Transcript Chapter 17 Chi Square - Azusa Pacific University
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