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16
1.3
we might try n s = 8 and n t = 2, which gives σ = ------ + ------------ = 2.08 and σ = 1.44, at a cost of 8($20) +
2
8 2 × 8
2(8)($20) = $480.
Comments

The variance components analysis is an effective method for quantifying the sources of variation in an
experimental program. Obviously, this kind of study will be most useful if it is done at the beginning
of a monitoring program. The information gained from the analysis can be used to plan a cost-effective
program for collecting and analyzing samples. Cost-effective means that both the cost of the measurement
program and the variance of the measurements produced are minimized.
It can happen in practice that some estimated variance components are negative. Such a nonsense
result often can be interpreted as a sign of the variance component being zero or having some insignificant
positive value. It may result from the lack of normality in the residual errors (Leone and Nelson, 1966).
In any case, it should indicate that a review of the data structure and the basic assumptions of the
components of the variance test should be made.

References
Box, G. E. P. (1998). “Multiple Sources of Variation: Variance Components,” Qual. Eng., 11(1), 171–174.
Box, G. E. P., W. G. Hunter, and J. S. Hunter (1978). Statistics for Experimenters: An Introduction to Design,
Data Analysis, and Model Building, New York, Wiley Interscience.
Davies, O. L. and P. L. Goldsmith (1972). Statistical Methods in Research and Production, 4th ed., rev., New
York, Hafner Publishing Co.
Edelman, D. A. (1974). “Three-Stage Nested Designs with Composited Samples,” Technometrics, 16(3),
409–417.
Edland, Steven D. and Gerald van Belle (1994). “Decreased Sampling Costs and Improved Accuracy with
Composite Sampling,” in Environmental Statistics, Assessment, and Forecasting, C. R. Cothern and N. P.
Ross (Eds.), Boca Raton, FL, CRC Press.
Elder, R. S. (1980). “Properties of Composite Sampling Procedures,” Technometrics, 22(2), 179–186.
Finucan, H. M. (1964). “The Blood-Testing Problem,” Appl. Stat., 13, 43–50.
Hahn, G. J. (1977). “Random Samplings: Estimating Sources of Variability,” Chemtech, Sept., pp. 580–582.
Krueger, R. C. (1985). Characterization of Three Types of Foundry Waste and Estimation of Variance Com-
ponents, Research Report, Madison, WI, Department of Civil and Environmental Engineering, The
University of Wisconsin–Madison.
Kussmaul, K. and R. L. Anderson (1967). “Estimation of Variance Components in Two-Stage Nested Design
with Composite Samples,” Technometrics, 9(3), 373–389.
Leone, F. C. and L. S. Nelson (1966). “Sampling Distributions of Variance Components. I. Empirical Studies
of Balanced Nested Designs,” Technometrics, 8, 457–468.
Mack, G. A. and P. E. Robinson (1985). “Use of Composited Samples to Increase the Precision and Probability
of Detection of Toxic Chemicals,” Am. Chem. Soc. Symp. on Environmental Applications of Chemo-
metrics, J. J. Breen and P. E. Robinson (Eds.), Philadelphia, PA, 174–183.
Rajagopal, R. and L. R. Williams (1989). “Economics of Compositing as a Screening Tool in Ground Water
Monitoring,” GWMR, Winter, pp. 186–192.

Exercises
25.1 Fish Liver. The livers of four fish were analyzed for a bioaccumulative chemical. Three
replicate measurements were made on each liver. Using the following results, verify that the
variance between fish is significantly greater than the analytical variance.
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