Chi-Square Analysis (Ch.8) Purpose. Purpose. Examples

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1 Chi-Square Analysis (Ch.8) Chi-square test of association (contingency) x tables rxc tables Post-hoc Interpretation Running SPSS Windows CROSSTABS Chi-square test of goodness of fit Purpose Chi-square test of association X associations (i.e., relation between two dichotomous variables) Examples (m/f) x Experience of physical aggression in past year (yes/no) First Language (English / Not English) x Getting Question on Test Correct (Correct/Incorrect) Purpose Chi-square test of association RxC associations (i.e., more categories than x) Examples Socioeconomic Status x Vehicle Brand Age Group x Preferred Music Genre 1

2 Example of a X Testing the association between Beer Consumption and Null hypothesis No association Proportion of cases in one cell to the marginal (e.g. 44/70) = proportion of the marginal on the other variable to the total (e.g., 54/4) M F Drink Beer Yes No Example of a X: Calculating Expected Values That is: H 0 : C E11 1 = 1 R T E = 70 4 E 11 = 36.3 M Drink Beer Yes 44 (36.3) No (17.7) 54 Once we have calculated one expected value, the others follow: E1 = = 33.7 F 6 (33.7) 70 4 (16.3) 34 4 Example of a X: Calculating Chi- Square value χ = ( Observed E ) xpected E xpected Drink Beer Yes No ( ) ( ) χ = =.3 df = (r-1) (c-1) = 1 critical value at.01 = 6.64 (see Table E in book) Report: χ (1) =.3, p <.01 M F 44 (36.3) 6 (33.7) 70 (17.7) 4 (16.3)

3 Example of a X: Test of Proportion In the case of a x, instead of a Chi-square test, you could use a test of proportion. For example we could compare the Proportion of male beer drinkers (44/54=.815) and female beer drinkers (6/=.). M p1 p Z = n1 + n p = 1 1 pq + N1+ N N1 N q = 1 p F p = Z = 1 1 Z = 3.04, p <.01 (.673)(.37) + 54 =.58 Drink Beer Yes No Assumptions of Chi-Square Test Sampling distributions of the O-E deviations is normal Potential problem if expected values are really small Data points must be independent of each other A subject contributes only once to the frequency count What to do with small expected frequencies? Yates correction (not recommended) Cochran s rule All expected frequencies greater than 1 No more than 0% should be less than 5 For example in a X, if you have one cell with expected frequency smaller than five (1/4 = 5%), you have violated Cochran s rule Collapse cells when possible (i.e., combine categories) 3

4 χ Distribution for different dfs Using SPSS The data can be in forms: By Category gender beer frequency : male = 1, female = Beer: yes = 1, no = Using SPSS or, by subject (would be 4 rows) male = 1, female = yes = 1, no = gender beer

5 Using SPSS Note: if you input the data this way, you must do the following in the Data Window: Data Weight cases by (freq var) Using SPSS Analyze Descriptive Statistics Crosstabs In Crosstabs, click on Chi-square under Statistics Observed, Expected, and Unstandardized s under Cells Using SPSS 5

6 Using SPSS * Beer Crosstabulation Total male female Beer yes no Total Using SPSS Pearson Chi-Square Continuity Correction a Likelihood Ratio Fisher's Exact Test Linear-by-Linear Association N of Valid Cases Chi-Square Tests Asymp. Sig. Value df (-sided).55 b a. Computed only for a x table Exact Sig. (-sided) Exact Sig. (1-sided) b. 0 cells (.0%) have expected count less than 5. The minimum expected count is R x C Example χ with variables having > levels first step is the same might want to do post hoc tests to further understand the association Look at table and describe the association (focus on large residuals) Or pick out specific cells ( x ) and test Or collapse cells to make a x and test 6

7 R x C Example: Adjusting the Type I Error Rate Make adjustment for increased chance of Type I error in posthoc tests Can use Bonferroni adjustment (when constructing a x table from existing cells k = # of x tables that can be made from a r x c table r! c! k = * !(r-)!!(c-)! use α =.05/k R x C Example: Obtained and Expected Frequencies residenc * yr_study Crosstabulation yr_study residenc Total res myself home-parents roommate spouse-partner first second third fourth Total R x C Example: Chi-Square Test Chi-Square Tests Asymp. Sig. Value df (-sided) Pearson Chi-Square a Likelihood Ratio Linear-by-Linear Association N of Valid Cases 379 a. 0 cells (.0%) have expected count less than 5. The minimum expected count is

8 R x C Example: Examine the Cells residenc * yr_study Crosstabulation yr_study residenc Total res myself home-parents roommate spouse-partner first second third fourth Total R x C Example: Examine the Cells residenc * yr_study Crosstabulation residenc res myself home-parents roommate spouse-partner Total yr_study first second third fourth Total Conclusion: A large proportion of the Chi square can be explained by the Fact that there is a very large proportion of first year students who live in residence O E E 39.5 = = 39.3% Contribution of cell (first year-residence) Chi square value x Posthoc: Examine Specific Contrast Extract a x table of interest or Collapse categories to form a x table (the example follows this second approach) In SPSS you can use the command RECODE to form new categories I did a x analysis in which I collapse all non-first year students into one category and all non-residence living students I did this in the syntax menu using the following commands: recode yr_study (1=1) ( thru hi = ) into year. recode residenc (1=1) ( thru hi = ) into resid. execute. 8

9 x Posthoc: Expected and Obtained Frequencies resid * year Crosstabulation resid Total res other housing year first year -4 years Total x Posthoc Bonferroni Adjustment k =! r! c! ( r )!!( c )! 5X 4X 3X X1 4X 3X X1 k = X1X 3X X1 X1X X1 k = 60 α = =.0008 In our example, we collapsed a number of categories. Therefore, we would not use the above adjustment. Gardner indicates that there are no specific meaningful Bonferonni adjustment when categories are collapsed and suggests at a minimum to use a Type I error rate of.01 x Posthoc: Chi-Square Test Pearson Chi-Square Continuity Correction a Likelihood Ratio Fisher's Exact Test Linear-by-Linear Association N of Valid Cases Chi-Square Tests Asymp. Sig. Value df (-sided) b Exact Sig. (-sided) Exact Sig. (1-sided) a. Computed only for a x table b. 0 cells (.0%) have expected count less than 5. The minimum expected count is

10 Chi Square Test of Goodness of Fit How closely a set of obtained frequencies compares to expected frequencies (based on theory or previous information) A significant test indicates badness of fit Use the same formula: χ = ( Observed Expected ) E xpected Example (Goodness of fit) You conduct a study to evaluate the frequency of alcohol consumption of university students. You want to determine whether your distribution differs from previous findings suggesting the following distribution: Category Never < Once per month 1-3 times per month Once per week More than once per week* % *I collapsed three categories (-3 times per week, 4-6 times per week, and every day) Example (Goodness of fit) Category % Obtained Expected Never < Once per month times per month Once per week More than once per week*

11 Example (Goodness of fit) Obtained Expected χ = ( Observed E ) xpected E xpected ( 15 ) ( ) χ = =.66 df = number of categories - 1 = 4 Gardner recommends Type I error rate of.0 Critical value at.0 = 5.99 Reject null of good fit: χ(4) =.66 p <.0 11

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