By various methods of selecting variables, 7 good models are considered with ...

[backcolor=rgba(232, 233, 238, 0.3)]By various methods of selecting variables, 7 good models are considered with following numerical results: 7. Table 1 Model Rad Terms R2 Cp 142.5 1 0.666 0.635 9.0771 X4 0.675 0.645 138.7 8.9639 3 X1, 12 0.979 0.974 2.7 2.4065 X1, X4 0.972 0.967 5.5 2.7343 X1, 12, 13 0.982 0.976 3.0 2.3121 X1, 12, 14 0.982 0.976 3.0 2.3087 X1,..., x4 0.982 0.974 5.0 2.4460 a) Choose the 'best' model from the above 7 models and give a suitable reason. b) The first two models in Table 1 are first two best models with one independent variable x2 and x4, respectively. If we consider the model with r2 and x4 as independent variables, we have R2 = 0.68, R = 0.616, and s = 9.321. It shows that this model is much worse than the model with r1 and r2 and the model with x1 and x4 as independent variables in Table 1. Explain this phenomenon. c) What is Rrea? Explain. d) Do you think R2ed from model 7 is lower or higher than Rd from model 3? Motivate your answer. pred
Expert Solution


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Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts for you. To get the remaining sub-part solved please repost the complete question and mention the sub-parts to be solved.

Given Information:

Consider the given Table:

Model	Terms	R2R2	R2adjR2adj	Cp	s
1	x2	0.666	0.635	142.5	9.0771
2	x4	0.675	0.645	138.7	8.9639
3	x1,x2	0.979	0.974	2.7	2.4065
4	x1,x4	0.972	0.967	5.5	2.7343
5	x1,x2,x3	0.982	0.976	3.0	2.3121
6	x1,x2,x4	0.982	0.976	3.0	2.3087
7	x1,…,x4	0.982	0.974	5.0	2.446

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a. Consider the criterion of adjusted R2 and s for the best Model:

• The model with the largest adjusted R2 would be considered as the best model.
• The model with the smallest value of s would be considered as the best model.
  

From the given Table-1, the largest adjusted R2is 0.976 and the smallest value of s is 2.3087. Based on the adjusted R2 and s criterion, the best model is the model with three independent variables x1,x2, and x4. Hence, the best model is Model 6.

b. Consider the model with predictors x2, and x4. For this model we have:

       <span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="R2=0.68,&#xA0;R2adj=0.616&#xA0;and&#xA0;s=9.321" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R2=0.68, R2adj=0.616 and s=9.321R2=0.68, R2adj=0.616 and s=9.321

Consider the values for the model with the predictors <span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="x1&#xA0;and&#xA0;x2" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">x1 and x2x1 and x2 and the model with <span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="x1&#xA0;and&#xA0;x4" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">x1 and x4x1 and x4

Model	Terms	R2R2	R2adjR2adj	Cp	s
3	x1,x2	0.979	0.974	2.7	2.4065
4	x1,x4	0.972	0.967	5.5	2.7343

The given new model with the predictors <span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="x2&#xA0;and&#xA0;x4" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">x2 and x4x2 and x4 is worse than the model with the predictors <span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="x1&#xA0;and&#xA0;x2" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">x1 and x2x1 and x2 and the model with the predictors <span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="x1&#xA0;and&#xA0;x4" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">x1 and x4x1 and x4 because if we consider the criterion of the adjusted R2 and s. The values of adjusted R2 and s  for the model with predictors <span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="x2&#xA0;and&#xA0;x4" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">x2 and x4x2 and x4 are low as compared to the model with the predictors <span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="x1&#xA0;and&#xA0;x2" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">x1 and x2x1 and x2 and the model with the predictors <span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="x1&#xA0;and&#xA0;x4" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">x1 and x4x1 and x4. Hence, the given model with the predictors <span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="x2&#xA0;and&#xA0;x4" role="presentation" style="box-sizing: inherit; display: inline; font-weight: normal; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">x2 and x4x2 and x4 is worse.

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c. <span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="R2pred" role="presentation" style="box-sizing: inherit; display: inline; font-style: normal; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R2predR2pred: The predicted R-square is a measure of how effectively a regression model predicts responses for new observations. The predicted R-square is used to determine whether or not the model fits the original data. The value of the predicted R-square can be negative and it is always lower than the R-squared value. The predicted R-square can be calculated by using the formula:

      <span class="MathJax" id="MathJax-Element-16-Frame" tabindex="0" data-mathml="R2pred=1-Predicted&#xA0;residual&#xA0;sum&#xA0;of&#xA0;squaressum&#xA0;of&#xA0;squares&#xA0;total" role="presentation" style="box-sizing: inherit; display: inline; font-style: normal; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R2pred=[1−Predicted residual sum of squaressum of squares total]R2pred=1-Predicted residual sum of squaressum of squares total