Chapter 17 Chi-Square and other Nonparametric Tests

James A. Van Slyke Azusa Pacific University

Distinctions between Parametric and Nonparametric Tests

 Parametric tests (e. g. t, z) depend substantially on population characteristics  Nonparametric tests depend minimally on population parameters   Fewer requirements Often referred to as

distribution free

tests  Advantages for parametric tests   Generally more powerful and versatile Generally robust to violations of the test assumptions (sampling differences)

Chi-Square ( χ

2

) Single Variable Experiments

 Often used with nominal data  Allows one to test if the observed results differ significantly from the results expected if H 0 were true

Chi-Square ( χ

2

) Single Variable Experiments

 Computational formula 

obt

2   

f o



f e f e

 2 where

f o

 the observed frequency in the cell

f e

  = summation over all cells

Chi-Square ( χ

2

) Single Variable Experiments

 Evaluation of Chi-Square obtained     Family of Curves Vary with degrees of freedom k-1 degrees of freedom where k equals the number of groups or categories The larger the discrepancy between the observed and expected results the larger the value of chi-square, the more unreasonable that H 0 is accepted

If

 2

obt

  2

crit

, reject H

0

Chi-Square: Test of independence between two variables

 Used to determine whether two variables are related  Contingency table  This is a two-way table showing the contingency between two variables  The variables have been classified into mutually exclusive categories and the cell entries are frequencies

Chi-Square: Test of independence between two variables

 Null hypothesis states that the observed frequencies are due to random sampling from a population  This population has proportions in each category of one variable that are the same for each category of the over variable  Alternative hypothesis is that these proportions are different

Chi-Square: Test of independence between two variables

 Calculation of chi-squared for contingency tables 

obt

2   

f o



f e f e

 2   fe can be found by multiplying the marginals (i.e. row and column totals lying outside the table) and dividing by N Sum (f o – f e ) 2 /f e for each cell

Chi-Square: Test of independence between two variables

 Evaluation of chi-square   df = number of f o scores that are free to vary While at the same time keeping the column and row marginals the same  Equation  df = (r – 1)(c – 1)  Where r = number of rows in the contingency table  c = number of columns in the contingency table

If

 2

obt

  2

crit

, reject H

0