### Using the Lorenz Curve to Characterize Risk Predictiveness and Etiologic Heterogeneity

FREE APP - get all of the videos on this channel on your phone. Put socialgamenews.info Economics%20Diagrams in your phone web browser and follow. Section 4 reviews the measurement and calculation of the. Gini coefficient. calculated from a Lorenz curve to indicate the level of inequality in a The Gini coefficient is a ratio between 0 and 1, where 0 implies that each. The Gini coefficient is often used to measure income inequality. Here, 0 The Gini coefficient is defined as a ratio of the areas on the Lorenz curve diagram.

Both the Lorenz curve and Gini coefficient have been primarily utilized in the economic and social sciences over the last century. In recent years, however, these methods have also seen applications in other areas such as medical and health services research. For example, the Lorenz curve has been used to describe patterns of drug use.

They infer that there are not many heavy users of insulin, but, in contrast, there is a group of heavy users of opioid analgesics. The Lorenz curve and Gini coefficient have also been used to explore the distribution of health professionals in relation to the population distribution of patients.

### Measures of Distributional Inequality

Chang and Halfon [ 4 ] examined the pediatrician-to-child ratios in the 50 states and showed a fourfold difference between the states with the highest Maryland and lowest Idaho ratios. The authors used Lorenz curve analyses to show that the concentration was greater among pediatricians than among all physicians, and that during the period —, despite a 46 per cent increase in the number of pediatricians nationwide, there was essentially no change in the national distribution, as evidenced by a minimal change in the Gini coefficient.

Similar kinds of studies have been conducted by Brown [ 5 ] in Alberta who examined and compared the distributions of various kinds of health practitioners, and by Kobayashi and Takaki [ 6 ] in Japan who examined the distribution of general physicians across municipal entities. As is described in further detail below, the estimation of both the Lorenz curve and the Gini coefficient involves ranking the units of observation on the basis of some quantity of interest and then estimating cumulative proportions.

## Lorenz curve

When there is error or variation in the measurement of the quantity of interest, an analysis that does not account for this error or variation may incorrectly rank the units and result in upwardly biased estimates for the concentration in the Lorenz curve and Gini coefficient. This situation can arise in many different circumstances. For instance, the Lorenz curve has traditionally been used to study the distribution of income in a population.

- Associated Data
- 1. INTRODUCTION
- Lorenz curve and Gini coefficient

In this case, depending upon the study design, there may be variation in the measurement of income from a number of possible sources including error in the reported income i. In this article we are particularly interested in the bias that may occur in a specific type of data configuration that can occur frequently in practice.

This configuration is nested in that members of the population, or experimental units, are the primary units of analysis, but within each experimental unit there are multiple observations that are aggregated to form the outcome whose distribution is of interest.

For example, using Lorenz curves and Gini coefficients Prakasam and Murthy [ 7 ] look at couples within states in India as the unit of analysis to explore the acceptance of family planning methods for different levels of literacy. Within this framework, our motivation for this work arises from a study of whether black patient visits to physicians are concentrated among a select group of physicians.

Described in more detail below, this analysis involves ranking physicians the primary units of analysis on the basis of the proportion of patient visits to each physician that were made by black patients that is, the patient visits constitute the multiple observations to be aggregated within physician. In this study, one observes the number of black patient visits in a sample of patient visits for each physician.

To understand the problem, consider the fact that the goal of this research is to evaluate the extent to which care of black patients is concentrated within the population of physicians.

Consider, for example, the extreme hypothesis that patients receive care from physicians randomly, i. In other words, the degree of concentration increases as the error variance increases.

To our knowledge, this problem has been explicitly recognized by a limited number of investigators. Lee proposed the generation of bootstrap samples to reorder the experimental components while using the original data to estimate the Gini coefficient.

Pham-Gia and Turkkan [ 12 ] proposed a parametric approach in the context of a study of inequality in income distributions when the incomes are measured with error, and in principle this has the potential to resolve the bias issue. They derive theoretical results for obtaining the distribution of the true income and the corresponding Lorenz curve assuming that the observed income and error are independent and follow either beta or gamma distributions.

We demonstrate that, although the Lorenz curve represents the distribution of predicted risks in a population at risk for the disease, in fact it can be estimated from a case—control study conducted in the population without the need for information on absolute risks. We explore two different estimation strategies and compare their statistical properties using simulations. The Lorenz curve is a statistical tool that deserves wider use in public health research.

The measure that is most widely used, the area under the receiver operating curve, was initially developed for diagnostic tests and is suitable for predicting a binary event.

It is based on the specificity and sensitivity of the risk score, and it measures the probability that a randomly selected diseased subject has a higher predicted risk than a randomly selected non-diseased subject. The net reclassification index evaluates the extent to which patients are reclassified when using a new predictive rule compared to an old one, conditional on disease status. Another popular type of measure to assess prediction accuracy is the error of prediction, or Brier score, which quantifies the distance between the prediction and the actual outcome.

If so, the prediction rule is considered to be well calibrated. However, we feel that the ideal graphical tool for representing this distribution is an old fashioned tool in statistics, the Lorenz curve. We believe that it is especially useful in the context of disease prevention because it maps out what public health policy investigators need to know.

That is, it tells us how much disease burden will occur in any given proportion of the population with risks above a chosen threshold. Very few investigators have made use of the Lorenz curve to illustrate public health concepts, though there have been some notable exceptions.

For example, Green et al. The Lorenz curve was initially developed by economists to characterize the distribution of the wealth or income among individuals, and it is frequently used to compare social inequalities between countries.