October 27, 2009
Readings:
Chapter 12. Bivariate Data Analysis (Johnson and Reynolds) (pp 431462)
Chapter 6 Tests of Significance and Measures of Association (Pollock)
(144151)
Chapter 10 Doing Your Own Political Analysis (Pollock Workbook)
Bring your CD's to Class from the Pollock workbook
Measures of Association with Nominal Variables
ChiSquare enables you to estimate whether a relationship exists, but how do you
know how strongly the variables are related? This is done through a battery of statistical tests that measure both the
magnitude of the relationship (how strong is it) and the significance of the
relationship/measure. Level of
Measurement Measure of
Association Range Characteristics Nominal Variables Lambda 0  1.0 may underestimate, but a
PRE measure Phi 0 1.0 Use for a 2x2 table only and is
chisquare based Cramer's V 0 1.0 chisquare based and the
compliment to PHI.
Measures of association summarize efficiently the
existence, and strength of the relationship.
LAMBDA this measures the strength of the relationship between two nominal (or a nominal/ordinal) variables.
A lambda value measures how much better one can predict the value of each case on the dependent variable if one knows the value of the independent variable.
I can convert my Lambda score into a percent and say that by knowing the independent variable I can explain that percent of the dependent variable.
Lambda ranges from 0 (no relation) to 1 (a perfect relationship)
Relationship strengths With Lambda
.000 to .10 none
.10.20 very weak
.20.30 moderate
.30.40 moderatestrong
.40 and above there is a strong relationship
Problems With Lambda
Lambda can Underestimate relationships, even when there are
significant chisquare values. This is especially true when you have many values
clustered around 1 response. When this is the case, use ......
Cramers V similar to lambda, except that one cannot use it as a PRE measure. It only describes the strength of the relationship. You should try to use lambda, unless you believe one of your variables is skewed. As with Lambda we go from 0 to 1.
.00 no effect
.10 small effect
.30 medium effect
.50 large effect
Phi only if you have a 2x2 table.
Here we can say with a .808
Cramer's V, that we have a very strong relationship between our independent and
dependent variables.
How to do it in
PASW
Analyze
Descriptive
CrossTabs
Click on the Statistics Tab
Select the Proper Statistic and continue with the crosstabs!
Here is a case where Lambda may be Underestimating
COMPARING TWO ORDINAL VARIABLES
If you are using nominal level data, you must use nominal level measures of association. The exception to this is if you are using dichotomous variables, you can treat these as ordinal!
MANY OF THE RELATIONSHIPS YOU WILL BE TESTING INVOLVE ORDINAL VARIABLES. Tests with Ordinal variables allow:
strength
significance
direction This allows us to determine if we have a positive or negative relationship between our variables.
When we have two ordinal variables the measures that we usually use are Kendall's Tau, Goodman and Kruskal's Gamma and Somer's d..
all three are PRE measures.
each has a possible range of 1 to 1. As we approach 1 or 1 the stronger our relationship. A value of 0 (zero), means no relationship.
A positive number indicates a positive relationship, a negative value means a negative relationship.
In variables that have ordinal variables, we can determine if we have a positive or negative relationship between our variables. This gives us greater statistical power than simply saying that there is a relationship.
Level of Measurement 
Measure of Association 
Range 
Characteristics 
Two Ordinal Variables 
Gamma 
1.0 to 1.0 
Tends to be generous 
Kendall's Tau B 
1.0 to 1.0 
For square tables 

Kendall's Tau C 
1.0 to 1.0 
For rectangular 

Somer's D  1.0 to 1.0  If you know the D.V. 
When to choose? Each one provides different estimates of the value. .
When to use Tau b or Tau c
use Tau b if the table is square the same number of rows as columns
use Tau c if the table is not square.
Guidelines
Under .1 very weak
.10 and .19 weak
.20 and .29 moderate
.30 or above strong
When to use Gamma
if you believe the Taus are underestimating
if you want to show off. The beer muscles of ordinal measures
under .2 pretty weak
.3.5 moderate
above .5 strong
When to use Somer's d
when you want to test a specific dependent variable
use the same guidelines of strength as Tau
The important thing is that you choose one not all. Use either all Tau's, Gamma's or dscores!
A POSITIVE RELATIONSHIP
changes in the independent variable are associated with changes in the same direction for the dependent variable. As the numbers in the categories of the i.v. increase, the numbers in the category of the d.v increases. The table below demonstrates this
As people believe abortion should be less restrictive, 1===>4. They are more likely to vote for Gore (0 to 1).
A NEGATIVE RELATIONSHIP
This exists when changes in the independent variable are associated with changes in the reverse direction in the dependent variable. As the numbers increase in the i.v., the numbers in the category of the dependent variable decrease!
As the independent variable increases, the dependent variable decreases.
As people disapprove of Bush More (14) , they are less likely to support Rick Perry (1 to 0)
The Tests of Significance
How the output looks
IGNORE THE ASYMP STD. ERROR AND THE APPROX T.!!!
WHAT THIS MEANS
The negative sign means that as the independent variable increases, the dependent variable decreases!
If you use Gamma: By knowing the independent variable, you can explain 88.3 of the dependent variable.
If you use TAUC (Rectangle) by knowing the independent variable, you can explain 77.8% of the dependent variable.
If you use TAUB (Square) we do not have one because we are dealing with a rectangular table.
Somer's D on the same test
A negative value does not mean a weak relationship. LOOK AT THE ABSOLUTE VALUE!!!!
From the 80's: In 1984, Ronald Reagan asked Americans if they were "better off now, than they were 4 years ago". The assumption here is that those who were would reward Reagan with 4 more years. Below is a crosstab of this relationship.From 2006
Opinions on Iraq and vote choice
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William Smith
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