The Analysis of Covariance

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The Analysis of Covariance

• Chapter 16

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Example 16.14
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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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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 ?
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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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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
• Variable selection

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Analysis of Covariance

• Another Approach

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Unequal Slopes
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Unequal slopes - Another way to write the model
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Origin in the Anova Table

• What being tested is
• All the intercepts all equal
• this, in most cases, is extreme extrapolation
• who cares about X0
• EMM asks that the lines are equal for X the
  mean of X
• You might want to choose a different X
• SPSS does not allow Post Hoc Tests

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Weight and OriginWeight In the Anova Table

• Weight - if the sums of the slopes is 0
• OriginWeight is testing is the deltas are zero
• If yes, then all slopes are equal

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Example

• In this example the slopes are not all equal -
  why?
• What are the three slopes?
• We need the parameter output

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