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Multivariate Correlational
Research (Part 2)
Dr. Stefanie Drew
[email protected]
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Review: The three causal rules
• Is there covariance?
• Is there temporal precedence?
• Are there third variables that could explain the
relationship?
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Advanced Correlational Techniques
Multivariate Techniques
• Longitudinal Designs
▫ Establishing temporal precedence
• Multiple Regression Design
▫ Ruling out third variable
• “Pattern and Parsimony” Approach
▫ Results of variety of correlational studies all
Note* In all techniques,
support causal theory
variables are measured not
manipulated
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Multivariate Design
• Involves more than two measured variables
• Widely used, especially when experiments are
not an option
Variable 1
Variable 2
Variable 3
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Multiple Regression
• Multiple Regression: statistical
technique used to test for the
influence of third variables
• Conducting multivariate
correlational study
Image adapted from pn.bmj.com/content/7/4/259.extract on March 19, 2014
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Barros et al., 2009:
Examining Recess Time and Behavior
• Two key variables
▫ Recess time: number of
minutes of recess
▫ Behavior problems:
standardized measurement
teacher rates
• Not binary correlation, because
measuring additional variable
with intention to look at
interrelationships
▫ Number of economically
disadvantaged children
▫ Public or private classrooms
▫ Class size
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What is the overall point of multiple
regression analysis?
• Test whether association of
•
interest still true when statistically
controlling for suspected third
variables
▫ Question: Does the relationship
between recess and behavior
problems stay within different
levels of income?
Control for: holding a potential
third variable steady while
investigating the association
between two other variables
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The Process
• Researchers examine third variables
▫ E.g. economic disadvantage to see if it correlates with other two
variables of interest
• Researcher must examine the data when controlling for
other variable
▫ E.g. when they take into account relationship between behavior
and income, does recess still have an effect on behavior
• Similar to identifying subgroups
▫ E.g. does bivariate relationship remain across all subgroups
Look within an income bracket, and see if behavior and recess
correlated
Look at next income bracket, see if behavior and recess correlated.
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Barros et al., 2009
Examining Recess Time and Behavior
• Find overall relationship is negative
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• Overall relationship is negative
Possible Outcome
• ALSO…
▫ Classrooms with poorest children ($)
have more
behavior problems
and shorter
recess
▫ Classrooms with wealthiest children
($$$) have fewer
behavior
problems and longer
recess
• If only look at one subset, _do_
still see negative relationship between
recess and behavior
• Conclusion: relationship between
recess and behavior still there when
controlling for income
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• Overall relationship is negative
• ALSO…
Alternative Outcome ▫
Classrooms with poorest children
behavior
($) have more
problems and shorter
recess
▫ Classrooms with wealthiest
children ($$$) have
fewer
behavior problems and longer
recess
• If only look at one subset, do not
still see negative relationship
between recess and behavior
• Conclusion: relationship between
recess and behavior not there
when controlling for income
▫ Income was the third variable
responsible for relationship
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Using Regression
• Testing whether a key relationship remains
when statistically controlling suspected third
variable
• Which of the two possible outcomes best
describes the relationship between recess and
behavior?
vs.
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Terminology
• Criterion variables (a.k.a. dependent variables): the
variable interested in understanding/predicting
▫ Example: behavior problems
• Predictor variables (a.k.a. independent variables):
the remaining variables measured in analysis
▫ Example: amount of recess and levels of income
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Regression tables
• Criterion variable typically goes in top
row or title
• Column for Beta Values
▫ (beta, β or standardized beta)
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Beta
• A.k.a. β , standardized beta
• One beta for each predictor variable
• Can compare (within same table) different
predictor variables to each other based on their
β values
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Beta vs r
• Similar to “r” in terms of relationship direction
▫ When other predictor variables are controlled for
+β: positive relationship between predictor and criterion variables
-β: negative relationship between predictor and criterion variables
β = 0: no relationship
• Similar to “r” in terms of relationship strength
▫ The larger β, the stronger the relationship
▫ The smaller β, the weaker the relationship
• Different from “r”
▫ No clear indication on effect size. Why?
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b values
• A.k.a. unstandardized coefficient
• Similar to β in terms of relationship direction
▫ When other predictors are controlled for
+b: positive relationship between
-b: negative relationship between
• Cannot compare two bs within same table. Why?
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A second look at Regression tables
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What about significance?
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Additional Predictors
• β indicates relationship present
between predictor and criterion
variables when all other
predictors held constant
• What is the purpose of adding
additional predictors?
▫ Helps control “third” variables
at same time
▫ Can look at other predictor
variables and get an idea of
other contributing factors
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Summary
Measurement
Association
r bivariate correlations + r: positive association
- r: negative association
r = 0: no association
β multiple regression
+ β: positive association
(when other predictors controlled for)
- β: negative association
(when other predictors controlled for)
β = 0: no association
(when other predictors controlled for)
b multiple regression
+ b: positive association
(when other predictors controlled for)
- b: negative association
(when other predictors controlled for)
b= 0: no association
(when other predictors controlled for)
Can compare
within table?
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A note about popular press…
• Studies may report a simple relationship but use
phrases
▫ “have been controlled for”
▫ “taking into account”
▫ “correcting for”
• Phrases indicate using multiple regression
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Multiple Regression
• Basically… is the association still present when
you break it down into subgroups?
• Problems
▫ Controls third variable problem, but does not
account for temporal precedence
▫ Can only control third variables researchers
measure (e.g. not unknown third variables)
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Pattern and Parsimony
• Converging evidence from
several studies
▫ Variety of methods used
• Using most parsimonious
explanation for an entire
pattern of data
• Example: smoking and lung
cancer
▫ Multiple studies looking at
the toxicity of cigarettes
Study
2
Study
1
Study
3
Single Theory
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Another note about popular press…
• Journalists may only report results of recent
study (don’t address full pattern)
• Selectively representing research (without giving
full context)
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Multivariate Design and Four Validities
• Construct
▫ How well is each variable measured?
• External
▫ Do results generate to other populations?
▫ Were samples taken randomly?
• Statistical
▫ What is the effect size?
▫ Is it statistically significant?
▫ Are there subgroups, outliers, curvilinear relationships?
• Internal
▫ Get closer to causal statements
Longitudinal designs establish temporal precedence
Multiple regression designs help rule out third variables
Exit Ticket
EXIT