1. Chi Square

2. A Non-Parametric Test  Uses nominal data e.g., sex, eye color, name of favorite baseball team e.g., sex, eye color, name of favorite baseball team  By tallying or tabulating frequencies for each category http://personal.centenary.edu/~bcarothe/stats/stats17.htm

3. Chi-Square For Goodness of Fit  “A nonparametric hypothesis test used with one nominal variable” (p. 586, Nolan & Heinzen)  Asks if the observed frequencies differ significantly from the hypothesized distribution  In other words, does the data “fit” the expected distribution  A significant chi-square indicates poor fit i.e., different distribution than hypothesized i.e., different distribution than hypothesized

4. Chi-Square for Independence  “Is a nonparametric test used with two nominal variables” (p. 586, Nolan & Heinzen)  Used to determine if two nominal variables are related to each other or if they are independent of one another

5. Observed Frequencies  Also known as your experimental values or tally scores!

6. Expected Frequencies  Expected frequencies are your theoretical values and derived from the null hypothesis  fe = (fc)(fr) N fe = expected frequencies fe = expected frequencies fc = total frequencies in the column fc = total frequencies in the column fr = frequencies in the row fr = frequencies in the row N = total N = total

7. Steps for Goodness of Fit  1. Set your hypotheses H0: The data follows a specified distribution. (You specify the distribution) H0: The data follows a specified distribution. (You specify the distribution) H1: The data does not follow a specified distribution H1: The data does not follow a specified distribution  2. Set your criteria Set alpha level Set alpha level df = (j-1); where j is the number of categories df = (j-1); where j is the number of categories Look up X 2 crit (chi square) from the Chi Square Distribution chart in the appendix Look up X 2 crit (chi square) from the Chi Square Distribution chart in the appendix

8. Steps for Goodness Cont’d  3. Test statistic X 2 observed= ∑ [ (fo-fe) 2 /fe ] X 2 observed= ∑ [ (fo-fe) 2 /fe ] Fe = your theorized frequencies Fe = your theorized frequencies Make table with these columns: Make table with these columns: Category, fo, fe, fo-fe, (fo-fe) 2, (fo-fe) 2 /feCategory, fo, fe, fo-fe, (fo-fe) 2, (fo-fe) 2 /fe X 2 = sum of the last column, (fo-fe) 2 /feX 2 = sum of the last column, (fo-fe) 2 /fe  4.APA & conclusion X 2 (df, N= ) =X 2 observed, p alpha, sig or ns X 2 (df, N= ) =X 2 observed, p alpha, sig or ns

9. Example: Goodness of Fit  1. In our class of 60 students,we want to hypothesize how many students are born in NY, in the US but out side of NY and outside of the US. H0: The data follows a specified distribution of 20 students in each category. (You can use any variation such as, 10, 40 and 10, as long as it adds up to 60) H0: The data follows a specified distribution of 20 students in each category. (You can use any variation such as, 10, 40 and 10, as long as it adds up to 60) H1: The data does not follow a specified distribution. H1: The data does not follow a specified distribution.  2. alpha = 0.05 df = (j –1)  (3-1) = 2 X 2 crit = 5.992

10. Goodness Example Cont’d  3.  4. X 2 (2, N=60) = 21.9*, p<.05, sig. The data does not follow the specified distribution of 20 students in each category. Students born in NY are more likely to be present in our class.

11. Steps for Independence  1.Set your hypotheses H0: There is no relationship between the two variables H0: There is no relationship between the two variables H1: There is a relationship between the two variables H1: There is a relationship between the two variables  2.Set your criteria Set alpha level Set alpha level df = (r-1)(c-1); where r is number of rows and c is number of columns df = (r-1)(c-1); where r is number of rows and c is number of columns Look up X 2 crit Look up X 2 crit

12. Steps for Independence Cont’d  3. Make Charts and calculate X 2 !

13. Steps for Independence Cont’d  4. Cramer’s V Tells you the size of the effect Tells you the size of the effect Use the numbers for Cohen’s d for Cramer’s V Use the numbers for Cohen’s d for Cramer’s V  5. APA & Conclusion X 2 (df, N= ) =X 2 observed, p alpha, sig or ns, Craver’s V =, effect size X 2 (df, N= ) =X 2 observed, p alpha, sig or ns, Craver’s V =, effect size

15. Example: Independence  Is there a relationship between, NY and outside NY residents, and team favored, NY Mets v. NY Yankees?  1. H0: There is no relationship between the two variables H1: There is a relationship between the two variables H1: There is a relationship between the two variables  2.Set your criteria Alpha = 0.05 Alpha = 0.05 df = (r-1)(c-1)  (2-1)(2-1) = 1 df = (r-1)(c-1)  (2-1)(2-1) = 1 X 2 crit = 3.841 X 2 crit = 3.841

16. Example: Independence cont’d  3.

17. Example: Independence cont’d  4. Effect size is small Effect size is small

18. Example: Independence Cont’d  5. X 2 (1, N= 122 ) = 22.098*, p<.05, sig, Craver’s V = 0.18, small effect. A chi- square test of independence indicated that there is a significant relationship between where a person resided (in NY or outside of NY) and favored NY teams.