Question

1. Fill in the blanks in the following tables. The column labeled “Seq SS” represents “sequential sums of squares” (measures the reduction in the SS when a term is added to a model that contains only the terms before it), while the column labeled “Adj SS” represents “adjusted sums of squares” (measures the reduction in the SS for each term relative to a model that contains all of the remaining terms).

[Hint: The t-statistics in the Coefficients table assume all other predictors are included in the model, so if we square these we get the F-statistics in the ANOVA table based on Adjusted Sums of Squares.]

Source	df	Seq SS	Adj SS	F-statistic based on Adj SS	p-value based on Adj SS
Regression	3	8208.9		39.38	0.000
X1	1	5453.4	1772.2	25.50	0.000
X2	1	2551.7
X3	1			2.93	0.090
Error	93			----	-------
Total	96	14671.5	14671.5	----	-------

Coefficients

Term	Coef	SE coef	t-statistic	p-value
Constant	-15.71	4.60	-3.42	0.001
X1	2.638	0.522	5.05	0.000
X2	0.5108	0.0842
X3	0.0106	0.00620	1.71	0.090

2. Calculate SSR(X₃|X₁), that is the sequential sum of squares obtained by adding X₃ to a model already containing only the predictor X₁. Show your work.
3. Explain in words what is measured by the quantity calculated in the previous part.
4. Discuss the conceptual difference between the sequential sum of squares (Seq SS) and adjusted sum of squares (Adj SS) in terms of the predictor X₂. For this data, what are the numerical values of these sums of squares for the predictor X₂?
5. Calculate the value of an F-statistic for testing H₀: β₂ = β₃ = 0 within the model Y_i = β₀ + β₁ X_i,1 + β₂ X_i,2 + β₃ X_i,3 + ε_i. It is not necessary to carry out the test – just calculate the value of F. Show your work.
6. Calculate the value of the coefficient of partial determination R²_Y,2|1.
7. Write a sentence that interprets the value calculated in the previous part.

Answer 1

Given that

a) It is already solved in the question

b) The sequential sum of squares tells us how much the SSE declines after we add another variable to the model that contains only the variables preceding it. By contrast, the adjusted sum of squares tells us how much the SSE declines after we add another variable to the model that contains every other variable.

So if you start with zero predictors and add X1, SS Regression increases by 5434.4. Then if you also add X3, the SS Regression increases by an additional 203.8. These are the sequential SS, which add up to the total SS Regression of 8208.9 if you add X2 too.

Thus we can say SSR(X3|X1) = 203.8