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
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
X2
X1
Y
Controlling for Other Explanatory Variables
• Categorical (Qualitative) Variables:
– Partial Tables: Contingency Tables showing X1-Y
relationship, separately for each level of X2
• Numeric (Quantitative) Variables:
– Mean and Std. Deviation of Responses (Y) versus
groups (X1), separately for each level of X2
– Regression of Y on X1, controlling for level of X2
(Multiple Regression)
Types of Multivariate Relationships
• Chain Relationships: X1 leads to changes in
(causes) X2 which in turn leads to changes (causes) Y.
X1 has an indirect effect on Y through the
intervening variable X2. X1-Y association vanishes
after controlling X2
X1  X2  Y
• Multiple Causes: X1 and X2 each have a direct effect
on Y. They can also have direct and indirect effects:
X1
Y
X2
X1
Y
X2
Types of Multivariate Relationships
• Suppressor Variables: No association appears
between X1 and Y until we control X2
• Statistical Interaction: The statistical association
between X1 and Y depends on the level of X2
X2
X1
Y
• Simpson’s Paradox: When direction of association
between Y and X1 is in opposite direction for all levels of
X2 as the direction of association when not controlling X2
Other Inferential Issues
• Sample Size: When controlling for X2, the sample
sizes can be quite small and you may not obtain
statistical significance for the X1-Y association
(lack of power)
• Categorization: When X2 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)