REGRESSION ANALYSISWITH A SIMULATION SPIN

BASICS & NOTATION Input parameters 1, 2, …, n Values of each denoted X1, X2, Xn

For each setting of X1, X2, Xn observe a Y Each set (X1, X2, Xn ,Y) is one observation As we vary the X-values, Y changes in a

linear (scaled proportional) manner Some of the X’s don’t matter much,

some are key

BASICS

nnxxY ...110

Assumptions• e is independent from sample to sample• e is independent of the X’s• e ~N(0, s2)

So we will examine the “noise”

MOTIVATING EXAMPLE: Close Air Support

Troops patrol their assigned area Discover targets for destruction from

the air Call for CAS May need an aircraft with laser-

designation-capable weapons May have a time deadline Have a distance from the FARP to the

target Effects measured on 1..100 scale

EXPIRATION DEADLINES

DAMAGE SCORE vs EXPIRATION DEADLINE

0 20 40 60 80 100 120 140 160 180 2000

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REGRESSION OUTPUT(Excel)Regression Statistics

Multiple R 0.985R Square 0.97

Adjusted R Square 0.97

Standard Error 4.744Observations 100

ANOVA

df SS MS FSignificanc

e FRegression 1 71324 71324 3169 2E-76Residual 98 2206 22.51Total 99 73530

Coefficient

sStandard

Error t Stat P-value Lower 95% Upper 95%Intercept 10.74 0.66 16.28 1E-29 9.43 12.05X Variable 1 0.551 0.01 56.29 2E-76 0.532 0.571

Y= 10.7 + .55 EXP

Test for b = 0

REGRESSION LINE

0 20 40 60 80 100 120 140 160 180 2000

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} ERROR

SERIAL RESIDUALS

0 20 40 60 80 100 120

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

MULTIPLE REGRESSION Look at all of the independent

variables Builds the complex multidimensional

function in n-space

MULTIPLE REGRESSION Regression Statistics

Multiple R 0.999985

R Square 0.99997

Adjusted R Square 0.999969

Standard Error 0.157982

Observations 100

ANOVA

df SS MS F Significance F

Regression 3 79706.99 26569 1064529 6.9E-217

Residual 96 2.396011 0.024958

Total 99 79709.39

CoefficientsStandard

Error t Stat P-value Lower 95% Upper 95%

Intercept 0.392709 0.041394 9.487206 1.88E-15 0.310543 0.474874

X Variable 1 0.812893 0.031872 25.50533 1.52E-44 0.749629 0.876158

X Variable 2 0.185655 0.000587 316.3272 1.1E-146 0.18449 0.18682

X Variable 3 0.535199 0.000303 1768.023 2E-218 0.534598 0.5358

Y=.39 + .81 LAZ + .19 DIST + .54 EXP

REGRESSION DIAGNOSTICS

Residuals that depend on one of the X’s

Residuals that have different variance at different values of an X