Aron, Aron, Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), 2005...

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

Chapter 11

Chi-Square Tests Chi-Square Tests and Strategies and Strategies When Population When Population Distributions Are Distributions Are Not NormalNot Normal


Chi-Square Tests

• Hypothesis testing procedure for nominal variables– Focus on number of people/items in each category (e.g.,

hair color, political party, gender)• Compare how well an observed distribution fits an

expected distribution• Expected distribution can be based on

– A theory– Prior results– Assumption of equal distribution across categories


Chi-Square Statistic

• Compares observed frequency distribution to expected frequency distribution– Compute difference between observed and

expected and square each one– Weight each by its expected frequency– Sum them

EEO 2

2 )(


Chi-Square Distribution

• Compare obtained chi-square to a chi-square distribution

• Does mismatch between observed and expected frequency exceed what would be expected by chance alone?


Chi-Square Test for Goodness of Fit

• Single nominal variable• Degrees of freedom = number of categories

minus 1


例：计算机测试，成绩可分为好、中、差 3种。某班有学生 48人，成绩为好的有 24人，中的有 12人，差的有 12人，问这 3种成绩的人数是否有显著不同？解： E=48*1/3=16人ⅹ2=(24-16)2/16+(12-16)2/16+(12-16)2/16=6df=3-1=2，查ⅹ 2分布表 df=2时，ⅹ 2

0.05=5.99，故 3 种成绩的人数有显著的不同。


• Contingency table– Lists number of

observations for each combination of categories

– To determine expected frequencies…

Chi-Square Test for Independence

• Two nominal variables– Independence means

no relation between variables

– To determine degrees of freedom…

)(CNRE

)1)(1( RowsColumn NNdf


好中差合计男 8 5 6 19

女 16 7 6 29

合计 24 12 12 48


Key Assumption for Chi-square Test

• No one individual can be counted in more than one cell.

• In other words, each score must not have any special relation to any other score.


Effect Size for Chi-Square

• For 22 chi-square, effect size is the phi coefficient– Same as the correlation between two nominal

variables• Small = .10• Medium = .30• Large = .50

• For larger tables…

N

2

))(( sCramer'

Smaller

2

dfN


Data Transformations

• Assumption of parametric tests, that populations follow a normal curve– Sometimes violated– Ceiling or floor effects

• Can transform data (e.g., square root)– Makes distribution more normal– Preserves order of scores but not mean, SD


Data Transformation Example

• Reaction time data before a square root transformation…


Data Transformation Example

• Reaction time data after a square root transformation.– Distribution is closer to

a normal curve– Mean and SD are

changed, but order or scores is preserved


Rank-order Tests

• Another strategy for non-normal distributions is to covert scores to ranks– Can then use special rank-order or

“nonparametric” tests– Each parametric test has a corresponding

nonparametric test (e.g., Wilcoxon rank-sum test in place if a t test for independent means)

• Can also use conventional, parametric tests on ranks without much loss of accuracy

Aron, Aron, Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), 2005...

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