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Nonparametric Techniques
Is used when having serious violations of distribution assumptions or not normal
Appropriate for data measured on scales that are not interval or ratio.
Selection of nonparametric techniques are: Chi-square testsMann-Whitney test Wilcoxon signed-rank testKruskal-Wallis testFriedman testSpearman’s rank-order correlation
Chi-square Tests
2 Main types
3 assumptions to deal before conducting chi-square tests:1) Random sampling2) Independence of observations3) Size of expected frequencies
Chi-square test for goodness of fit
For analysis of a single categorical variable
Chi-square test for independence or relatedness
For analysis of the relationship between 2 categorical variables
Chi-square test for Goodness of Fit
used to compare observed and expected frequencies in each category.
sample size is usually small
Chi-square test for Goodness of Fit Steps to conduct chi-square test for goodness
of fit:1) Select the Data menu2) Click on Weight Cases to open the dialogue box3) Click on the Weight cases by radio button4) Select the relevant variable and move to
Frequency Variable5) Then, select Analyze menu6) Click on Nonparametric Tests and then Chi
Square 7) Select the required variable to move into Test
Variable List box
Goodness of fit chi square Output file will look like this:
You can see from the output that the chi-square value is no significant (p > .05).
Interpreting Chi square test for Goodness of Fit
ExampleColor preference of 150 people, p < 0.05Category
ColorObserved
FrequenciesExpected
FrequenciesYellow 35 20%
Red 50 30%
Green 30 10%
Blue 10 10%
White 25 30%
Chi-square requires that you use numerical values, not percentage or ratios.
Chi-square should not be calculated if the expected value in any category is less than 5.
Category Color
Observed Frequencies
Expected Frequencies
Yellow 35 30
Red 50 45
Green 30 15
Blue 10 15
White 25 45
Color preference of 150 people
Calculate chi-square
2 = Chi-square
O = Observed frequency
E = Expected frequency
k = number of categories, groupings, or possible outcomes
Category Color
O E (O-E) (O-E)2 (O-E)2
EYellow 35 30 5 25 0.83
Red 50 45 5 25 0.56
Green 30 15 15 225 15
Blue 10 15 -5 25 1.67
White 25 45 -20 400 8.89
Calculate chi-square
2 = 26.95
Calculate Degrees of freedom (df)
df = N – 1
Refers to the number of values that are free to vary after restriction has been placed on data.
Defined as N- 1, the number in the group minus one restriction.
= 5 – 1= 4
Critical 2 values2 = 26.95 , df = 4 , p < 0.05
If chi-square value is bigger than critical value, reject null hypothesis.
If chi-square value is smaller than critical value, fail to reject null hypothesis.
Critical 2
Chi-square Test for Relatedness or Independence
Used to evaluate group differences when the test variable is nominal, dichotomous, ordinal, or grouped interval.
A test of the influence or impact that a subject’s value on one variable has on the same subject’s value for a second variable.
Steps to conduct chi-square test for goodness of fit:1. Select the Analyze menu2. Click on Descriptive Statistics and then Crosstabs3. Select a row and column variable to move into the
respective box4. Click on Statistics command pushbutton to open
Crosstabs: Statistics subdialogue box5. Click on the Chi-square check box then Continue6. Click on the Cells subdialogue box7. In the Counts box, click on the Observed and
Expected check boxes8. In the Percentages box, click on the Row, Column
andTotal check boxes9. Click on Continue and then OK.
Chi-square Test for Relatedness or Independence
Interpreting Chi square test for Relatedness or IndependenceExample
H0 : The two categorical variables are independent.H1. : The two categorical variables are related.
Incidence of three types of malaria in three tropical regions.
Calculate expected frequency
e = expected frequency
c = frequency for that column
r = frequency for that row
n = total number of subjects in study
Calculate Degrees of freedom (df)
df = (r-1)(c-1) = (3-1)(3-1)
= (2)(2)= 4
r = number of categories in the row variable
c = number of categories in the column variable
Find critical 2 values
2 = 125.516 , df = 4 , p < 0.05
Chi-square value is bigger than critical chi-square value, reject null hypothesis.
Critical 2
REFERENCES Green, S. B., Salkind, N. J., & Akey, T. M. (2000). Using SPSS
for Windows: Analyzing and understanding data (2nd ed.). New Jersey: Prentice Hall.
Coakes, S. J., Steed, L., & Ong, C. (2010). SPSS:analysis without anguish: version 17.0 for Windows (Version 17.0 ed.). McDougall Street, Milton, Qld: John Wiley & Sons Australia, Ltd.
Hinkle, Wiersma, & Jurs. Chi-square test for goodness of fit. Retrieved from http://www.phy.ilstu.edu/slh/chi-square.pdf
Penn State Lehigh Valley. Chi-square test. Retrieved 9 March, 2011, from http://www2.lv.psu.edu/jxm57/irp/chisquar.html
Maben, A. F. Chi-square test. Retrieved from http://www.enviroliteracy.org/pdf/materials/1210.pdf
Bench, M. Interpreting the chi-square test. Retrieved 9 March, 2011, from http://www.mathbench.umd.edu/mod106_chisquare/page10.htm