Ch. 21. Subsampling designs in industry: Statistical inference for variance components

Publisher Summary In many industrial problems, variability among the units can be attributed to various sources that are hierarchical in their occurrences. As the sources of variability are hierarchical in nature, the estimation of corresponding variance components would also require successive subsampling. This successive subsampling results in a “hierarchical” or “nested” design. In such designs, the sources of variability or factors are nested and this nesting is determined by the sampling design. As unit-to-unit variability, the lowest in hierarchy, is always assumed to be present, it is usually termed as the “random error.” Apart from random error, there are three other factors, and such a design is often termed as a “three-stage nested design.” A p -stage nested design is defined similarly. Factors can be fixed or random. However, by the very nature of subsampling, once a factor is assumed random, all factors that are lower in hierarchy are also deemed random. Because nested designs are often employed with the objective of estimating various sources of variability, all the factors are usually chosen as random. This chapter discusses nested designs in the random effects framework, as this is the most common situation in industrial problems.

[1]  L. S. Nelson.  Column: Technical Aids: Using Nested Designs: II. Confidence Limits for Standard Deviations , 1995 .

[2]  I. Olkin,et al.  A numerical procedure for finding the positive definite matrix closest to a patterned matrix , 1991 .

[3]  D. Gaylor,et al.  Variance Component Testing in Unbalanced Nested Designs , 1974 .

[4]  Y. Ojima.  Control Chart Method for Analyzing Staggered Nested Data , 1997 .

[5]  O. L. Davies,et al.  Statistical methods in research and production , 1958 .

[6]  R. Cole.  Staggered nested designs to identify hierarchical scales of variability , 2001 .

[7]  A. I. Khuri.  Exact tests for random models with unequal cell frequencies in the last stage , 1990 .

[8]  Robert L. Mason,et al.  Statistical Design and Analysis of Experiments , 2003 .

[9]  Calyampudi Radhakrishna Rao,et al.  Linear Statistical Inference and its Applications , 1967 .

[10]  D. Naik,et al.  Statistical tests for random effects in staggered nested designs , 1995 .

[11]  F. M. Speed,et al.  Computing Expected Mean Squares , 1980 .

[12]  S. R. Searle.  Linear Models , 1971 .

[13]  Lynn Roy Lamotte,et al.  On Non-Negative Quadratic Unbiased Estimation of Variance Components , 1973 .

[14]  R. Khattree.  Some practical estimation procedures for variance components , 1998 .

[15]  A. Khuri,et al.  Variance Components Analysis: A Selective Literature Survey , 1985 .

[16]  Satterthwaite Fe.  An approximate distribution of estimates of variance components. , 1946 .

[17]  R. C. Bose,et al.  Essays in probability and statistics , 1971 .

[18]  Ronald D. Snee,et al.  Graphical Analysis of Process Variation Studies , 1983 .

[19]  Optimal tests for nested designs with circular stationary dependence , 1994 .

[20]  Ravindba Khattbee,et al.  Optimum tests for bandom effects in unbalanced nested designs , 1990 .

[21]  Kathleen G. Purdy.  Confidence intervals for variance components , 1998 .

[22]  Ravindra Khattree,et al.  Computer Programs: A Computer Program to Estimate Variance Components in Staggered Nested Designs , 1998 .

[23]  Inequality restricted linear selection indices , 1988 .

[24]  L. S. Nelson.  Column: Technical Aids: Using Nested Designs: I. Estimation of Standard Deviations , 1995 .

[25]  E. Lehmann.  Testing Statistical Hypotheses , 1960 .

[26]  Hardeo Sahai,et al.  The Analysis of Variance , 2000 .

[27]  On robustness of tests for random effects in balanced nested designs , 1989 .

[28]  Optimum tests for treatments when blocks are nested and random , 1992 .

[29]  Y. Ojima.  General formulae for expectations, variances and covariances of the mean squares for staggered nested designs , 1998 .

[30]  Anthony N. Pettitt,et al.  Sampling Designs for Estimating Spatial Variance Components , 1993 .

[31]  L. S. Nelson.  Variance Estimation Using Staggered, Nested Designs , 1983 .

[32]  T. W. Anderson.  ASYMPTOTIC THEORY FOR PRINCIPAL COMPONENT ANALYSIS , 1963 .

[33]  L. Calvin,et al.  A Sampling Design With Incomplete Dichotomy , 1961 .

[34]  Calyampudi R. Rao,et al.  Linear Statistical Inference and Its Applications. , 1975 .

[35]  C. E. Lemke,et al.  A Method of Solution for Quadratic Programs , 1962 .

[36]  T. A. Bancroft,et al.  Statistical Theory in Research , 1952 .

[37]  D. Lawley.  TESTS OF SIGNIFICANCE FOR THE LATENT ROOTS OF COVARIANCE AND CORRELATION MATRICES , 1956 .

[38]  James R. Smith,et al.  The Use and Analysis of Staggered Nested Factorial Designs , 1981 .

[39]  Robert L. Mason,et al.  Estimation of variance components in staggered nested designs , 1997 .

[40]  H. Lenz,et al.  Frontiers in Statistical Quality Control 2. , 1984 .

[41]  NONNEGATIVE ESTIMATION OF VARIANCE COMPONENTS: A MODIFICATION TO HENDERSON'S ANOVA METHODOLOGY , 1999 .

[42]  P. Rao.  Variance components estimation : mixed models, methodologies and applications , 1997 .

[43]  Kenny Q. Ye.  Statistical Tests for Mixed Linear Models , 2000, Technometrics.

[44]  M. Oliver,et al.  Determining the spatial scale of variation in soil radon concentration , 1995 .

[45]  Yoshikazu Ojima,et al.  Generalized staggered nested designs for variance components estimation , 2000 .

[46]  Richard Roland Prairie,et al.  Optimal designs to estimate variance components and to reduce product variability for nested classifications , 1962 .

[47]  R. Khattree,et al.  STATISTICAL TESTS FOR NESTED DESIGNS UNDER PARTIAL BALANCE , 2002 .