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Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall 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
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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
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
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
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
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 )(
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
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?
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Chi-Square Test for Goodness of Fit
• Single nominal variable• Degrees of freedom = number of categories
minus 1
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
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。
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
• 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
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
好 中 差 合计男 8 5 6 19
女 16 7 6 29
合计 24 12 12 48
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
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.
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
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
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
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
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Data Transformation Example
• Reaction time data before a square root transformation…
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
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
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
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