1. Model Comparison: some basic concepts
Lecture 9
Review of Lecture 8
Model Comparison: some basic concepts
12/7/2018
ST3131, Lecture 9

2. Review of Lecture 8: Properties of the LS estimates
1). Linearity
2). Unbiased
3). BLUE estimators (Best Linear Unbiased Estimators)
4) Normality
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ST3131, Lecture 9

3. Variances of the Estimates
Noise variance estimator and Standard Errors
12/7/2018
ST3131, Lecture 9

4. Review of Lecture 8: Inferences for Individual Coefficients
Hypothesis Testing
Confidence Interval
12/7/2018
ST3131, Lecture 9

5. Full Model: include ALL predictor variables of interest
Model Comparison
Full Model: include ALL predictor variables of interest
Reduced Model: include PART of the predictor variables of interest
Or the regression coefficients satisfy some specified constraints
Full Model+
Two Advantages: 1) Reduced Models are simple.
2) Emphasize some of the parameters/effects/factors
12/7/2018
ST3131, Lecture 9

6. H0: Reduced model is adequate
Nested Models: A set of models are said to be nested if they can be obtained from a larger model as special cases. For example, the following models are nested:
Model Testing:
H0: Reduced model is adequate
H1: Full model is adequate
Remark: Reduced Models are simpler than Full Models since the former contain fewer parameters. If Reduced Models are verified to be adequate, there is no reason why we should NOT use them.
12/7/2018
ST3131, Lecture 9

7. Common Cases for Model Testing
Case 1: ALL NON-intercept coefficients are zero
Case 2: SOME of the coefficients are zero
Case 3: SOME of the coefficients are EQUAL to each other
Case 4: Other specified CONSTRAINTS on coefficients
The resulting models are Reduced Models of the Full Models. We will develop a UNIFIED TEST for the above cases. The MAIN IDEA is to compare the SQUAREs explained by the EXTRA Coefficients in the Full Models with the SSE under the Full Model. We need two concepts for this Unified Test: Squares explained by the extra coefficients, and the degrees of freedom.
12/7/2018
ST3131, Lecture 9

8. Let SSE(F) be the SSE of the Full Model, SSE(R) be the SSE of the Reduced Model. Then
This is because the Full Model uses MORE Coefficients than the Reduced Model. The SQUAREs explained by the EXTRA coefficients in the Full Model are
Note that the SST for the Reduced Model and the Full Model are the , why?
12/7/2018
ST3131, Lecture 9

9. Degree of Freedom: Statistic T has k degrees of freedom if T can be written as the Squared Sum of k INDEPENDENT Normal Random Variables with mean zero and the same variance. For example,
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ST3131, Lecture 9

10. Now we are ready to define the Unified Test: F-test:
H0: Reduced Model is adequate
H1: Full Model is adequate
Use F-statistic
F is the RATIO of the Average Squares explained by the EXTRA dfs in the Full Model to the Average Squares unexplained by the Full Model ( the noise variance estimator).
12/7/2018
ST3131, Lecture 9

11. When F-value is large, i. e
When F-value is large, i.e., the average squares explained by the extra dfs in the Full Model is large compared to the average squares unexplained by the Full Model, we may think the Reduced Model is not adequate. That is, we reject H0 when
or when the P-value
Notice that
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ST3131, Lecture 9

12. After-class Questions: What is the main purpose for model comparison?
Why can we use F-test to compare different models?
Can we use R-square or adjusted R-square to compare different models?
12/7/2018
ST3131, Lecture 9