Gérard Genette's Evolving Narrative Poetics John Pier

Author of what is perhaps the most perennial treatise of the early years of narratology and whose influence is among the most pervasive, Gérard Genette produced a narrative theory in a context that rendered that theory unique. Appearing at a time when structural linguistics, a shaky doctrine at best, was called on to act as a "pilot science" for the social sciences, Genette's Narrative Discourse filled a number of gaps in the emerging approach to the study of narrative. Where other studies were focused largely on devising formal models of the "story" level, Narrative Discourse turned toward the signifying level, "discourse." Where an unbridged territory lay between the deeper levels and the surface level, Narrative Discourse explored the various relations between the narrated story, the signifying narrative text and the narrating act. And where the new paradigms were neglectful of the more traditional questions of narrative theory-point of view, narrator-character discourse, narrative time, the status of the narrator-Narrative Discourse integrated them into the narratological debate. Not the least of its achievements was to have provided an innovative terminology for narrative devices, a terminology which, for instance, adopted the term "diegesis" from film theory in order to mark off the presence or absence of the narrator in the narrated world (homodiegetic vs. heterodiegetic) from the relation of the narrating act to the narrated events (extradiegetic vs. intradiegetic), or, to take another example, a terminology that drew attention to iterative narration (saying one time what occurred several times) as a form of syllepsis that shares with analepsis and prolepsis the feature of "taking together." All in all, the achievement of Narrative Discourse was to set a new standard by providing a comprehensive and wellarticulated approach to describing and analyzing the texture of narrative as a self-regulating system.

Many of the terms inaugurated by Genette have long been household words of the theoretical and critical discourse about narrative, even beyond paradigms that do not consciously adhere to his system. Although this theory together with its terminology and the welcome synthetic treatment of familiar narrative devices and techniques offered by Narrative Discourse have contributed to the book's success, the theory and its application have been submitted to various critiques. In part, these critiques have resulted from but also contributed to a certain codification of the system, as revealed in school manuals and elsewhere by use of its terminology as mere labels, either devoid of its theoretical import, misapplied to the textual organization it is meant to describe, or appropriated in ways that are incompatible with its original intent.

The reply to these critiques came with the publication of Narrative Discourse Revisited in 1983. Rather than summarize the pros and cons of the various arguments put forth, what I would like to point out here is that a number of the questions debated in this book and carried on by other researchers in other forums might never have been raised, or they would have been approached from a different angle, had Narrative Discourse never appeared. What would the state of research on perspective and point of view be today had Genette, in introducing the notion of focalization, not drawn a distinction between "who sees?" and "who speaks?" Contested, modified and reformulated, Genette's focalization in fact opened up new lines of investigation: re-conceptualizing of the relations between speaker and focalizer thanks to fine-grained linguistic analyses of the expression of subjectivity; the necessity of accounting for focalization at both micro-and macro-level; the impact of focalization that varies within a single sentence; etc. Another example of a theoretical issue for which the stage was set by Genette but whose ramifications were to be explored in other research contexts is voice. Finding grammatical person a questionable criterion by which to classify narrators and thus adopting the opposition homodiegetic/heterodiegetic, which allows for degrees of narrator presence in the story, he identified a space for borderline, mixed and ambiguous narrators (Revisited 104). It was some years later (in 1994) that Monika Fludernik, in a special issue of Style devoted to second-person narrative, took fuller account of this form, noting for example the propensity of second-person narrators for present-tense narration, their tendency toward a certain complicity with the narratee and the affinity of such narration with interior monologue-features that are in no way incompatible with Genettian narratology.

Other examples could of course be cited, but the point I wish to make is that although Genette's narrative theory might appear to be dated from the vantage point of postclassical narratology, this may not be the case when it is viewed in accordance with what it itself purports to be: a study of the specificity of narrative within the scope of an open poetics. Now, from this perspective, it would be fitting to reconsider Narrative Discourse, not as a paradigm case of classical narratology with its rage for taxonomies, binarisms and closed and sterile formalisms, but as a method that navigates between the particular (in this case, Proust's Remembrance of Things Past) and the general to which all organized bodies of knowledge, including the theory of literature, aspire. What David Herman, in "Sciences of the Text," observed about Roland Barthes's pre-S/Z narrative theory would then also hold true for Genette and his other contemporaries, namely that narratological theories of the classical period did not have at their disposal the resources of discourse analysis, text grammars and methods for oral and conversational storytelling which were in a state of gestation at the time, but that they prefigured developments that were to take form at a later time. This being the case, and unlike several of the more recent theories of narrative that draw on the postulates of possible worlds logic or on those of the cognitive sciences or theories that model themselves after the methodologies of sociolinguistics or artificial intelligence, Genette's approach does not seek to incorporate other scientific paradigms, but rather to position itself within poetics.

It is in fact rather astonishing to read in the introduction of Narrative Discourse that the analysis of Proust's novel will be conducted, not with structural linguistics and its categories as its theoretical basis, but by analogy with rhetorical expansion of the traditional grammatical-and not linguistic-categories of the verb: tense, mood and person (29).

So the question that now arises is this: if Genette's narrative poetics is to be included under classical narratology, where lies the fault line between classical and postclassical narratology? According to Herman's well-known discussion, postclassical narratology, synonymous with narratologies (of which poststructuralist narratology is but one variety), is characterized by "a plurality of models for narrative analysis," and it incorporates classical narratology as one of its "moments" ("Introduction" 1). It tests the limits and possibilities of its structuralist ancestor, enriching it with more inclusive and open-ended paradigms and methodologies, but with the same goal of developing rigorous descriptive and explanatory models. Narratology, once "a subfield of structuralist literary theory," has since expanded its scope to include "any principled approach to the study of narratively organized discourse" (27 n. 1). Ansgar Nünning, for his part, draws a sharper line between the two when he opposes structuralist ("classical") narratology to the new ("postclassical") narratologies: the one is text-centered, focuses on narrative langue, textual "features" and "properties," emphasizes formalist description and taxonomies, and constitutes a (relatively) unified (sub)discipline, etc.; the other is context-oriented, focuses on narrative parole and the dynamics of the reading process, favors thematic readings and ideologically-charged evaluations, and represents an interdisciplinary project of heterogeneous approaches, etc. ("Narratology" 243 -44).

The transition between the two phases can perhaps best be summarized in Martin Kreisworth's phrase: "the narrative turn." Overall, the narrative turn stems from two factors.

First is the ubiquity of narrative, the fact that narrative informs all aspects of human experience and knowledge and is thus present in all disciplines, calling for a transdisciplinary approach to narrative theory that embraces, not only literature and the humanities, but also jurisprudence, medicine, the social and even the natural sciences. Second is the "export" of narratological concepts and methods to other literary genres (transgeneric narratology) as well as to non-literary and non-linguistic narratives (transmedial narratology) and even an appeal to narratology as a "master discipline." It is this expansion of research agendas that has provided postclassical narratology with its dynamism and that has also spawned a host of socalled compound or "hyphenated" narratologies-some twenty-five, by Nünning's count, not to speak of a nearly equal number of applications of varying narratological relevance.

Needless to say, the risk inherent in "Narratology's Centrifugal Forces," as Jackson G. Barry has put it, is to dilute the very principles that were the hallmark of the original theory and its driving forces, although I hasten to add that many first-rate postclassical narratologists have not fallen prey to this risk and continue to work in the spirit of discrimination and rigor that characterize narratological research.

In contrast to the research agendas growing out of the narrative turn, the object of Genette's narrative poetics is the specificity of narrative, not its ubiquity. The theory seeks to describe the functioning of narrative in a delicate balance between the particular and the general rather than to elaborate a model valid for narrative in other fields or to incorporate paradigms borrowed from other disciplines-a situation which seems to have colored the postclassical view of Genette. For my part, I shall not take issue with an author for failing to do what he never intended to do in the first place. More pertinently, it is worth pointing outand this leads to one of the main points I wish to make-that references to Genette in published research on narrative theory over the past few decades are limited almost exclusively to Narrative Discourse, Narrative Discourse Revisited and a small number of earlier essays plus, occasionally, a few isolated pages of Fiction and Diction. All too frequently left out of account by narrative theorists is Genette's important work on transtextuality, poetics, aesthetics and the theory of art that succeeded the narratological studies and whose implications for narrative theory are rarely discussed. The proper context in which to place Genettian narratology, it seems to me, is not a transdisciplinary theory of narrative cultivated by postclassical narratology, but an open poetics.

Included in the French edition of Narrative Discourse, but not in the English translation, are two essays, "Critique et poétique" and "Poétique et histoire," that stake out the early stage of Genette's conceptual framework for narrative theory. Where the object of criticism is the work in its singularity, its closure, that of poetics is a general theory of literary forms, the combinatory possibilities of discourse, such that the singular work is but a particular case beyond which other possibilities can be foreseen or deduced: not the "real" of literary discourse, but its "virtual"-"literariness," to adopt the Russian formalist term.

Historically, poetics studies not the succession of works but transformations, in other words, permanence and change: for this reason, the historical object of literature is "the elements transcending works and constitutive of the literary game that, in a word, we shall call forms: rhetorical codes, narrative techniques, poetic structures, etc." (Figures III 18 my translation).

These conceptual distinctions are fundamental, for they cast light on Genette's characteristically typological style of reasoning, his "tabular poetics," as Christine Montalbetti put it in a fine volume devoted precisely to Genette's "open poetics": if the aim of his narrative theory is to highlight the functioning of narrative, its "immanence," the accent of poetics, which embraces both the actual and the virtual, falls on "transcendence" of the text, the extension and exploration of fields of investigation.

Transcendence is a key notion in Genette's poetics and aesthetics that will be more fully developed in The Work of Art, it being emphasized that in this system transcendence is in no way to be associated with mysticism or with expression of the sublime. The object of poetics is not the singular text but, as maintained in The Architext and further developed in Palimpsests, "textual transcendence of the text": "all that sets a text in a relationship, whether obvious or concealed, with other texts" (Palimpsests 1). This textual transcendence, also called "transtextuality," takes the form of five types of relations, extending from the inclusion of one text in another (intertextuality) to the most abstract or implicit relation such as genre (architextuality) and including the critical relation (metatextuality), transformation of one text by another (hypertextuality, studied in Palimpsests) and the "intermediate zone" lying between texts and the surrounding social discourse (paratextuality, studied in Paratexts). The transtextual dimension of Genette's poetics, almost never referred to by postclassical narratologists, addresses a number of topics that are currently under debate. To take one example, the recent transgeneric applications of narratology, predicated on the "export" of narratological concepts to lyrical poetry and drama, are examined in Palimpsests, but in a wholly different light. The dramatization of a narrative text (or narrativization of a play) involves a specific type of hypertextual transformation, a change of mode in the sense of Plato or Aristotle, or "transmodalization." What occurs in such cases is not the extension of narratological principles and paradigms to cover another genre, but a form of textual transcendence in which a given mode of representation is transformed into another with, of course, the technical consequences that such a transformation entails. On another plane, paratextuality, broken down into peritext, which surrounds the text within the volume, includes both editorial features (title page, cover, composition, etc.) and epitext, lying outside the text properly speaking, either public (e.g. interviews) or private (journals, rough drafts, correspondence, etc.). Paratextuality forms a privileged crossroads of pragmatic and strategic of negotiation, contributing to a "community of interest" that allows a book, literary or otherwise, to be presented as such. It is notable that the "toolkit" conceptions of narratology propounded by some, focused on the textual properties of narrative and thus on textual immanence, are apt to totally overlook the broader picture of the various forms of transtextual relations and thus the crucial role that poetics can play in the theory of narrative.

It can be concluded, then, that when considered transtextually, the typological reasoning that underlies Genette's tabular poetics should not be reduced to a "boxing in" of the literary text, as poststructuralists would be inclined to say, but is rather a means by which the text transcends its immanence: "the text interests me (only) in its textual transcendence" (Architext 81). This being the case, texts in their materiality have no particular status as such, but acquire one only when apprehended in a web of transtextual relations. Genette is thus led to inquiring further into the literariness of literature, in other words (to quote Jakobson), into "the differentia specifica of verbal art in relation to other arts and in relation to other kinds of verbal behavior" (Jakobson 350; qtd. in Fiction 2). For Genette, the pertinent question is to determine how it is that an aesthetic function is attributed to certain verbal objects. This matter is taken up in Fiction and Diction, where three aspects of literariness are investigated: its regimes, its criteria, its modes.

There exist two regimes of literariness, one constitutive or essentialist, independent of any judgment (a sonnet is a sonnet, a novel is a novel, etc.), the other conditional, subject to individual aesthetic judgment (an autobiography appreciated for the eloquence of its style, a novel that revolutionizes narrative technique, etc.). These regimes intersect with two criteria: thematic (what the text is "about") and rhematic (the type of discourse exemplified: poetry or prose). Regime and criterion combine to form two modes of literariness: fiction, defined by the imaginary nature of its objects; and diction, characterized by its formal features. The common trait between the two modes is their transitivity, fiction being pseudo-referential and diction a form of discourse in which "signification is inseparable from verbal form" (25).

When looked at in the broader perspective of Genette's poetics, it can thus be seen that textual immanence is transcended on the one hand by various types of transtextual relations, and on the other by overlapping dimensions and facets of literariness.

In the same volume, Genette takes up the relations between fictional and factual narrative, a question largely ignored by classical narratology due to its bias toward eighteenthto twentieth-century prose fiction. From the perspective of the categories established in Narrative Discourse, he finds no radical differences between the two forms in strictly narratological terms. To better apprehend the relations between them, he thus modifies the criteria somewhat, firstly by introducing two antithetical views of fictional and factual narrative: that of John Searle, for whom there exists "no textual, syntactical, or semantic property that will identify a text as a work of fiction" (Searle 65;qtd. in Fiction 57), as opposed to Käte Hamburger (followed by Dorrit Cohn), who focuses on the textual "indexes" of fictional narrative, notably markers of the characters' subjectivity such as free indirect speech. Genette arrives at a balanced compromise as to the applicability of these two positions, adopting a "gradualist" stance on the relations between fictional and factual narrative (thus setting himself off from Cohn's "separatist" approach): heterodiegetic fictional narrative, for example, is often a mimesis of history or of chronicles, while the new journalism draws freely on the techniques of literary fiction (Fiction 81). The other important innovation of this article is that it introduces a new factor into narrative voice: the author. Genette works out the five possible functional relations of identity and non-identity between author, narrator and character that come into play in narratives. Identity between the three characterizes autobiography, just as non-identity between them marks heterodiegetic fiction.

On the other hand, identity of narrator and character but non-identity of either with the author signals homodiegetic fiction. Overall, the non-identity of author and narrator defines fictional narrative and their identity, going so far as to assimilate the narrator into the author, is characteristic of factual narrative. But the system is in no way constrained to these general postulates, due in part to the fact that introduction of the author into a narrative is as much a matter of paratextuality as it is of narratology. Thus, Caesar's The Gallic Wars, in the third person, is an autobiography whereas Yourcenar's Mémoires d'Hadrien, in which narrator and character coincide but not narrator and author, is the autobiography of a historical figure by a fictitious narrator.

The discussion of fictional and factual narrative exemplifies the evolution of Genette's narrative theory away from a "restricted narratology"-the study of narrative with a particular focus on its specificity-toward an open poetics (Fiction 56). The general framework of this development is confirmed by other factors including the integration of transtextual relations into the system as well as by a differentiated analysis of the aesthetics of literature-its literariness. And this leads to the general aesthetic theory elaborated in The Work of Art and

The Aesthetic Relation, a theory already sketched out in a chapter of Palimpsests devoted to "hyperaesthetic practices" and touching on non-linguistic types of transformation in painting and music. With this extension of the field of investigation, however, comes the realization that aesthetic questions exceed the boundaries of poetics and that literariness covers a range of phenomena too narrow to fully address the relations between aesthetics and art (Fiction x, 29). And indeed, one of the principal innovations of The Work of Art is to shift the emphasis away from literariness to "artistry" (from littérarité to articité) in an effort, not to define the beautiful or to provide a classification of the arts (the traditional concerns of aesthetics), but to analyze the artistic relation, a particular type of the more general aesthetic relation, at the same time amplifying patterns of reasoning already present in Genette's earlier work in poetics.

The first of the two volumes explores the modes of existence of the work of art's immanence and transcendence. Adopting with a number of revisions a distinction formulated by Nelson Goodman, Genette explores the two regimes of immanence. The autographic regime consists in the materiality of artistic objects which lend themselves to direct sensorial perception: "things" such as an original painting or a carved sculpture which are either unique and subject to forgery or multiple and open to reproduction (cast sculpture, printmaking); and "events," corresponding roughly to the performing arts. The allographic regime covers forms of art whose object or event is ideal. Where in the autographic regime there is no distinction between immanence and physical manifestation, in the allographic regime immanence emerges from the constitutive features established by convention or common to the iteration of multiple manifestations through execution or (de)notation (cf. type/token): for the literary text, recitation (diction) and inscription; for the musical text, performance and score; for stage production, performance and stage directions. Thus an edition of Ulysses, whether mass produced or pirated, remains Ulysses, provided the copy is "orthographically" correct. In practice, the allographic regime engages the recipient in a process aimed at identifying the constitutive properties immanent to the ideal object or event, sifting them out from contingent properties-a process of "allographic reduction." As a mental operation, this process underlies the attitude of reception that contributes to determining the artistic object.

Transcendence consists in all the ways the physical object or event or the ideal object or event of immanence are "exceeded." Where given works are concerned, the autographic and allographic regimes are mutually exclusive; as modes, however, transcendence and immanence are compatible and complementary while the modes of transcendence, of which there are three, can coexist. The first type of transcendence, plural immanences, results from The regimes of immanence and their modes of transcendence constitute the modes of existence of the work of art, but not its modes of action. The latter is the realm of two types of relation which, in aesthetic theory, are often made to coincide, but that Genette is careful to separate: the aesthetic relation, consisting in "an emotional response (of appreciation) to an attentional object, whatever it might be, considered with regard to its aspect-or rather an attentional object constituted by the aspect of an object, whatever it might be" (Aesthetic 222); and the artistic relation, resulting from the ascription of an aesthetic relation to a natural or a man-made object (artifact) and whose locus is the interaction between the aesthetic and the technical. When aesthetic objects ("something that is [now] the object of aesthetic attention," meaning that "it is not the object that makes the relation aesthetic, but the relation that makes the object aesthetic" [10 -11]) and artifacts overlap, the result is a work of art, defined early in the first volume as follows: "a work of art is an intentional aesthetic object, or, what comes to the same thing, a work of art is an artifact (or human product) with an aesthetic function" (Work 4 emphasis original). Key to this definition is the concept of intention, which comes in two forms. Production of any object in the human sphere proceeds from an intention in the strong sense, while reception of that object hinges on intention in the weaker sense, notably, attention to one or another of the object's aspects such that an attempt by the receiver, justified or not, to "seek out" in or to ascribe to an object an aesthetic function bestows on that object the status of a work of art. A similar duality, reflecting a subjective relativism at the level of aesthetic appreciation, is affirmed in a reiteration of the definition of a work of art in the second volume when it is noted that interpretation can be either objective and ontological ("in order to be a work of art, an object must in fact proceed from an aesthetic intention") or subjective and functional ("in order to function as a work of art, an object must be received as proceeding from an aesthetic intention") (Aesthetic 215). This distinction translates respectively into artistries of the constitutive type and artistries of the attentional type and is the equivalent, in aesthetics, of what Genette had earlier described as the constitutive and the conditional regimes of literariness. And finally, the argument can be viewed as a reformulation of Goodman's proposal to replace the question "What is art?" with "When is art?" by asking: "When is there an aesthetic relation?" This enables Genette to conclude, from Goodman's implicit and unavowed position, that being a work of art and functioning as a work of art are conditions independent of one another, with the consequence that the artistic relation is not fully dependent on the objective artistic status of an object (210 -15).

It will not escape the attention of readers that Genette's two works on aesthetics and the artistic relation make no mention of Narrative Discourse or Narrative Discourse Revisited or that the premises and descriptive procedures of his structuralist narratology are supplanted by frequent references to analytical philosophy. The shift in emphasis, however, may not represent an abandonment of earlier concerns that it might seem to at first sight. As suggested throughout this commentary, a constant conceptual patterning can be traced in the course of Genette's evolution, extending from his early insistence on separating the general from the particular in poetics and literary criticism to other categories aimed at highlighting various aspects of literary and artistic phenomena: constitutive/conditional literariness; textuality/transtextuality; immanence/transcendence; artistic/aesthetic relation. This being the case, Genette's writing after his narratological studies can be regarded, in one of its facets, as a way of specifying and developing the theoretical framework of his narrative theory.

One might be tempted, in light of these considerations, to conclude with a transition from classical to postclassical narratology, that is, to return to Herman's observation, the passage from narratology as "a subfield of structuralist literary theory" to "any principled approach to the study of narratively organized discourse"-a response, in effect, to Kreiswirth's "narrative turn." By and large, Genette's followers and critics alike have approached his narratological studies along the lines of narrative immanence, giving little consideration if any to the modes of textual transcendence by which narratives are defined interrelatedly with other narratives and other forms of (mainly) artistic expression or to the broader implications for narrative theory of aesthetic and artistic relations. To the extent that Genette has incorporated non-literary aesthetic phenomena and philosophical aesthetics into the scope of his research, he can be said to work in a spirit of transdisciplinarity comparable to that of postclassical narratology. On the other hand, his primary focus, since the 1980s, has not been "narratively organized discourse" or the adoption of new methodologies brought to bear on the study of narrative, but, among other things, how certain objects, natural but in particular artifactual, are endowed with aesthetic or artistic functions. From this perspective, Genette's narrative theory should not be regarded as or regarded only as "a subfield of structuralist literary theory," but as a theory of a particular kind of cultural exchange whose constitutive categories are determined by or governed by the interaction of aesthetic attention and appreciation. Such a context for narrative theory does not appear to have been explored by postclassical narratology. This, at least, is what can be concluded from Nünning's two-page inventory, where no heading is to be found for "Narrative Poetics" or "Narratology and Aesthetics" and where it is among "Philosophical Narrative Theories" under the heading "Narratology and Theories of Fictionality," in an apparent reference to Fiction and Diction, that Genette's name appears. Genette himself has not undertaken a re-examination of his

  the author's intention to produce a new version, but not a new work, as illustrated by adaptations and revisions, for example. Partial manifestations are a form of transcendence that occur either in fragments of incomplete works or indirectly (a hypertext that offers a glimpse of its hypotext, a sound recording of a musical performance). And finally, the plural work results either from physical transformation (the changes that have occurred to the Parthenon over time) or, more particularly, from a multiplicity of receptions, from the functional plurality of works. As a metaphor of the plural work, Genette mentions a story by Borges in which two identical texts of Don Quixote, one by Cervantes, the other by Pierre Ménard three hundred years later, are constitutive of different works: different interpretations of the same literal meaning governed by widely separated historical and interpretational contexts.


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Poincaré's neo-Kantian conventionalism towards an Einsteinian empiricism with constitutive principles.

 Schlickiana 5, Berlin, Parerga, 2010, pp. 191-203.

Henri Poincaré's conventionalist philosophy of geometry looms large in Moritz Schlick's early writings on the theory of relativity. But as Tom Ryckman (2005, p. 52) points out, Poincaré's star begins to pale for Schlick in the early 1920s, when his philosophy takes a decidedly empiricist turn. This turn of events may naturally be viewed as a by-product of the much-celebrated confirmation of Einstein's general theory of relativity by the British eclipse expeditions of 1919.

While the significance of the empirical success of Einstein's general theory of relativity on Schlick's philosophy can hardly be doubted, Schlick's defection from Poincaré's conventionalist philosophy of space may have an additional source: dissatisfaction with Poincaré's theory of relativity. My discussion of Schlick's view of Poincaré's theory begins with a review of the difference between Einstein's and Poincaré's theories, that turns on the form of light-waves as judged by observers in inertial frames of reference. I summarize the evolution of Poincaré's philosophy of geometry in the early years of relativity theory, which Schlick ignored throughout his life, and in the second section of the paper, I recall Schlick's discussion of Poincaré's views on the relativity of space, in which Schlick focused on similitude relations, at the expense of relations of covariance. I then take up Lindemann's preface to the English translation of Schlick's Raum und Zeit in der gegenwärtigen Physik, where the issue of the shape of light-shells in inertial frames is raised. It is this very issue, I suggest, which distinguished most sharply Poincaré's theory of relativity from Einstein's special theory of relativity, and may have prompted Schlick to move away from 1 Einstein's light-sphere, Poincaré's light-ellipsoid

The compatibility of Einstein's postulates of relativity and light-speed invariance followed for [START_REF] Einstein | Zur Elektrodynamik bewegter Körper[END_REF] from an argument which may be summarized as follows.1 Let a spherical light-wave propagate from the common coordinate origin of two inertial frames designated k and K at time t = τ = 0. In system K the wave spreads with velocity c such that the wavefront is expressed as:

x 2 + y 2 + z 2 = c 2 t 2 . ( 1
)
To obtain the equation of the wavefront in frame k moving with velocity v with respect to K, we apply a certain transformation of coordinates from K to k to the equation ( 1) and find:

ξ 2 + η 2 + ζ 2 = c 2 τ 2 . (2)
Since (1) goes over to (2), Einstein observed, the light-wave that is spherical in K is likewise spherical in k, it propagates with the same velocity c, and consequently, "our two basic principles are mutually compatible" (Einstein 1905, § 3, p. 901). Henri Poincaré was quick to grasp the idea that the principle of relativity could be expressed mathematically by transformations that form a group. This fact had several immediate consequences for Poincaré's understanding of relativity. Notably, Poincaré identified invariants of the Lorentz transformation directly from the fact that the transformation may be construed as a rotation about the coordinate origin in four-dimensional space (with one imaginary axis). Any transformation of the Lorentz group, he noted further, may be decomposed into a dilation and a linear transformation leaving invariant the quadratic form x 2 + y 2 + z 2 -t 2 , where light velocity is rationalized to unity (Poincaré 1906, § 4).

Somewhat curiously, for one who had contributed to discussions on the so-called Riemann-Helmholtz-Lie problem of space, Poincaré avoided drawing consequences for the foundations of geometry from the "new mechanics" of the Lorentz group, with one exception. He observed that measurement of length had implied the displacement of solids considered to be rigid, and yet:

[T]hat is no longer true in the current theory, if we admit the Lorentzian contraction. In this theory, two equal lengths are, by definition, two lengths that are spanned by light in the same lapse of time. (Poincaré 1906, p. 132).

Light-waves, in other words, constituted the new standard of both temporal and spatial measurement. But how was one to go about measuring lengths with Lorentzcontracted rods? Poincaré's measurement problem called for a solution, and shortly, Poincaré provided one. In lectures delivered at the Sorbonne in 1906-1907, he interpreted the Lorentz transformation with respect to a geometric figure representing the surface of a light-wave, which I will refer to as a "light-ellipsoid", following Darrigol's coinage (1995). The light-ellipsoid is characteristic of Poincaré's approach to kinematics, illustrating it on four separate occasions, with minor variations, during the final six years of his life, from 1906 to 1912.2 Like Einstein, Poincaré considered electromagnetic radiation to be the only physical phenomenon not subject to Lorentz-contraction. In his first philosophical commentary on relativity theory, he drew a series of consequences for the philosophy of phenomenal space, during which he invoked a thought-experiment, borrowed from Delboeuf, which proceeded as follows. Let all objects undergo the same expansion overnight; in the morning, the unsuspecting physicist will not notice any change. Poincaré likened his fantasy of an overnight spatial expansion to the relativity of moving bodies in contemporary physics, in that Lorentz's theory admitted a contraction of bodies in their direction of motion with respect to the ether. In the latter case, Poincaré similarly disallowed detection of the contraction, due to compensating effects on measuring instruments (Poincaré 1908, pp. 96-100). Also like Einstein, Poincaré admitted the principle of observational equivalence among inertial observers. He retained, however, a semantic distinction between true and apparent quantities, corresponding respectively to quantities measured in a frame at absolute rest, and those measured in a frame in uniform motion with respect to the absolutely-resting frame. 3 To convey his meaning, Poincaré called up an observer in uniform motion with respect to a frame K, considered to be at rest. The observer in motion is at rest with respect to a frame k, in which all measuring rods of length are contracted in the direction of their motion with respect to the ether, according to Lorentz contraction:

= γ -1 , γ = 1 1 -v 2 /c 2 ,
where designates the length of the rod in frame K, v is the velocity of frame k with respect to K, and c is the velocity of light, a universal constant. Physicists at rest in k can correct for the Lorentz-contraction of their rulers due to their velocity v with respect to K; the correction factor for terrestrial observers was calculated by Poincaré to be on the order of 5 • 10 -9 . Provided we neglect any motion of the Sun with respect to the ether, Poincaré's measurement protocol allows us to ascertain the "true" dimensions of objects in motion with respect to the ether frame K, measured by co-moving observers. To see better how this protocol might work in practice, imagine a material sphere, clamped to a workbench. When measured at rest with respect to K, an orthogonal projection of the sphere is a circle (Fig. 1a). Since measuring rods are contracted in the sense of the observer's motion, the "true" form of this sphere is a flattened ellipsoid. We multiply the measured diameter by the above-mentioned correction factor to find the length of the ellipsoid's minor axis. In a direction orthogonal to the sphere's motion with respect to K, neither measuring rod nor sphere is contracted, such that the measured diameter of the sphere is equal to the length of the major axis of the ellipsoid. When projected orthogonally onto a plane parallel to the motion of the sphere, the flattened ellipsoid has the shape of an ellipse, as shown in Fig. 1c, where the eccentricity is greatly exaggerated for the sake of illustration. Not all objects in motion are subject to Lorentz-contraction, the unique exception being electromagnetic waves. Light-waves propagate isotropically in the ether, according to Poincaré, and consequently, if instead of a material sphere we had a spark generator at rest in K, the resulting light shell has the form of a sphere, as measured by observers at rest with respect to K (Fig. 1b).

When the same light shell is measured in the moving frame k with concrete rods, at a certain moment of apparent time t determined by light-synchronized clocks at rest in k, the shell is naturally found to be spherical. Knowing that the measuring rods are actually Lorentz-contracted, we correct for the contraction in the same manner as in the case of the material sphere, and realize that the light shell has the form of an ellipsoid of revolution, the major-axis of which is aligned with the direction of motion of k with respect to K. Orthogonal projection of the corrected form of the light-shell on a plane parallel to the motion of k is shown in Fig. 1d (again, with eccentricity corresponding to a frame velocity approaching that of light). In his Sorbonne lectures, Poincaré employed the light-ellipsoid in pursuit of two objectives. First, he showed that length and time measurements are transitive for observers in uniform motion, by imagining a light source in uniform motion v, that passes through the coordinate origin O at time t 0 = 0. At a later time t, the light-wave originating at t 0 and propagating in all directions with speed c has a spherical wavefront of radius OH = ct. At an instant of apparent time t , an observer at rest in frame k measures the light-sphere's radius with a concrete rod, that she knows to be contracted in her direction of motion, and finds the light shell to be perfectly spherical. She corrects her measurements by a Lorentz factor, and realizes that the "true" locus of light coincides with an ellipsoid of rotation elongated in her direction of motion with respect to K. The exact dimensions of the light-ellipsoid depend on the moment of apparent time t at which the length measurements are performed. However, the form of the light-ellipsoid is the same for all apparent times (t > 0), in that the ellipsoidal eccentricity e is a constant function of frame velocity v: e = v/c. In Poincaré's hands, the light-ellipsoid was a powerful tool. From the suppositions of Lorentz contraction and invariance of light-waves in inertial frames, he demonstrated that the apparent time of moving observers is a linear function of apparent displacement, and consequently, that space and time measurements are transitive among frames moving uniformly with respect to the ether. It allowed contraction factor him to derive the Lorentz transformation, making his light-ellipsoid section the first graphical illustration of kinematic relations in the new mechanics. 4 The source of the stark contradiction between Einstein's and Poincaré's views of the form of a light-wave lies in their variant protocols for length measurement.

γ = 1/ 1 -v 2 /c
Instead of considering all inertial frames to be equivalent with respect to space and time measurement, as recommended by Einstein, Poincaré employed a privileged frame. In any non-privileged inertial frame, bodies in motion are contracted, and time intervals are dilated with respect to the privileged frame. Like Einstein, Poincaré considered light-waves to be the only objects not subject to Lorentz contraction. In his first philosophical commentary on relativity theory, he proposed a thought-experiment, which proceeds as follows: let all objects undergo the same expansion overnight; in the morning, the unsuspecting physicist will not notice any change. The worlds of last night and this morning are then, as Poincaré writes, "indiscernible". Up to this point in Poincaré's parable, there is no link to the principle of relativity, since all objects are at relative rest in his imaginary universe. In what follows, however, Poincaré likens the overnight expansion to the relativity of moving bodies:

In both cases, there can be no question of absolute magnitude, but [only] of the measurement of magnitude by means of some instrument; this instrument may be a meter-stick, or a segment spanned by light; we measure only the relation of magnitude to instrument; and if this relation is altered, we have no means of knowing whether it is the magnitude or the instrument that has changed. (Poincaré 1908, p. 100) According to Lorentz's electron theory, all bodies contract in their direction of motion with respect to the ether, but the contraction escapes detection in principle, because of compensating effects on our measuring instruments. 5 Schlick was impressed with Poincaré's argument, which he rehearsed in his philosophical treatise on the theory of relativity, Raum und Zeit in der gegenwärtigen Physik (1917, reed. Engler and Neuber, eds., 2006). What Schlick focused upon was the objectivity of length measurement:

If we, for instance, assumed that the dimensions of all objects are lengthened or shortened in one direction only, say that of the Earth's 4 Details are supplied in a forthcoming article in Einstein Studies. My reconstruction of the light-ellipsoid differs from that of [START_REF] Damour | [END_REF], in that length measurement is carried out by comoving observers at an instant of apparent time t , instead of true time t. [Note added 2013-12-19: My views on this topic have changed due to archival finds in 2012 (see Walter 2014).]

5 For an insightful review of Lorentz's electron theory, see [START_REF] Darrigol | The electron theories of Larmor and Lorentz: a comparative study[END_REF].

axis, we should again not notice this transformation, although the shape of bodies would have changed completely, spheres becoming ellipsoids of rotation, cubes becoming parallelepipeds, and indeed perhaps very elongated ones. (Schlick 1920b, p. 26;cf. Engler and Neuber, eds., 2006, p. 202) Where Poincaré took care to distinguish between the principle of similitude and the principle of relativity, Schlick wiped out any such distinction. We recall that for Poincaré, just as for Einstein, light-waves are not subject to Lorentzian contraction.

In fact, according to Poincaré, if it were not for the invariance of the form of light-waves with respect to uniform motion, there would be no talk of Lorentz contraction at all:

If the wave-surfaces of light were subject to the same deformations as material bodies, we would never have noticed the Lorentz-FitzGerald deformation. (Poincaré 1908, p. 100) This is a crucial observation for (Minkowski 1909), stigmatized the low evidensory status of the contraction in the Lorentz's electron theory as a "gift from above". As Minkowski's spacetime theory would have it, the phenomenon of Lorentz contraction is a simple manifestation of the spacetime metric. Such an evolution in Schlick's understanding appears all the more likely, in light of the fact that Poincaré's own views on the relativity of space and time evolved after 1907. In 1912, for example, he formulated a response of sorts to Minkowski's Cologne lecture, entitled, quite naturally, "Space and time". Most notably, Poincaré allowed the symmetry group of mechanics to define the concepts of space and time. What this amounted to, in Poincaré's conventionalist scheme, was the adoption of spacetime as a convention, where earlier, he considered the geometry of phenomenal space to be conventional. According to Minkowski, empirical and theoretical considerations forced scientists to adopt Minkowski spacetime. Poincaré retorted, in effect, that scientists are free to choose between two spacetime conventions, and that his own preference was for Galilei spacetime, rather than Minkowski spacetime [START_REF] Walter | Hypothesis and convention in Poincaré's defense of Galilei spacetime[END_REF]. Like other philosophers, however, Schlick did not acknowledge Poincaré's twelfth-hour embrace of spacetime conventionalism.

2 Poincaré's philosophy of relativity and Schlick's empiricist turn

One consequence of adopting the convention of either Galilei or Minkowski spacetime is that the geometry of phenomenal space is fixed by this choice, and the spatial geometry in both cases is that of Euclid. And while Poincaré recognized this consequence, he did not acknowledge that it mooted his pre-relativist philosophy of geometry. As for Schlick, the immediate consequence of his neglect of Poincaré's switch to conventional spacetime is that his version of Poincaré's philosophy of space remained that of the author of La science et l'hypothèse (1902), and not "L'espace et le temps" (1912).

In his essay, "Die philosophische Bedeutung des Relativitätsproblems," for instance, Schlick rehearsed Poincaré's pre-relativist doctrine of space, according to which there is no fact to the matter of the geometry of phenomenal space (Schlick 1915, 150). Among the readers of Schlick's paper was Albert Einstein. Schlick sent him a copy of his paper, prompting a quick reply on 14 December, 1915.

Einstein fully approved the view of relativity presented by Schlick, remarking that "There is nothing in your exposition with which I find fault."6 Further correspondence between Schlick and Einstein bears witness to Einstein's early admiration for Schlick's conventionalist reading of general relativity, which Schlick defended in opposition to the neo-Kantian interpretations of the Marburg School. For example, in the original edition of Raum und Zeit in der Gegenwärtigen Physik, Schlick limited the role of empirical facts to that of a helpful guide, informing us only "whether it is more practical to employ Euclidean geometry or a non-Euclidean geometry in physical descriptions of nature" (Engler and Neuber, eds., 2006, p. 211). The first inkling in Schlick's published work that something is not right about Poincaré's theory of relativity comes not from Schlick himself, but from the scientist who introduced his work to English-language readers, the Oxford physicist Frederick A. Lindemann. The key philosophical achievement of the special theory of relativity, Lindemann observed, was that the descriptions of events depend on the state of motion of the observer. This spatio-temporal relativity was not just another philosopher's dream, or theorist's convention for Lindemann, but a view forced upon scientists:

The reasons which force this conclusion upon the physicist may be made clear by considering what will be the impression of two observers passing one another who send out a flash of light at the moment at which they are close together. The light spreads out in a spherical shell, and it might seem obvious, since the observers are moving relatively to one another, that they cannot both remain at the center of this shell. The celebrated Michelson-Morley experiment proves that each observer will conclude that he does remain at the center of the shell. The only explanation for this is that the ideas of length and time of the one observer differ from those of the other. (Lindemann, in Schlick 1920b, p. iv.) Contrary to Lindemann's assertion, an alternative explanation for the Michelson-Morley experiment had been proposed by

  FIGURE 1. Illustration of Poincaré's view of the "true" form of a fast-moving material sphere and a light shell in frames K and k, at rest (a, b), and in uniform motion (c, d), respectively, with respect to the quiescent ether. Measurements are performed by observers at rest with respect to frame K (a, b), or k (c, d).

  Figure 2: A section of Poincaré's light ellipsoid (with modern notation).

  Fynn Ole Engler and Mathias Iven, eds, Moritz Schlick: Ursprünge und Entwicklungen seines Denkens,

	Moritz Schlick's reading
	of Poincaré's theory of relativity
	Scott A. Walter *

  Poincaré, as mentioned above. Lindemann's observation that light-spheres remain light-spheres in special relativity was something of a commonplace among relativists by 1910. It may, however, have led Schlick to realize that Poincaré and Einstein differed over the shape of light-waves for inertial observers, and that this difference was significant for the philosophy of space and time. Beyond the latter conflict between Poincaré's and Einstein's theories of relativity, the question remains of what relation, if any, Schlick noticed between these theories of physics and Poincaré's philosophy of geometry. Lindemann's preface notwithstanding, there is no compelling evidence that Schlick was aware of the difference between Poincaré's theory of relativity and Einstein's special theory of relativity. As we saw above, Schlick was quite familiar with Poincaré's discussion of the relation between the new mechanics and conventionalist philosophy of geometry, but did not see fit to comment. Some of Schlick's contemporaries drew such conclusions, beginning with Minkowski. In 1908, Minkowski argued that only the four-dimensional world is real, thereby implicitly contradicting Poincaré's geometric conventionalism. Twelve years later, Schlick agreed wholeheartedly with Minkowski's spacetime realism, paraphrasing the mathematician in the pages of Die Naturwissenschaften: These remarks of Schlick's are not, however, explicitly tied to Poincaré. As mentioned above, in 1912 Poincaré explained in essence that the theory of relativity is wholly compatible with Galilean kinematics, and that furthermore, Galilei spacetime is more convenient than Minkowski spacetime. Schlick's remarks on the subject in 1921 suggest a disagreement with Poincaré's view; they do not establish that Poincaré's conventionalism was his target. As for general relativity, Poincaré died a full year before Einstein began work on a relativistic theory of gravitation based on variably-curved (pseudo-Riemannian) spacetime. In a much-commented lecture delivered to the Prussian Academy of Science in 1921,[START_REF] Einstein | Zur Elektrodynamik bewegter Körper[END_REF] engaged with Poincaré's philosophy of geometry. On this occasion, Einstein acknowledged that Poincaré's philosophy of geometry, according to which the facts can never decide the question of the geometry of phenomenal space, is correct, "sub specie aeterni". He also opined that the current state of theoretical physics was such that there was little choice but to admit notions such as rigid rods and ideal clocks, which have no exact referent in reality. Einstein's remark applies to both the special and general theories of relativity, and would appear to constitute strong support for the conventionalist interpretation of these theories, as laid out in Schlick's writings of the time. At about this time, however, Schlick began to move away from conventionalism, toward a new empiricism. It appears unlikely that Einstein was the source of Schlick's dissatisfaction with conventionalism in the early 1920s, in light of his remarks to the Prussian Academy. Perhaps Lindemann's praise of an Einsteinian version of special relativity led Schlick to reconsider the relation between Poincaré's new mechanics and his philosophy. Whatever the source of Schlick's change of heart may be, his embrace of a Minkowskian spacetime ontology marks a turning point in his relation to Poincaré's conventionalist philosophy of geometry.

	with Euclidean geometry. 8
	Everything real is four-dimensional; three-dimensional bodies are just
	as much mere abstractions as are lines or surfaces. 7
	Schlick's awareness of the problem posed by Poincaré's theory of relativity for
	Poincaré's conventionalist philosophy is similarly suggested by an observation
	made the following year, to the effect that the special theory of relativity is "ir-
	reconcilable" with Galilean kinematics, just as general relativity is irreconcilable

The notation has been modernized. For technical details, see my study of light-spheres in relativity, to appear in a forthcoming volume of Einstein Studies, edited by David Rowe. The content of this section draws largely on the latter study.

A description of the light-ellipsoid appears in Science et méthode(1908, p. 239).

The notion of an absolutely-resting frame remained abstract for Poincaré, who later embraced the conventionality of spacetime, preferring Galilei spacetime over Minkowski spacetime[START_REF] Walter | Hypothesis and convention in Poincaré's defense of Galilei spacetime[END_REF].

Cited byHoward (1984, p. 618).

"Alles Wirkliche ist vierdimensional; dreidimensionale Körper sind genau so gut bloße Abstraktionen wie Linien oder Flächen."Schlick (1920a, p. 474).

"Nun ist die Spezielle Relativitätstheorie mit den Sätzen der Galileischen Kinematik, die Allgemeine außerdem noch mit den Sätzen Euklids unvereinbar."Schlick (1921, p

. 99).


Heungsun Hwang

Yoshio Takane

Art Hur Tenenhaus

AN ALTERNATIVE ESTIMATION PROCEDURE FOR PARTIAL LEAST SQUARES PATH MODELING

Since it s incept ion, part ial least squares pat h modeling has suff ered from t he absence of a single opt imizat ion crit erion for est imat ing comp onent weight s. A new est imat ion procedure is proposed t o address t his enduring issue. T he proposed procedure aims t o minimize a single least squares crit erion for est imat ing component weight s under b ot h Mode A and Mode B. An alt ernat ing least squares algorit hm is develop ed t o minimize t he crit erion. T his procedure provides quit e similar or ident ical solut ions t o t hose obt ained from exist ing Lohmöller's algorit hm in real and simulat ed dat a analyses. T he prop osed procedure can serve as an alt ernat ive t o t he exist ing one in t hat it is well-grounded in t heory as well as performs comparably in pract ice.

Int r od u ct ion

Part ial least squares pat h modeling (P LSP M) [START_REF] Wold | Est imat ion of principal comp onent s and relat ed met hods by it erat ive least squares[END_REF][START_REF] Wold | Nonlinear it erat ive part ial least squares (NIPALS) modeling: Some current development s[END_REF][START_REF] Wold | Soft modeling: T he basic design and some ext ensions[END_REF][START_REF] Lohmöller | Latent variable path modeling with partial least squares[END_REF]) is a long-st anding approach t o st ruct ural equat ion modeling. In paramet er est imat ion, t his approach adopt s a st rat egy of est imat ing a lat ent variable as a component or weight ed composit e of indicat ors. In t his regard, P LSP M can be considered a component -based approach t o st ruct ural equat ion modeling [START_REF] Tenenhaus | Comp onent -based st ruct ural equat ion modelling[END_REF]. It carries out two main st ages sequent ially t o est imat e paramet ers. T he first st age est imat es latent variables as component s, which requires t he est imat ion of component weight s. T his st age uses an iterative algorit hm t o est imat e t he component weight s. T he second st age est imat es remaining paramet ers in measurement and st ruct ural models (i.e., pat h coeffi cient s and/ or loadings) by means of ordinary linear regression. T hat is, pat h coeffi cient s are est imat ed by regressing each dependent lat ent variable on it s explanat ory lat ent variables, whereas loadings are est imat ed by regressing indicat ors on t heir corresponding lat ent variables. T he second st age is t hus non-it erat ive, which is based on t he lat ent variables obt ained from t he first st age. Accordingly, t he first st age is t he most crucial estimat ion procedure in P LSP M [START_REF] Hanafi | P LS pat h modeling: comput at ion of lat ent variables wit h t he est imat ion mode B[END_REF]. [START_REF] Lohmöller | Latent variable path modeling with partial least squares[END_REF] algorit hm is best known for t he first st age and implement ed int o most software programs for P LSP M, including LVP LS [START_REF] Lohmöller | LVPLS program manual[END_REF], P LS Graph [START_REF] Chin | PLS-Graph user's guide ver sion 3.0[END_REF], Smart P LS [START_REF] Ringle | SmartP LS 2.0 (beta)[END_REF], and XLSTAT (Addinsoft , 2009). As will be explained in more det ail in Sect ion 2, t his algorit hm repeat s two st eps, called int ernal and ext ernal est imat ion. In t he int ernal est imat ion st ep, a so-called inner est imat e or inner component is obt ained for each lat ent variable under diff erent schemes such as cent roid, fact orial, and pat h weight ing. In t he ext ernal est imat ion st ep, component weight s for each block of indicat ors are est imat ed in two diff erent ways called Mode A and Mode B.

It is not known which crit erion t he Lohmöller algorit hm aims t o opt imize by repeat ing t he two st eps (e.g., [START_REF] Coolen | Least squares pat h analysis wit h opt imal scaling[END_REF][START_REF] Wold | T he ML and P LS t echniques for modeling wit h lat ent variables: Hist orical and comparat ive Asp ect s[END_REF]. A few at t empt s have been made t o address t his issue. For example, [START_REF] Hanafi | P LS pat h modeling: comput at ion of lat ent variables wit h t he est imat ion mode B[END_REF] present ed associat ion-maximizat ion crit eria for t he cent roid and fact orial schemes under Mode B (also see [START_REF] Tenenhaus | Regularized generalized canonical correlat ion analysis[END_REF]. To our knowledge, nevert heless, no single opt imizat ion crit erion is yet available for t he algorit hm, which includes bot h Mode A and Mode B as special cases. T he lack of a single opt imizat ion crit erion makes it diffi cult t o evaluat e t he algorit hm [START_REF] Mcdonald | P at h analysis wit h comp osit e variables[END_REF].

In t his paper, we propose an alt ernat ive procedure for t he first est imat ion st age of P LSP M. T he proposed procedure aims t o minimize a single least squares crit erion for est imat ing component weight s under bot h Mode A and Mode B. An alt ernat ing least squares (ALS) algorit hm is used t o minimize t he crit erion, which repeat s t he same two st eps used in t he Lohmöller algorit hm. A ma jor diff erence is t hat t he ALS algorit hm updat es t he inner est imat es and component weight s opt imally by minimizing t he least squares crit erion. Consequent ly, t he proposed procedure is well-defined in a least squares sense.

T he paper is organized as follows. In Sect ion 2, we provide a brief descript ion of t he exist ing Lohmöller algorit hm. In Sect ion 3, we provide a det ailed account of t he proposed procedure. In Sect ion 4, we invest igat e t he performance of t he proposed and ext ant procedures t hrough t he analyses of real and simulat ed dat a. In t he final sect ion, we discuss implicat ions of t he proposed procedure.

E x ist in g P LSP M A lgor it h m

We briefly describe t he Lohmöller algorit hm. Refer t o [START_REF] Tenenhaus | P LS P at h Modeling[END_REF] for a fuller descript ion of t he algorit hm.

Let η j denot e an N by 1 vect or of t he j t h lat ent variable (j = 1, . . . , J ), where N is t he number of individuals. Let X j denot e an N by P j mat rix consist ing of a block of indicat ors associat ed wit h η j . Let w j denot e a P j by 1 vect or of component weight s assigned t o X j . In P LSP M, convent ionally, bot h indicat ors and lat ent variables are assumed t o be st andardized, such t hat t hey have zero means and unit variances (e.g., η j η j = N ). However, t hey are t o be normalized here, so t hat t heir lengt h is equal t o one (e.g., η j η j = 1). T his normalizat ion makes t he exposit ion of equat ions simpler while producing ident ical est imat es of weight s, pat h coeffi cient s, and loadings. T he individual scores of st andardized lat ent variables can always be obt ained by mult iplying t heir normalized scores by √ N .

T he Lohmöller algorit hm begins by choosing arbit rary init ial values for w j and comput ing η j = X j w j . T hen, it repeat s t he following two st eps t o est imat e w j and η j . St ep 1 (int ernal est imat ion): Updat e t he inner est imat e for η j . T he inner est imat e, denot ed here by f j , is a weighed composit e of t he lat ent variables connect ed t o η j in a given st ruct ural model. Such connect ed lat ent variables cont ain t hose aff ect ing η j as well as t hose being aff ect ed by η j . The inner estimate takes the general form as follows.

f j = Q j q= 1 e j q η q , (1)
where e j q is a scalar value, called t he inner weight , which is assigned t o each of t he Q j lat ent variables (η q 's) t hat are connect ed t o η j . As shown in (1), updat ing t he inner est imat e amount s t o updat ing it s inner weight s, given lat ent variables. T hree diff erent ways, so-called schemes, are available for t he calculat ion of t he inner weight s: cent roid [START_REF] Wold | Soft modeling: T he basic design and some ext ensions[END_REF], fact orial [START_REF] Lohmöller | Latent variable path modeling with partial least squares[END_REF], and pat h weight ing. In t he cent roid scheme, e j q 's are t he signs of t he correlat ions between η q 's and η j . In t he fact orial scheme, e j q 's are t he correlat ions between η q 's and η j . In t he pat h weight ing scheme, e j q 's are t he regression coeffi cient s of η j on η q 's if η j is a dependent variable, whereas t hey are t he correlat ions between η q 's and η j if η j is an explanat ory variable. T he pat h weight ing scheme is recommended over t he ot her schemes because it t akes int o account bot h direct ions and magnit udes of t he relat ionships between lat ent variables [START_REF] Esposit O Vinzi | P LS pat h modeling: from foundat ions t o recent development s and op en issues for model assessment and improvement[END_REF].

Figure 1 displays a prot otype, st ruct ural model t o illust rat e t he first st ep. T his model consist s of four lat ent variables (J = 4). For t he prot otype model, t he inner est imat e for each of t he four lat ent variables is given as f 1 = e 13 η 3 f 2 = e 23 η 3 f 3 = e 31 η 1 + e 32 η 2 + e 34 η 4 f 4 = e 43 η 3

(2)

As explained above, t he inner weight s for t hese inner est imat es are calculat ed based on which scheme is chosen. For example, if t he pat h weight ing scheme is adopt ed, e 31 and e 32 are t he regression coeffi cient s of η 3 on η 1 and η 2 , because η 1 and η 2 are explanat ory variables for η 3 , whereas e 34 are t he correlat ion between η 3 and η 4 , because η 3 is an explanat ory variable for η 4 . All t he ot her inner weight est imat es are simply correlat ions between two connect ed lat ent variables, because all lat ent variables are normalized and t he regression coeffi cient of one lat ent variable on t he ot her is equivalent t o t he correlat ion between t hem. St ep 2 (ext ernal est imat ion): Updat e w j . T here are two ways of est imat ing component weight s on t he basis of t he nat ure of t he measurement model: Mode A and Mode B. Mode A is known t o be more suit able for reflect ive indicat ors, whereas Mode B is for format ive indicat ors (e.g., [START_REF] Tenenhaus | P LS P at h Modeling[END_REF]. Specifically, under Mode A, w j is updat ed by regressing X j on f j , as follows.

w j = X j f j (f j f j ) -1 .
(3) Under Mode B, w j is updat ed by regressing f j on X j , as follows.

w j = (X j X j ) -1 X j f j . (4)
Subsequent ly, η j is updat ed by η j = X j w j , and normalized such t hat η j η j = w j X j X j w j = 1. T his normalizat ion can be done by mult iplying w j by (w j X j X j w j ) -1/ 2 , indicat ing t hat t he eff ect of (f j f j ) -1 in (3) will be cancelled out . Consequent ly, under Mode A, w j can be updat ed simply by w j = X j f j .

(5)

T he above st eps are repeat ed unt il no subst ant ial diff erences occur between t he previous and current weight est imat es for all J blocks of indicat ors. A summary of t his algorit hm is provided in t he Appendix.

As st at ed earlier, it is unknown which opt imizat ion crit erion t he Lohmöller algorit hm seeks t o maximize or minimize under Mode A and Mode B. In t he next sect ion, we propose a single least squares crit erion t hat is t o be consist ent ly minimized for est imat ing component weight s under bot h modes.

T h e P r op osed E st im a t ion P r oced u r e for P LSP M

Let H = [η 1 , . . . , η J ] denot e an N by J matrix consisting of all J lat ent variables. Let ε j denot e a J by 1 vect or consist ing of Q j inner weight s for t he Q j lat ent variables connect ed t o η j , and of J -Q j zeros for t he remaining unconnect ed lat ent variables. T hen, let f j = H ε j denot e an N by 1 vect or of t he inner est imat e for η j . For example, in t he prot otype model depicted in Figure 1 e 32 , 0, e 34 ] , andε 4 = [0, 0, e 43 , 0] .

, H = [η 1 , η 2 , η 3 , η 4 ], ε 1 = [0, 0, e 13 , 0] , ε 2 = [0, 0, e 23 , 0] , ε 3 = [e 31 ,
We propose a least squares crit erion for est imat ing all weight s under Mode A, as follows.

Minimize

φ A = J j = 1 SS(X j -f j w j ), (6)
sub ject t o η j η j = 1, where SS(M ) = trace(M M ) for any mat rix M . T his crit erion appears similar t o a blockwise join loss funct ion for principal component analysis (Gifi, 1990, p. 152), where a vect or of ob ject scores is replaced by t he inner est imat e. We propose a least squares crit erion for est imat ing all weight s under Mode B, as follows.

Minimize

φ B = J j = 1 SS(f j -X j w j ), (7)
sub ject t o η j η j = 1. Crit erion (7) may be viewed as a blockwise meet loss version (Gifi, 1990, p. 167) of t he covariance-maximizat ion crit erion for regularized generalized canonical correlat ion analysis [START_REF] Tenenhaus | Regularized generalized canonical correlat ion analysis[END_REF]. Let α j denot e a binary value t hat indicat es which mode is used for updat ing t he component weight s for t he j t h block of indicat ors. T hat is, α j = 1 if Mode A is used, and α j = 0 if Mode B is used. We t hen develop a single opt imizat ion crit erion for t he P LSP M algorit hm by combining ( 6) and ( 7), as follows.

Minimize φ = J j = 1 α j SS(X j -f j w j ) + J j = 1 (1 -α j )SS(f j -X j w j ), (8)
sub ject t o η j η j = 1. T his crit erion subsumes ( 6) and ( 7) as special cases by set t ing all α j 's t o one or zero, respect ively. Moreover, it can be used for est imat ing t he weight s for each block of indicat ors under eit her Mode A or Mode B by set t ing t he corresponding α j t o one or zero, respect ively.

We develop an ALS algorit hm t o minimize (8). T his algorit hm begins by assigning arbit rary init ial values t o w j and obt aining η j = X j w j . T hen, it alt ernat es t he following two st eps. St ep 1 (int ernal est imat ion): Updat e f j for fixed w j . T his st ep reduces t o updat ing t he inner weight s in ε j , given lat ent variables. It is equivalent t o minimizing

φ j = α j SS(X j -H ε j w j ) + (1 -α j )SS(H ε j -η j ). ( 9
)
Let e j denot e a Q j by 1 vect or consist ing of non-zero inner weight s only. Let Γ j denot e an N by Q j mat rix formed by eliminat ing t he columns of H corresponding t o any zero element s in ε j . T hen, minimizing (9) is equivalent t o minimizing

φ j = α j SS(X j -Γ j e j w j ) + (1 -α j )SS(Γ j e j -η j ). ( 10
)
By solving 1 2 ∂φ j ∂e j = 0, the least squares estimate of e j is obt ained as

e j = α j w j w j Γ j Γ j + (1 -α j )Γ j Γ j -1 Γ j η j . (11)
T hen, f j is updat ed by f j = H ε j , where ε j is const ruct ed from t he est imat e of e j . St ep 2 (ext ernal est imat ion): Updat e w j for fixed f j . T his is equivalent t o minimizing

φ j = α j SS(X j -f j w j ) + (1 -α j )SS(f j -X j w j ). ( 12
)
Not e t hat in (12), f j does not involve w j because η j is not connect ed wit h it self.

By solving 1 2 ∂φ j ∂w j = 0, the least squares estimate of w j is obt ained as

w j = α j f j f j I + (1 -α j )X j X j -1 X j f j , (13)
where I is an ident ity mat rix of size P j . Subsequent ly, η j is updat ed by η j = X j w j , and normalized. We repeat t he two st eps unt il t he diff erence in t he values of (8) between t he previous and current it erat ions decreases below a pre-det ermined t hreshold (e.g., .00001). A summary of t he ALS algorit hm is also present ed in t he Appendix.

A few remarks concerning t he ALS algorit hm are in order. First , it is easily seen t hat if Mode A is used or equivalent ly α j = 1, (13) reduces t o (3) and ( 5), whereas if Mode B is used or α j = 0, (13) reduces t o (4). T his indicat es t hat t he algorit hm deals wit h Mode A and Mode B as special cases. Second, in t he first step, the estimat es of t he inner weight s are obt ained in such a way t hat t hey minimize a least squares crit erion, condit ionally upon t he est imat es of component weight s. T hus, we may call t he st ep t he "least squares scheme." On t he ot her hand, it is uncert ain which crit erion t he exist ing schemes seek t o opt imize except for a few special cases [START_REF] Hanafi | P LS pat h modeling: comput at ion of lat ent variables wit h t he est imat ion mode B[END_REF][START_REF] Tenenhaus | Regularized generalized canonical correlat ion analysis[END_REF]. T hird, t he ALS algorit hm defines convergence as t he decrease in t he value of t he opt imizat ion crit erion (8) beyond a cert ain t hreshold, whereas t he Lohmöller algorit hm defines convergence as a sort of equilibrium, i.e., t he point at which no subst ant ial diff erence occurs between t he previous and current est imat es of weight s, because it does not involve an opt imizat ion crit erion. Last ly, at least in t heory, a t hird type of mode can be considered by t aking any value of α j between 0 and 1. For example, by specifying α j = .1, t he second t erm of t he crit erion can have a great er influence on t he est imat ion of component weight s. However, in pract ice, it is not yet clear what such types of mode connot e and whet her using t hem is sensible subst ant ively.

E m p ir ica l C om p ar ison s

In t his sect ion, we compare t he proposed procedure t o t he ext ant procedure based on t he Lohmöller algorit hm, using real and simulat ed dat a.

Real Data Analysis

We applied t he proposed and ext ant procedures t o fit t he American cust omer sat isfact ion index (ACSI) model [START_REF] Fornell | T he American cust omer sat isfact ion index: Nat ure, purp ose, and findings[END_REF] t o a consumer-level dat aset collect ed in 2002. T his dat aset consist s of t he responses of 774 consumers t o t he service unit s (e.g., police, garbage pick-up services, et c.) wit hin t he US sect or of public administ ration.

T he ACSI model specifies t he relat ionships among ant ecedent and consequent lat ent variables of cust omer sat isfact ion. As depict ed in Figure 2, t he ACSI model includes fourt een indicat ors: x 1 = cust omer expect at ions about overall quality, x 2 = customer expect at ions about reliability, x 3 = cust omer expect at ions about cust omizat ion, x 4 = overall quality, x 5 = reliability, x 6 = cust omizat ion, x 7 = price given quality, x 8 = quality given price, x 9 = overall cust omer sat isfact ion, x 10 = confirmat ion of expect at ions, x 11 = dist ance t o ideal product or service, x 12 = formal or informal complaint behavior, x 13 = repurchase int ent ion, and x 14 = price t olerance. T he measures and scales of t hese indicat ors are available in [START_REF] Fornell | T he American cust omer sat isfact ion index: Nat ure, purp ose, and findings[END_REF]. T he ACSI model also involves six lat ent variables t hat underlie t he fourt een indicat ors, as follows: CE = cust omer expect at ions, P Q = perceived quality, P V = perceived value, CS = cust omer sat isfact ion, CC = cust omer complaint s, and CL = cust omer loyalty. x 8

x 9

x 1 0 x 1 1

x 1 2

x 1 3

x 1 4

F igure 2: T he American cust omer sat isfact ion index model. No residual t erms are displayed.

We used Smart P LS [START_REF] Ringle | SmartP LS 2.0 (beta)[END_REF] t o implement t he ext ant procedure in combinat ion wit h t he pat h weight ing scheme. As displayed in Figure 2, t he ACSI model assumes t hat all indicat ors are reflect ive. T his suggest s t hat Mode A should be more appropriat e for est imat ing weight s.

Tables 1 and2 present t he est imat es of weight s, loadings, and pat h coeffi cient s obt ained from t he proposed and ext ant procedures under Mode A. As shown in t he t ables, bot h procedures result ed in quit e similar paramet er est imat es, leading t o t he same int erpret at ions.

Simulated Data Analysis

We furt her compared t he performance of t he proposed and ext ant procedures based on simulat ed dat a. In part icular, we focused on how similarly t he proposed and ext ant procedures would perform under two diff erent models.

Simulation 1

Figure 3 displays t he st ruct ural equat ion model considered in t he first simulat ion st udy, along wit h it s unst andardized and st andardized paramet er values. In t his model, t hree lat ent variables were specified, each of which underlay t hree indicat ors. Individual-level mult ivariat e normal dat a were drawn from N (0, Σ ), where Σ is t he implied populat ion covariance mat rix derived based on t he unst andardized paramet er values in t he framework of covariance st ruct ure analysis (e.g., J öreskog, 1970). T his indicat es t hat t he lat ent variables in t he model were assumed t o be equivalent t o common fact ors.

We considered t hree diff erent levels of sample size (N = 25, 100, 400). Five hundred samples were generat ed at each sample size. We used t he same init ial values per sample for t he proposed and ext ant procedures. In t he model, all indicat ors were reflect ive, so t hat we used Mode A for bot h procedures. T he pat h weight ing scheme was employed for t he ext ant procedure. P LSP M provides st andardized paramet er est imat es. Table 3 present s t he bias, st andard deviat ion, and mean square error of each st andardized paramet er est imat e obt ained from t he two procedures. As shown in t he t able, t he paramet er est imat es of bot h procedures shared t he same propert ies. In general, t heir loading est imat es were posit ively biased, whereas t heir pat h coeffi cient s were negat ively biased. As st at ed above, in t his st udy, t he simulat ed dat a were generat ed under t he assumpt ion t hat a lat ent variable was equivalent t o a common fact or. Under t his assumpt ion, P LSP M is known t o yield biased est imat es (e.g., Dijkst ra, 2010) because it regards lat ent variables as component s rat her t han common fact ors. T he st andard deviat ions of t he loading and pat h coeffi cient est imat es decreased wit h sample size. T he mean square errors of t hese est imat es became closer t o zero wit h sample size. Not ably, all t he paramet er est imat es obt ained from bot h procedures exhibit ed quit e similar biases, st andard deviat ions, and mean square errors across all sample sizes. T his indicat es t hat t he proposed procedure result ed in virt ually ident ical paramet er est imat es as t hose from t he ext ant one.

As discussed in Sect ion 3, t echnically, t he proposed procedure allows a compromise between Mode A and Mode B by t aking t he value of α j between 0 and 1. As a reviewer suggest ed, we have invest igat ed t he eff ect of adopt ing such a t hird type of mode on paramet er est imat ion. Specifically, we applied t he proposed procedure under α j = .5, so t hat Mode A and Mode B cont ribut ed simult aneously t o obt aining est imat es. As shown in Table 3, t his case t ended t o produce less biased est imat es part icularly in small samples, whereas it t ended t o yield larger st andard deviat ions of t he est imat es. Consequent ly, it s est imat es t ended t o show larger mean square errors t han t hose obt ained under Mode A. T hus, at least in t his st udy, adopt ing α j = .5 was of lit t le benefit over using Mode A in est imat ing paramet ers. Alt hough permit t ing a compromise between t he two convent ional modes is a t echnically novel feat ure, as st at ed earlier, it is unclear what such a compromise indicat es subst ant ively, when it can be useful, and how the value of α j can be chosen. T he first simulat ion st udy was useful t o evaluat e how similarly t he proposed and ext ant procedures performed. Nonet heless, t his st udy may be somewhat t oo simple in t hat it involved only t hree blocks of reflect ive indicat ors and assumed t he same correlat ions among each block of indicat ors. T hus, we conduct ed anot her simulat ion st udy, which considered bot h format ive and reflect ive indict ors as well as diff erent correlat ions among each block of indicat ors. Specifically, we used t he model specified in [START_REF] Ringle | On t he use of format ive measurement sp ecificat ions in st ruct ural equat ion modeling: A Mont e Carlo simulat ion st udy t o compare covariance-based and part ial least squares model est imat ion met hodologies[END_REF] for t he second simulat ion st udy. Figure 4 displays t he model given in [START_REF] Ringle | On t he use of format ive measurement sp ecificat ions in st ruct ural equat ion modeling: A Mont e Carlo simulat ion st udy t o compare covariance-based and part ial least squares model est imat ion met hodologies[END_REF], along wit h it s paramet er values. [START_REF] Ringle | On t he use of format ive measurement sp ecificat ions in st ruct ural equat ion modeling: A Mont e Carlo simulat ion st udy t o compare covariance-based and part ial least squares model est imat ion met hodologies[END_REF] did not provide populat ion residual variances. Inst ead, t hey provided t he populat ion correlat ion mat rix of indicat ors, derived based on t he specified model (see Table 5 in [START_REF] Ringle | On t he use of format ive measurement sp ecificat ions in st ruct ural equat ion modeling: A Mont e Carlo simulat ion st udy t o compare covariance-based and part ial least squares model est imat ion met hodologies[END_REF]. We generat ed mult ivariat e normal dat a, using t he correlat ion mat rix.

x 1 4 .1

x 1 5

x 1 6 x 1 7

x 1 9 x 1 8

x 1 2 x 1

x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9

x 1 0 x 1 1 x 1 3 4 5 1 3

. As in t he first simulat ion st udy, we considered t hree diff erent levels of sample size (N = 25, 100, 400). Five hundred samples were generat ed at each sample size. We used t he same init ial values per sample for t he proposed and ext ant procedures. Mode A was applied for est imat ing t he weight s for reflect ive indicat ors, whereas Mode B was used for est imat ing t hose for format ive indicat ors. T he pat h weight ing scheme was employed for t he ext ant procedure.

Table 4 provides t he bias, st andard deviat ion, and mean square error of each st andardized paramet er est imat e obt ained from t he two procedures. T he paramet er est imat es of bot h procedures showed t he same behaviors, alt hough it was somewhat diffi cult t o charact erize t hem clearly. For example, some weight est imat es for format ive indicat ors were negat ively biased, ot her est imat es were posit ively biased, and t he ot hers were biased in diff erent direct ions over sample size. Conversely, all loading est imat es were posit ively biased regardless of sample size. T wo est imat es of pat h coeffi cient s were negat ively biased, whereas one est imat e was posit ively biased, across sample sizes. It was diffi cult t o explain where t hese biases came from because [START_REF] Ringle | On t he use of format ive measurement sp ecificat ions in st ruct ural equat ion modeling: A Mont e Carlo simulat ion st udy t o compare covariance-based and part ial least squares model est imat ion met hodologies[END_REF] did not discuss explicit ly whet her t heir populat ion correlat ion mat rix was generat ed based on t he assumpt ion t hat t he lat ent variables were equivalent t o common fact ors as in t he first st udy. T he st andard deviat ions and mean square errors of all paramet er est imat es decreased wit h sample size. Import ant ly, all t he paramet er est imat es obt ained from bot h procedures involved quit e similar biases, st andard

C on clu sion

We proposed an alt ernat ive est imat ion procedure for est imat ing component weight s in P LSP M. From t echnical perspect ives, t his procedure has several advant ages over t he ext ant one. First , it adopt s a single opt imizat ion crit erion t o est imat e t he weight s under bot h Mode A and Mode B. T hus, t his addresses t he enduring issue of lack of a single opt imizat ion crit erion in P LSP M. Second, t he proposed procedure applies an ALS algorit hm t o minimize t he single crit erion. T his algorit hm has been proven t o converge [START_REF] De Leeuw | Addit ive st ruct ure in qualit at ive dat a: An alt ernat ing least squares met hod wit h opt imal scaling feat ures[END_REF]. In cont rast , convergence of t he ext ant algorit hm has not been fully proven except for t he cases of dealing wit h only one or two lat ent variables [START_REF] Hanafi | P LS pat h modeling: comput at ion of lat ent variables wit h t he est imat ion mode B[END_REF][START_REF] Henseler | On t he convergence of t he part ial least squares pat h modeling algorit hm[END_REF]. T hird, t he proposed procedure est imat es t he inner weight s opt imally in a least squares sense. On t he ot her hand, in t he ext ant procedure, it is unclear how t he exist ing schemes were derived and in what sense t heir est imat es of t he inner weight s are opt imal. Last ly, t he least squares criterion (8) can serve as a vehicle for furt hering t echnical ext ensions of P LSP M. For example, mult icollinearity among a block of indicat ors can have a negat ive influence on t he est imat ion of component weight s under Mode B [START_REF] Esposit O Vinzi | P LS pat h modeling: from foundat ions t o recent development s and op en issues for model assessment and improvement[END_REF][START_REF] Tenenhaus | Regularized generalized canonical correlat ion analysis[END_REF]. To address t his issue, we may int egrat e a ridge penalty int o (8), as follows. φ = J j = 1 α j SS(X jf j w j ) + J J = 1

(1 -α j ) (SS(f j -X j w j ) + λ j SS(w j )), ( 14) where λ j is a block wise ridge paramet er. Moreover, (8) can be minimized in combinat ion wit h opt imal scaling (e.g., [START_REF] Gifi | Nonlinear multivariate analysis[END_REF][START_REF] Young | Quant it at ive analysis of qualit at ive dat a[END_REF]. T his nonlinear ext ension can be of use in dealing wit h discret e indicat ors. Besides t hese t echnical implicat ions, t he proposed procedure was found t o provide quite comparable paramet er est imat es t o t hose obt ained from t he ext ant one in a real Appendix: A summary of t he Lohmöller and ALS algorit hms.

T he Lohmöller algorit hm T he ALS algorit hm S t e p 0 ( I n it ia liza t io n ) For j = 1, . . . , J choose t he j t h arbit rary weight vect or (w 0 j ) η 0 j = X j w 0 j X j w 0 j E nd For s = 0, 1, 2, . . . . (unt il convergence)

St ep 1 ( I n t er n a l E st im a t ion ) For j = 1, . . . , J

f s j = Q j
q= 1 e j q η s q , where e j q is calculat ed as follows: For t he cent roid scheme, e j q = sign(corr(η s j , η s q )) For t he fact orial schem e, e j q = corr(η s j , η s q ) For t he pat h weight ing scheme, e j q = corr(η s j , η s q ), if η j aff ect s η q ω j q , ot herwise where ω j q is t he qt h element of t he regression coeffi cient s of η j on ηq 's. End St ep 2 ( E x t er n a l E st im a t ion ) For j = 1 . . . J w s+ 1 j = X j f s j (f s j f s j ) -1 , if Mode A w s+ 1 j = (X j X j ) -1 X j f s j , if Mode B η s+ 1 j = X j w s + 1 j X j w s + 1 j End Check if

J j = 1 P j p= 1
(w s j p -w s+ 1 j p ) < .00001. If not , go back t o St ep 1. E nd S t e p 0 ( I n it ia liza t io n ) For j = 1, . . . , J choose t he j t h arbit rary weight vect or (w 0 j ) η 0 j = X j w 0 j X j w 0 j E nd For s = 0, 1, 2, . . . . (unt il convergence)

St ep 1 ( I n t er n a l E st im a t ion ) For j = 1, . . . , J

α j = 1, if Mode A α j = 0, if Mode B f s j = Q j
q= 1 e j q η s q , where e j q is t he qt h elem ent of e s j = (α j w s j w s j Γ s j Γ s j + (1-α j )Γ s j Γ s j ) -1 Γ s j η s j

End

St ep 2 ( E x t er n a l E st im a t ion ) For j = 1 . . . J α j = 1, if Mode A α j = 0, if Mode B w s+ 1 j = (α j f s j f s j I + (1 -α j )X j X j ) -1 X j f s j , η s+ 1 j = X j w s + 1 j X j w

K

  ey W ords and Phrases: P art ial least squares pat h modeling, Mode A, Mode B, schemes, opt imizat ion crit erion, alt ernat ing least squares. * McG ill Universit y * * University of Vict oria * * * Cent raleSup elec-L2S, UMR CNR S 8506 and Bioinform at ics/ Biost at ist ics P lat form IHU-A-ICM Mail Address: heungsun.hwang@mcgill.ca T he aut hors wish t o t hank Claes Fornell for providing t he ACSI dat a. R equest s for reprint s should b e sent t o: Heungsun Hwang, Depart m ent of P sychology, McG ill University, 1205 Dr. P enfield Avenue, Mont real, Q C, H3A 1B1, Canada. Tel: 514-398-8021.

Table 1 :

 1 T he est imat es of weight s and loadings of t he ACSI model obt ained from t he proposed and ext ant proceduresfor P LSP M.

	Lat ent	Indicat or	Weight est imat es P rop osed Ext ant	Loading est imat es P rop osed Ext ant
	CE	x 1	.4447	.4523	.8651	.8679
		x2	.4375	.4310	.8772	.8750
		x3	.3219	.3207	.7189	.7179
	P Q	x4	.4042	.4048	.9336	.9328
		x 5 x 6	.4114 .2986	.4034 .3072	.9325 .8004	.9303 .8045
	P V	x 7	.4251	.4229	.8024	.8012
		x8	.7060	.7080	.9332	.9339
	CS	x9	.3851	.3855	.9388	.9387
		x10	.3480	.3414	.9232	.9216
		x 11	.3487	.3550	.9097	.9113
	CC	x 12	1.000	1.000	1.000	1.000
	CL	x 13 x 14	.5827 .4812	.5827 .4813	.9507 .9268	.9507 .9268

Table 2 :

 2 T he est imat es of pat h coeffi cient s of t he ACSI model obt ained from t he prop osed and ext ant procedures for P LSP M.

		P rop osed	Ext ant
	CE → P Q	.5822	.5819
	CE → P V	.1220	.1230
	CE → CS	.0330	.0353
	P Q → P V	.6469	.6466
	P Q → CS	.6707	.6668
	P V → CS	.2656	.2676
	CS → CC	-.4000	-.4002
	CS → CL	.5824	.5831
	CC → CL	-.0976	-.0972

  T he st ruct ural equat ion model sp ecified for t he first simulat ion st udy. St andardized paramet ers are given in parent heses.

	.51	.51	.51	.51	.51		.51	.51	.51	.51
	x 1	x 2	x 3	x 4	x 5		x 6	x 7	x 8	x 9
	1 (.7)	1 (.7)	1 (.7)	1 (.7)	1 (.7)	(.7)	1	1 (.7)	1 (.7)	1 (.7)
		1	.6 (.6)		2		.6 (.6)		3
	.49
				.3136				.3136
	F igure 3:

Table 3 :

 3 T he bias, st andard deviat ion (SD), and mean square error (MSE) of each paramet er est imat e obt ained from t he proposed and ext ant procedures for P LSP M in t he first simulat ion st udy. P P 1 : P rop osed procedure under αj = 1; PP 2 : P rop osed procedure under α j = .5; EP : Ext ant procedure.

	P aramet ers N	P P 1	Bias P P 2	EP	P P 1	SD P P 2	EP	P P 1	MSE P P 2	EP
	Loading 1	25	.0842	.0052	.0842	.1590	.2900	.1633	.0324	.0841	.0338
	(.7)	100	.1067	.0957	.1067	.0537	.0970	.0536	.0143	.0186	.0143
		400	.1120	.1092	.1120	.0250	.0448	.0250	.0132	.0139	.0132
	Loading2	25	.0977	-.0110	.0973	.1242	.2959	.1241	.0250	.0877	.0249
	(.7)	100	.1064	.0930	.1064	.0519	.0994	.0518	.0140	.0185	.0140
		400	.1103	.1078	.1103	.0256	.0467	.0256	.0128	.0138	.0128
	Loading3	25	.0766	-.0021	.0775	.1573	.3137	.1543	.0306	.0984	.0298
	(.7)	100	.1106	.0869	.1106	.0490	.1073	.0489	.0146	.0191	.0146
		400	.1121	.1068	.1121	.0236	.0468	.0236	.0131	.0136	.0131
	Loading4	25	.1016	.0672	.1023	.1164	.1813	.1139	.0239	.0374	.0234
	(.7)	100	.1079	.1013	.1079	.0442	.0762	.0440	.0136	.0161	.0136
		400	.1110	.1098	.1110	.0210	.0326	.0209	.0128	.0131	.0128
	Loading5	25	.1042	.0675	.1039	.1136	.1774	.1139	.0237	.0360	.0238
	(.7)	100	.1092	.1112	.1092	.0461	.0702	.0460	.0141	.0173	.0140
		400	.1121	.1113	.1121	.0215	.0315	.0214	.0130	.0134	.0130
	Loading6	25	.0992	.0617	.1006	.1098	.1993	.1039	.0219	.0435	.0209
	(.7)	100	.1077	.1001	.1077	.0464	.0757	.0465	.0138	.0158	.0138
		400	.1120	.1114	.1120	.0215	.0308	.0215	.0130	.0134	.0131
	Loading7	25	.0938	.0013	.0951	.1569	.2969	.1449	.0334	.0882	.0300
	(.7)	100	.1097	.0930	.1097	.0489	.1044	.0489	.0144	.0195	.0144
		400	.1097	.1117	.1097	.0229	.0453	.0229	.0126	.0145	.0126
	Loading8	25	.0757	-.0127	.0764	.1837	.2888	.1835	.0395	.0835	.0395
	(.7)	100	.1055	.0844	.1055	.0540	.0983	.0539	.0140	.0168	.0140
		400	.1114	.1048	.1114	.0242	.0453	.0242	.0130	.0130	.0130
	Loading9	25	.0605	-.0015	.0615	.2162	.2788	.2182	.0504	.0777	.0514
	(.7)	100	.1084	.0966	.1084	.0488	.1066	.0488	.0141	.0207	.0141
		400	.1125	.1068	.1125	.0224	.0464	.0224	.0132	.0135	.0132
	P at h 1	25	-.1024	-.0664	-.1021	.1655	.1854	.1635	.0379	.0388	.0372
	(.6)	100	-.1555	-.1382	-.1554	.0801	.0808	.0799	.0306	.0256	.0305
		400	-.1531	-.1451	-.1531	.0405	.0396	.0405	.0251	.0226	.0251
	P at h 2	25	-.1091	-.0703	-.1091	.1576	.1769	.1573	.0368	.0362	.0366
	(.6)	100	-.1461	-.1300	-.1461	.0812	.0762	.0810	.0279	.0227	.0279
		400	-.1500	-.1493	-.1500	.0396	.0401	.0396	.0241	.0239	.0241
	4.2.2 Simulation 2

Table 4 :

 4 T he bias, st andard deviat ion (SD), and mean square error (MSE) of each paramet er est imat e obt ained from t he proposed and ext ant procedures for P LSP M in t he second simulat ion st udy.

	P aramet ers	N	Bias P rop osed	Ext ant	P rop osed	SD	Ext ant	MSE P rop osed	Ext ant
	Weight 1	25	-.1443	-.1442	.2812		.2811	.0999	.0998
	(.1)	100	-.1461	-.1461	.1206		.1206	.0359	.0359
		400	-.1577	-.1579	.0575		.0575	.0282	.0282
	Weight 2	25	-.0773	-.0771	.2355		.2356	.0614	.0615
	(.2)	100	-.0772	-.0770	.1149		.1149	.0192	.0191
		400	-.0655	-.0654	.0545		.0545	.0073	.0072
	Weight 3	25	-.1354	-.1356	.2630		.2631	.0875	.0876
	(.1)	100	-.1391	-.1393	.1148		.1148	.0325	.0326
		400	-.1338	-.1340	.0519		.0519	.0206	.0207
	Weight 4	25	.0298	.0297	.2252		.2252	.0516	.0516
	(.6)	100	.0614	.0615	.0763		.0763	.0096	.0096
		400	.0687	.0688	.0366		.0366	.0061	.0061
	Weight 5	25	.2674	.2675	.2026		.2026	.1125	.1126
	(.4)	100	.3090	.3089	.0724		.0724	.1007	.1007
		400	.3103	.3102	.0366		.0367	.0976	.0976
	Weight 6	25	.0294	.0299	.4651		.4649	.2172	.2171
	(.4)	100	.1421	.1426	.3178		.3176	.1212	.1212
		400	.2276	.2280	.1591		.1589	.0771	.0772
	Weight 7	25	-.1872	-.1867	.4839		.4835	.2691	.2686
	(.6)	100	.0410	.0410	.3089		.3087	.0971	.0970
		400	.1298	.1295	.1492		.1491	.0391	.0390
	Weight 8	25	.0705	.0701	.5018		.5017	.2568	.2566
	(.1)	100	-.0989	-.0991	.3496		.3494	.1320	.1319
		400	-.1255	-.1259	.2161		.2159	.0625	.0625
	Weight 9	25	-.2287	-.2288	.5146		.5149	.3171	.3175
	(.4)	100	-.1280	-.1281	.5045		.5046	.2709	.2710
		400	.1415	.1411	.4416		.4416	.2150	.2149
	Weight 10	25	-.1332	-.1333	.6262		.6261	.4099	.4098
	(.3)	100	-.1598	-.1597	.5498		.5498	.3278	.3278
		400	.0085	.0090	.4136		.4137	.1711	.1712
	Weight 11	25	-.1197	-.1194	.6373		.6371	.4204	.4202
	(.2)	100	-.1571	-.1567	.6042		.6039	.3897	.3893
		400	-.3426	-.3424	.4836		.4834	.3513	.3510
	Weight 12	25	-.1170	-.1169	.5816		.5814	.3520	.3517
	(.2)	100	-.1023	-.1022	.5153		.5152	.2760	.2759
		400	-.0963	-.0963	.4253		.4253	.1902	.1901
	Weight 13	25	-.3089	-.3088	.5303		.5302	.3767	.3765
	(.4)	100	-.3531	-.3531	.4463		.4462	.3238	.3238
		400	-.4782	-.4784	.3686		.3687	.3645	.3648
	Loading 1	25	.1596	.1599	.0177		.0174	.0258	.0259
	(.8)	100	.1623	.1623	.0074		.0074	.0264	.0264
		400	.1624	.1625	.0036		.0036	.0264	.0264
	Loading 2	25	.2347	.2346	.0265		.0268	.0558	.0558
	(.7)	100	.2364	.2367	.0137		.0136	.0561	.0562
		400	.2370	.2373	.0062		.0062	.0562	.0563

deviat ions, and mean square errors of all paramet er est imat es across all sample sizes, indicat ing t hat t he two procedures yielded almost ident ical paramet er est imat es.

  End Check if φ s -φ s+ 1 < .00001. If not , go back t o St ep 1. E nd

s + 1 j