Q3.5: Model interpretation – Applied Regression Modeling, 3rd edition

Answer the following questions based on Section 3.5: Model interpretation.

Consider the following data:

X₁	0.5	0.8	1.0	1.7	1.4	1.3	1.8	1.2	1.1	0.9
X₂	0.7	0.4	0.3	0.2	1.0	0.6	0.5	0.4	0.8	0.9
X₃	0.5	0.4	0.3	0.9	0.6	0.8	0.8	0.5	0.9	0.7

Use statistical software to fit a multiple linear regression model with response variable, Y, and predictor variables, X₁, X₂, and X₃. Use the fitted model to answer the following questions.

1.

What conclusion do we draw from the global usefulness test p-value of 0.0002 at a 5% significance level?

There is no significant linear association between Y and any of X1, X2 or X3.

There is a significant linear association between Y and all of X1, X2 and X3.

There is a significant linear association between Y and at least one of X1, X2 and X3.

There is a significant linear association between Y and at least two of X1, X2 and X3.

2.

What conclusion do we draw from the X₁ regression parameter p-value of 0.025 at a 5% significance level?

There is a significant linear association between Y and X1.

There is a significant linear association between Y and X1, holding X2 and X3 constant.

There is a significant linear association between Y and X1, ignoring X2 and X3.

There is a significant linear association between Y and X1, when X2 and X3 increase.

3.

How do we interpret the estimated regression parameter, b₁-hat = 0.742?

We expect Y to be 0.742 when X1 = 0.

We expect Y to increase by 0.742 units for each 1-unit increase in X1.

We expect Y to increase by 0.742 units for each 1-unit increase in X1, holding X2 and X3 constant.

We expect Y to increase by 0.742 units for each 1-unit increase in X1, when X2 and X3 are both 0.

4.

How do we interpret the regression standard error, s = 0.2175?

21.75% of the variation in Y can be accounted for by a multiple linear regression association between Y and (X1, X2, X3).

21.75% of the variation in (X1, X2, X3) can be accounted for by a multiple linear regression association between Y and (X1, X2, X3).

We can expect to predict Y to an accuracy of approximately ± 0.2175 (at a 95% confidence level).

We can expect to predict Y to an accuracy of approximately ± 0.4350 (at a 95% confidence level).

5.

How do we interpret the coefficient of determination, R² = 0.9564?

95.64% of the variation in Y can be accounted for by a multiple linear regression association between Y and (X1, X2, X3).

95.64% of the variation in (X1, X2, X3) can be accounted for by a multiple linear regression association between Y and (X1, X2, X3).

We can expect to predict Y to an accuracy of approximately ± 0.9564 (at a 95% confidence level).

We can expect to predict Y to an accuracy of approximately ± 1.9128 (at a 95% confidence level).

Applied Regression Modeling

3rd edition

Iain Pardoe