Direction:The Invisible Player

Robert E. JohnsonDept. of Mathematical Sciences

VCU

Based on - Johnson and Herr, “Direction: The Invisible Player”, ASA Proceedings of the Statistical Education Section, 48-52, 1993.

Workload Related to Anesthesiology Service

• Dependent Variable– WORKLOAD: man-hours

• Independent Variables– CASES: number of surgical cases– ELIGIBLE: rate of service eligibility per 1000

patients

Source: Procedures and Analyses for Staffing Standards Development: Data/Regression Analysis Handbook. San Diego, CA: Navy Manpower and Material Analysis Center, 1979. [From: Myers, Classical and Modern Regression with Applications, PWS-Kent Publishing Co., Boston, MA, 1990 (pages 381-383)].

CorrelationsWorkload/Anesthesiology Service

Cases Eligible

Workload 0.980 0.971(<0.001) (<0.001)

Regression AnalysisWorkload/Anesthesiology Service

Analysis of Variance Section

Sum of Mean ProbSource DF Squares Square F-Ratio LevelModel 2 1.457E+07 7284372.7 124.752 <0.001Error 9 525517.4 58390.82Total(Adj.) 11 1.509E+07

Root Mean Square Error 241.642 R-Squared 0.965

Regression AnalysisWorkload/Anesthesiology Service

Regression Equation Section

Indep. Regression Standard T-Value ProbVariable Coefficient Error (Ho: B=0) LevelIntercept 137.2353 115.0564 1.193 0.264Cases 2.960778 1.214657 2.438 0.038Eligible 3.085881 2.919934 1.057 0.318

CorrelationsWorkload/Anesthesiology Service

Cases Eligible

Workload 0.980 0.971(<0.001) (<0.001)

Cases 0.975(<0.001)

cZcX

ZX c

XZ c

X=Cases; Z=Eligible; Y=Workload

cY

Growth Rate in Experimental Rats

• Dependent Variable– GROWTH: growth rate

• Independent Variables– DOSE: dosage of a dietary supplement– DOSESQ: dosage squared

Hypothetical Data Source: SAS Institute Inc., SAS/STAT User’s Guide, Version 6, Fourth Edition, Volume 2, SAS Institute Inc., Cary, NC, 1989 (page 1438).

Correlations Growth Rate in Experimental Rats

Dose DoseSq

Growth .186 .359(0.608) (0.309)

Regression AnalysisGrowth Rate in Experimental Rats

Analysis of Variance Section

Sum of Mean ProbSource DF Squares Square F-Ratio LevelModel 2 655.706 332.853 51.555 <0.001Error 7 45.1938 6.45626Total(Adj.) 9 710.9 78.9889

Root Mean Square Error 2.541 R-Squared 0.936

Regression Analysis Growth Rate in Experimental Rats

Regression Equation Section

Indep. Regression Standard T-Value ProbVariable Coefficient Error (Ho: B=0) LevelIntercept 35.65744 5.617927 6.3471 4E-04Dose 5.262896 0.558022 9.4313 3E-05DoseSq 0.12767 0.012811 9.966 2E-05

Correlations Growth Rate in Experimental Rats

Dose DoseSq

Growth .186 .359(0.608) (0.309)

Dose 0.983(<0.001)

cZ

cX

cY

ZX c

XZ c

X=Dose; Z=DoseSq; Y=Growth

Unique Contribution of an Independent Variableto the Total Variation (CSS)

“When independent variables are correlated, there is no unique sum of squares which can be ascribed to an independent variable as reflecting its effect in reducing the total variation in Y. The reduction in total variation ascribed to an independent variable must be viewed in the context of the other independent variables in the model…”

- Neter & Wasserman

Neter and Wasserman, Applied Linear Statistical Models,Richard D. Irwin, Inc., 1974 (page 253).

cZcX

XZ c

Contribution of an Independent Variableto the Total Variation (CSS)

cYType 3:

Partial SSThe SS ascribed to the direction in the XZc-plane orthogonal to X.

This is the reduction in

error SS when Z joins X in the

model.

Type 1 SSThe SS

ascribed to the direction of Z.

This is the regression SS when Z is the only variable in

the model.

cZcX

XZ c

Contribution of an Independent Variableto the Total Variation (CSS)

cY

W

cX

XZ c

Contribution of an Independent Variableto the Total Variation (CSS)

cY W XZc-plane

W

cX

XZ c

Contribution of an Independent Variableto the Total Variation (CSS)

cYGiven Xc and

the XZc-plane, there is no

unique direction in the

plane which can be

ascribed to this SS.

W XZc-plane

cX

Variance Inflation Factor (VIF)

cX̂ cY

cZcX

Variance Inflation Factor (VIF)

cX̂

cY

ab

)cos1(1

2

abVIF

VIF: Variance Inflation Factor

2

2

2

1

2

( )1

1( )

1

i

x i

n

x ii

ii

VarSS R

SS X X

VIFR

β

β