Crosstab 2 – Measures of Association

Crosstab 2 – Measures of Association POLS 300 Butz

Crosstab • Crosstabulations are appropriate for examining relationships between variables that are nominal, ordinal, or dichotomous. • Displays joint distribution of two variables.

Measures of Association • Measures of association summarize efficiently the existence, direction, and strength of a relationship between two variables. • A single summary number.

Measures of Association • Most measures of association range from –1 to +1. • Closer the value to 0, the weaker the relationship. • + and – indicate the direction of the relationship.

Measures of Association for Crosstabulations • Purpose – to determine if nominal/ordinal variables are related in a crosstabulation • At least one nominal variable • Lamda • Chi-Square • Cramer’s V • Two ordinal variables • Tau (tau b, tau c) • Gamma

Lamda • Requires one nominal variable. • Lambda is designed to indicate whether the values of one variable tend “to cluster” with certain values of the other variable • so that knowing a case’s value for the independent variable would help one predict the case’s value for the dependent variable.

Lamda • Lambda= (E without– E with) E without • E withoutstands for the number of errors made in predicting or guessing the dependent variable without knowledge of the independent variable. (Rule 1) • E with is the number of errors with knowledge of the independent variable. (Rule 2)

Lamda • The logic is that if the two variables are related, having knowledge of the independent variable should provide (predict) knowledge of the dependent variable. • Lambda results are a measure of “proportional reduction in error” (PRE).

Lamda

Lamda – Rule 1 (knowledge of dependent variable – partisanship - only)

Lamda – Rule 2(knowledge of independent variable and dependent variable)

Lamda –Calculation of Errors • Errors w/Rule 1: .60 * 100 = 60 • Errors w/Rule 2: 16 + 10 + 14 + 10 = 50 • Lamda =(Errors R1 – Errors R2)/Errors R1 • Lamda = (60-50)/60=10/60=.17

Lamda • PRE measure • Ranges from 0-1 • 0 – means that introducing the IV does NOT reduce errors in predicting DV • > 0 – represents the “percent reduction in error” in predicting DV when IV is introduced!

Relationships between Ordinal Variables • There are several measures of association appropriate for relationships between ordinal variables • Gamma, Tau-b, Tau-c, Somer’s d • All are based on identifying concordant, discordant, and tied pairs of observations

Pairs • Many of the measures for ordinal variables are based on pairing the data. • Concordant pair: one case has higher/lower values on both variables than the other case. • Discordant pair: one case is lower on one of the variables but higher on the other variable. • Tied pair: both observations are tied on at least one of the variables.

Concordant Pairs:Ideology and Voting • Ideology - conserv (1), moderate (2), liberal (3) • Voting - never (1), sometimes (2), often (3) • Consider two hypothetical individuals in the sample with scores • Individual A: Ideology=1, Voting=1 • Individual B: Ideology=2, Voting=2 • Pair A&B are considered a concordant pair because B’s ideology score is greater than A’s score, and B’s partisanship score is greater than A’s score

Concordant Pairs (cont’d) • All of the following are concordant pairs • A(1,1) B(2,2) • A(1,1) B(2,3) • A(1,1) B(3,2) • A(1,2) B(2,3) • A(2,2) B(3,3) • Concordant pairs are consistent with a positive relationship between the IV and the DV (ideology and voting)

Discordant Pairs • All of the following are discordant pairs • A(1,2) B(2,1) • A(1,3) B(2,2) • A(2,2) B(3,1) • A(1,2) B(3,1) • A(3,1) B(1,2) • Discordant pairs are consistent with a negative relationship between the IV and the DV (ideology and voting)

Identifying Concordant Pairs - Obs/cell * Obs. In cells below & to the right - #Concordant = 80(70 +10 +20 + 80) + 20(20 + 80) + 10(10 + 80) + 70(80) == 22,900

Identifying Discordant Pairs • Obs/cell * Obs. In cells above & to the right • #Discordant = 20(10 +10) + 0(10 +10 + 70 +10) + 70(10) + 20(10 + 10) = 1,500

Gamma • Gamma is calculated by identifying all possible pairs of individuals in the sample and determining if they are concordant or discordant • Gamma = (#C - #D) / (#C + #D)

Interpreting Gamma • Gamma = 21400/24400 =.88 • Gamma ranges from -1 to +1 • More concordant pairs will lead to a positive statistic (positive relationship) • More discordant pairs leads to a negative statistic (negative relationship) • Gamma does not account for tied pairs • Tau (b and c) and Somer’s d account for tied pairs in different ways

Identifying Tied Pairs (Row Variable) - Obs/cell * Obs. In cells remaining in row - #Tied1 = 80(10 + 10) + 10(10) + 20(70 + 10) + 70(10) + 0(20 + 80) + 20(80) == 5,600

Identifying Tied Pairs (Column Variable) - Obs/cell * Obs. In cells remaining in column - #Tied2 = 80(20 + 0) + 20(0) + 10(70 + 20) + 70(20) + 10(10 + 80) + 10(80) == 5,600

Tau B and C • Tau b is suitable for square tables, same number of rows and columns. • Tau c is suitable for non-square tables.

Tau B Calculation • 21,400/ sqrt {(24,400 + 5,600)(24,400 + 5,600)} = • 21,400/ 30,000 = .71 • Strong, Positive Relationship

Square tables: Non-Square tables:

Crosstab 2 – Measures of Association