1. Incremental Partitioning of Variance (aka Hierarchical Regression)
Incremental Partitioning=التقسيم التدريجي. Hierarchical Regression=الانحدار الهرمي
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2. What is hierarchical regression?
Use with survey any thing except experimental data.
SRM 625 Applied Multiple Regression, Hutchinson

3. Why is it used?
To test effects of one or more IV's after controlling for one or more other IV's
To determine the amount of variance a particular variable (or set of variables) explains, above and beyond what another variable (or set of variables) already explains
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4. How does this differ from examination of part/partial correlations in a standard regression analysis?
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5. When to use hierarchical regression
When your purpose is explanation
When you have a solid theoretical basis for determining
which variables to control
order of entry into the equation
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6. When should you not use hierarchical regression?
When your purpose is prediction
When you are interested in testing relative importance of predictors
Why is this inappropriate for hierarchical regression?
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7. Determining order of variable entry
Based on theory or prior research
e.g., prior research suggests a particular variable may "cause" both Y and your IV of interest
Based on logic
e.g., you would not control for a variable which occurs after your primary IV
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8. What is wrong with the logic of this analysis?
Example:
A researcher wishes to determine if certain dispositional
or personality characteristics account for whether or not
freshmen drop out of school. He administers several
personality inventories during the 1st week of class. At
the end of the freshman year he administers a survey to
measure college experiences. He then conducts a hierarchical
regression to determine if dispositional characteristics
explain a significant proportion of variance in likelihood
of dropping out, after controlling for college experiences.
What is wrong with the logic of this analysis?
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9. Other considerations in determining order of entry
Typically the variable(s) of primary interest is(are) entered last
Order should not be based on "giving variables a chance" by entering first
Order should not be based on examination of bivariate correlations
SRM 625 Applied Multiple Regression, Hutchinson

10. Sets of variables may be used either as control variables or as variables of primary interest
Use sets when you have conceptually and empirically related IV's
e.g., subscales on a personality inventory
Use sets when you have non-causally correlated exogeneous variables
e.g., SES and school resources
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12. How does one conduct a hierarchical regression analysis?
Run the regression analysis in a series of steps
A hierarchical regression requires a minimum of two steps
Step 1 enter control variable(s)
Step 2 enter variable(s) of primary interest
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13. Then conduct a test of the increment (or change) in R2 at step 2
i.e., this is a test of the increase in R2 produced by the additional variable(s) above and beyond what the control variables explained
Hierarchical regression may involve > 2 steps, in which case successive improvements in R2 are tested at each step
Step 2: how much R2 increased.
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14. Determining the increment in R2
The increment in R2 is the same as a squared part correlation
i.e., R2 change = R2 step 2 - R2 step 1 or
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15. increment added at step 3 = R2 step 3 - R2 step 2 or
Same general procedure applies when you have > 2 steps in the analysis
increment added at step 3 = R2 step 3 - R2 step 2 or
SRM 625 Applied Multiple Regression, Hutchinson

16. Likewise, the increment in R2 added by a “set” of variables is tested using squared, part correlations.
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17. Testing the increment in R2
There are two general approaches to testing the R2 at each step
They differ in choice of error term to use
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18. Approach I: Using Model I Error
Model I error uses the error term at the step being tested, with subsequent IV's included in the error term (i.e., based on the full model)
The error term and its degrees of freedom change at each step as variables are added into the equation
SRM 625 Applied Multiple Regression, Hutchinson

19. For example, for a 2-step analysis with two variables at
step 1 and one variable added at step 2, you would
conduct two F tests -- one for the control variables and
one for the IV of primary interest.
Step 1:
Step 2:

20. How would you more generally express the F test equations when using the model 1 error approach (regardless of the number of variables at each step)?
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21. Approach II: Using Model II Error
Model II error uses the error term from the last step (i.e., full model) after all variables have been entered
All tests of increments in R2 are based on the same mean square error and degrees of freedom
Analogous to error term in ANOVA where all IV's are tested against a single error term
SRM 625 Applied Multiple Regression, Hutchinson

22. Using the same example on Slide 19, how would you
conduct the two F tests -- one for the control variables
and one for the primary IV of interest, using the model
II error (aka "pure" error)?
Step 1:
Step 2:

23. Advantages of Model II approach
Removes systematic sources of variance from the error term
Generally more powerful than the Model I approach since error term will tend to be smaller
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24. Disadvantages of Model II approach
Can be less powerful due to loss of error degrees of freedom if IV's at subsequent steps add only small increments to R2, and there are numerous IV's in the overall model
Usually requires hand calculation since not available in most standard statistical packages
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25. Advantages of Model I approach
Can be more powerful if increments in R2 are trivial and overall model contains many variables
Easily obtained in standard statistical packages
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26. Disadvantages of Model I approach
May violate assumption of random error if IV's left in the error term are significant
Generally less powerful since error term contains variance due to subsequent IV's
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27. How do you choose between the two approaches?
If IV's are trivial, and numerous IV's are included in the model, use the Model I approach
If IV's add substantial increments to R2 and/or the model does not have many IV's use the Model II approach
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28. Bottom line: take the approach which has greater power for your variables and data set
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29. Other cautions and issues
Increment in R2 will depend not only upon the correlation between the variable(s) of interest and Y, but also upon how correlated the independent variables are with each other
Why?
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30. cautions – cont’d
Results of hierarchical regression should not be used to compare increments in R2 across samples
R2 is sample specific and is affected by a number of factors including variability and reliability of the variables
Reliability=trust
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