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Multivariate Relationships Goal: Show a causal relationship between two variables (X Y) Elements of a cause-and-effect relationship: –Association between variables (based on methods we’ve covered this semester) –Correct time order (X occurs before Y) –Elimination of alternative explanations (variable Z that acts on both X and Y, making them appear to be associated) –Anecdotal evidence does not rule out causality

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Controlling for Other Variables Observational studies: Researchers are unable to control levels of variables and may only observe them as they occur in nature Statistical Control: Identifying individuals (cases) by their level of an alternative explanatory (control) variable (although not assigning subjects the levels) Spurious Association: When both variables of interest are dependent on a third variable, and their association vanishes when controlling for the other variable X2X2 X1X1 Y

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Controlling for Other Explanatory Variables Categorical (Qualitative) Variables: –Partial Tables: Contingency Tables showing X 1 -Y relationship, separately for each level of X 2 Numeric (Quantitative) Variables: –Mean and Std. Deviation of Responses (Y) versus groups (X 1 ), separately for each level of X 2 –Regression of Y on X 1, controlling for level of X 2 (Multiple Regression)

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Types of Multivariate Relationships Chain Relationships: X 1 leads to changes in (causes) X 2 which in turn leads to changes (causes) Y. X 1 has an indirect effect on Y through the intervening variable X 2. X 1 -Y association vanishes after controlling X 2 X 1 X 2 Y Multiple Causes: X 1 and X 2 each have a direct effect on Y. They can also have direct and indirect effects: X1X1 X2X2 Y X1X1 X2X2 Y

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Types of Multivariate Relationships Suppressor Variables: No association appears between X 1 and Y until we control X 2 Statistical Interaction: The statistical association between X 1 and Y depends on the level of X 2 X1X1 X2X2 Y Simpson’s Paradox: When direction of association between Y and X 1 is in opposite direction for all levels of X 2 as the direction of association when not controlling X 2

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Other Inferential Issues Sample Size: When controlling for X 2, the sample sizes can be quite small and you may not obtain statistical significance for the X 1 -Y association (lack of power) Categorization: When X 2 is quantitative there can be many partial tables/associations, with few observations. Multiple regression models help avoid this problem. Comparing Measures: Often we wish to compare estimates of a parameter across levels of the control variable. Can use 2-sample z-test (ch. 7)

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10. Introduction to Multivariate Relationships Bivariate analyses are informative, but we usually need to take into account many variables. Many explanatory.

10. Introduction to Multivariate Relationships Bivariate analyses are informative, but we usually need to take into account many variables. Many explanatory.

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