Table Of ContentJan-Bernd Lohmoller
Latent Variable
Path Modeling with
Partial Least Squares
Springer-Verlag Berlin Heidelberg GmbH
Dr. Jan-Bernd Lohmoller
Free University Berlin
Dept. of Political Science
(Otto-Suhr-Institute)
Ihnestra13e 21
D-I000 Berlin 33, FRG
Min Ollern toudacht
ISBN 978-3-642-52514-8
CIP-Titelaufnahme der Deutschen Bibliothek
Lohmăller, Jan-Bernd:
Latent variable path model ing with partial least squares / Jan
Bernd Lohmăller.
ISBN 978-3-642-52514-8 ISBN 978-3-642-52512-4 (eBook)
DOI 10.1007/978-3-642-52512-4
This work is subject ta copyright. AII rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translatioI), reprinting, reuse of illustrations, re citation, broad
casting, reproduction on microfilms Of in other ways, and starage in data banks. Duplication of this
publication or parts thereof is on1y permitted under the provisions of the German Copyright Law of
September 9, 1965, in its version of June 24,1985, and a copyright fee must always be paid. Violations
fali under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1989
Originally published by Physica-Verlag Heidelberg in 1989
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of
a specific statement, that such names are exempt from the relevant protective laws and regu1ations and
therefore free for general use.
TEX typeset by J. B. L.
712017130-543210
Preface
I am greatly indepted to my teachers - and collegues - Professors Heinz Mandl,
Rolf Oerter, Rainer B. Pelka, and Jiirgen W. Falter, who gave me unceasing
encouragement in my work through their useful! questioning and their confronting
me with many problems of a challenging nature over a period of many years.
Since 1978 when I first heard a lecture by Professor Herman Wold, I have
been excited and captivated by the potentialities preferred by his PLS modeling
method. In the ensuing years of collaboration I not only was introduced into and
motivated by his scholarly life-style but also personally moved by his attentive
and generous attitude. Thanks to his constant inspiration, criticism and encour
agement and his "Advice to a Young Scientist" he became what I am proud to
call my Doktorvater.
Experience with a particular statistical method derives from various sources,
namely (a) mathematical reasoning, (b) programming, (c) application and (d)
teaching. All four types of experience leave their marks on the language, style,
presentation, and type of reasoning in this dissertation: (a) An attempt has been
made to make the mathematical reasoning the core of this monograph. (b) The
program PLS 1.8 (Lohmoller 1981a) documents how I understand PLS and is
understandable to every computer but not, unfortunately, to every reader. So it
has to be explained avoiding computer jargon. (c) With every new application
new insight is obtained as to what the method is able to do and what not, which
may, or may not, require changes in the program and the mathematical treatment.
(d) Teaching proves to be the most demanding test for a method. Personally, I
don't mind if this monograph is all of "too" mathematical, computerized, applied
and didactic at the same time.
Summary
Model construction may be seen as the very essence of scientific work. Within
the frame of reference of a model theoretical and empirical content are kept apart
in order to prove that they match one another (Sc.l.l). Because of their simple
algebraic tractability linear-relation models are often favoured. They are called
path models, owing to their graphical representation as path or arrow diagrams.
Path models with latent variables (LVP models) are applicable to a broad range
of subject matters (Sc.7).
Relations in LVP models may or may not be subject to a causal interpretation
(Sc.l.3). For operative use in most applications it is sufficient to specify predic
tors (Sc.3.1), which leads on to estimation by Wold's Partial Least Squares (PLS)
method. The introductory chapter (Sc.l) is followed by a description of the esti
mation method, Wold's basic method (Sc.2.1 ), as well as of the extended method
(Sc.2.2, Sc.2.3).
The core of this monograph (Sc.3) is devoted to the statistical foundation of
the PLS method, in Sc.3.1 by a recapitulation of Wold's predictor specification
for one relation; in Sc.3.2 and Sc.3.3 by a reconstruction of the psychometric
factor score problem from the PLS theory; in Sc.3.4 by a unifying view on several
traditional two-block models like Canonical analysis; in Sc.3.5 by the introduction
of the Split Principal Component theorem as a basis for reflections on properties
of multi-block models. The most recent developments by Manfred Giang (1988)
are not integrated into this monograph.
The extension of PLS modeling to contingency tables (Sc.4) and three-mode
data cubes (Sc.6) raises the question (Sc.1.3) as to whether individual scores (case
values) are essential to the model (applied model, data structure model), ores
sential only to the estimation an merely incidential to the theory (abstract model,
covariance structure model). PLS is data-structure oriented, while the ML esti
mation method of LISREL is covariance-structure oriented (Sc.5). The distinction
of individual vs. general, of applied vs. abstract, of data vs. covariance modeling
gives a rationale for the choice between PLS and ML estimation method. The
ensuing difference between PLS and ML estimates are presented and explained in
Sc.5.5.
Contents
Preface 5
Summary 6
Table of Contents 7
List of Tables 11
1 Basic Principles of Model Building 13
1.1 Empirical and Theoretical Concepts 13
1.1.1 Levels of Theory and Data . 13
1.2 Causation and Prediction ..... . 16
1.3 Data Modeling vs. Covariance Modeling 22
1.4 Notation ................. . 23
2 The Basic and the Extended PLS Method 27
2.1 Wold's Basic Method of Soft Modeling . 28
2.1.1 Model Specification ... . 28
2.1.2 The Basic PLS Algorithm ... . 30
2.1.3 Extensions and Properties ... . 31
2.2 The Extended Method: Specification and Properties 31
2.2.1 Formal Specification . . . . . . . . . . . 31
2.2.2 Deductive Properties of the LVP Model 32
2.2.3 Inductive Properties of the LVP Model . 33
2.2.4 Specification of an LVP Model 35
2.3 Estimation in the Extended Method . . 37
2.3.1 Partial Least Squares ..... . 37
2.3.2 LS Modules for PLS Algorithms 39
2.3.3 The Inner Weighting Modes . . . 41
2.3.4 Patterned Orthogonalization of LVs 43
2.3.5 Weights and Loadings . . . . . . . . 46
2.3.6 Three Solutions for Conflicting Constraints 47
2.4 Assessment of Results . . . . . . . . . . 49
2.4.1 Information from the Model . . . 49
2.4.2 Five Predictions, Five Residuals 50
2.4.3 Fit Indices . . . . . . . . . . . . 52
8 Table of Contents
2.4.4 Reliability Indices . . . . . . . . . . . . . 53
2.4.5 Some Advice . . . . . . . . . . . . . . . . 55
2.5 Application: Home Environment and Intelligence 56
3 Foundations of Partial Least Squares 63
3.1 Conditional Expectation and Predictor Specification 63
3.1.1 The Notion of Conditional Expectation . 64
3.1.2 Properties of the Conditional Expectation 67
3.1.3 The Linear Expectation 69
3.1.4 Predictor Specification . . . . . . . . . 72
3.1.5 Estimation .............. . 73
3.1.6 Interlocking Conditional Expectation . 75
3.1. 7 Eigenvalue Problem and Power Method 77
3.1.8 Convergence of Power Algorithm 80
3.2 Principal and Other Components 81
3.2.1 Concepts and Notations 82
3.2.2 Aggregates, Composites 86
3.2.3 Factors ....... . 86
3.2.4 Components . . . . . . 88
3.2.5 Principal Components . 91
3.2.6 The Principal Components Model 94
3.3 Factor Score Estimation . . . . . . 99
3.3.1 Properties of Factor Scores ... . 99
3.3.2 Factor Estimation ModeB .... . 102
3.3.3 Factor Estimation ModeB, Reestimated Loadings . 104
3.3.4 Factor Estimation ModeA . . . . . . . . . . . . . . 106
3.3.5 Factor Estimation ModeA, Reestimated Loadings. 108
3.3.6 Summary of Factor Estimation 109
3.4 Predictive Two-Block Models ...... . 110
3.4.1 The Two-Block Factor Model .. . 111
3.4.2 The Canonical Correlation Model . 112
3.4.3 The Principal Predictor Model . . 117
3.4.4 The Interbattery Factor Model . . 121
3.4.5 Fortier's Simultaneous Linear Prediction Model . 124
3.4.6 The MIMIC Model . 125
3.4. 7 Discussion . . . . . . . . . . . . 127
3.5 Split Principal Components ..... . 128
3.5.1 Hierarchical Component Model 128
3.5.2 Splitting Principal Components . 133
3.5.3 Horst's Maximum Variance Algorithm 137
3.5.4 The Principle of Constant Proportionality . 137
·3.5.5 Principal Component, One Variable Omitted. 139
3.5.6 Hierarchical Component Model, One Block Omitted 141
3.5. 7 Applications of the Split PC Theorem . . . . 144
3.6 Split Multiple Regression .............. . 147
3.6.1 Split Multiple Regression and PLS Approach 147
Table of Contents 9
3.6.2 How Great is the Loss in R2? 149
3.6.3 Conclusions and Recommendations . . . . . . . . . . . . . . 152
3.7 Uncorrelated Dimensions in Generalized Canonical Correlation Anal-
ysis ....................... . 153
4 Mixed Measurement Level Multivariate Data 155
4.1 Categorical Variables and LS Methods . . . . . 155
4.1.1 Super Contingency Tables ....... . 156
4.1.2 Canonical Analysis of Contingency Tables 162
4.1.3 Principal Components of Contingency Tables 165
4.1.4 Categorical Scaling .............. . 167
4.1.5 LV Path Analysis of Super Contingency Tables 169
4.2 Mixture of Categorical and Interval-Scaled Variables 172
4.2.1 The Mixed Product Moment Matrix 172
4.2.2 One Categorical Predictor .. 173
4.2.3 Two Categorical Predictors 175
4.2.4 One Categorical Predictand . 177
4.2.5 More Variables ....... . 180
4.3 Application: SES and Educational Aspiration 181
4.4 Different Slopes in Different Groups . . . . . 185
4.4.1 MV Path Models with Product Variables 185
4.4.2 LV Path Models with Product Variables . 189
4.4.3 Metric of Product Variables . . . . . . . . 190
4.5 Application: TV Consumption and Fear of Crime . 193
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . 197
5 Predictive vs. Structural Modeling: PLS vs. ML 199
5.1 Covariance vs. Data Structure Models ... . 200
5.2 Scored and Unscored LVs .......... . 204
5.3 Consistency and Bias in a Two-Block Model. 209
5.4 The Interpretation of Consistency at Large 213
5.5 Some PLS - LISREL Comparisons . . . . . . 216
5.6 The PLS Solution of the Identification Problem . 222
5.6.1 Restriction for Scale Unambiguity (RSU) 222
5.6.2 Restrictions for Identifiability 225
5.6.3 Identifiability in PLS Model . . . . . . . . 225
6 Latent Variables Three-Mode Path (LVP3) Analysis 227
6.1 Three-Way Data Models ............... . 227
6.2 The Kronecker Principal Component (KPC) Model . 228
6.3 The Three-Mode LV Path (LVP3) Model 232
6.4 Special Cases and Properties . . . . . 233
6.5 The PLS Estimation of LVP3 Models . 235
6.6 Application: Longitudinal Data . 236
6. 7 Concluding Remarks . . . . . . . . . . 239
10 Table of Contents
7 PLS Programs and Applications 241
7.1 PLS Programs . . . . . . . . . . . . . . . . . 241
7.2 Applications . . . . . . . . . . . . . . . . . . . 242
7.2.1 Applications to Non-Individual Data . 242
7.2.2 Applications in Psychological and Educational Research 245
Bibliography 249
Author Index 273
Subject Index 277
