Making Comparisons All hypothesis testing follows a common

Making Comparisons • All hypothesis testing follows a common logic of comparison • Null hypothesis and alternative hypothesis – mutually exclusive – exhaustive • “Republicans have higher income than Democrats”? – Descriptive, relational, and causal

Experimental Design • Draw a random sample • Manipulate the independent variable through treatment or intervention • Random assignment into experimental and control groups • Control (keep constant) other outside factors • Observe the effect on the dependent variable

Methods of Making Comparisons Independent Variable Categorical measures (nominal or ordinal) Continuous measures (interval or ratio) Categorical Cross-Tabulation measures (Chapter 10) & (Chapter 7) Chi(nominal or Logistic Regression square ordinal) Dependent Variable Continuous Compare Means & (Chapter 8) measures (Chapter 9) Correlation & (interval or Dummy Variables Linear Regression ratio)

Inferences about Sample Means • Hypothesis testing is an inferential process • Using limited information to reach a general conclusion • Observable evidence from the sample data • Unobservable fact about the population • Formulate a specific, testable research hypothesis about the population

Null Hypothesis • no effect, no difference, no change, no relationship, no pattern, no … • any pattern in the sample data is due to random sampling error

Errors in Hypothesis Testing • Type I Error – A researcher finds evidence for a significant result when, in fact, there is no effect (no relationship) in the population. – The researcher has, by chance, selected an extreme sample that appears to show the existence of an effect when there is none. – The p-value identifies the probability of a Type I error.

Cross-tabulation • Relationship between two (or more) variables – Joint frequency distribution – Contingency table • Observations should be independent of each other – One person’s response should tell us nothing about another person’s response • Mutually exclusive and exhaustive categories

Cross-tabulation • If the null hypothesis is true, the independent variable has no effect on the dependent variable • The expected frequency for each cell Male Female Total Pro- ? ? 2 Anti- ? ? 8 Total 5 5 10

Expected Frequency of Each Cell • Expected frequency in the ith row and the jth column ……… (Eij) • Total counts in the ith row ……… (Ti) • Total counts in the jth column ……… (Tj) • Total counts in the table ……… (N)

• Observed frequencies: Male Female Total Pro- 0 2 2 Anti- 5 3 8 Total 5 5 10 Male Female Total Pro- 1 1 2 Anti- 4 4 8 Total 5 5 10 • Expected frequencies:

Chi-square (X 2) • For each cell, calculate: • (observed frequency - expected frequency)2 expected frequency • Add up the results from all the cells

Cross-Tabulation Independent Variable Nominal measures Lambda Cramer’s V gamma Lambda Kendall’s tau-b Ordinal measures Cramer’s V Kendall’s tau-c Somer’s d Nominal measures Dependent Variable Lambda Cramer’s V Ordinal measures

Measures of Association • Symmetrical measures of association – e. g. Kendall’s tau-b and tau-c • Asymmetrical measures of association – e. g. lambda and Somer’s d • Directional measures of association – e. g. Somer’s d • PRE measures of association – e. g. lambda and Somer’s d