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Model selection

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Model selection

Stepwise regression

- A common problem is that there is a large set of candidate predictor variables.
- (Note: The examples herein are really not that large.)
- Goal is to choose a small subset from the larger set so that the resulting regression model is simple, yet have good predictive ability.

- 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

- Stepwise regression: Enter and remove predictors, in a stepwise manner, until there is 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.

- Start with no predictors in the “stepwise model.”
- At each step, enter or remove a predictor based on partial F-tests (that is, the t-tests).
- Stop when no more predictors can be justifiably entered or removed from the stepwise model.

- Specify an Alpha-to-Enter (αE = 0.15) significance level.
- Specify an Alpha-to-Remove (αR = 0.15) significance level.

- Fit each of the one-predictor models, that is, regress y on x1, regress y on x2, … regress y on xp-1.
- The first predictor put in the stepwise model is the predictor that has the smallest t-test P-value (below αE = 0.15).
- If no P-value < 0.15, stop.

- Suppose x1 was the “best” one predictor.
- Fit each of the two-predictor models with x1 in the model, that is, regress y on (x1, x2), regress y on (x1, x3), …, and y on (x1, xp-1).
- The second predictor put in stepwise model is the predictor that has the smallest t-test P-value (below αE = 0.15).
- If no P-value < 0.15, stop.

- Suppose x2 was the “best” second predictor.
- Step back and check P-value for β1 = 0. If the P-value for β1 = 0 has become not significant (above αR = 0.15), remove x1 from the stepwise model.

- Suppose both x1 and x2 made it into the two-predictor stepwise model.
- Fit each of the three-predictor models with x1 and x2 in the model, that is, regress y on (x1, x2, x3), regress y on (x1, x2, x4), …, and regress y on (x1, x2, xp-1).

- The third predictor put in stepwise model is the predictor that has the smallest t-test P-value (below αE = 0.15).
- If no P-value < 0.15, stop.
- Step back and check P-values for β1 = 0 andβ2 = 0. If either P-value has become not significant (above αR = 0.15), remove the predictorfrom the stepwise model.

- The procedure is stopped when adding an additional predictor does not yield a t-test P-value below αE = 0.15.

Predictor Coef SE Coef T P

Constant 81.479 4.927 16.54 0.000

x1 1.8687 0.5264 3.55 0.005

Predictor Coef SE Coef T P

Constant 57.424 8.491 6.76 0.000

x2 0.7891 0.1684 4.69 0.001

Predictor Coef SE Coef T P

Constant 110.203 7.948 13.87 0.000

x3 -1.2558 0.5984 -2.10 0.060

Predictor Coef SE Coef T P

Constant 117.568 5.262 22.34 0.000

x4 -0.7382 0.1546 -4.77 0.001

Predictor Coef SE Coef T P

Constant 103.097 2.124 48.54 0.000

x4 -0.61395 0.04864 -12.62 0.000

x1 1.4400 0.1384 10.40 0.000

Predictor Coef SE Coef T P

Constant 94.16 56.63 1.66 0.127

x4 -0.4569 0.6960 -0.66 0.526

x2 0.3109 0.7486 0.42 0.687

Predictor Coef SE Coef T P

Constant 131.282 3.275 40.09 0.000

x4 -0.72460 0.07233 -10.02 0.000

x3 -1.1999 0.1890 -6.35 0.000

Predictor Coef SE Coef T P

Constant 71.65 14.14 5.07 0.001

x4 -0.2365 0.1733 -1.37 0.205

x1 1.4519 0.1170 12.41 0.000

x2 0.4161 0.1856 2.24 0.052

Predictor Coef SE Coef T P

Constant 111.684 4.562 24.48 0.000

x4 -0.64280 0.04454 -14.43 0.000

x1 1.0519 0.2237 4.70 0.001

x3 -0.4100 0.1992 -2.06 0.070

Predictor Coef SE Coef T P

Constant 52.577 2.286 23.00 0.000

x1 1.4683 0.1213 12.10 0.000

x2 0.66225 0.04585 14.44 0.000

Predictor Coef SE Coef T P

Constant 71.65 14.14 5.07 0.001

x1 1.4519 0.1170 12.41 0.000

x2 0.4161 0.1856 2.24 0.052

x4 -0.2365 0.1733 -1.37 0.205

Predictor Coef SE Coef T P

Constant 48.194 3.913 12.32 0.000

x1 1.6959 0.2046 8.29 0.000

x2 0.65691 0.04423 14.85 0.000

x3 0.2500 0.1847 1.35 0.209

Predictor Coef SE Coef T P

Constant 52.577 2.286 23.00 0.000

x1 1.4683 0.1213 12.10 0.000

x2 0.66225 0.04585 14.44 0.000

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

- Do not jump to the conclusion …
- that all the important predictor variables for predicting y have been identified, or
- that all the unimportant predictor variables have been eliminated.

- Many t-tests for βk= 0are conducted in a stepwise regression procedure.
- The probability is high …
- that we included some unimportant predictors
- that we excluded some important predictors

- The final model is not guaranteed to be optimal in any specified sense.
- The procedure yields a single final model, although in practice there are often several equally good models.
- It doesn’t take into account a researcher’s knowledge about the predictors.

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

The regression equation is

PIQ = 111 + 2.06 MRI - 2.73 Height

Predictor Coef SE Coef T P

Constant 111.28 55.87 1.99 0.054

MRI 2.0606 0.5466 3.77 0.001

Height -2.7299 0.9932 -2.75 0.009

S = 19.51 R-Sq = 29.5% R-Sq(adj) = 25.5%

Analysis of Variance

Source DF SS MS F P

Regression 2 5572.7 2786.4 7.32 0.002

Error 35 13321.8 380.6

Total 37 18894.6

Source DF Seq SS

MRI 1 2697.1

Height 1 2875.6

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

Predictor Coef SE Coef T P

Constant -13.667 2.647 -5.16 0.000

Age 0.70162 0.04396 15.96 0.000

Weight 0.90582 0.04899 18.49 0.000

BSA 4.627 1.521 3.04 0.008

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

Error 16 3.06 0.19

Total 19 560.00

Source DF Seq SS

Age 1 243.27

Weight 1 311.91

BSA 1 1.77

- Stat >> Regression >> Stepwise …
- Specify response and all possible predictors.
- If desired, specify predictors that must be included in every model. (This is where researcher’s knowledge helps!!)
- Select OK. Results appear in session window.