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MEASURES OF ASSOCIATION CHI SQUARE, CORRELATION, REGRESSION Research Methods

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Relationships Between Two (or More) Variables I Association: The relationship to which two variables covary Direction: Positive, Negative, or No relationship Magnitude: Coefficient of Association: (0,1) or (-1,1) Significance (Statistical Significance)

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Relationships Between Two (or More) Variables II Significance (Statistical Significance): The likelihood that the association is due to chance and is not true of the population Or (inversely) level of confidence in the results We dont expect to find the exact value of a parameter, but something close (better as the random sample is larger)

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Measures of Association (very abbreviated) Nominal data: Lambda (Statistical significance: Chi-square) (Reduction in error: Tau-b) Ordinal data: Gamma (Statistical significance: Chi-square or Gamma) (Reduction in error: Tau) Interval or Ratio data: Pearson Correlation Coefficient (Statistical significance: t or Z) (Explained variance or reduction in error: R 2 )

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Multivariate Analysis: Three or More Variables Control for the effects of other factors: Value of DV is probably not influenced solely by the IV of interest. What else influences DV? Alternative rival hypotheses: To persuade your audience, show that your theory and evidence is better than, or adds to, other possible explanations Causality (earlier in semester)

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Regression Example: Size of Legislatures How large should a legislature be? …no political problem is less susceptible of a precise solution than that which relates to the number most convenient for a representative legislature The Federalist, No. 55 Theory suggests legislature size should be a function of population Stigler (1972): logged population Taagepera (1972), Taagepera and Shughart (1989): cube root of population

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Simple Regression

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How Large Should a State Legislature Be? H0: Si = Ŝ Representative bodies fulfill functions and, for a certain level or type of government, all legislative chambers should be of the same size (S). Regardless of the polity (i) represented, an optimal size for a legislature exists for given functions. H1: Si = f( Pi ) Chamber size is positively correlated with population (P) of the polity (through some transformation). H2: Ci = f( Xi ) Chamber size varies with regard to characteristics of the polity represented. These characteristics include population, political culture, geography, ethnic or other cleavages, social mobilization, the economy, and interest diversity.

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Size of Legislatures

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Simple Regression

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Factors that May Contribute to Legislature Size

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Multiple Regression

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Crosstabulation Example: Gender and Party Is there a Gender Gap between two parties? Which is the dependent variable and which is the independent variable? Compute percentages across the dependent variable, that is, sum to 100% for each independent variable.

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Partisanship by Gender

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Partisanship by Gender: Significance and Association

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Crosstabulation Example: Ideology and Party Are the two parties ideological distinct? Which is the dependent variable and which is the independent variable? [Here, party=f(ideology), not all would agree!] Again, compute percentages across the dependent variable, that is, sum to 100% for each independent variable.

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Partisanship by Ideology

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An Example: Simpsons Paradox I

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Simpsons Paradox II

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Same Example: Spurious Relationship I

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Spurious Relationship II

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Causality Correlation is not causation Null hypothesis can be rejected, not proven or accepted Causality can be rejected, not proven or accepted Two variables may show a relationship or association. But does one cause the other?

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Suggesting Causality Covariation or Association Demonstrated Time Order Shown Causal Linkage Explained Alternative Explanations Eliminated

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