Title: Using Variance Component Models to Compare Variability of Manufacturing Processes
Abstract: A common experiment employed in industrial settings is the
repeatability and reproducibility (R&R) study. One objective of such
an experiment is to determine if a measurement procedure or instrument
is adequate for monitoring a process. Such a study may also be described
as a gauge capability analysis. If the measurement error is small relative
to total process variation, then the measurement procedure is deemed "adequate".
R&R studies typically employ a two-factor crossed design with the factors
"parts" and "operators". Measures of interest are functions of the variance
components of this model. Several methods have been proposed for constructing
confidence intervals for these measures of variability. These studies are
summarized for cases where operators are either fixed or random. Results
from a study to compare the variability of two measurement processes are
Title: Probability Matching Priors For One-Way Unbalanced Random Effects Models
Abstract: This paper considers development of noninformative priors
for unbalanced one-way random effects models. The main focus is on the
development of probability matching priors (that is those for which the
coverage probabilities of Bayesian credible intervals match asymptotically
their frequentist counterparts) when the parameter of interest is the variance
ratio. Under a suitable orthogonal reparametrization (Cox and Reid, 1987,
JRSS B), it is possible to characterize the class of first order probability
matching priors. However, it is shown that there does not exist any second
order probability matching prior in this case. Also, it is shown that the
one-at-a-time reference prior of Berger and Bernardo works well, and is
fairly robust with respect to the choice of prior parameters.
Title: Confidence Intervals for Variance Components
Abstract: The random-effect mean squares from an unweighted means ANOVA
(UANOVA) have been exploited by a number of writers to form confidence
intervals for variance components. Except for a few special cases, these
results are limited to models where a UANOVA can be defined. For a two
variance component model, we show how this limitation can be eliminated
by defining two mean squares that have properties similar to the mean squares
from a random one-way UANOVA. These two mean squares can then be used for
variance component inference in the same way that mean squares from a UANOVA
are utilized. Also, several examples are investigated that illustrate some
of the limitations using the UANOVA mean squares. An alternative to the
UANOVA mean squares, which seems to have better performance characteristics,
Title: Non-Conjugate Bayesian Analysis Of One Parameter Item Response Models
Abstract: We present a unified Bayesian approach for the analysis of one-parameter item response models. A necessary and sufficient condition is given for the propriety of posteriors under improper priors with nonidentifiable likelihoods. Posterior distributions for item and subject parameters may be improper when the sum of the binary responses for an item or subject takes its minimum or maximum possible value. When the item parameters have a flat prior but the item totals do not fall at a boundary value, we prove the propriety of the Bayesian joint posterior under some sufficient conditions on the joint (proper) distribution of the subject parameters. The methods are implemented using Markov chain Monte Carlo and illustrated with an example from a cross-over study comparing three medical treatments.
The Bayesian methods are compared against the usual generalized mixed
linear models. We also provide some results related to the analysis of
binary matched pairs data.
Title: Bayesian Random Regression Models for Longitudinal Binary Data with Applications to Dairy Cattle Breeding
Abstract: Our concern is genetic analysis of a disease, mastitis, observed
longitudinally in dairy cows. Data were 3341 test-day records from 329
first-lactation Holstein cows scored for absence/presence of mastitis at
14, 30 days (d) of lactation, and every 30d thereafter. Susceptibility
was related to a Gaussian process. Given the parameters, the probability
of the data for each cow was the product of probabilities at each test-day.
The conditional probability of infection at time t for each cow was a normal
integral, its argument being a function of fixed and random effects, and
of time. Models for the latent variable were: 1) year-month of test + a
5-parameter linear regression + breeding value (BV) of cow + environmental
effect peculiar to all records of a cow + residual. 2) As before, excluding
BV, but with a 5-parameter random regression for each cow. 3) Hierarchical
model of a 3-parameter regression for each cow. Computations were by Gibbs
sampling, with data augmentation. Model 1 posterior mean of heritability
was .05. Model (2) heritabilities were, e.g., .27 (14d), .03 (120d), and
.07 (305d). Model 3 heritabilities were .57 (14d), .06 (120d) and .19 (305d).
Bayes factors were: .011 (Model 1/Model 2), .017 (1/3) and 1.535 (2/3).
Model can be extended: fitting structured covariance matrices for residuals,
thick-tailed distributions and oscillatory components.
Title: Modeling the Power Function For a Variance Component Under Heterogeneous Error Variances
Abstract: Traditional analysis of variance (ANOVA) tests are based on
the assumption that the error variances are homogeneous. Departure
from such an assumption combined with a certain degree of imbalance, in
the data set under consideration, can have a considerable effect on the
performance of a given ANOVA test. In this paper, we consider the
effect of heterogeneity of the error variances on the power of the usual
concerning the among-group variance component for an unbalanced random
one-way model. The power is modeled empirically using generalized
linear models techniques. The purpose of the proposed modeling approachis
to provide added insight into the combined effects of heterogeneity of
variances and data imbalance on the test. This includes, in particular,
the ability to monitor changes in the power values, and to detect influential
error variances, which can be helpful in increasing robustness of
Title: Jackknife Estimation of Variance Components
Abstract: A general theory of jackknifying M-estimators is considered
for a general mixed model which includes mixed linear normal model and
mixed logistic model. Our M-estimators cover the modified profile MLE (e.g.,
REML estimators), penalized MLE, or M-estimators not associated with a
maximization process (e.g., the method of moment estimators). We also present
the related mean square error estimation of empirical best predictor of
a general mixed effect. Simulation results show that our method is quite
robust against departure from normality. Our method is illustrated using
a real life data.
Title: Optimal Design for Variance Components and Design Implementation
Abstract: Since variance components are nonlinear parameters in a linear
model, classical optimal designs for estimating variance components in
the m-stage nested model depend on the values of the parameters. We derive
optimal local and Bayesian designs for estimating variance components in
the nested model. These designs are then compared to designs in the quality
control literature. Tree structures are used to allow computation and visualization
of the resulting designs.
Title: The One-Way Random Model: Some New Problems and Applications
Abstract: In the talk, I will present some new applications of the one-way
random effects model. The applications include the assessment of occupational
exposure to contaminants, and testing bioequivalence. These applications
give rise to some new hypotheses testing problems, and confidence and tolerance
interval problems, involving the one-way random model. I will give a review
of these problems and the available solutions.
Title: Estimation of MSE of EBLUP Estimator under General Mixed ANOVA Models
Abstract:Estimation of a linear combination of fixed effects and realized
values of random effects in general mixed ANOVA models is of considerable
interest in practice; for example, in animal breeding experiments and small
area estimation. Empirical BLUP estimators are obtained by replacing the
variance components by consistent estimators; in particular ML, REML or
invariant quadratic estimators. We obtain a second order approximation
to MSE of EBLUP and study its accuracy. An estimator of MSE , correct to
second order, is also obtained. We show that the MSE estimator when ML
is used involves an extra term that is absent in the case of REML and invariant
quadratic estimators of variance components. The special case of a balanced
two-way crossed random effects model with interaction is used to illustrate
Title: Nonnegative Estimators for Variance Components Utilizing Prior Information
Abstract: For the variance components of the one-way Analysis of variance
model, nonnegative estimators that can utilize prior information are developed
from the Analysis of Variance, Unweighted Sums of Squares, Minimum Variance
Quadratic Unbiased Estimation and related procedures. Biases and Mean Square
Errors of these estimators are evaluated and their sensitivity to the prior
values are examined. Analogous procedures for the One-way Analysis of Covariance
Model are also examined.
Title: Variance Components for Dependent Qualitative Responses
Abstract: The concept of variance components for qualitative and semi-quantitative
traits will be explored. Measures of heterogeneity for categorical responses
will be discussed with reference to genetic analyses. Issues of decomposability
will be discussed and connections with generalized mixed models explored
with a view toward interpretation and applicability.
Title: Nonnegative estimation of variance components in multivariate unbalanced mixed models with two variance components
Abstract: In this talk we address the problem of nonnegative estimation
of variance components in multivariate unbalanced mixed models with two
variance components. This generalizes the work in Mathew/Niyogi/Sinha (JMA,
1994). We will illustrate our results with an example.
Title: Variance component models for mapping human quantitative trait genes: a brief survey
Abstract: With the ready availability of high-resolution genetic maps
and efficient high-throughput genotyping, considerable attention is now
being given to mapping genes contributing to quantitative traits in humans.
These include traits related to heart disease, alcoholism, and many biochemical
and physiological traits. Because the genetic basis of such traits is usually
completely unknown, so-called allele-sharing methods are popular in these
contexts, as they make very few specific assumptions concerning the genotype-phenotype
relationship. One of the simplest such involves sib-pairs, and I will focus
on them in my talk. My aim is to give a brief partial survey of the use
of variance components for mapping quantitative traits using allele-sharing
Title: On recent estimation approaches for variance components in generalized linear mixed models
Abstract: In view of the cumbersome and often intractable numerical
integrations required for a full likelihood analysis, several suggestions
have been made recently for approximate inference in generalized linear
mixed models. For example, we refer to the penalized quasi-likelihood (PQL)
approach of Breslow and Clayton (1993, JASA), bias corrected PQL approach
of Breslow and Lin (1995, Biometrika), hierarchical likelihood approach
of Lee and Nelder (1996, JRSSB). But as shown by Sutradhar and Qu (1998,
CJS) (see also Jiang (1998, JASA), these approaches do not yield consistent
estimators for the variance components of the random effects of the mixed
model. In this talk I will discuss a small variance based likelihood approximation
technique which yields consistent estimators of the variance components
both for binary and Poisson mixed models, which are important special cases
of the generalized linear mixed model. I will also outline a simulation
based general likelihood procedure as an extension of Jiang's (1998) simulation
based method of moments for the estimation of the variance components.
Title: Non-Conjugate Bayesian Analysis of Variance Component Models
Abstract: We consider the usual normal linear mixed model for variance
components from a Bayesian viewpoint. With conjugate priors and balanced
data, Gibbs sampling is easy to implement. However, simulating from full
conditionals can become difficult for the analysis of unbalanced data with
possibly non-conjugate priors, thus leading one to consider alternative
Markov chain Monte Carlo schemes. We propose and investigate a method for
posterior simulation based on an independence chain. The method is customized
to exploit the structure of the variance component model, and it works
with arbitrary prior distributions. As a default reference prior, we consider
a version of Jeffreys prior based on the integrated (``restricted'') likelihood.
We demonstrate the ease of application and flexibility of this approach
in several familiar settings, even in the presence of unbalanced data.
This work is joint with Rob Kass, Carnegie Mellon University.
Title: Optimal Assembled Designs with Variance Components
Abstract: An assembled design is a hybrid design which places
a nested design at each design point in a crossed design. This allows
for the estimation of both main and interaction effects as well as variance
components that arise from nested random factors such as lot number and
batch number. Assembled designs are also discussed for the estimation
of dispersion effects on variance components. Optimal assembled designs
are constructed and discussed for many practical situations.
Title: On Mixed AMMI Models for Investigating Genotype-Environment Interaction
Abstract: Additive main effects multiplicative interaction (AMMI) models
provide a tool for analyzing genotype-environment interaction in plant
breeding. Typically, they have been used in a fixed effects model framework
for the analysis of complete genotype by environment data sets. Assuming
environments as random effects, multiplicative interaction models can be
estimated in a mixed model framework for incomplete data. Under normality,
parameter estimates are obtained by maximum likelihood based procedures.
A factor analytic covariance matrix has been used to model the structure
of the multiplicative interaction terms within an environment. Both fixed
and mixed AMMI models are discussed. Biplots under both approaches are
presented for a complete set of data from a plant breeding experiment.
Although the differing procedures to obtain the biplots under both approaches,
they showed same interaction pattern.
Title: A Simple General Method for Constructing Confidence Intervals for Functions of Variance Components
Abstract: A single simple general method is proposed for constructing
confidence intervals for arbitrary functions of variance components in
balanced normal theory models. The method produces the commonly known
exact (Chi-Squares and F distribution based) confidence intervals for expected
mean squares and ratios of them. The concept of "surrogate variables"
is introduced as part of the description of the method. "Equal-tail" and
"shortest-length" confidence intervals from this method can be easily computed
using Monte Carlo simulations. The two-way random effects model without
interaction is considered. Using computer simulations, it is shown that
the proposed method produces intervals that maintain the nominal
confidence level and have comparable or smaller average lengths than
those produced by the best existing methods.
Author: Hartless, Glen, James G. Booth, and Ramon C. Littell
Title: Local influence diagnostics for the prediction of random effects in a linear mixed model
Abstract: The local influence approach to diagnostics developed by Cook
(1986) provides a general method for assessing how slight perturbations
to problem formulation can influence maximum likelihood (ML) inference.
In this poster, Cook's likelihood displacement measure is extended to the
problem of prediction through the use of the profile predictive likelihood.
It is shown that application to the linear mixed model with known variance
is straightforward. In particular, we will demonstrate the use of three
perturbation schemes for assessing influence on the estimation of fixed
effects, random effects and linear combinations thereof. The three schemes
perturb the error variance, random effects variance and the response, respectively.
The techniques are illustrated with an example.
Title: Variance Comonents in disassesmbly and reassembly experiments
Abstract: One of the main demands of a production process is stability.
The typical modern industrial product is a complicated combination of different
types of components. This research is stimulated by the problem of detecting
a component that systematically causes degradation and variance of the
product. The basic problem is to find components that have major influence
on the variance of assembled products in order to stabilize the process,
reduce variance and improve quality. Taguchi (1987) proposed so called
disassembly and reassembly experiments. In the simple case when we examine
only two components, A and B, the two components together
with the time factor yield a three factor experiment (where each factor
has m levels) with only m2 observations. Such
a design is known as the Latin Square or m3-1 design.
Since component units used in the experiment design were sampled from a
large population of units, it is natural to consider the factors A
and B as random effects. The influence of each factor is naturally
measured by its variance component. The larger the variance component of
the factor the stronger its effect on the process variance. Thus, we may
test the significance of each factor by estimating the variance components
from the data. We present the model resulting from this experiment as a
special case of mixed linear model and compared several estimators for
the variance components. We also suggested and compared several ways to
extend the basic Latin square design for disassembly and reassembly experiments.
Title: Some Facts that Illustrate the Connection Between REML and the Cholesky Decomposition
Abstract: A well known fact is that the Cholesky decomposition leads
to computation of the likelihood function for REML. The Cholesky decomposition
is an interesting data mining tool for treating large, sparse and symmetric
matrices, and when used to compute the REML likelihood function the availability
of derivatives by backward differentiation permits maximum likelihood estimation.
It is less well known, but when singular variance-covariance matrices are
encountered (such as with Kalman filtering) and the mixed model equations
can not be formed, the Cholesky decomposition continues to lead to likelihood
evaluation. However, the Cholesky decomposition must be applied to a matrix
that is symmetric and unfortunately indefinite. Better decomposition algorithms
are available to factorize indefinite matrices (such as the Bunch-Parlett
method), but these approaches pre-assume existence of the Cholesky decomposition
as an intrinsic part of likelihood evaluation. Alternatively, even when
the Cholesky algorithm fails because of the property of indefiniteness,
the aborted calculations continue to permit likelihood evaluation for REML.
This poster describes some of the matrix facts that have lead to the above
Title: Comparison of Variance Component Estimation Methods for Clutch Pattern in Laying Hens
Abstract: A study was carried out to investigate the efficiency of different
variance component estimation methods to determine the genetic parameters
for clutch traits in laying hens (sequence length and delay). Most of variance
component estimation methods have been developed to solve the problems
in animal breeding application. Recently a wide array of methods have been
developed to estimate variance components. For instance, Henderson Method's,
ML (Maximum Likelihood), REML (Restricted Maximum Likelihood), DFREML (Derivative
Free Restricted Maximum Likelihood), MIVQUE (Minimum Variance Quadratic
Unbiased Estimation),Bayesian Method, Gibbs Sampling. In this study we
compared a mostly used methods in animal breeding studies such as TYPE
I and TYPE III of Henderson's Method, ML, REML, DFREML, MIVQUE, MINQUE0.
Data were obtained from a commercial sire line in Turkey. A total 1980
animal with 43 sire and approximately 8 dam per sire was used. The hens
were housed in individual cages and selected based on records made between
22 and 40 week of age. Eggs were collected daily and individually. At the
end of the period, hens have more than ten pause day were excluded from
the data set. Each sequence length consists of consecutively laid egg number
and delay was calculated pause day between sequence lengths. Mean of sequence
length and delay were obtained 7.41 and 1.13 respectively. The heritability
estimates of sequence length were 0.11 (TYPE I), 0.12 (TYPE III), 0.13
(ML), 0.13 (REML), 0.40 (DFREML), 0.14 (MIVQUE) and 0.10 (MINQUE0). The
heritability estimates of delay was 0.03 for all estimation methods except
DFREML was 0.05. Because of including pedigree information DFREML can explain
the variability and relationship better.
Last modified: Wed Dec 29 08:52:36 EST 1999