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Variable selection and model building

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Variable selectionand model building

Part II

- A common situation is that there is a large set of candidate predictor variables.
- (Note: The examples herein are not really that large.)
- Goal is to choose a small subset from the larger set so that the resulting regression model is simple and useful:
- provides a good summary of the trend in the response
- and/or provides good predictions of response
- and/or provides good estimates of slope coefficients

- Stepwise regression: Enter and remove predictors, in a stepwise manner, until no justifiable reason to enter or remove more.
- Best subsets regression: Select the subset of predictors that do the best at meeting some well-defined objective criterion.

- The list of candidate predictor variables must include all the variables that actually predict the response.
- There is no single criterion that will always be the best measure of the “best” regression equation.

Best subsets regression

…. or all possible subsets regression

- Consider all of the possible regression models from all of the possible combinations of the candidate predictors.
- Identify, for further evaluation, models with a subset of predictors that do the “best” at meeting some well-defined criteria.
- Further evaluate the models identified in the last step. Fine-tune the final model.

- Response y: heat evolved in calories during hardening of cement on a per gram basis
- Predictor x1: % of tricalcium aluminate
- Predictor x2: % of tricalcium silicate
- Predictor x3: % of tetracalcium alumino ferrite
- Predictor x4: % of dicalcium silicate

- If there are p-1 possible predictors, then there are 2p-1 possible regression models containing the predictors.
- For example, 10 predictors yields 210 = 1024 possible regression models.
- A best subsets algorithm determines best subsets of each size, so that candidates for a final model can be identified by researcher.

- Different criteria quantify different aspects of the regression model, so can lead to different choices for best set of predictors:
- R-squared
- Adjusted R-squared
- MSE (or S = square root of MSE)
- Mallow’s Cp
- (PRESS statistic)

- R2 can only increase as more variables are added.
- Use R-squared values to find the point where adding more predictors is not worthwhile, because it yields a very small increase in R-squared.
- Most often, used in combination with other criteria.

Best Subsets Regression: y versus x1, x2, x3, x4

Response is y

x x x x

Vars R-Sq R-Sq(adj) C-p S 1 2 3 4

1 67.5 64.5 138.7 8.9639 X

1 66.6 63.6 142.5 9.0771 X

2 97.9 97.4 2.7 2.4063 X X

2 97.2 96.7 5.5 2.7343 X X

3 98.2 97.6 3.0 2.3087 X X X

3 98.2 97.6 3.0 2.3121 X X X

4 98.2 97.4 5.0 2.4460 X X X X

- Makes you pay a penalty for adding more predictors.
- According to this criterion, the best regression model is the one with the largest adjusted R-squared.

- According to this criterion, the best regression model is the one with the smallest MSE.
- Adjusted R-squared increases only if MSE decreases, so the adjusted R-squared and MSE criteria yield the same models.

Best Subsets Regression: y versus x1, x2, x3, x4

Response is y

x x x x

Vars R-Sq R-Sq(adj) C-p S 1 2 3 4

1 67.5 64.5 138.7 8.9639 X

1 66.6 63.6 142.5 9.0771 X

2 97.9 97.4 2.7 2.4063 X X

2 97.2 96.7 5.5 2.7343 X X

3 98.2 97.6 3.0 2.3087 X X X

3 98.2 97.6 3.0 2.3121 X X X

4 98.2 97.4 5.0 2.4460 X X X X

Mallow’s Cp statistic

- Cp estimates the size of the bias introduced in the estimates of the responses by having an underspecified model (a model with important predictors missing).

- If there is no bias, the expected value of the observed responses and the expected value of the predicted responses both equal μY|x.
- Fitting the data with an underspecified model, introduces bias, , into predicted response at the ith data point.

no bias

bias

Weight = -1.22 + 0.283 Height + 0.111 Water, MSE = 0.017

Weight = -4.14 + 0.389 Height, MSE = 0.653

- Because of bias, variance in the predicted responses for data point i is due to two things:
- random sampling variation
- variance associated with the bias

Sum the two variance components over all n data points to obtain a measure of the total variation in the predicted responses:

If there is no bias, Γp achieves its smallest value, p:

So, Γp seems to be a good measure of an underspecified model:

The best model is simply the one with the smallest value of Γp. We even know that the theoretical minimum of Γp is p.

If we know the population variance σ2, we can estimate Γp:

where MSEp is the mean squared error from fitting the model containing the subset of p-1 predictors (p parameters).

But we don’t know σ2. So, estimate it using MSEall, the mean squared error obtained from fitting the model containing all of the predictors.

- Estimating σ2 using MSEall :
- assumes that there are no biases in the full model with all of the predictors, an assumption that may or may not be valid, but can’t be tested without additional information.
- guarantees that Cp = p for the full model.

- Subset models with small Cpvalues have a small total (standardized) variance of prediction.
- When the Cp value is …
- near p, the bias is small (next to none),
- much greater than p, the bias is substantial,
- below p, it is due to sampling error; interpret as no bias.

- For the largest model with all possible predictors, Cp= p (always).

- Identify subsets of predictors for which the Cp value is near p (if possible).
- The full model always yields Cp= p, so don’t select the full model based on Cp.
- If all models, except the full model, yield a large Cp not near p, it suggests some important predictor(s) are missing from the analysis.
- When more than one model has a Cp value near p, in general, choose the simpler model or the model that meets your research needs.

Best Subsets Regression: y versus x1, x2, x3, x4

Response is y

x x x x

Vars R-Sq R-Sq(adj) C-p S 1 2 3 4

1 67.5 64.5 138.7 8.9639 X

1 66.6 63.6 142.5 9.0771 X

2 97.9 97.4 2.7 2.4063 X X

2 97.2 96.7 5.5 2.7343 X X

3 98.2 97.6 3.0 2.3087 X X X

3 98.2 97.6 3.0 2.3121 X X X

4 98.2 97.4 5.0 2.4460 X X X X

The regression equation is

y = 62.4 + 1.55 x1 + 0.510 x2 + 0.102 x3 - 0.144 x4

Source DF SS MS F P

Regression 4 2667.90 666.97 111.48 0.000

Residual Error 8 47.86 5.98

Total 12 2715.76

The regression equation is y = 52.6 + 1.47 x1 + 0.662 x2

Source DF SS MS F P

Regression 2 2657.9 1328.9 229.50 0.000

Residual Error 10 57.9 5.8

Total 12 2715.8

The regression equation is

y = 62.4 + 1.55 x1 + 0.510 x2 + 0.102 x3 - 0.144 x4

Source DF SS MS F P

Regression 4 2667.90 666.97 111.48 0.000

Residual Error 8 47.86 5.98

Total 12 2715.76

The regression equation is y = 103 + 1.44 x1 - 0.614 x4

Source DF SS MS F P

Regression 2 2641.0 1320.5 176.63 0.000

Residual Error 10 74.8 7.5

Total 12 2715.8

Best Subsets Regression: y versus x1, x2, x3, x4

Response is y

x x x x

Vars R-Sq R-Sq(adj) C-p S 1 2 3 4

1 67.5 64.5 138.7 8.9639 X

1 66.6 63.6 142.5 9.0771 X

2 97.9 97.4 2.7 2.4063 X X

2 97.2 96.7 5.5 2.7343 X X

3 98.2 97.6 3.0 2.3087 X X X

3 98.2 97.6 3.0 2.3121 X X X

4 98.2 97.4 5.0 2.4460 X X X X

The regression equation is

y = 71.6 + 1.45 x1 + 0.416 x2 - 0.237 x4

Predictor Coef SE Coef T P VIF

Constant 71.65 14.14 5.07 0.001

x1 1.4519 0.1170 12.41 0.000 1.1

x2 0.4161 0.1856 2.24 0.052 18.8

x4 -0.2365 0.1733 -1.37 0.205 18.9

S = 2.309 R-Sq = 98.2% R-Sq(adj) = 97.6%

Analysis of Variance

Source DF SS MS F P

Regression 3 2667.79 889.26 166.83 0.000

Residual Error 9 47.97 5.33

Total 12 2715.76

The regression equation is

y = 48.2 + 1.70 x1 + 0.657 x2 + 0.250 x3

Predictor Coef SE Coef T P VIF

Constant 48.194 3.913 12.32 0.000

x1 1.6959 0.2046 8.29 0.000 3.3

x2 0.65691 0.04423 14.85 0.000 1.1

x3 0.2500 0.1847 1.35 0.209 3.1

S = 2.312 R-Sq = 98.2% R-Sq(adj) = 97.6%

Analysis of Variance

Source DF SS MS F P

Regression 3 2667.65 889.22 166.34 0.000

Residual Error 9 48.11 5.35

Total 12 2715.76

The regression equation is

y = 52.6 + 1.47 x1 + 0.662 x2

Predictor Coef SE Coef T P VIF

Constant 52.577 2.286 23.00 0.000

x1 1.4683 0.1213 12.10 0.000 1.1

x2 0.66225 0.04585 14.44 0.000 1.1

S = 2.406 R-Sq = 97.9% R-Sq(adj) = 97.4%

Analysis of Variance

Source DF SS MS F P

Regression 2 2657.9 1328.9 229.50 0.000

Residual Error 10 57.9 5.8

Total 12 2715.8

Stepwise Regression: y versus x1, x2, x3, x4

Alpha-to-Enter: 0.15 Alpha-to-Remove: 0.15

Response is y on 4 predictors, with N = 13

Step 1 2 3 4

Constant 117.57 103.10 71.65 52.58

x4 -0.738 -0.614 -0.237

T-Value -4.77 -12.62 -1.37

P-Value 0.001 0.000 0.205

x1 1.44 1.45 1.47

T-Value 10.40 12.41 12.10

P-Value 0.000 0.000 0.000

x2 0.416 0.662

T-Value 2.24 14.44

P-Value 0.052 0.000

S 8.96 2.73 2.31 2.41

R-Sq 67.45 97.25 98.23 97.87

R-Sq(adj) 64.50 96.70 97.64 97.44

C-p 138.7 5.5 3.0 2.7

Best Subsets Regression: PIQ versus MRI, Height, Weight

Response is PIQ

H W

e e

i i

M g g

R h h

Vars R-Sq R-Sq(adj) C-p S I t t

1 14.3 11.9 7.3 21.212 X

1 0.9 0.0 13.8 22.810 X

2 29.5 25.5 2.0 19.510 X X

2 19.3 14.6 6.9 20.878 X X

3 29.5 23.3 4.0 19.794 X X X

Stepwise Regression: PIQ versus MRI, Height, Weight

Alpha-to-Enter: 0.15 Alpha-to-Remove: 0.15

Response is PIQ on 3 predictors, with N = 38

Step 1 2

Constant 4.652 111.276

MRI 1.18 2.06

T-Value 2.45 3.77

P-Value 0.019 0.001

Height -2.73

T-Value -2.75

P-Value 0.009

S 21.2 19.5

R-Sq 14.27 29.49

R-Sq(adj) 11.89 25.46

C-p 7.3 2.0

Best Subsets Regression: BP versus Age, Weight, ...

Response is BP

D

u

W r S

e a P t

i t u r

A g B i l e

g h S o s s

Vars R-Sq R-Sq(adj) C-p S e t A n e s

1 90.3 89.7 312.8 1.7405 X

1 75.0 73.6 829.1 2.7903 X

2 99.1 99.0 15.1 0.53269 X X

2 92.0 91.0 256.6 1.6246 X X

3 99.5 99.4 6.4 0.43705 X X X

3 99.2 99.1 14.1 0.52012 X X X

4 99.5 99.4 6.4 0.42591 X X X X

4 99.5 99.4 7.1 0.43500 X X X X

5 99.6 99.4 7.0 0.42142 X X X X X

5 99.5 99.4 7.7 0.43078 X X X X X

6 99.6 99.4 7.0 0.40723 X X X X X X

Stepwise Regression: BP versus Age, Weight, BSA, Duration, Pulse, Stress Alpha-to-Enter: 0.15 Alpha-to-Remove: 0.15

Response is BP on 6 predictors, with N = 20

Step 1 2 3

Constant 2.205 -16.579 -13.667

Weight 1.201 1.033 0.906

T-Value 12.92 33.15 18.49

P-Value 0.000 0.000 0.000

Age 0.708 0.702

T-Value 13.23 15.96

P-Value 0.000 0.000

BSA 4.6

T-Value 3.04

P-Value 0.008

S 1.74 0.533 0.437

R-Sq 90.26 99.14 99.45

R-Sq(adj) 89.72 99.04 99.35

C-p 312.8 15.1 6.4

The regression equation is

BP = - 12.9 + 0.683 Age + 0.897 Weight

+ 4.86 BSA + 0.0665 Dur

Predictor Coef SE Coef T P VIF

Constant -12.852 2.648 -4.85 0.000

Age 0.68335 0.04490 15.22 0.000 1.3

Weight 0.89701 0.04818 18.62 0.000 4.5

BSA 4.860 1.492 3.26 0.005 4.3

Dur 0.06653 0.04895 1.36 0.194 1.2

S = 0.4259 R-Sq = 99.5% R-Sq(adj) = 99.4%

Analysis of Variance

Source DF SS MS F P

Regression 4 557.28 139.32 768.01 0.000

Residual Error 15 2.72 0.18

Total 19 560.00

The regression equation is

BP = - 13.7 + 0.702 Age + 0.906 Weight + 4.63 BSA

Predictor Coef SE Coef T P VIF

Constant -13.667 2.647 -5.16 0.000

Age 0.70162 0.04396 15.96 0.000 1.2

Weight 0.90582 0.04899 18.49 0.000 4.4

BSA 4.627 1.521 3.04 0.008 4.3

S = 0.4370 R-Sq = 99.5% R-Sq(adj) = 99.4%

Analysis of Variance

Source DF SS MS F P

Regression 3 556.94 185.65 971.93 0.000

Residual Error 16 3.06 0.19

Total 19 560.00

The regression equation is

BP = - 16.6 + 0.708 Age + 1.03 Weight

Predictor Coef SE Coef T P VIF

Constant -16.579 3.007 -5.51 0.000

Age 0.70825 0.05351 13.23 0.000 1.2

Weight 1.03296 0.03116 33.15 0.000 1.2

S = 0.5327 R-Sq = 99.1% R-Sq(adj) = 99.0%

Analysis of Variance

Source DF SS MS F P

Regression 2 555.18 277.59 978.25 0.000

Residual Error 17 4.82 0.28

Total 19 560.00

- Stat >> Regression >> Best subsets …
- Specify response and all possible predictors.
- If desired, specify predictors that must be included in every model.
- (Researcher’s knowledge!)

- Select OK. Results appear in session window.

Model building strategy

- Decide on the type of model needed
- Predictive: model used to predict the response variable from a chosen set of predictors.
- Theoretical: model based on theoretical relationship between response and predictors.
- Control: model used to control a response variable by manipulating predictor variables.

- Decide on the type of model needed
- Inferential: model used to explore strength of relationships between response and predictors.
- Data summary: model used merely as a way to summarize a large set of data by a single equation.

- Decide which predictor variables and response variable on which to collect the data.
- Collect the data.

- Explore the data
- Check for outliers, gross data errors, missing values on a univariate basis.
- Study bivariate relationships to reveal other outliers, to suggest possible transformations, to identify possible multicollinearities.

- Randomly divide the data into a training set and a test set:
- The training set, with at least 15-20 error d.f., is used to fit the model.
- The test set is used for cross-validation of the fitted model.

- Using the training set, fit several candidate models:
- Use best subsets regression.
- Use stepwise regression (only gives one model unless specifies different alpha-to-remove and alpha-to-enter values).

- Select and evaluate a few “good” models:
- Select based on adjusted R2, Mallow’s Cp, number and nature of predictors.
- Evaluate selected models for violation of model assumptions.
- If none of the models provide a satisfactory fit, try something else, such as more data, different predictors, a different class of model …

- Select the final model:
- Compare competing models by cross-validating them against the test data.
- The model with a larger cross-validation R2 is a better predictive model.
- Consider residual plots, outliers, parsimony, relevance, and ease of measurement of predictors.