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 (χ2) Single Variable
Experiments


Often used with nominal data
Allows one to test if the observed results
differ significantly from the results expected if
H0 were true
Chi-Square (χ2) Single Variable
Experiments

Computational formula
 obt  
2
 fo  fe 
2
fe
where
f o  the observed frequency in the cell
f e  the expected frequency in the cell (if H 0 were true)
 = summation over all cells
Chi-Square (χ2) 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 H0 is accepted
If 
2
obt

2
crit
, reject H0
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


 fo  fe 
2
fe
fe can be found by multiplying the marginals (i.e. row
and column totals lying outside the table) and dividing
by N
Sum (fo – fe)2/fe for each cell
Chi-Square: Test of independence
between two variables

Evaluation of chi-square



df = number of fo 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 H0