1. Non-parametric Measures of Association

2. Chi-Square Review Did the | organization| split | Type of leadership for organization this year? | Factional Weak or Divided Strong Unitary| Total ------------+--------------------------------------------+---------- No split | 33 195 835 478 | 1,541 Split | 27 0 16 5 | 48 ------------+--------------------------------------------+---------- Total | 60 195 851 483 | 1,589 Pearson chi2(3) = 377.2845 Pr = 0.000

3. Non-Parametric Statistics Most are based on the chi-square statistic and are used to look at the relationship between two ordinal or nominal variables, allowing us to control for: – # of categories – Sample size The statistic you want depends upon whether you have nominal or ordinal variables and how many categories the variables have.

4. Measures of Association: Non-parametric Tests for Nominal StatisticUsed forBoundsFormula Phi (  ) 2x2 tables0 and 1 The Contingency Coefficient (C) Square tables (2x2, 3x3, etc.) Vary depending upon the # of columns/rows Cramer’s VAny0 and 1

5. Measures of Association: Non-parametric Tests for Ordinal StatisticUsed forBoundsFormula Gamma (γ)Any-1 and 1 Kendall’s Tau-b (τ b )Only for square tables (2x2, 3x3, etc.) -1 and 1 Kendall’s Tau-c (τ c )Any-1 and 1 Sommer’s dAny, but it is aysmmetric so you must identify your dependent and independent variables -1 and 1

6. Practice Problem Did the | organization| split | Type of leadership for organization this year? | Factional Weak or Divided Strong Unitary| Total ------------+--------------------------------------------+---------- No split | 33 195 835 478 | 1,541 Split | 27 0 16 5 | 48 ------------+--------------------------------------------+---------- Total | 60 195 851 483 | 1,589 Pearson chi2(3) = 377.2845 Pr = 0.000 likelihood-ratio chi2(3) =. Cramér's V = 0.4873 gamma = -0.6873 ASE = 0.084 Kendall's tau-b = -0.1659 ASE = 0.027

7. Lambda ( ) Can be used to look at the relationship between two nominal variables. Based on a logic of making a proportional reduction of error Asymmetric measure

8. Calculating Lambda ( )- 1 Y/XProtestantCatholicTotal Approve105060 Disapprove20 40 Total3070100

9. Calculating Lambda ( )- 2 Y/XProtestantCatholic Approve060 Disapprove400

10. Calculating Lambda ( )- 3 Y/XProtestantCatholic Approve070 Disapprove300

11. Calculating Lambda ( )- 4 We went from 40 errors to 30 errors between tables 2 and 3 by knowing the person’s religion. To calculate lambda, = E₁ - E₂/ E₁, where – E₁ = the smallest expected value of errors when we don’t know the categories of the independent variable (i.e., the smallest frequency of the dependent variable) – E₂ = the smallest expected number of errors when we know the categories of the independent variable (i.e., the smallest frequency of the independent variable) = 40-30/40 =10/40 =0.25

12. Interpreting Lambda ( ) If E₁ = E₂ ( =0) then knowledge of the independent variable does not help at all in error reduction—the two variables are independent. If E₂ = 0 ( =1) then knowledge of the independent variable reduces error to zero, i.e., the two variables are “perfectly dependent.”