Reducing costs in linear regression 

David Causeur & Thierry Dhorne 
Laboratory of Applied Statistics (SABRES) 
University of South Brittany 
Rue Yves Mainguy, Tohannic 
F­56000 Vannes, France 
email : causeur@iu­ 

Methodological aspects of grading


Experimental constraints

Experimental costs :

Multiplicity of grading systems : 


Main references

Reduction of experimental costs :


Double sampling : 

Connioee, D. and Moran, M.A. (1972)

Double sampling with regression in comparative studies of carcass composition.

Biometrics 28, 1011­ 1023. 

Cook, G.L., Jones, D.W. and Kempster, A.J. (1983)

A note on a simple criterion for choosing among sample joints for use in double sampling.

Animal Production 36, 493­495. 

Engel B. and Walstra P. (1991) Increasing precision or reducing expense in regression by using information from a concomitant variable. 
Biometrics 47, 13­20. 

Refinements : 

Causeur, D. and Dhorne, T. (1998)

Finite­sample properties of a multivariate extension of double­regression.

Biometrics. 54 (4), 299­309. 

Causeur, D. (1998)

Plan d'échantillonnage  á plusieurs phases pour la réduction des coûts expérimentaux en régression linéaire.

Revue de Statistique Appliquée. XLVI (4), 59­73. 

Multiplicity of grading systems :

Causeur, D. and Dhorne, T. (2000)

Using surrogate predictors in linear regression models.

Submitted to Biometrika

Main parts of the talk



Experimental costs in linear regression experiments

Sampling design for linear regression experiments

Double sampling for reduction of costs

Naïve sampling design 


Auxiliary covariate


Double sampling design 



Practical properties of Z


Double sampling procedure


Random sub sampling or selection


Estimation procedure






Double sampling properties

    Unbiased procedure


        Efficiency of the procedure


        Comparison with OLS efficiency 







Optimization of the double sampling design 

E.C. protocol : ''120 carcasses'' constraint 


Objective function to be optimized



Multiple auxiliary covariates

Multiple phase sampling plan


    To take into account the different costs of the auxiliary covariates

Monotone sampling design


Multiple phase sampling plan : example

Number of auxiliary covariates : 7 

Nr of covariates Optimal plan  Cost Reduction (%)
0 120 45600 0
1 (76,245) 34795 23.70
2 (69,216,398) 32275 29.22
3 (64,195,195,377) 31125 31.74
4 (53,134,209,209,209) 29155 36.06
5 (48,153,178,178,178,241) 27575 39.53
6 (47,141,157,157,157,217,337) 26570 41.73
7 (46,61,140,155,155,155,215,333)  26430 42.04


Optimal subsets :



Experimental constraints

Multiplicity of new grading systems 

Multiplicity of prediction formulae

Multiplicity of linear regression experiments with measurements of Y

Main practical problems

Sampling procedure


Methodological proposals

    How do these methods interact with PLS, PCR, ... ?

Practical proposals