Transcript Chi Square
STAT 3120 Statistical Methods I Lecture 8 Chi-Square STAT3120 – Chi Square Dependent Variable Independent (predictor) Variable Statistical Test Comments Quantitative Categorical T-TEST (one, two or paired sample) Determines if categorical variable (factor) affects dependent variable; typically used for experimental or planned change studies Quantitative Quantitative Correlation /Regression Analysis Test establishes a regression model; used to explain, predict or control dependent variable Categorical Categorical Chi-Square Tests if variables are statistically independent (i.e. are they related or not?) STAT3120 – 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 STAT3120 – Chi Square Using this data, we could create the following 2x2 matrix: Promoted Not Promoted Total Male 40 80 120 Female 10 70 80 Total 50 150 200 STAT3120 – Chi Square Now, a few questions… 1) From the data, what is the probability of being promoted? 2) Given that you are MALE, what is the probability of being promoted? 3) Given that you are promoted, what is the probability that you are MALE? 4) Given that you are FEMALE, what is the probability of being promoted? 5) Given that you are promoted, what is the probability that you are female? STAT3120 – 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 ChiSquare. This is the test used to determine if two 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 Develop the appropriate hypothesis statements and testing matrix for the gender/promotion data. STAT3120 – 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). You can see an example of the distribution on pg 641. The Χ2 test statistic calculation can be found on page 640. 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. STAT3120 – Chi Square Going back to our example, the data provided is “observed”: Promoted Not Promoted Total Male 40 80 120 Female 10 70 80 Total 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”… STAT3120 – 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 Male Female Total Not Promoted Total 120*.25 = 30 120*.75 = 90 120 80*.25 = 20 50 80*.75 = 60 150 80 200 Notice that the marginal values did not change…only the interior values changed. STAT3120 – 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)+((7060)2/60) = 3.33+1.11+5+1.67 = 11.11 This is conceptually equivalent to a t-statistic or a z-score. STAT3120 – Chi Square To determine if this is in the rejection region, we must determine the df and then use the table on page 732. 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? STAT3120 – Chi Square From the book Outliers, Malcolm Glidewell makes the point that the month in which a boy is born will determine his probability of playing in the NHL. The months of birth for players in the NHL are on the next page… (data taken from http://sports.espn.go.com/espn/page2/story?pa ge=merron/081208) STAT3120 – Chi Square January February March April May June July August September October November December 51 46 61 49 46 49 36 41 36 34 33 30 Now, if there is NO relationship between birth month and playing hockey, what SHOULD the distribution of months look like? Lets do this one in EXCEL… Note that this is technically referred to as a “goodness of fit” test – where we are assessing if the actual distribution “fits” what would be expected. STAT3120 – Chi Square Practice Problems for Chi-Square: 15.55 15.56 15.57 15.58 For all of these, identify the hypothesis statements, the testing matrix, and the decision.