# Partial area under ROC curve

Command: | Statistics ROC curves Partial area under ROC curve |

## Introduction

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

### The partial area under the ROC curve for a prespecified specificity interval

The partial area **pAUC** for a prespecified specificity 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 prespecified specificity interval (Zhou et al., 2002).

### The partial area under the ROC curve for a prespecified sensitivity interval

For calculating the partial area under the ROC curve for a prespecified 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 pAUC_{x} (Walter, 2005).

Therefore, analogous to the definitions for the pAUC for a specificity interval, the partial area **pAUC** for a prespecified 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 prespecified sensitivity interval (Zhou et al., 2002).

### Methodology

MedCalc calculates the pAUC non-parametricly 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):

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 sensitivity 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 specificity 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 prespecified sensitivity interval are analogous to the results for a specificity 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.