Linear Models for Unbalanced Data

by Searle, Shayle R.

• ISBN13: 9780470040041
• ISBN10: 0470040041
• Edition: 1st
• Format: Paperback
• Copyright: 2006-03-17
• Publisher: Wiley-Interscience

Summary

WILEY-INTERSCIENCE PAPERBACK SERIES The Wiley-Interscience Paperback Series consists of selected books that have been made more accessible to consumers in an effort to increase global appeal and general circulation. With these new unabridged softcover volumes, Wiley hopes to extend the lives of these works by making them available to future generations of statisticians, mathematicians, and scientists. "[This book] provides an excellent discussion of the methodology and interpretation of linear models analysis of unbalanced data (data having unequal numbers of observations in the subclasses), generally without matrices?the author does an excellent job of emphasizing the more practical nature of the book. Highly recommended for graduate and undergraduate libraries." '?"Choice "This is a very comprehensive text, aimed at both students studying linear-model theory and practicing statisticians who require an understanding of the model-fitting procedures incorporated in statistical packages?This book should be considered as a text for college courses as it provides a clearly presented and thorough treatment of linear models. It will also be useful to any practicing statistician who has to analyze unbalanced data, perhaps arising from surveys, and wishes to understand the output from model-fitting procedures and the discrepancies in analysis from one recognized package to another." '?"Biometrics This newly available and affordably priced paperback version of Linear Models for Unbalanced Data offers a presentation of the fundamentals of linear statistical models unique in its total devotion to unbalanced data and its emphasis on the up-to-date cell means model approach to linear models for unbalanced data. Topic coverage includes cell means models, 1-way classification, nested classifications, 2-way classification with some-cells-empty data, models with covariables, matrix algebra and quadratic forms, linear model theory, and much more.

Author Biography

<b>SHAYLE R. SEARLE</b>, PhD, is Professor Emeritus of Biometry at Cornell University. He is the author of Linear Models, Matrix Algebra Useful for Statistics, and Variance Components, all published by Wiley. <p>

Table of Contents

An Up-Dated Viewpoint: Cell Means Models	1	(16)
Statistics and computers	1	(2)
Balanced and unbalanced data	3	(7)
Factors, levels, effects and cells	3	(2)
Balanced data	5	(1)
Special cases of unbalanced data	5	(2)
Unbalanced data	7	(1)
A summary	8	(2)
Cell means models	10	(3)
Statistical computing packages	13	(1)
Hypothesis testing	14	(3)
Basic Results for Cell Means Models: The 1-Way Classification	17	(44)
An example	17	(1)
The model and model equations	18	(3)
The model equations	18	(1)
The model	19	(2)
Estimation	21	(2)
Expected values and sampling variances	23	(1)
Estimating E(yij) and future observations	24	(1)
Estimating the error variance	25	(1)
Reductions in sums of squares: the R(.) notation	26	(3)
Partitioning the total sum of squares	29	(3)
For the analysis of variance	29	(2)
The coefficient of determination	31	(1)
The best linear unbiased estimator (BLUE)	32	(3)
Linear functions	32	(1)
Definition of BLUE	32	(1)
Verification: μi is the BLUE of μi	32	(2)
Derivation of μi as the BLUE	34	(1)
A linear function of BLUEs is a BLUE	34	(1)
Normality assumptions	35	(3)
The yijs are normal	35	(1)
The BLUE, μi, is normal	35	(1)
μi and σ2 are independent	36	(1)
SSRm/σ2 is distributed as Χ2a-1	36	(1)
SSE/2 is distributed as Χ2N-a	37	(1)
SSRm and SSE are independent	37	(1)
F-statistics	37	(1)
The analysis of variance table	38	(2)
A summary of arithmetic	38	(1)
Tests of hypotheses	39	(1)
Linear combinations of cell means	40	(13)
Confidence intervals	40	(2)
Hypothesis tests	42	(1)
One-part hypotheses	42	(1)
Two-part hypotheses	43	(2)
Fitting the mean	45	(1)
Other forms of a grand mean	46	(1)
Contrasts	47	(1)
Orthogonal contrasts for balanced data	47	(3)
Orthogonal contrasts for unbalanced data	50	(3)
The overparameterized model	53	(1)
Appendix	54	(4)
Proof that SSRm/σ2 ~ Χ2a-1 (Section 2.10d)	54	(3)
Properties of δi (Section 2.12g)	57	(1)
Summary	58	(1)
Model and estimation	58	(1)
Sums of squares and F-statistics	58	(1)
Exercises	59	(2)
Nested Classifications	61	(17)
Notation	61	(2)
The model	63	(1)
Estimation	63	(2)
Confidence intervals, and one-part hypotheses	65	(1)
Analysis of variance	65	(8)
For the cell means	65	(2)
Cell means within levels of the primary factor	67	(1)
Means for the levels of the primary factor	68	(2)
An overall mean	70	(1)
Summary	70	(1)
Using the hypotheses in the analysis of variance	71	(2)
More than two factors	73	(4)
Models	73	(1)
Sums of squares	74	(1)
Hypotheses tested by analysis of variance sums of squares in the fixed effects model	75	(2)
Exercises	77	(1)
The 2-Way Crossed Classification with All-Cells-Filled Data: Cell Means Models	78	(54)
Notation	78	(3)
The model	81	(1)
Estimation	81	(1)
Confidence intervals, and one-part hypotheses	82	(2)
Analysis of variance	84	(2)
A standard analysis	84	(1)
An alternative approach	85	(1)
Unweighted means	86	(6)
Definitions	87	(1)
Hypothesis: equality of row means	88	(1)
Numerator sums of squares	88	(1)
A general hypothesis: row means equal	89	(2)
A similar result for columns	91	(1)
A caution	91	(1)
Least squares means	91	(1)
Weighted means	92	(3)
Weighting by cell frequencies	92	(1)
Evaluation	93	(1)
A general weighting	94	(1)
Similar results for columns	94	(1)
Interactions	95	(4)
A meaning of interaction	95	(1)
A measure of interaction	96	(1)
The number of interactions in a grid	97	(1)
A consequence of interactions	98	(1)
Testing a hypothesis of no interactions	99	(18)
A no-interaction model	100	(1)
Estimation in the no-interaction model	101	(2)
Reduction in sum of squares	103	(1)
The F-statistic	104	(1)
Consequences of testing for interactions	105	(1)
Estimating means	105	(1)
Estimating the residual variance	106	(1)
Other F-statistics	107	(2)
Partitioning the total sum of squares	109	(2)
Analysis of variance	111	(2)
Hypotheses in the analysis of variance table	113	(3)
Balanced data	116	(1)
Computer output	117	(1)
The no-interaction model	117	(7)
Estimating means	118	(2)
F-statistics	120	(1)
Analysis of variance	121	(2)
Hypotheses in the analysis of variance table	123	(1)
Appendix	124	(3)
Summary	127	(2)
Model and estimation	127	(1)
Means of cell means	127	(1)
Interactions	127	(1)
The no-interaction model	128	(1)
Analysis of variance for the with-interaction model	128	(1)
Using the no-interaction model	128	(1)
Exercises	129	(3)
The 2-Way Classification with Some-Cells-Empty Data: Cell Means Models	132	(37)
Preliminaries	132	(4)
Model	132	(1)
Estimation	132	(1)
Linear functions of cell means	133	(1)
Estimating the residual variance	134	(1)
Confidence intervals	134	(1)
Example 1	134	(1)
Analysis of variance	135	(1)
Estimability: an introduction	136	(3)
Row means of cell means	137	(1)
Interactions	138	(1)
Connected data	139	(6)
Basic ideas	139	(1)
A geometric algorithm	140	(1)
Separation of disconnected data	141	(2)
Estimability of contrasts, and of all cell means	143	(1)
Connectedness for several factors	144	(1)
Testing for interactions	145	(9)
Fitting the model	145	(3)
The reduction in sum of squares for the no-interaction model	148	(1)
Testing the hypothesis of no-interaction	148	(1)
What hypothesis?	149	(2)
Specifying the hypothesis corresponding to R (μij\μi,τj)	151	(2)
Using the no-interaction model	153	(1)
Analysis of variance	154	(4)
Weighted squares of means analysis	155	(1)
The all-cells-filled analysis of Table 4.8	155	(2)
Disconnected data	157	(1)
Subset analyses for with-interaction models	158	(7)
Difficulties	158	(1)
What is of interest?	158	(1)
The investigator's role	158	(1)
A procedure	159	(1)
Subset analyses	160	(1)
Interactions	160	(1)
Examples	161	(2)
Difficulties with subsets	163	(1)
The investigator	164	(1)
Summary	165	(1)
Model and estimation	165	(1)
No-interaction model	165	(1)
Testing for no interactions	166	(1)
Using the no-interaction model	166	(1)
Exercises	166	(3)
Models with Covariables (Analysis of Covariance): The 1-Way Classification	169	(43)
The single slope model	171	(19)
Model	171	(1)
Estimation of means	172	(1)
Estimation and unbiasedness	172	(1)
Sampling variances and covariances	173	(2)
Estimating residual variance	175	(1)
Between- and within-classes sums of squares and products	176	(2)
Partitioning the total sum of squares	178	(3)
The equal-class-effects sub-model: a single line	181	(2)
Fitting the single-line model	183	(1)
Partitioning SST	183	(1)
Establishing tests of hypotheses	183	(2)
Two examples of Χ2-variables	185	(2)
Analysis of variance	187	(1)
Tests of intercepts	188	(2)
The multiple slopes model: intra-class regression	190	(19)
The model	190	(1)
Estimation of means	191	(1)
Estimation and unbiasedness	191	(1)
Sampling variances and covariances	192	(1)
Estimating residual variance	192	(2)
Partitioning the total sum of squares	194	(4)
The equal-class-effects sub-model: a pencil of lines	198	(1)
Fitting the model	198	(2)
Partitioning SST	200	(1)
Partitioning R (μ, bi)	200	(3)
Analysis of variance	203	(2)
Tests of intercepts	205	(4)
Appendix	209	(1)
Summary	209	(1)
Single slope model	210	(1)
Multiple slopes model	210	(1)
Exercises	210	(2)
Matrix Algebra and Quadratic Forms (A Prelude to Chapter 8)	212	(29)
Matrix algebra	212	(8)
Notation	212	(1)
The rank and trace of a matrix	213	(1)
Eigenvalues and eigenvectors	214	(1)
Idempotent matrices	214	(1)
Summing vectors and J matrices	214	(1)
Positive definite and allied matrices	215	(1)
Dispersion matrices	215	(1)
Generalized inverse matrices	216	(1)
Generalized inverses of X'X	217	(2)
Calculating a generalized inverse	219	(1)
Solving linear equations	220	(4)
Solutions	220	(1)
An invariance property	221	(3)
Partitioning X'X	224	(2)
Generalized inverses	224	(1)
Rank	225	(1)
Non-central Χ2 and f	226	(4)
Normal distributions	226	(2)
Χ2-distributions	228	(1)
F-distributions	228	(2)
Quadratic forms	230	(3)
Mean and variance	231	(1)
The Χ2-distribution of a quadratic form	232	(1)
Independence of two quadratic forms	232	(1)
Hypothesis testing	233	(5)
A general procedure	233	(1)
Alternative forms of a hypothesis	234	(1)
Scrutinizing θ' A θ	234	(1)
Using expected values	234	(1)
Too many statements	235	(1)
Using eigenvectors of A	235	(3)
Exercises	238	(3)
A General Linear Model	241	(85)
The model	243	(3)
Normal equations and their solutions	246	(8)
The general case	246	(1)
Examples	247	(6)
The 1-way classification	253	(1)
Using a solution to the normal equations	254	(5)
Mean, and dispersion matrix, of β	254	(1)
Estimating E(y)	255	(1)
Residual sum of squares	256	(2)
Estimating residual error variance	258	(1)
Partioning the total sum of squares	259	(3)
Reduction in sum of squares	259	(1)
Sum of squares due to the mean	260	(1)
Coefficient of determination	261	(1)
Partitioning the model	262	(12)
More than one kind of parameter	262	(1)
Estimation	263	(1)
Sums of squares	264	(1)
Examples	265	(1)
The 1-way classification with covariate	265	(1)
The 1-way classification	265	(2)
Expected sums of squares	267	(1)
The ``invert part of the inverse'' algorithm	267	(1)
For the full rank case	267	(2)
For the non full rank case---a caution	269	(3)
Extended partitioning	272	(1)
Partitioning the total sum of squares	272	(1)
Summary	273	(1)
F-statistics from partitioned models	274	(8)
Error sums of squares	275	(1)
Reductions in sums of squares	275	(1)
Two sums of squares	275	(1)
Χ2-properties	276	(1)
Independence of SSE	276	(1)
Independence of each other	277	(1)
Summary	278	(1)
A general hypothesis	278	(1)
Specific hypotheses	279	(2)
The full rank case	281	(1)
Examples	281	(1)
The 1-way classification	281	(1)
The 1-way classification with a covariate	281	(1)
Estimable functions	282	(6)
Invariance to solutions of the normal equations	282	(2)
Definitions	284	(1)
BLU estimation	285	(1)
Confidence intervals	286	(1)
Other properties	286	(1)
Basic estimable functions	287	(1)
Full rank models	288	(1)
Summary	288	(1)
The general linear hypothesis	288	(15)
A general form	289	(1)
The F-statistic	290	(3)
The hypothesis H: K'β = 0	293	(1)
Estimation under the hypothesis	294	(1)
Calculating Q	294	(2)
Analysis of variance	296	(1)
Non-testable hypotheses	296	(2)
Partially testable hypotheses	298	(1)
Independent and orthogonal contrasts	299	(1)
An example (Example 4)	299	(1)
The general case	300	(1)
A necessary condition	301	(2)
Contrasts are linearly independent of the mean	303	(1)
Restricted models	303	(9)
Using restrictions explicitly	304	(1)
General methodology	305	(1)
Non-estimable restrictions	306	(1)
A solution vector	306	(1)
Estimable functions	307	(1)
Hypothesis testing	308	(1)
Estimable restrictions	309	(1)
A solution vector	309	(1)
Estimable functions	310	(1)
Hypothesis testing	310	(1)
The full rank model	311	(1)
Application to a cell means model	312	(3)
Four methods of estimation	315	(2)
Summary	317	(3)
Appendix	320	(2)
Estimation under the hypothesis (Section 8.8d)	320	(1)
The likelihood ratio test	320	(2)
Exercises	322	(4)
The 2-Way Crossed Classification: Overparameterized Models	326	(58)
The with-interaction model	326	(14)
The model	326	(1)
Relationships to cell means	327	(1)
Reparameterization	328	(3)
Estimable functions	331	(1)
Estimation	332	(1)
Hypothesis testing	332	(1)
Analysis of variance	333	(2)
Model equations	335	(3)
The normal equations	338	(1)
Solving the normal equations	339	(1)
A zero sum of squares	339	(1)
The without-interaction model	340	(12)
The model	340	(1)
Relationships to cell means	341	(1)
Reparameterization	341	(1)
Estimable functions	341	(1)
Estimation	342	(1)
Hypothesis testing	342	(1)
Analysis of variance	343	(1)
Model equations	344	(1)
The normal equations	344	(1)
Solving the normal equations	344	(4)
Sampling variances	348	(1)
Estimating cell means and contrasts	349	(3)
Sums of squares for the overparameterized model	352	(6)
R (μ), R (α|μ) and R (β|μ)	352	(1)
R (β|μ,α), and R (α|μ,β)	352	(2)
R (Φ|μ, α, β)	354	(1)
Example (continued)	355	(2)
Analyses of variance	357	(1)
Models with Σ-restrictions	358	(15)
The no-interaction model	359	(1)
Solutions	360	(1)
Analysis of variance sums of squares	360	(1)
Another sum of squares	361	(1)
Associated hypotheses	362	(1)
The invert-part-of-the-inverse algorithm	362	(1)
The with-interaction model	363	(1)
All-cells-filled data	363	(4)
Some-cells-empty data	367	(6)
Other restrictions	373	(1)
Constraints on solutions	373	(5)
Solving the normal equations	373	(2)
A general algorithm	375	(1)
Other procedures	375	(1)
Constraints and restrictions	376	(2)
Exercises	378	(6)
Extended Cell Means Models	384	(32)
Multi-factor data: basic results	384	(3)
The model	384	(1)
Estimation	385	(1)
Residual variance	385	(1)
Estimable functions	385	(1)
Hypothesis testing	386	(1)
Analogy: the 1-way classification	386	(1)
Some-cells-empty data	386	(1)
Multi-factor data: all-cells-filled data	387	(9)
The 3-way crossed classification	388	(1)
Computing, for Σ-restricted, overparameterized models	389	(2)
Example	391	(4)
More than three factors	395	(1)
Main-effects-only models	396	(4)
Models with not all interactions	400	(15)
The 2-way classification	400	(1)
Unrestricted models	401	(1)
Restricted models	402	(1)
Examples	402	(3)
The general case	405	(2)
Estimability	407	(1)
All cells filled	407	(1)
Some cells empty	407	(1)
Hypothesis testing	408	(1)
Examples of estimable functions	409	(1)
The 2 x 2 case with all cells filled	409	(1)
The 2 x 2 case with an empty cell	410	(1)
A 3-factor case, with no 3-factor interactions	410	(4)
A 3-factor case with two sets of interactions absent	414	(1)
Conclusions	414	(1)
Exercises	415	(1)
Models with Covariables: The General Case and Some Applications	416	(41)
A traditional description	416	(3)
A linear model description	419	(12)
A general model	420	(1)
Estimation	421	(1)
Normal equations	421	(1)
Assumptions on the covariates	421	(1)
Estimators	422	(1)
Dispersion matrices of estimators	423	(1)
Analysis of variance	424	(1)
Hypothesis testing	424	(1)
Some general hypotheses	424	(2)
Associated hypotheses in the analysis of variance	426	(1)
Examples: 1-way classification, one covariable	426	(1)
The single slope model	426	(3)
The multiple slopes model	429	(1)
Other hypotheses about slopes	430	(1)
Confirming associated hypotheses of Chapter 6	431	(7)
The single slope model (Table 6,4)	432	(1)
Sums of squares	432	(1)
Associated hypotheses	432	(2)
The multiple slopes model: Tables 6.6 and 6.7	434	(1)
Models	434	(2)
Sums of squares	436	(1)
Associated hypotheses	436	(2)
Usefulness of the hypotheses	438	(1)
The 1-way classification: restricted overparameterized models	438	(3)
The 1-way classification: two covariates	441	(6)
Basic tools and notation	441	(1)
Single slope for each covariate	442	(2)
Intra-class slopes and a single slope	444	(1)
Intra-class slopes for each covariate	445	(2)
The 2-way classification: single slope models for one covariate	447	(4)
The with-interaction model	447	(1)
The no-interaction model	448	(3)
The 2-way classification: multiple slope models	451	(3)
Interaction models, with unbalanced data	451	(1)
Intra-row slopes for one covariate	451	(1)
Intra-row slopes for one covariate, intra-column for another	452	(1)
Intra-row slopes for each of two covariates	452	(1)
Intra-row plus intra-column slopes for one covariate	452	(1)
No-interaction models with balanced data	452	(1)
Intra-row slopes for one covariate	453	(1)
Intra-row slopes for one covariate, intra-column for another	453	(1)
Intra-row slopes for each of two covariates	454	(1)
Intra-row plus intra-column slopes for one covariate	454	(1)
Exercises	454	(3)
Comments on Computing Packages	457	(27)
Sums of squares output	458	(3)
Σ-restrictions	460	(1)
Weighted squares of means	461	(1)
Patterns of filled cells	461	(1)
Balanced data	461	(1)
Four sets of output from SAS GLM	461	(1)
Sums of squares from SAS GLM	461	(4)
Type I	462	(1)
Type II	462	(1)
Type III	463	(1)
Type IV	463	(2)
Estimable functions in SAS GLM output	465	(6)
Examples	465	(1)
Output is parameter labels and coefficients	465	(1)
Estimable function obtained from output	466	(1)
Estimable function provides the hypothesis	467	(1)
Summary	468	(1)
Comment	468	(1)
Verification	469	(1)
A general result	469	(1)
Examples	470	(1)
Solution vector output	471	(7)
BMDP2V	472	(1)
GENSTAT ANOVA	472	(1)
SAS GLM	472	(1)
Using a generalized inverse	472	(1)
The solution vector	472	(1)
Standard errors and t-statistics output	473	(1)
Estimating estimable functions	474	(1)
Other features	474	(1)
SAS HARVEY	475	(2)
SPSS ANOVA	477	(1)
A faulty computing algorithm	478	(2)
Output for analysis with covariates	480	(4)
Mixed Models: A Thumbnail Survey	484	(33)
Introduction	484	(6)
An example	484	(2)
A general description	486	(3)
Estimation	489	(1)
Estimating variance components from balanced data	490	(5)
Example	491	(1)
Merits	492	(1)
Broad applicability	492	(1)
Unbiasedness	492	(1)
Sampling variances	493	(1)
Demerits	494	(1)
Negative estimates	494	(1)
Distributional properties	495	(1)
Estimating variance components from unbalanced data	495	(12)
General ANOVA methodology	495	(3)
Merits	498	(1)
Demerits	498	(1)
Henderson's Method I	498	(2)
Henderson's Method II	500	(1)
Henderson's Method III	500	(2)
ML (Maximum Likelihood)	502	(2)
REML (Restricted Maximum Likelihood)	504	(2)
MINQUE (Minimum Norm Quadratic Unbiased Estimation)	506	(1)
MIVQUE (Minimum Variance Quadratic Unbiased Estimation)	507	(1)
I-MINQUE (Interative MINQUE)	507	(1)
Prediction of random variables	507	(3)
An example: the 1-way classification	510	(4)
The model	510	(1)
Analysis of variance	511	(1)
Testing the mean	511	(2)
Prediction	513	(1)
Estimating variance components	514	(1)
Exercises	514	(3)
References	517	(5)
Statistical Tables	522	(5)
List of Tables and Figures	527	(6)
Index	533