.0009943883 0.4199052
## Cov2 0.0010145903 0.4284360
## Cov3 0.0006604938 0.2789100
```

```
## Multi-Reader Multi-Case Analysis of Variance
## Data: VanDyke
## Factor types: Random Readers and Random Cases
## Covariance method: jackknife
##
## Experimental design: factorial
##
## Obuchowski-Rockette variance component and covariance estimates:
##
## Estimate Correlation
## reader 0.0049747475 NA
## treatment:reader 0.0007828283 NA
## Error 0.0023681257 NA
## Cov1 0.0009943883 0.4199052
## Cov2 0.0010145903 0.4284360
## Cov3 0.0006604938 0.2789100
##
##
## ANOVA global test of equal treatment binary_sens:
## MS(T) MS(T:R) Cov2 Cov3 Denominator F df1
## 1 0.02390123 0.001802469 0.00101459 0.0006604938 0.003572952 6.689493 1
## df2 p-value
## 1 15.71732 0.02008822
##
##
## 95% CIs and tests for treatment binary_sens pairwise differences:
## Comparison Estimate StdErr df CI.Lower CI.Upper t
## 1 1 - 2 -0.09777778 0.03780451 15.71732 -0.1780371 -0.0175185 -2.586405
## p-value
## 1 0.02008822
##
##
## 95% treatment binary_sens CIs (each analysis based only on data for the
## specified treatment):
## Estimate MS(R) Cov2 StdErr df CI.Lower CI.Upper
## 1 0.8222222 0.009135802 0.001646465 0.05893747 14.456811 0.6961876 0.9482568
## 2 0.9200000 0.005086420 0.000382716 0.03741657 7.575855 0.8328691 1.0000000
```

Special statistical methods are needed in MRMC analyses to estimate covariances between performance metrics from different readers and tests when cases are treated as a random sample and are rated by more than one reader or evaluated with more than one test. For this estimation, the package provides the DeLong method (DeLong, DeLong, and Clarke-Pearson 1988), jackknifing (Efron 1982), and an unbiased method (Gallas, Pennello, and Meyers 2007). The applicability of each depends on the study design as well as the performance metric being analyzed. DeLong is appropriate for a balanced factorial design and empirical ROC AUC, jackknifing for any design and metric, and unbiased for any design and empirical ROC AUC.

Covariance Method | Study Design | Metric | Function |
---|---|---|---|

DeLong | Factorial | Empirical ROC AUC | `DeLong()` |

Jackknife | Any | Any | `jackknife()` |

Unbiased | Any | Empirical ROC AUC | `unbiased()` |

Jackknifing is the default covariance method for `mrmc()`

. Others can be specified with its `cov`

argument.

```
## DeLong method
est <- mrmc(
empirical_auc(truth, rating), treatment, reader, case, data = VanDyke,
cov = DeLong
)
```

```
## Multi-Reader Multi-Case Analysis of Variance
## Data: VanDyke
## Factor types: Random Readers and Random Cases
## Covariance method: DeLong
##
## Experimental design: factorial
##
## Obuchowski-Rockette variance component and covariance estimates:
##
## Estimate Correlation
## reader 0.0015364254 NA
## treatment:reader 0.0002045840 NA
## Error 0.0007921325 NA
## Cov1 0.0003420090 0.4317573
## Cov2 0.0003395265 0.4286234
## Cov3 0.0002358497 0.2977402
##
##
## ANOVA global test of equal treatment empirical_auc:
## MS(T) MS(T:R) Cov2 Cov3 Denominator F df1
## 1 0.004796171 0.0005510306 0.0003395265 0.0002358497 0.001069415 4.484854 1
## df2 p-value
## 1 15.06611 0.05123303
##
##
## 95% CIs and tests for treatment empirical_auc pairwise differences:
## Comparison Estimate StdErr df CI.Lower CI.Upper
## 1 1 - 2 -0.04380032 0.0206825 15.06611 -0.0878671960 0.0002665519
## t p-value
## 1 -2.117747 0.05123303
##
##
## 95% treatment empirical_auc CIs (each analysis based only on data for the
## specified treatment):
## Estimate MS(R) Cov2 StdErr df CI.Lower CI.Upper
## 1 0.8970370 0.003082629 0.0004775239 0.03307642 12.59597 0.8253461 0.9687280
## 2 0.9408374 0.001304602 0.0002015292 0.02150464 12.56530 0.8942155 0.9874592
```

```
## Unbiased method
est <- mrmc(
empirical_auc(truth, rating), treatment, reader, case, data = VanDyke,
cov = unbiased
)
```

```
## Multi-Reader Multi-Case Analysis of Variance
## Data: VanDyke
## Factor types: Random Readers and Random Cases
## Covariance method: unbiased
##
## Experimental design: factorial
##
## Obuchowski-Rockette variance component and covariance estimates:
##
## Estimate Correlation
## reader 0.0015365290 NA
## treatment:reader 0.0002077588 NA
## Error 0.0007883925 NA
## Cov1 0.0003416706 0.4333762
## Cov2 0.0003390650 0.4300713
## Cov3 0.0002356148 0.2988547
##
##
## ANOVA global test of equal treatment empirical_auc:
## MS(T) MS(T:R) Cov2 Cov3 Denominator F df1
## 1 0.004796171 0.0005510306 0.000339065 0.0002356148 0.001068281 4.489614 1
## df2 p-value
## 1 15.03418 0.0511618
##
##
## 95% CIs and tests for treatment empirical_auc pairwise differences:
## Comparison Estimate StdErr df CI.Lower CI.Upper
## 1 1 - 2 -0.04380032 0.02067154 15.03418 -0.0878519409 0.0002512968
## t p-value
## 1 -2.118871 0.0511618
##
##
## 95% treatment empirical_auc CIs (each analysis based only on data for the
## specified treatment):
## Estimate MS(R) Cov2 StdErr df CI.Lower CI.Upper
## 1 0.8970370 0.003082629 0.0004771788 0.0330712 12.58802 0.8253526 0.9687214
## 2 0.9408374 0.001304602 0.0002009512 0.0214912 12.53391 0.8942323 0.9874424
```

By default, readers and cases are treated as random effects by `mrmc()`

. Random effects are the appropriate designations when inference is intended for the larger population from which study readers and cases are considered to be a random sample. Either, but not both, can be specified as fixed effects with the `fixed()`

function in applications where study readers or cases make up the entire group to which inference is intended. When readers are designated as fixed, `mrmc()`

test results additionally include reader-specific pairwise comparisons of the diagnostic tests as well as mean estimates of the performance metric for each reader-test combination.

```
## Fixed readers
est <- mrmc(
empirical_auc(truth, rating), treatment, fixed(reader), case, data = VanDyke
)
```

```
## Multi-Reader Multi-Case Analysis of Variance
## Data: VanDyke
## Factor types: Fixed Readers and Random Cases
## Covariance method: jackknife
##
## Experimental design: factorial
##
## Obuchowski-Rockette variance component and covariance estimates:
##
## Estimate Correlation
## reader 0.0015349993 NA
## treatment:reader 0.0002004025 NA
## Error 0.0008022883 NA
## Cov1 0.0003466137 0.4320314
## Cov2 0.0003440748 0.4288668
## Cov3 0.0002390284 0.2979333
##
##
## ANOVA global test of equal treatment empirical_auc:
## MS(T) Cov1 Cov2 Cov3 Denominator X2 df
## 1 0.004796171 0.0003466137 0.0003440748 0.0002390284 0.0008758604 5.475953 1
## p-value
## 1 0.01927984
##
##
## 95% CIs and tests for treatment empirical_auc pairwise differences:
## Comparison Estimate StdErr CI.Lower CI.Upper z
## 1 1 - 2 -0.04380032 0.01871748 -0.08048591 -0.00711473 -2.340075
## p-value
## 1 0.01927984
##
##
## 95% treatment empirical_auc CIs (each analysis based only on data for the
## specified treatment):
## Estimate Var(Error) Cov2 StdErr CI.Lower CI.Upper
## 1 0.8970370 0.0010141028 0.0004839618 0.02428971 0.8494301 0.9446440
## 2 0.9408374 0.0005904738 0.0002041879 0.01677632 0.9079564 0.9737183
##
##
## Reader-specific 95% CIs and tests for empirical_auc pairwise differences (each
## analysis based only on data for the specified reader):
## Reader Comparison Estimate StdErr CI.Lower CI.Upper z
## 1 1 1 - 2 -0.02818035 0.02551213 -0.078183215 0.021822507 -1.1045864
## 2 2 1 - 2 -0.04653784 0.02630183 -0.098088476 0.005012792 -1.7693768
## 3 3 1 - 2 -0.01787440 0.03120965 -0.079044180 0.043295388 -0.5727202
## 4 4 1 - 2 -0.02624799 0.01729129 -0.060138290 0.007642316 -1.5179891
## 5 5 1 - 2 -0.10016103 0.04405746 -0.186512066 -0.013809995 -2.2734182
## p-value
## 1 0.26933885
## 2 0.07683102
## 3 0.56683414
## 4 0.12901715
## 5 0.02300099
##
##
## Single reader 95% CIs:
## empirical_auc treatment reader StdErr CI.Lower CI.Upper
## 1 0.9196457 1 1 0.0301255164 0.8606008 0.9786907
## 2 0.8587762 1 2 0.0363753335 0.7874818 0.9300705
## 3 0.9038647 1 3 0.0282594118 0.8484773 0.9592522
## 4 0.9731079 1 4 0.0173388332 0.9391244 1.0000000
## 5 0.8297907 1 5 0.0417201720 0.7480206 0.9115607
## 6 0.9478261 2 1 0.0221416887 0.9044292 0.9912230
## 7 0.9053140 2 2 0.0298151099 0.8468775 0.9637506
## 8 0.9217391 2 3 0.0297673065 0.8633963 0.9800820
## 9 0.9993559 2 4 0.0007213348 0.9979421 1.0000000
## 10 0.9299517 2 5 0.0262023046 0.8785961 0.9813073
```

```
## Fixed cases
est <- mrmc(
empirical_auc(truth, rating), treatment, reader, fixed(case), data = VanDyke
)
```

```
## Multi-Reader Multi-Case Analysis of Variance
## Data: VanDyke
## Factor types: Random Readers and Fixed Cases
## Experimental design: factorial
##
## Obuchowski-Rockette variance component and covariance estimates:
##
## Not applicable because cases are fixed
##
##
## ANOVA global test of equal treatment empirical_auc:
## MS(T) MS(T:R) F df1 df2 p-value
## 1 0.004796171 0.0005510306 8.704 1 4 0.04195875
##
##
## 95% CIs and tests for treatment empirical_auc pairwise differences:
## Comparison Estimate df StdErr CI.Lower CI.Upper t
## 1 1 - 2 -0.04380032 4 0.01484629 -0.08502022 -0.00258042 -2.950254
## p-value
## 1 0.04195875
##
##
## 95% treatment empirical_auc CIs (each analysis based only on data for the
## specified treatment):
## Estimate MS(R) StdErr df CI.Lower CI.Upper
## 1 0.8970370 0.003082629 0.02482994 4 0.8280981 0.9659760
## 2 0.9408374 0.001304602 0.01615303 4 0.8959894 0.9856854
```

**MRMCaov** supports factorial, nested, and partially paired study designs. In a factorial design, one set of cases is evaluated by all readers and tests. This is the design employed by the VanDyke study as indicated by its dataset `case`

identifier values which appear within each combination of the `reader`

and `treatment`

identifiers. Designs in which a different set of cases is evaluated by each reader or with each test can be specified with unique codings of case identifiers within the corresponding nesting factor. Example codings for these two nested designs are included in the `VanDyke`

dataset as `case2`

and `case3`

. The `case2`

identifiers differ from reader to reader and thus represent a study design in which cases are nested within readers. Likewise, the `case3`

identifiers differ by test and are an example design of cases nested within tests. Additionally, the package supports partially paired designs in which ratings may not be available on all cases for some readers or tests; e.g., as a result of missing values. Nested and partially paired designs require specification of jackknife (default) or unbiased as the covariance estimation method.

`## Case identifier codings for factorial and nested study designs`

```
## Observation
## Factor 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
## reader 1 1 2 2 3 3 4 4 5 5 1 1 2 2 3 3 4
## treatment 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
## case 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2
## case2 1.1 1.1 2.1 2.1 3.1 3.1 4.1 4.1 5.1 5.1 1.2 1.2 2.2 2.2 3.2 3.2 4.2
## case3 1.1 2.1 1.1 2.1 1.1 2.1 1.1 2.1 1.1 2.1 1.2 2.2 1.2 2.2 1.2 2.2 1.2
## Observation
## Factor 18 19 20 21 22 23 24 25 26 27 28 29 30
## reader 4 5 5 1 1 2 2 3 3 4 4 5 5
## treatment 2 1 2 1 2 1 2 1 2 1 2 1 2
## case 2 2 2 3 3 3 3 3 3 3 3 3 3
## case2 4.2 5.2 5.2 1.3 1.3 2.3 2.3 3.3 3.3 4.3 4.3 5.3 5.3
## case3 2.2 1.2 2.2 1.3 2.3 1.3 2.3 1.3 2.3 1.3 2.3 1.3 2.3
```

`## ... with 1110 more observations`

```
## Cases nested within readers
est <- mrmc(
empirical_auc(truth, rating), treatment, reader, case2, data = VanDyke
)
```

```
## Multi-Reader Multi-Case Analysis of Variance
## Data: VanDyke
## Factor types: Random Readers and Random Cases
## Covariance method: jackknife
##
## Experimental design: cases nested within reader
##
## Obuchowski-Rockette variance component and covariance estimates:
##
## Estimate Correlation
## reader 1.293517e-03 NA
## treatment:reader 9.213005e-05 NA
## Error 8.079682e-04 NA
## Cov1 3.490676e-04 0.4320314
## Cov2 0.000000e+00 0.0000000
## Cov3 0.000000e+00 0.0000000
##
##
## ANOVA global test of equal treatment empirical_auc:
## MS(T) MS(T:R) Cov2 Cov3 Denominator F df1 df2 p-value
## 1 0.004796171 0.0005510306 0 0 0.0005510306 8.704 1 4 0.04195875
##
##
## 95% CIs and tests for treatment empirical_auc pairwise differences:
## Comparison Estimate StdErr df CI.Lower CI.Upper t
## 1 1 - 2 -0.04380032 0.01484629 4 -0.08502022 -0.00258042 -2.950254
## p-value
## 1 0.04195875
##
##
## 95% treatment empirical_auc CIs (each analysis based only on data for the
## specified treatment):
## Estimate MS(R) Cov2 StdErr df CI.Lower CI.Upper
## 1 0.8970370 0.003082629 0 0.02482994 4 0.8280981 0.9659760
## 2 0.9408374 0.001304602 0 0.01615303 4 0.8959894 0.9856854
```

```
## Cases nested within tests
est <- mrmc(
empirical_auc(truth, rating), treatment, reader, case3, data = VanDyke
)
```

```
## Multi-Reader Multi-Case Analysis of Variance
## Data: VanDyke
## Factor types: Random Readers and Random Cases
## Covariance method: jackknife
##
## Experimental design: cases nested within treatment
##
## Obuchowski-Rockette variance component and covariance estimates:
##
## Estimate Correlation
## reader 1.642585e-03 NA
## treatment:reader 9.078969e-05 NA
## Error 8.058382e-04 NA
## Cov1 0.000000e+00 0.0000000
## Cov2 3.455973e-04 0.4288668
## Cov3 0.000000e+00 0.0000000
##
##
## ANOVA global test of equal treatment empirical_auc:
## MS(T) MS(T:R) Cov2 Cov3 Denominator F df1 df2
## 1 0.004796171 0.0005510306 0.0003455973 0 0.002279017 2.104491 1 68.42325
## p-value
## 1 0.1514363
##
##
## 95% CIs and tests for treatment empirical_auc pairwise differences:
## Comparison Estimate StdErr df CI.Lower CI.Upper t
## 1 1 - 2 -0.04380032 0.03019283 68.42325 -0.10404242 0.01644178 -1.450686
## p-value
## 1 0.1514363
##
##
## 95% treatment empirical_auc CIs (each analysis based only on data for the
## specified treatment):
## Estimate MS(R) Cov2 StdErr df CI.Lower CI.Upper
## 1 0.8970370 0.003082629 0.0004861032 0.03320586 12.79430 0.8251827 0.9688914
## 2 0.9408374 0.001304602 0.0002050913 0.02158730 12.75962 0.8941114 0.9875634
```

A single-reader multi-case (SRMC) analysis involves a single readers of multiple cases to compare reader performance metrics across two or more diagnostic tests. An SRMC analysis can be performed with a call to `srmc()`

.

SRMC Function

`srmc(response, test, case, data, cov = jackknife)`

DescriptionReturns an

`srmc`

class object of data that can be used to estimate and compare reader performance metrics in a single-reader multi-case statistical analysis.

Arguments

`response`

: object defining true case statuses, corresponding reader ratings, and a reader performance metric to compute on them.`test`

,`case`

: variables containing the test and case identifiers for the`response`

observations.`data`

: data frame containing the response and identifier variables.`cov`

: function`jackknife`

,`unbiased`

, or`DeLong`

to estimate reader performance metric covariances.

The function is used similar to `mrmc()`

but without the `reader`

argument. Below is an example SRMC analysis performed with one of the readers from the `VanDyke`

dataset.

```
## Subset VanDyke dataset by reader 1
VanDyke1 <- subset(VanDyke, reader == "1")
## Compare ROC AUC treatment means for reader 1
est <- srmc(binormal_auc(truth, rating), treatment, case, data = VanDyke1)
```

```
## Call:
## srmc(response = binormal_auc(truth, rating), test = treatment,
## case = case, data = VanDyke1)
##
## Positive truth status: 1
##
## Response metric data:
##
## # A tibble: 2 × 2
## N data$binormal_auc $treatment $reader
## <dbl> <dbl> <fct> <fct>
## 1 114 0.933 1 1
## 2 114 0.951 2 1
##
## ANOVA Table:
##
## Df Sum Sq Mean Sq
## treatment 1 0.00010393 0.00010393
## reader 0 0.00000000 0.00000000
## treatment:reader 0 0.00000000 0.00000000
##
##
## Obuchowski-Rockette error variance and covariance estimates:
##
## Estimate Correlation
## Error 0.0008371345 NA
## Cov1 0.0004275594 0.5107416
## Cov2 0.0000000000 0.0000000
## Cov3 0.0000000000 0.0000000
```

```
## # A tibble: 2 × 3
## Group$reader $treatment a b
## <fct> <fct> <dbl> <dbl>
## 1 1 1 1.70 0.537
## 2 1 2 1.85 0.503
```

```
## Single-Reader Multi-Case Analysis of Variance
## Data: VanDyke1
## Factor types: Fixed Readers and Random Cases
## Covariance method: jackknife
##
## Experimental design: cases nested within reader
##
## Obuchowski-Rockette variance component and covariance estimates:
##
## Estimate Correlation
## Error 0.0008371345 NA
## Cov1 0.0004275594 0.5107416
## Cov2 0.0000000000 0.0000000
## Cov3 0.0000000000 0.0000000
##
##
## 95% CIs and tests for treatment binormal_auc pairwise differences:
## Comparison Estimate StdErr CI.Lower CI.Upper z p-value
## 1 1 - 2 -0.01765763 0.0286208 -0.07375337 0.03843810 -0.616951 0.537267
##
##
## Single reader 95% CIs:
## binormal_auc treatment reader StdErr CI.Lower CI.Upper
## 1 0.9331609 1 1 0.03348342 0.8675346 0.9987872
## 2 0.9508186 2 1 0.02351871 0.9047228 0.9969144
```

A single-test and single-reader multi-case (STMC) analysis involves a single reader of multiple cases to estimate a reader performance metric for one diagnostic test. An STMC analysis can be performed with a call to `stmc()`

.

STMC Function

`stmc(response, case, data, cov = jackknife)`

DescriptionReturns an

`stmc`

class object of data that can be used to estimate a reader performance metric in a single-test and single-reader multi-case statistical analysis.

Arguments

`response`

: object defining true case statuses, corresponding reader ratings, and a reader performance metric to compute on them.`case`

: variable containing the case identifiers for the`response`

observations.`data`

: data frame containing the response and identifier variables.`cov`

: function`jackknife`

,`unbiased`

, or`DeLong`

to estimate reader performance metric covariances.

The function is used similar to `mrmc()`

but without the `test`

and `reader`

arguments. In the following example, an STMC analysis is performed with one of the tests and readers from the `VanDyke`

dataset.

```
## Subset VanDyke dataset by treatment 1 and reader 1
VanDyke11 <- subset(VanDyke, treatment == "1" & reader == "1")
## Estimate ROC AUC for treatment 1 and reader 1
est <- stmc(binormal_auc(truth, rating), case, data = VanDyke11)
```

```
## # A tibble: 1 × 2
## a b
## <dbl> <dbl>
## 1 1.70 0.537
```

```
## binormal_auc StdErr CI.Lower CI.Upper
## 0.93316094 0.03348342 0.86753465 0.99878723
```

ROC curves can be estimated, summarized, and displayed apart from a multi-case statistical analysis with the `roc_curves()`

function. Supported estimation methods include the empirical distribution (default), binormal model, and binormal likelihood-ratio model.

ROC Curves Function

`roc_curves(truth, rating, groups = list(), method = "empirical")`

DescriptionReturns an

`roc_curves`

class object of estimated ROC curves.

Arguments

`truth`

: vector of true binary case statuses, with positive status taken to be the highest level.`rating`

: numeric vector of case ratings.`groups`

: list or data frame of grouping variables of the same lengths as`truth`

and`rating`

.`method`

: character string indicating the curve type as`"binormal"`

,`"binormalLR"`

,`"empirical"`

, or`"trapezoidal"`

.

A single curve can be estimated over all observations or multiple curves estimated within the levels of one or more grouping variables. Examples of both are given in the following sections using variables from the `VanDyke`

dataset referenced inside of calls to the `with()`

function. Alternatively, the variables may be referenced with the `$`

operator; e.g., `VanDyke$truth`

and `VanDyke$rating`

. Resulting curves from `roc_curves()`

can be displayed with the `print()`

and `plot()`

functions.

Multiple group-specific curves can be obtained from `roc_curves()`

by supplying a list or data frame of grouping variables to the `groups`

argument. Groups will be formed and displayed in the order in which grouping variables are supplied. For instance, a second grouping variable will be plotted within the first one.

Estimated parameters for curves obtained with the binormal or binormal likelihood-ratio models can be extracted as a data frame with the `parameters()`

function.

```
## Binormal curves
curves_binorm <- with(VanDyke, {
roc_curves(truth, rating,
groups = list(Treatment = treatment, Reader = reader),
method = "binormal")
})
```

```
## Warning in binormal_params(data$truth, data$rating, ...):
## Binormal curve fit has AUC = 1 due to lack of interior points.
## Consider fitting an empirical curve instead.
```

```
## # A tibble: 10 × 3
## Group$Treatment $Reader a b
## <fct> <fct> <dbl> <dbl>
## 1 1 1 1.70 0.537
## 2 2 1 1.85 0.503
## 3 1 2 1.40 0.561
## 4 2 2 1.66 0.447
## 5 1 3 1.74 0.635
## 6 2 3 1.62 0.488
## 7 1 4 1.93 0.202
## 8 2 4 Inf 1
## 9 1 5 1.06 0.464
## 10 2 5 1.73 0.422
```

Estimates for different parameterizations of the binormal likelihood-ratio model are additionally returned and include those of the binormal model and the simplification of Pan and Metz (1997; Metz and Pan 1999) as well as those of the bi-chi-squared model (Hillis 2017).

```
## Binormal likelihood-ratio curves
curves_binormLR <- with(VanDyke, {
roc_curves(truth, rating,
groups = list(Treatment = treatment, Reader = reader),
method = "binormalLR")
})
```

```
## Warning in binormalLR_params(data$truth, data$rating, ...):
## Likelihood ratio binormal curve fit has AUC = 1 due to lack of interior points.
## Consider fitting an empirical curve instead.
```

```
## # A tibble: 10 × 4
## Group$Treatment $Reader Metz$d_a $c bichisquared…¹ $theta binorma…²
## <fct> <fct> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 1 2.13 -0.298 3.42 1.71e+ 0 1.71e+0
## 2 2 1 2.35 -0.321 3.79 1.70e+ 0 1.87e+0
## 3 1 2 1.73 -0.281 3.17 1.32e+ 0 1.40e+0
## 4 2 2 0.00700 -0.791 73.3 3.48e- 7 4.99e-3
## 5 1 3 0.0140 -0.746 47.0 2.22e- 6 9.99e-3
## 6 2 3 2.08 -0.330 3.94 1.23e+ 0 1.65e+0
## 7 1 4 0.000417 -0.932 797. 1.09e-10 2.95e-4
## 8 2 4 Inf 0 1 Inf Inf
## 9 1 5 0.895 -0.508 9.37 5.94e- 2 6.66e-1
## 10 2 5 2.02 -0.553 12.1 2.18e- 1 1.49e+0
## # … with 1 more variable: binormal$b <dbl>, and abbreviated variable names
## # ¹bichisquared$lambda, ²binormal$a
```

Points on an ROC curve estimated with `roc_curves()`

can be extracted with the `points()`

function. True positive rates (TPRs) and false positive rates (FPRs) on the estimated curve are returned for a given set of sensitivity or specificity values or, in the case of empirical curves, the original points. ROC curve points can be displayed with `print()`

and `plot()`

.

ROC Points Function

`## Method for class 'roc_curves'`

`points(x, metric = "specificity", values = seq(0, 1, length = 101), ...)`

`## Method for class 'empirical_curves'`

`points(x, metric = "specificity", values = NULL, which = "curve", ...)`

DescriptionReturns an

`roc_points`

class object that is a data frame of false positive and true positive rates from an estimated ROC curve.

Arguments

`x`

: object from`roc_curves()`

for which to compute points on the curves.`metric`

: character string specifying`"specificity"`

or`"sensitivity"`

as the reader performance metric to which`values`

correspond.`values`

: numeric vector of values at which to compute ROC curve points, or`NULL`

for default empirical values as determined by`which`

.`which`

: character string indicating whether to use curve-specific observed values and 0 and 1 (`"curve"`

), the combination of these values over all curves (`"curves"`

), or only the observed curve-specific values (`"observed"`

).

```
## Extract points at given specificities
curve_spec_pts <- points(curves, metric = "spec", values = c(0.5, 0.7, 0.9))
print(curve_spec_pts)
```

```
## # A tibble: 30 × 3
## Group$Treatment $Reader FPR TPR
## * <fct> <fct> <dbl> <dbl>
## 1 1 1 0.1 0.862
## 2 1 1 0.3 0.908
## 3 1 1 0.5 0.935
## 4 2 1 0.1 0.843
## 5 2 1 0.3 0.966
## 6 2 1 0.5 0.978
## 7 1 2 0.1 0.747
## 8 1 2 0.3 0.821
## 9 1 2 0.5 0.872
## 10 2 2 0.1 0.821
## # … with 20 more rows
```

```
## Extract points at given sensitivities
curve_sens_pts <- points(curves, metric = "sens", values = c(0.5, 0.7, 0.9))
print(curve_sens_pts)
```

```
## # A tibble: 30 × 3
## Group$Treatment $Reader FPR TPR
## * <fct> <fct> <dbl> <dbl>
## 1 1 1 0.0116 0.5
## 2 1 1 0.0246 0.7
## 3 1 1 0.254 0.9
## 4 2 1 0 0.5
## 5 2 1 0 0.7
## 6 2 1 0.337 0.9
## 7 1 2 0.0130 0.5
## 8 1 2 0.0543 0.7
## 9 1 2 0.609 0.9
## 10 2 2 0 0.5
## # … with 20 more rows
```

A mean ROC curve from multiple group-specific curves returned by `roc_curves()`

can be computed with the `means()`

function. Curves can be averaged over sensitivities, specificities, or binormal parameters (Chen and Samuelson 2014). Averaged curves can be displayed with `print()`

and `plot()`

.

ROC Means Function

`## Method for class 'roc_curves'`

`mean(x, ...)`

`## Method for class 'binormal_curves'`

`mean(x, method = "points", ...)`

DescriptionReturns an

`roc_points`

class object.

Arguments

`x`

: object from`roc_curves()`

for which to average over the curves.`method`

: character string indicating whether to average binormal curves over`"points"`

or`"parameters"`

.`...`

: optional arguments passed to`points()`

, including at which`metric`

(`"sensitivity"`

or`"specificity"`

) values to average points on the ROC curves.

```
## Average sensitivities at given specificities (default)
curves_mean <- mean(curves)
print(curves_mean)
```

```
## # A tibble: 20 × 2
## FPR TPR
## * <dbl> <dbl>
## 1 0 0
## 2 0 0.402
## 3 0.0145 0.686
## 4 0.0290 0.762
## 5 0.0435 0.790
## 6 0.0580 0.802
## 7 0.0725 0.813
## 8 0.101 0.835
## 9 0.116 0.844
## 10 0.130 0.852
## 11 0.159 0.862
## 12 0.188 0.872
## 13 0.319 0.904
## 14 0.362 0.912
## 15 0.435 0.925
## 16 0.638 0.956
## 17 0.667 0.961
## 18 0.696 0.966
## 19 0.913 0.992
## 20 1 1
```

```
## Average specificities at given sensitivities
curves_mean <- mean(curves, metric = "sens")
print(curves_mean)
```

```
## # A tibble: 23 × 2
## FPR TPR
## * <dbl> <dbl>
## 1 0 0
## 2 0.00698 0.511
## 3 0.00804 0.556
## 4 0.00899 0.578
## 5 0.00995 0.6
## 6 0.0109 0.622
## 7 0.0214 0.667
## 8 0.0280 0.689
## 9 0.0358 0.711
## 10 0.0450 0.733
## # … with 13 more rows
```

The reader performance metrics described previously for use with `mrmc()`

and related functions to analyze multi and single-reader multi-case studies can be applied to truth and rating vectors as stand-alone functions. This enables estimation of performance metrics for other applications, such as predictive modeling, that may be of interest.

AUC, partial AUC, sensitivity, and specificity are estimated below with an empirical ROC curve. Estimates with binormal and binormal likelihood-ratio curves can be obtained by replacing `empirical`

in the function names with `binormal`

and `binormalLR`

, respectively.

`## [1] 0.9229791`

```
## Partial area for specificity from 0.7 to 1.0
empirical_auc(VanDyke$truth, VanDyke$rating, partial = "spec", min = 0.70, max = 1.0)
```

`## [1] 0.2499923`

```
## Partial area for sensitivity from 0.7 to 1.0
empirical_auc(VanDyke$truth, VanDyke$rating, partial = "sens", min = 0.70, max = 1.0)
```

`## [1] 0.2262129`

`## [1] 0.8812346`

`## [1] 0.94434`

Sensitivity and specificity for binary ratings are available with the `binary_sens()`

and `binary_spec()`

functions as demonstrated in the next example based on a binary rating created from the numeric one in the `VanDyke`

dataset.

```
## Create binary classification
VanDyke$binary_rating <- VanDyke$rating >= 3
## Sensitivity
binary_sens(VanDyke$truth, VanDyke$binary_rating)
```

`## [1] 0.8711111`

`## [1] 0.8478261`

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