Analysis of Covariance

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Analysis of Covariance

• Qualitative variables
• Quantitative variables

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Example Kutner Ch 22 table 1
of populations Sampled??
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Run regression 3 times

• Once for each promotion
• Save output

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Simple Model
Covariate
Common Slope
Factor Level
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Simple Model
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Hypothesis testing

• Simple Model

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SPSS
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SPSS Simple Model
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SPSS - Simple
What is being tested ?
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Estimates
eq1 Predicted sales 17.368
.899prev_sales
eq2 Predicted sales 12.292
.899prev_sales
eq3 Predicted sales 4.391
.899prev_sales
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Compare Output

• Did we same results??
• What about Sigma?

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More Complex Model
Covariate
Different Slopes
Factor Level
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Complex Model
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Hypothesis testing

• Complex Model

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SPSS - Complex
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SPSS - Complex
What is being tested?
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SPSS Complex
What are the prediction equations ? Same result
as when we did the 3 separately?
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Which model is appropriate?

• Simple ??
• Complex ??
• We do not know at this point

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Need to test
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Matrix Setup

• Design Matrix
• H matrix

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L Matrix

• /lmatrix betas all 0 0 0 1 -1 0
• all 0 0 0 1 0 -1

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Additional topics

• Expected Marginal Means
• Test at some other X
• RSQ
• Design Matrix

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Problems Areas

• Multi-colinearity
• Problem Points
• Non-constant variance as a function of the
  independents bp test on design matrix
• Variable selection

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Additional Topics In ANCOVA
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Subset Selection In ANACOV

• Much more complex
• Uses - stepwise

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Steps

• Setup Model
• Get Design Matrix
• Merge design with original data
• Fit Y versus factors save residuals - r
• Fit multivariate (factorscovariates) versus
  factors save residuals rr
• Fit r versus rr use stepwise
• Take final model Use ANCOVA

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"Statistical Issues Associated with Analysis of
Weight Change in Animals."

• or
• Some Things You Might Not Know

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Two Major Topics Today

• Weight Gain (Pre-Post test before-after
  baseline data) very simple approach
• Analysis of Covariance What is Really Being
  Tested?

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Weight Gain
Treatment
BEFORE X
AFTER Y
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Usual Analysis
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Example - Milliken Johnsonpre_post.sav

• Cholesterol Study
• Pre and Post
• 4 Diets

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Data
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Results - Usual
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Trouble In River City ??

• Not So Fast General Approach
• Analysis of Covariance

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From the General to the Usual
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Results - RSQUARED
Usual - R Squared .131 (Adjusted R Squared
.038)
General - R Squared .542 (Adjusted R Squared
.409)
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Usual Prediction Model
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General Prediction Model
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Computer Model - SPSS
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More Results - General
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What is Being Tested ? Depends Upon The Computer
Package!!
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DietPre_Ch
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Intercept
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Pre_Ch
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A Different Look