psych 100a - lecture 17-18

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Nonparametric Tests

Chapter 15

Revised by Jeffrey B. Henriques, Ph.D.

University of Wisconsin-Madison

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Nonparametric Statistics

• A special class of hypothesis tests

• Used when assumptions for parametric

tests are not met

• Like?!

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

When to Use Nonparametric Tests

• When the dependent variable is

nominal

• Examples?

• When the dependent variable is ordinal

• Examples?

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

When to Use Nonparametric Tests

• When the sample size is small

• Why?

• When underlying population is not

normal

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Limitations of Nonparametric Tests

• Cannot easily use confidence intervals

or effect sizes

• Have less statistical power than

parametric tests

• More likely to commit type II error

• Meaning?

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Chi-Square Test for Goodness of Fit

• When we have one nominal variable

• The six steps of hypothesis testing1. Identify the populations, distribution, and

assumptions

2. State the null and research hypotheses

3. Determine the characteristics of the

comparison distribution

4. Determine the critical value, or cutoff

5. Calculate the test statistic

6. Make a decision

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Formulae

c 2 =(O- E)2

E

é

ëê

ù

ûúå

1 rowrow kdf

1 columncolumn kdf

))((2 columnrowXdfdfdf

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Step 4: Determine the Critical Value

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Step 5: Calculate the Test Statistic

c 2 =(O-E)2

E

é

ëê

ù

ûúå = (20.571+ 20.571= 41.142

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Step 6: Make a Decision

41.14

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

VEGAS BABY!!!

Rolling dice at the Mirage

Lots of Snake Eyes coming up

Are the dice fixed?

Test with goodness of fit

Does our distribution FIT the expected

distribution

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenVEGAS BABY!!!

Expected distribution for 120 rolls if fair:

Each die(dice) has 1/6 chance

1/6 X 120 = 20 of each type

Expected Distribution =

[20,20,20,20,20,20]

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenVEGAS BABY!!!

Actual distribution for 120 rolls is:

[28,16,23,23,17,13]

Are these dice fair?

Use Goodness of fit χ2

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. HeinzenVEGAS BABY!!!

Determine critical χ2 value:

df = number of categories – 1

= 6-1 = 5

χ2 critical for df=5 is

11.07 from table

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Chi-Square Test for Independence

• The six steps of hypothesis testing1. Identify the populations, distribution, and

assumptions

2. State the null and research hypotheses

3. Determine the characteristics of the

comparison distribution

4. Determine the critical value, or cutoff

5. Calculate the test statistic

6. Make a decision

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

c 2 =(O- E)2

E

é

ëê

ù

ûúå = (2.218+0.837+ 2.197+0.829 = 6.081

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Step 6: Make a Decision

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Cramer’s V (phi)

• The effect size for chi-square test for

independence

))(( /

2

columnrowdfN

X

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Graphing Chi-Square Percentages

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

An Alternative Graph

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Relative Risk

• We can quantify the size of an effect

with chi square through relative risk,

also called relative likelihood

• By making a ratio of two conditional

proportions, we can say, for example,

that one group is three times as likely

to show some outcome or, conversely,

that the other group is one-third as

likely to show that outcome

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

HUH?!?!?

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Other Nonparametric Tests

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Spearman Rank-Order Correlation

Coefficient

• Quantifies the association between two

ordinal variables

• Calculating Spearman’s correlation

rs=1-

6( D2 )åN (N 2 -1)

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

A little example

• Is there a relationship between

• Level of “national competitiveness”

• Level of “national pride”

?

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

Essentials of Statistics for the Behavioral Sciences, 2nd EditionSusan A. Nolan Thomas E. Heinzen

rs=1-

6( D2 )åN (N 2 -1)

=1-6(98.5)

10(102 -1)=1-

591

10(99)=1-

591

990=1-0.597 = 0.403