Measures of Measures of AssociationAssociation

February 25, 2011February 25, 2011

ObjectivesObjectivesBy the end of this meeting, participants By the end of this meeting, participants

should be able to:should be able to:

a)a) Calculate ordinal measures of Calculate ordinal measures of association and discuss how association and discuss how appropriate each is for various types appropriate each is for various types of variables.of variables.

b)b) Calculate nominal measures of Calculate nominal measures of association.association.

c)c) Perform a chi-square significance test Perform a chi-square significance test of a cross tabulation.of a cross tabulation.

What is Association?What is Association?Association can be thought of in two general Association can be thought of in two general

different ways:different ways:a)a) Covariation- the extent to which two Covariation- the extent to which two

variables change togethervariables change together• Positive covariation- as one variable increases Positive covariation- as one variable increases

another variable increases as well (be aware of another variable increases as well (be aware of coding!!)coding!!)

• Negative covariation- as one variable increases Negative covariation- as one variable increases another variable decreases (or vice versa)another variable decreases (or vice versa)

• No relationship- changes in one variable have No relationship- changes in one variable have no systematic effect on another variableno systematic effect on another variable

What is Association?What is Association?b)b) Probability- how well can one variable Probability- how well can one variable

predict changes in another variablepredict changes in another variable

c)c) Most measures of association are Most measures of association are bounded between -1 and 1bounded between -1 and 1

Ordinal MeasuresOrdinal Measuresa)a) KendallKendall’’s tau (s tau (ττ))• Most commonly used ordinal measureMost commonly used ordinal measure• Measures covariation, meaning that it will have Measures covariation, meaning that it will have

the same value regardless of which variable is the same value regardless of which variable is the independent and the dependent variable (it the independent and the dependent variable (it is a symmetric measure)is a symmetric measure)

• The value of the measure ranges from -1 to 1The value of the measure ranges from -1 to 1• Generally speaking a value over .7 is Generally speaking a value over .7 is

considered strong, one between .3 and .7 considered strong, one between .3 and .7 moderate and less than .3 weak. This is a loose moderate and less than .3 weak. This is a loose standard and will differ depending on the state standard and will differ depending on the state of the literature. of the literature.

Ordinal MeasuresOrdinal Measuresb)b) More Specific TausMore Specific Taus• Tau-b- Will only reach a value of 1 if both Tau-b- Will only reach a value of 1 if both

variables have the same number of categoriesvariables have the same number of categories• Tau-c- Corrects for unequal number of Tau-c- Corrects for unequal number of

categories between the independent and categories between the independent and dependent variables (although remember the dependent variables (although remember the measure is symmetric).measure is symmetric).

c)c) SomerSomer’’s ds d• Is a measure of association based on the Is a measure of association based on the

differences in percentagesdifferences in percentages• d is a measure of how much the dependent d is a measure of how much the dependent

variable changes as a result of the independent variable changes as a result of the independent variable (this is an asymmetric measure). You variable (this is an asymmetric measure). You need to be careful that you are analyzing the need to be careful that you are analyzing the expected relationship rather than the oppositeexpected relationship rather than the opposite

Ordinal MeasuresOrdinal Measuresd)d) Goodman and KruskalGoodman and Kruskal’’s Gamma (s Gamma (γγ))• Computes whether two variables measure the Computes whether two variables measure the

same underlying dimensionsame underlying dimension• γγ is based on the logic of concordant (similar) and is based on the logic of concordant (similar) and

discordant (dissimilar) pairs discordant (dissimilar) pairs • γγ ranges from -1 to +1 ranges from -1 to +1

e)e) SpearmanSpearman’’s Rho (s Rho (ρρ) also known as the ) also known as the SpearmanSpearman’’s rank order coefficients rank order coefficient

d)d) This measure computes the association between This measure computes the association between two variables that are rank orders (like income)two variables that are rank orders (like income)

e)e) This measure should not be used with variables This measure should not be used with variables that that are ordered categories such as ideologyare ordered categories such as ideology

CaveatsCaveatsa) Ordinal level measures of association

assume a monotonic relationship.b) That is, as one variable increases, the

other variable will consistently increase or consistently decrease (but not both).

c) The relationship between age and voter turnout is not likely to be monotonic, therefore ordinal level measures of association are problematic.

d) Make sure that the categories are ordered in a logical way. This may mean excluding missing or rare cases (such as did not vote, third party, etc.) .

How to choose among the How to choose among the measures?measures?

a)a) When examining rank ordered When examining rank ordered variables, the choice is clearly variables, the choice is clearly SpearmanSpearman’’s Rho (s Rho (ρρ).).

b)b) What about other cases?What about other cases?• Tau is the most commonly used ordinal Tau is the most commonly used ordinal

measure of associationmeasure of association• Gamma should generally only be used Gamma should generally only be used

when examining a scalewhen examining a scale• Presenting more than one measure is an Presenting more than one measure is an

acceptable option provided that it aids acceptable option provided that it aids the readersthe readers’’ understanding. understanding.

Nominal Measure of AssociationNominal Measure of Associationa)a) This measure should be used for nominal This measure should be used for nominal

data or data where the categories cannot be data or data where the categories cannot be ordered.ordered.

b)b) Lambda (Lambda (λλ) ) • The most commonly used measure, lambda, is The most commonly used measure, lambda, is

asymmetric, having different values depending asymmetric, having different values depending on which variable is the independent and on which variable is the independent and dependent variable.dependent variable.

• Is more of predictive measure, it computes the Is more of predictive measure, it computes the proportional reduction in error (PRE). In other proportional reduction in error (PRE). In other words lambda estimates how much better your words lambda estimates how much better your prediction about the value of dependent variable prediction about the value of dependent variable is when you know the independent variable. is when you know the independent variable.

Significance TestsSignificance Testsa) We want to know whether the association

we observe is due to chance—whether we observe an association because of the sample we happened to draw—or whether the relationship between variables exists in the underlying population of interest.

b) Chi–Square Test (χ2)• Tests the null hypothesis that there is no

relationship between the two variables in the population. If we reject the null hypothesis, we conclude that the relationship between variables is statistically significant.

Chi–Square Test (χ2)

a)a) Once a Chi-Square is completed, using the Once a Chi-Square is completed, using the degrees of freedom a p value can be found. degrees of freedom a p value can be found.

b)b) Researchers almost want the lowest Researchers almost want the lowest possible p value or, in other words, the possible p value or, in other words, the lowest probability that the relationship lowest probability that the relationship between the two variables occurred by between the two variables occurred by chance.chance.

c)c) The standard value in the social science is The standard value in the social science is p values of less than .05 but values of less p values of less than .05 but values of less than .1 or .01 are not uncommon. than .1 or .01 are not uncommon. Whatever standard is chosen it should be Whatever standard is chosen it should be clearly marked on the table.clearly marked on the table.

Substantive CriteriaSubstantive Criteriaa)a) Significance tests, while important, are Significance tests, while important, are

not the sole determinant of good not the sole determinant of good research. research.

b)b) Small samples are more likely to be Small samples are more likely to be found not significant.found not significant.

c)c) It is important to determine whether the It is important to determine whether the predicted relationship is substantively predicted relationship is substantively important. important.

d)d) Recall that association is part of Recall that association is part of causation but not the same thing as causation but not the same thing as causation. causation.

Think AboutThink About

What is the difference between statistical significance and substantive importance?

For March 7For March 7a)a) Two separate homeworks for two separate Two separate homeworks for two separate

grades.grades.b)b) HW 1: Select two ordinal variables with a HW 1: Select two ordinal variables with a

possible relationship from the PS-ARE data. possible relationship from the PS-ARE data. (You may use the same two as last time.)(You may use the same two as last time.)• Recode the variables (if necessary) so that they are Recode the variables (if necessary) so that they are

both ordinal scales.both ordinal scales.• Compute the cross tabulation of the two variables Compute the cross tabulation of the two variables

with both frequencies and percentages.with both frequencies and percentages.• Does a chi-square test show a statistically Does a chi-square test show a statistically

significant relationship between the two variables?significant relationship between the two variables?• Find the most appropriate measure of association Find the most appropriate measure of association

and interpret it.and interpret it.

c)c) HW 2: Read WKB chapter 13HW 2: Read WKB chapter 13 and turn-in answers and turn-in answers to questions 2 & 3 on pp. 296-297.to questions 2 & 3 on pp. 296-297.