ANOVA Knowledge Assessment 1. In what situation should you use ANOVA (the F stat) instead of doing a t test? 2. What information does the F statistic give you? 3. For the ANOVA, the dependent variable should be what level of measurement? What about the IV? What about the IV? 4. Why is the F test called “exploratory”?

ANOVA Knowledge Assessment 1. In what situation should you use ANOVA (the F stat) instead of doing a t test? When your independent variable has 3 or more categories/attritbutes. When your independent variable has 3 or more categories/attritbutes. 2. What information does the F statistic give you? The F statistic tells you the ratio of between-group variance to within-group variance. The F statistic tells you the ratio of between-group variance to within-group variance. 3. For the ANOVA, the dependent variable should be what level of measurement? What about the IV? The dependent variable should be interval-ratio. The independent variable can be either nominal or ordinal. The dependent variable should be interval-ratio. The independent variable can be either nominal or ordinal. 4. Why is the F test called “exploratory”? Because a significant F statistic doesn’t allow you to identify which difference(s) in means are statistically significant. Because a significant F statistic doesn’t allow you to identify which difference(s) in means are statistically significant.

Contingency Tables (cross tabs)  Generally used when variables are nominal and/or ordinal Even here, should have a limited number of variable attributes (categories) Even here, should have a limited number of variable attributes (categories)  Some find these very intuitive…others struggle It is very easy to misinterpret these critters It is very easy to misinterpret these critters

Interpreting a Contingency Table  WHAT IS IN THE INDIVIDUAL CELLS? The number of cases that fit in that particular cell The number of cases that fit in that particular cell In other words, frequencies (number of cases that fit criteria)In other words, frequencies (number of cases that fit criteria) For small tables, and/or small sample sizes, it may be possible to detect relationships by “eyeballing” frequencies. For most.. For small tables, and/or small sample sizes, it may be possible to detect relationships by “eyeballing” frequencies. For most.. Convert to Percentages: a way to standardize cells and make relationships more apparentConvert to Percentages: a way to standardize cells and make relationships more apparent

Example 1  Is there is an even distribution of membership across 4 political parties?  (N=40 UMD students) CategoriesF% Republican1230% Democrat1435% Independent923% Green510%

Example 2  A survey of 10,000 U.S. residents  Is one’s political view related to attitudes towards police? What are the DV and IV? What are the DV and IV?  Convention for bivariate tables The IV is on the top of the table (dictates columns) The IV is on the top of the table (dictates columns) The DV is on the side (dictates rows). The DV is on the side (dictates rows).

Example 2 Continued Attitude Towards Police Political Party Total RepubDemocratLibertarianSocialist Favorable Unfav Total

The Percentages of Interest Attitude Towards Police Political Party Total RepubDemocratLibertarianSocialist Favorable2900 (60%) 2100 (54%) 180 (53%) 30 (52%) 5210 Unfav Total

The Test Statistic for Contingency Tables  Chi Square, or χ 2 Calculation Calculation Observed frequencies (your sample data)Observed frequencies (your sample data) Expected frequencies (UNDER NULL)Expected frequencies (UNDER NULL) Intuitive: how different are the observed cell frequencies from the expected cell frequencies Intuitive: how different are the observed cell frequencies from the expected cell frequencies Degrees of Freedom: Degrees of Freedom: 1-way = K-11-way = K-1 2-way = (# of Rows -1) (# of Columns -1)2-way = (# of Rows -1) (# of Columns -1)

CHI SQUARE  The most simple form of the Chi square is the one-way Chi square test Used to determine whether frequencies observed differ significantly from an even (expected under null) distributionUsed to determine whether frequencies observed differ significantly from an even (expected under null) distribution

Chi Square: Steps 1. Find the expected (under null hypothesis) cell frequencies 2. Compare expected & observed frequencies cell by cell 3. If null hypothesis is true, expected and observed frequencies should be close in value 4. Greater the difference between the observed and expected frequencies, the greater the possibility of rejecting the null

1-WAY CHI SQUARE  1-way Chi Square Example: There is an even distribution of membership across 4 political parties (N=40 UMD students) Find the expected cell frequencies ( F e = N / K) Find the expected cell frequencies ( F e = N / K) Categories FoFoFoFo FeFeFeFe Republican1210 Democrat1410 Independ.910 Green510

1-WAY CHI SQUARE  1-way Chi Square Example: There is an even distribution of membership across 4 political parties (N=40 UMD students) Compare observed & expected frequencies cell-by-cell Compare observed & expected frequencies cell-by-cell Categories FoFoFoFo FeFeFeFe f o - f e Republican12102 Democrat14104 Independ.910 Green510-5

1-WAY CHI SQUARE  1-way Chi Square Example: There is an even distribution of membership across 4 political parties (N=40 UMD students) Square the difference between observed & expected frequencies Square the difference between observed & expected frequencies Categories FoFoFoFo FeFeFeFe f o - f e (f o - f e ) 2 Republican Democrat Independ.9101 Green

1-WAY CHI SQUARE  1-way Chi Square Example: There is an even distribution of membership across 4 political parties (N=40 UMD students) Divide that difference by expected frequency Divide that difference by expected frequency Categories FoFoFoFo FeFeFeFe f o - f e (f o - f e ) 2 (f o - f e ) 2 /f e Republican Democrat Independ Green ∑=∑=∑=∑=4.6

Interpreting Chi-Square  Chi-square has no intuitive meaning, it can range from zero to very large As with other test statistics, the real interest is the “p value” associated with the calculated chi-square value As with other test statistics, the real interest is the “p value” associated with the calculated chi-square value Conventional testing = find χ 2 (critical) for stated “alpha” (.05,.01, etc.)Conventional testing = find χ 2 (critical) for stated “alpha” (.05,.01, etc.) Reject if χ 2 (observed) is greater than χ 2 (critical) Reject if χ 2 (observed) is greater than χ 2 (critical) SPSS: find the exact probability of obtaining the χ2 under the null (reject if less than alpha)SPSS: find the exact probability of obtaining the χ2 under the null (reject if less than alpha)

The Chi-Square Sampling Distribution (Assuming Null is True)

Interpreting χ 2 The old fashioned way  Chi square = 4.6  df (1-way Chi square) = K-1 = 3  X 2 (critical) (p<.05) = (from Appendix C)  Obtained (4.6) < critical (7.815)  Decision Fail to reject the null hypothesis. There is not a significant difference in political party membership at UMD Fail to reject the null hypothesis. There is not a significant difference in political party membership at UMD

2-WAY CHI SQUARE  For use with BIVARIATE Contingency Tables Display the scores of cases on two different variables at the same time (rows are always DV & columns are always IV) Display the scores of cases on two different variables at the same time (rows are always DV & columns are always IV) Intersection of rows & columns is called “cells” Intersection of rows & columns is called “cells” Column & row marginal totals (a.k.a. “subtotals”) should always add up to N Column & row marginal totals (a.k.a. “subtotals”) should always add up to N N=40 Packers Fan Vikings Fan TOTALS Like Brett Favre Don’t Like Favre TOTALS:202040

Null Hypothesis for 2-Way χ2 The two variables are independent The two variables are independent Independence: Independence: Classification of a case into a category on one variable has no effect on the probability that the case will be classified into any category of the second variableClassification of a case into a category on one variable has no effect on the probability that the case will be classified into any category of the second variable

N=40 Packers Fan Vikings Fan TOTALS Like Brett Favre 14 (10.5) 7 (10.5) 21 Don’t Like Favre 6 (9.5) 13 (9.5) 19 TOTALS: WAY CHI SQUARE  Find the expected frequencies F e = Row Marginal X Column Marginal F e = Row Marginal X Column Marginal N “Like Favre” Row = (21 x 20)/40 =420/40=10.5 “Don’t Like” Row = (19 x 20)/40 = 380/40= 9.5

2-WAY CHI SQUARE Compare expected & observed frequencies cell by cellCompare expected & observed frequencies cell by cell X 2 (obtained) = 4.920X 2 (obtained) = df= (r-1)(c-1) = 1 X 1 = 1df= (r-1)(c-1) = 1 X 1 = 1 X 2 (critical) = (Healey Appendix C)X 2 (critical) = (Healey Appendix C) Obtained > CriticalObtained > Critical CONCLUSION:CONCLUSION: Reject the null: There is a relationship between the team that students root for and their opinion of Brett Favre (p<.05). Reject the null: There is a relationship between the team that students root for and their opinion of Brett Favre (p<.05).

Chi Square – Example #2 Is quality of a school system significantly related to a community’s per capita income? Is quality of a school system significantly related to a community’s per capita income? Per Capita Income QualityLowHighTotals Low18624 High Totals302050

Chi Square – Example #2 First, calculate expected frequencies… Per Capita Income QualityLowHighTotals Low 18 (14.4) 6 (9.6) 24 High 12 (15.6) 14 (10.4) 26 Totals302050