Stats - Chapter 9 Fall 2017

X squared distribution

- 2 categorical variables - Size of x2 Indicates How Certain You Are There is Relationship - Tells us Nothing about the Strength of Relationship - Need "Measures of Association" to Indicate How Strong the Relationship Is

Interpreting Chi Square Statistic

- Can we conclude more than the distribution is uneven? - The larger the difference between fo and fe for a category --- The greater the contribution to Chi Square - Negative differences indicate lower f than expected - Positive differences indicate higher f than expected - So... reject the null hypothesis and conclude that "the responses are not evenly distributed across the four categories in the population." - Also...Conclude that more people are Satisfied and Very Satisfied with Police than expected

Degrees of Freedom

- Statistical "Room to Move" or - Values that are "Free to Vary in the Data" df = (n-1) df = (R-1)(C-1)

Chi Square Tests for Categorical Variables

1. Chi Square test with 1 Categorical Variable: "Goodness of Fit Test" - Test for Distribution of Frequencies Across Categories of 1 Categorical Variable 2. Chi Square test with 2 Categorical Variables: "Test for Independence" - Test for Statistically Significant Relationship Between 2 Categorical Variables

5 Steps in Hypothesis Testing

Example 2 Step 1: State Hypotheses - H0: χ2 = 0 No Relation between Criminal Record and Job Offer - H1: χ2 > 0 Sig. Relationship between Criminal Record and Job Offer Step 2: Select Probability Distribution - Chi-square Distribution Step 3: Select a Significance Level and a Critical Value - Alpha = .05 - df = (R-1)(C-1) (R: # of Rows; C: # of Columns) - Table E-4 with 1 df = 3.841 - Reject Null Hypothesis if χ2obt > 3.841 Step 4: Calculate the Test (Chi-Square) Statistic Step 5: χ2obt = 8.34 > 3.84: reject the null and conclude that "Criminal Record and Job Offer are Sig. Related"

5 STEPS IN HYPOTHESIS TESTING

Example: Step 1: State Hypotheses - H0: χ2 = 0 Observed f are evenly distributed across categories - H1: χ2 > 0 Observed f are unevenly distributed Step 2: Select Probability Distribution - Chi-square Distribution Step 3: Select a Significance Level and a Critical Value - Alpha = .05 - df = (k - 1) k = number of categories - Table E-4 with 3 df = 7.815 - Reject Null Hypothesis if χ2obt > 7.815 Step 4: Calculate the Test (Chi-Square) Statistic Step 5: χ2obt = 14 > 7.815: reject the null and conclude that "Responses are not evenly distributed across the 4 categories"

Hypothesis Test Statistic Chi Square Test

IV: Categorical DV: Categorical

Hypothesis Test Statistic Z/T Test

IV: Categorical - (2 categories) DV: Continuous

Hypothesis Test Statistic F-Test ANOVA

IV: Categorical - (3 or more categories) DV: Continuous

Hypothesis Test Statistic Regression

IV: Continuous - (or Categorical) DV: Continuous

Chi Square Distribution

Theoretical probability distribution - Used with 1 or 2 Categorical Variables - Right Tail Distribution, Always Positive (Always 1-tailed test!!!) - Shape Changes with 'degrees of freedom' (df) As df Increase - Chi-Square Distribution Approaches Z Distribution

Measures of Association for Categorical Data Statistic - Phi Coefficent (Φ)

Type of Variable: Nominal Number of Categories: 2x2 - Ranges from 0 to 1 - Rules of Thumb for Strength of Relationships 0 to .29 = Weak Relationship .3 to .59 = Moderate Relationship .6 to 1 = Strong Relationship

Measures of Association for Categorical Data Statistic - Cramer's V

Type of Variable: Nominal Number of Categories: RxC - Nominal Level - 2 or More Categories - Ranges from 0 to 1 - Takes into Account Extra Rows/Columns - But what if we have a nominal level variable with more than two categories? The book discusses how you can use the contingency coefficient, but a disadvantage to this statistics is that a perfect relationship is not always anchored by a maximum value of 1.0. Depending on the size of the table, a perfect relationship may be lower than 1.0. - A measure of association for nominal level variables that does not have this disadvantage is known as Cramer's V.

Measures of Association for Categorical Data Statistic - Yule's Q

Type of Variable: Ordinal Number of Categories: 2x2

Measures of Association for Categorical Data Statistic - Gamma (γ)

Type of Variable: Ordinal Number of Categories: RxC

Measures of Association for Categorical Data

Wide Range of Different Measures of Association: Depends on the (1) Type of Variable and (2) Number of Categories