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# Partial area under ROC curve

 Command: StatisticsROC curvesPartial area under ROC curve

## Introduction

The partial area under the ROC curve summarizes a portion of the ROC curve over a pre-specified interval of interest. This interval can be a specificity or sensitivity interval.

### The partial area under the ROC curve for a pre-specified specifity interval

The partial area pAUC for a pre-specified specifity interval is given by the area ABCD. The partial area corresponding to a random model is the area ABgh. The partial area corresponding to a perfect model is the area ABef.

The random model area ABgh is the minimum pAUC ($pAUC_{min}$), and the perfect model area ABef is the maximum pAUC ($pAUC_{max}$).

The partial area pAUC can be interpreted as the average sensitivity in the pre-specified specificity interval (Zhou et al., 2002).

### The partial area under the ROC curve for a pre-specified sensitivity interval

For calculating the partial area under the ROC curve for a pre-specified sensitivity interval, MedCalc takes the right border (Specificity = 0%; or False Positive Rate FPR = 1 ) as the baseline. The pAUC calculated this way is sometimes called the horizontal partial area under the curve pAUCx (Walter, 2005).

Therefore, analogous to the definitions for the pAUC for a specificity interval, the partial area pAUC for a pre-specified sensitivity interval is given by the area ABCD. The partial area corresponding to a random model is the area ABgh. The partial area corresponding to a perfect model is the area ABef.

Again, the random model area ABgh is the minimum pAUC ($pAUC_{min}$), and the perfect model area ABef is the maximum pAUC ($pAUC_{max}$).

The partial area pAUC can be interpreted as the average specificity in the pre-specified sensitivity interval (Zhou et al., 2002).

### Methodology

MedCalc calculates the pAUC non-parametrically using the trapezoidal method.

Confidence intervals are calculated using bootstrapping (Efron, 1987; Efron & Tibshirani, 1993).

### Standardized pAUC

The Standardized pAUC (pAUCs) is defined as (McCLish, 1989):

$$pAUCs = \frac {1} {2} \left( 1 + \frac {pAUC - pAUC_{min}} { pAUC_{max} - pAUC_{min} } \right)$$

The Standardized pAUC (pAUCs) has a maximum value of 1 and a minimum value 0.5. Therefore it allows to view the partial area on the same scale as the total area under the ROC curve.

## Required input

• Variable: select the variable of interest.
• Classification variable: select a dichotomous variable indicating diagnosis (0=negative, 1=positive).

If your data are coded differently, you can use the Define status tool to recode your data.

It is important to correctly identify the positive cases.

• Filter: (optionally) a filter in order to include only a selected subgroup of cases (e.g. AGE>21, SEX="Male").
• Options:
• Sensitivity or Specificity interval: select whether you want the partial area under the ROC curve for a defined sensitivy or specificity interval.
• From and To: the lower and upper value of the interval of interest. Common values are from 80%, 90%, 95% to 100%.
• Bootstrap Confidence Interval: select this option to calculate a confidence interval for the partial area under the ROC curve using the bootstrap technique (Efron, 1987; Efron & Tibshirani, 1993).
• Advanced: click this button to specify the bootstrap parameters: number of replications and random number seed.
• Select Display ROC curve window to obtain the graph in a separate window.

## Results - Specificity interval

The following report is displayed:

First MedCalc shows the parameters of the analysis.

Next, the following are reported:

• Area under curve: this is the total area under the ROC curve.
• The Partial Area (pAUC): the area under the ROC curve in the specified specifity interval (see image below).
• The 95% Bootstrap CI of pAUC is reported if the corresponding option has been selected.
• The Standardized Partial Area (pAUCs) as defined by McCLish (1989), see above.
• The 95% Bootstrap CI of pAUCs is reported if the corresponding option has been selected. If this confidence interval does not include 0.5, then it can be concluded that the pAUCs is significantly different from the random model.

## Results - Sensitivity interval

The results for the partial area under the ROC curve for a pre-specified sensitivity interval are anologous to the results for a specifiity interval.

## Literature

• Efron B (1987) Better Bootstrap Confidence Intervals. Journal of the American Statistical Association 82:171-185.
• Efron B, Tibshirani RJ (1993) An introduction to the Bootstrap. Chapman & Hall/CRC.
• McClish DK (1989) Analyzing a Portion of the ROC Curve. Medical Decision Making 9:190-195.
• Walter SD (2005) The partial area under the summary ROC curve. Statistics in Medicine 24:2025–40.
• Zhou XH, Obuchowski NA, McClish DK (2002) Statistical methods in diagnostic medicine. Wiley-Interscience.

## Statistical Methods in Diagnostic MedicineXiao-Hua Zhou, Nancy A. Obuchowski, Donna K. McClish

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Statistical Methods in Diagnostic Medicine provides a comprehensive approach to the topic, guiding readers through the necessary practices for understanding these studies and generalizing the results to patient populations.

Following a basic introduction to measuring test accuracy and study design, the authors successfully define various measures of diagnostic accuracy, describe strategies for designing diagnostic accuracy studies, and present key statistical methods for estimating and comparing test accuracy.