Bivariate Testing (Chi Square) HMI 7530 – Programming in R STATISTICS MODULE: Bivariate Testing (Chi Square) Jennifer Lewis Priestley, Ph.D. Kennesaw State University 1

STATISTICS MODULE Basic Descriptive Statistics and Confidence Intervals Basic Visualizations Histograms Pie Charts Bar Charts Scatterplots Ttests One Sample Paired Independent Two Sample Proportion Testing ANOVA Chi Square and Odds Regression Basics 2 2 2

STATISTICS MODULE: Chi Square When presented with categorical data, one common method of analysis is the “Contingency Table” or “Cross Tab”. This is a great way to display frequencies - For example, lets say that a firm has the following data: 120 male and 80 female employees 40 males and 10 females have been promoted 3

STATISTICS MODULE: Chi Square Using this data, we could create the following 2x2 matrix: Promoted Not Promoted Total Male 40 80 120 Female 10 70 50 150 200 4

STATISTICS MODULE: Chi Square What is the probability of: Selecting a female? Selecting someone who was promoted? Selecting a female GIVEN that the individual was promoted? Selecting someone who was promoted GIVEN that the individual was female? 5

STATISTICS MODULE: Chi Square The answers to these questions help us start to understand if promotion status and gender are related. Specifically, we could test this relationship using a Chi-Square. This is the test used to determine if two categorical variables are related. The relevant hypothesis statements for a Chi-Square test are: H0: Variable 1 and Variable 2 are NOT Related Ha: Variable 1 and Variable 2 ARE Related 6

STATISTICS MODULE: Chi Square The Chi-Square Test uses the Χ2 test statistic, which has a distribution that is skewed to the right (it approaches normality as the number of obs increases). The observed counts are provided in the dataset. The expected counts are the counts which would be expected if there was NO relationship between the two variables. 7

STATISTICS MODULE: Chi Square Going back to our example, the data provided is “observed”: Promoted Not Promoted Total Male 40 80 120 Female 10 70 50 150 200 What would the matrix look like if there was no relationship between promotion status and gender? The resulting matrix would be “expected”… 8

STATISTICS MODULE: Chi Square From the data, 25% of all employees were promoted. Therefore, if gender plays no role, then we should see 25% of the males promoted (75% not promoted) and 25% of the females promoted… Promoted Not Promoted Total Male 120*.25 = 30 120*.75 = 90 120 Female 80*.25 = 20 80*.75 = 60 80 50 150 200 Notice that the marginal values did not change…only the interior values changed. 9

STATISTICS MODULE: Chi Square Now, calculate the X2 statistic using the observed and the expected matrices: ((40-30)2/30)+((80-90)2/90)+((10-20)2/20)+((70-60)2/60) = 3.33+1.11+5+1.67 = 11.11 This is conceptually equivalent to a t-statistic or a z-score. 10

STATISTICS MODULE: Chi Square To determine if this is in the rejection region, we must determine the df. Df = (r-1)*(c-1)… In the current example, we have two rows and two columns. So the df = 1*1 = 1. At alpha = .05 and 1df, the critical value is 3.84…our value of 11.11 is clearly in the reject region…so what does this mean? 11

STATISTICS MODULE: Chi Square #here, the code is pretty simple… Xtab(var1~var2, data=data) Then a Chi Squared test: chisq.test(var1, var2, correct=FALSE) 12