Chi Square

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

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

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

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

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

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

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

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

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.

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

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

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

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 = X 2 crit = 3.841

Example: Independence cont’d  3.

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

Example: Independence Cont’d  5. X 2 (1, N= 122 ) = *, 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.