Why Design?Why Design?(why not just observe and model?)(why not just observe and model?)

Q: Why Experimental Design A: To avoid multicollinearity

Issues: (1) Testing joint importance versus individual significance

(2) Prediction versus modeling individual effects

(3) Collinearity (correlation among inputs)

Example: Hypothetical company’s sales Y depend on TV advertising X1 and Radio Advertising X2.

Y = 0 + 1X1 + 2X2 +e

Jointly critical (can’t omit both!!)

Two engine plane can still fly if engine #1 failsTwo engine plane can still fly if engine #2 failsNeither is critical individually

Data Sales; input store TV radio sales; (more code)cards; 1 869 868 9089 2 836 820 8290 (more data) 40 969 961 10130

proc g3d data=sales; scatter radio*TV=sales/shape=sval color=cval zmin=8000;run;

TV

Sales

Radio

Conclusion: Can predict well with just TV, just radio, or both!

SAS code: proc reg data=next; model sales = TV radio;

Analysis of Variance

Sum of MeanSource DF Squares Square F Value Pr > FModel 2 32660996 16330498 358.84 <.0001 (Can’t omit both)Error 37 1683844 45509Corrected Total 39 34344840

Root MSE 213.32908 R-Square 0.9510 Explaining 95% of variation in sales

Parameter Estimates

Parameter StandardVariable DF Estimate Error t Value Pr > |t|Intercept 1 531.11390 359.90429 1.48 0.1485TV 1 5.00435 5.01845 1.00 0.3251 (can omit TV)radio 1 4.66752 4.94312 0.94 0.3512 (can omit radio)

Estimated Sales = 531 + 5.0 TV + 4.7 radio with error variance 45509 (standard deviation 213).

TV approximately equal to radio so, approximately

Estimated Sales = 531 + 9.7 TV or

Estimated Sales = 531 + 9.7 radio

Regression

The REG ProcedureModel: MODEL1Dependent Variable: sales

Number of Observations Read 40Number of Observations Used 40

Analysis of Variance

Sum of MeanSource DF Squares Square F Value Pr > F

Model 2 32660996 16330498 358.84 <.0001Error 37 1683844 45509Corrected Total 39 34344840

Root MSE 213.32908 R-Square 0.9510Dependent Mean 9955 Adj R-Sq 0.9483Coeff Var 2.14291

Parameter Estimates

Parameter StandardVariable DF Estimate Error t Value Pr > |t|

Intercept 1 531.11390 359.90429 1.48 0.1485TV 1 5.00435 5.01845 1.00 0.3251radio 1 4.66752 4.94312 0.94 0.3512

Design

The REG ProcedureModel: MODEL1Dependent Variable: SALES

Number of Observations Read 40Number of Observations Used 40

Analysis of Variance

Sum of MeanSource DF Squares Square F Value Pr > F

Model 2 32641505 16320753 358.66 <.0001Error 37 1683699 45505Corrected Total 39 34325204

Root MSE 213.31990 R-Square 0.9509Dependent Mean 10300 Adj R-Sq 0.9483Coeff Var 2.07111

Parameter Estimates

Parameter StandardVariable DF Estimate Error t Value Pr > |t|

Intercept 1 530.72803 366.53079 1.45 0.1560TV 1 5.00492 0.25552 19.59 <.0001Radio 1 4.66742 0.25552 18.27 <.0001

Design matrix-1 for low level +1 for high 12 obs.

1 2 2

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1 0.15 0 0 0.10

1 1 1 1 0 0.15 0.10 0, ( ' )

1 1 1 1 0 0.10 0.15 0

1 1 1 1 0.10 0 0 0.15

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

X X X

1 2 2

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1 0.083 0 0 0

1 1 1 1 0 0.083 0 0, ( ' )

1 1 1 1 0 0 0.083 0

1 1 1 1 0 0 0 0.083

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

X X X

2(0.5( )) 0.15Var estimated effect

2(0.5( )) 0.08333Var estimated effect

High Low

High 5 1

Low 1 5

High Low

High 3 3

Low 3 3

X1 X2