Concept: Lorenz Curve

Introduction

The Lorenz Curve is a graphical display of the distribution of the cumulative percent of events by the cumulative percent of people in the population.

Application of the Lorenz Curve

In the Health Inequities deliverable (Martens et al., 2010), the Lorenz curve is a graphical display of the distribution of the cumulative percent of events by the cumulative percent of people in the five neighbourhood income quintiles in the population, by increasing income.

The horizontal axis (x-axis) of the curve displays the cumulative percent of people in the population (by increasing neighbourhood income quintile group) and the vertical axis (y-axis) displays the cumulative percent of events in the population. The Lorenz curve can be expressed as what percentage of the population represented by the neighbourhood income quintile holds what percentage of the events in the population.

Each neighbourhood income quintile represents approximately 20% of the Manitoba population, divided into rural or urban (Winnipeg and Brandon). In a perfectly equitable situation, one would expect that 20% of events (i.e., premature deaths, teenage pregnancies, etc.) would occur in each income quintile group: U1 would contribute 20% of all events in the population; U2 would contribute another 20% of all events in the population and so forth. As a reference, a line of equality is also displayed on the graph to indicate this perfectly equitable situation; however, most cases present some inequality between the percentage of events and the income quintiles of the population. A Lorenz curve is generated when at least one of the income quintiles that captures N% of the population does not contribute the same N% on the Y axis. If a larger proportion of events occur in lower neighbourhood income quintile groups, the Lorenz curve will bend above the line of equality; if a larger proportion of events occur in higher neighbourhood income quintile groups, the Lorenz curve will bend below the line of equality (Lorenz, 1905).

The total area lying in-between the line of equality and the Lorenz curve is known as the Gini coefficient; larger areas represent larger disparities between neighbourhood income groups and smaller areas represent smaller disparities between neighbourhood income groups. Please see the glossary term Gini coefficient for more information.

Overview: Approach to Generating the Lorenz Curve

This is the approach taken to generate the Lorenz Curve.

1. First you have to fit a model for the outcome of interest (say premature mortality). This model will be adjusted by sex and age - using a Poisson or Negative Binomial distribution.
   
2. From this adjusted model, you will get an adjusted rate.
   
3. Now for the Lorenz Cure:
   
   1. The Lorenz curve has an x-axis (cumulative proportion of the population - reported as the crude cumulative percent of the denominator) and a y-axis (cumulative proportion of the event - reported as the adjusted cumulative percent of the numerator).
      
      Using the adjusted rate from the model, calculate the adjusted numerator:

      adjusted numerator = adjusted rate * denominator

   2. Determine the proportion each income quintile contributes to the x and y -axis. (i.e.: U1 percent denominator = 514408/2639878 = 0.19, U1 adjusted numerator = 2730.21/8170.65 = 0.33)
      
   3. Add up the cumulative percents (each of the sums for the denominator and the adjusted numerator should = 1).
      
      Each percentage value determines a point on the Lorenz Curve. (Please see the table below for an example of how to calculate the points for the Lorenz Curve).
   
   
4. You can use the values in the Lorenz Cure to calculate its GINI Coefficient.

Example of a Lorenz Curve

An example of the Lorenz Curve from the Health Inequities deliverable (Martens et al., 2010) is presented in the following link: Example of the Lorenz Curve .