Title: Lambda
Course: UGQuants (SSPS) > Section: Association for Nominal and Ordinal Variables

Lambda

Lambda is a PRE measure we may use when both our Independent Variable and our Dependent Variable are nominal.

The formula for lambda is:

Where:

- F_IV is the sum of the largest cell frequencies in each category of the independent variable
- M_DV is the largest marginal total among all categories of the dependent variable
- N is the total number of cases

An example helps to illustrate how to calculate Lambda.

Let us imagine a situation where we know the partisan identification of respondents (Conservative, Labour and LibDem) and their preferred electoral system (first past the post, national list PR, single transferable vote).

 | Lib Dem | Labour | Conservative | Total
STV | 35 | 20 | 10 | 65
PR | 15 | 35 | 30 | 80
FPTP | 5 | 20 | 50 | 75
Total | 55 | 75 | 90 | 220

To calculate lambda we must:

1. Identify N. In this case this is the total number of respondents, or 220
2. Identify F_IV

To do this first we must identify our independent variable (IV). If we are looking at the impact of partisan identification on whether a respondent supports a a particular electoral system, we can probably assume that party identification is the independent variable (i.e. we might expect that the political party one supports helps shape one's view of the best kind of electoral system).

Next we must identify the categories of the independent variable. These are Conservative, Labour and Liberal Democrat. For each of these categories we have to identify the largest cell frequency and then add them up. So, within the Lib Dem category ,the largest cell frequency is 35 (STV). Within the Labour category the largest cell frequency is also 35 (PR) and within the Conservative category the largest cell frequency is 50 (FPTP). If we add this up we get 35+35+50, so F_IV = 120

1. Identify M_DV

Given we have chosen party identification as the IV then our Dependent Variable is the preferred electoral system.  We need to find the largest row total of any category in the DV. The largest total is 80 (PR), so M_DV = 80.

1. Next we must insert these figures into the formula:

Evaluate the results: we would reduce the error of prediction by 28.5% if we had knowledge of the independent variable.

Let us say for the sake of interest that we want to know whether we’d make better predictions if we treated views of electoral systems as the IV and partisan identification as the DV. One might make the case, for example, that how a voter feels about the electoral systems helps them to determine which party they will support.  If we recalculate Lambda using these figures, our F_IV would be 120 but the largest total of the DV categories would be 90. Plugging these figures into the formula we would get:

(120-90) / 220-90 = 30/130 = .230

In other words we’d reduce the errror of our predictions by 23% if we treated views of electoral systems as our independent variable.  We’re therefore better off using party identification as a predictor of support for electoral reform rather than using electoral reform as a predictor of party support.