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Measures of Association: Lambda and Gamma

We are skipping backward in the Adventures in Criminal Justice Research textbook to Chapter 8. Lambda and Gamma are what is referred to as Proportional Reduction in Error (PRE) measures of association. I like to think of them as statistics that tell me how much smarter I am because I know a little bit more information. Let’s say I was with a large group of people and attempting to guess everyone’s position on REQUIRING GUN PERMITS and they were all disguised (maybe dressed in drag) so that I could not tell who is female and who is male. I could guess their position on the death penalty, but I would only be able to use information I obtain from a simple frequency distribution that tells me that 83.0% favor REQUIRING GUN PERMITS and 17.0% oppose REQUIRING GUN PERMITS. So I would go along and guess that approximately 4/5ths of the group favored REQUIRING GUN PERMITS and approximately 1/5th of the group opposed REQUIRING GUN PERMITS. I’d be right some and wrong a lot. Now, let’s say I was able to determine who is male and who is female and then I guess who favors and opposes the death penalty. The accuracy of my guessing would be improved. Lambda and Gamma will tell me exactly how much more accurate I am by knowing that additional piece of information, SEX. The trick is to know when to use Lambda and when to use Gamma. Here is the rule. If both variables are at the nominal level of measurement, then use Lambda, otherwise use Gamma. Let’s use SPSS to calculate our PRE for the crosstab of SEX and FAVOR or OPPOSE REQUIRING GUN PERMITS. First, what level of measurement are the two variables? Sex is obviously a nominal variable as each person is either male or female and being in one category or another implies nothing about possession of more or less of the variable SEX. REQUIRING GUN PERMITS might be considered by some to be ordinal, but for the moment let’s consider it to be a nominal variable. Now, Click ANALYZE-DESCRIPTIVES-CROSSTABS-STATISTICS and check the box for Lambda. Now click CONTINUE and OK. SPSS will produce the following output:

We are interested only in the information in the Lambda area and only for the Symmetric information. We see a value of 0.101 for Lambda. The next piece of information of value for us is that in the column for Approximate Significance, 0.000. The decision making rule for Lambda is the same as for Chi-Square. If the Significance level is ≤ 0.05 then we can interpret the actual calculated value. If the Significance level is > 0.05, we cannot interpret the value of Lambda. In this situation we can interpret Lambda as the significance level tells us that in less if we take 1,000 samples all of the samples will produce a positive value, other than zero, for Lambda. This means that I will make 10.1% fewer errors guessing if an individual FAVORS or OPPOSES REQUIRING GUN PERMITS if I know the individual’s biological sex. My Proportional Reduction in Error (PRE) is 10.1%.

Now let’s assume that REQUIRING GUN PERMITS is an ordinal variable. That means we calculate Gamma. We do this the same way as we did for Lambda only we check the box next to Gamma. When we do this we see the following output:

How do we interpret this output for Gamma? First, we see the calculated value for Gamma of negative 0.529. The next number of importance to us is the number in the column under Approximate Significance. The decision making rule for Gamma is the same as for Lambda and Chi-Square. If the Significance level is ≤ 0.05 then we can interpret the actual calculated value. If the Significance level is > 0.05, we cannot interpret the value of Gamma. In this instance we are able to interpret the value of Gamma, 0.529. What does this number tell us?

Di re ctio na l Mea s ure s

.1 0 1 .0 2 0 4.7 3 9 .0 0 0 .0 0 0 .0 0 0 .

c .1 3 9 .0 2 8 4.7 3 9 .0 0 0 .0 4 6 .0 1 3 .0 0 0

d .0 4 6 .0 1 3 .0 0 0

Symmet ric FAVOR OR OPPOSE GUN PERMITS Depen d ent RESPO NDENTS SEX Depen d ent FAVOR OR OPPOSE GUN PERMITS Depen d ent RESPO NDENTS SEX Depen d ent

Lam bd a Goo d man an d Kru s kal tau

No min al by No minal

Va lu e

As ym p. Std . Error

a App ro x. T

b Ap pro x. Sig .

No t ass um in g t he n ull h yp ot hes is .

a. Us in g th e as ympt ot ic st and ard err or as s um in g t he nu ll h yp o th es is .

b . Cann ot b e co m pu ted be cau se th e asym pt otic s tan dard erro r equ als z ero.

c. Base d o n chi- s qu are ap pr ox ima tio n

d .

Sym m etric Me as ures - .5 29 .0 6 6 - 6.5 52 .0 0 0

9 66

Gam ma

Ord in al b y Ord in al N o f Valid Cas es

Valu e

As ym p. Std . Error

a App ro x. T

b Ap pro x. Sig .

No t ass umin g t he n ull hyp ot hes is .

a. Us in g th e as ympt ot ic st and ard erro r as s umin g th e nu ll hyp ot hes is .

b .

We will make 52.9% fewer errors attempting to guess the position on REQUIRING GUN PERMITS of a group of people if we first determine their sex, than if we do not know their sex. To see this demonstrated, let’s examine the crosstabulation of these two variables in the table below.

We can see that a larger percentage of females favor REQUIRING GUN PERMITS than do males, Epsilon = 16.2% (90.3% – 74.1% = 16.2%), and that a much lower percentage of males oppose REQUIRING GUN PERMITS than do females, Epsilon = 36.6% (25.9% - 9.7% = 35.6%). Therefore, if I first ascertain if my subject is male or female I will make 52.9% fewer errors guessing if the individual FAVORS or OPPOSES REQUIRING GUN PERMITS. My

Proportional Reduction in Error is 52.9%