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**Keywords:** Factor shifts; sequential data; unique solutions;
time series; spectral shifts; three-way or three-mode analysis;
Parafac; Parafac2; Tucker T3 and T2

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**Keywords:** Shifted factor analysis; latent position shift;
lag; multilinearity; principal component analysis; bilinear model;
quasi-ALS; uniqueness; identifiability

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**Keywords:** Shifted factor analysis; latent position shift;
lag; multilinearity; Parafac1; Parafac2; multiway analysis; quasi-ALS;
chromatography

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**Keywords:**
alternative medicine; complementary (natural) versus conventional (or
traditional, biomedical) treatment; intractible pain; factor analysis;
attitude toward health; psychological models; lay beliefs; PARAFAC;
3-way; multimode

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**Keywords:** diagonalization, sliced-array, stretched-array,
matricized, unfolding, multilinear and quasi-multilinear models,
Tucker, T1, T2, T3, Parafac/Candecomp, Parafac1, Parafac2, Paratuck2.

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**Keywords:** Linear and multilinear algebra; tensors; array notation; three-way models; n-way arrays; Tucker; T2; T3; Parafac / Candecomp

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**Keywords:** aluminium oxide; aluminium hydride; oxidation; parallel factor analysis; XPS

**Keywords:** Factor analysis; Trilinear; PARAFAC; Fluorescence spectroscopy; PMF3; TPALS; DTDMR

Issues addressed include the "corrupt scientific method", the "nil" hypothesis, when it seems most appropriate to test the null hypothesis of no effect, the proper interpretation of a confidence interval, the purpose of a significance test, whether meta-analysis should replace significance testing, and eight misinterpretations of significance testing (e.g., "the probability of rejecting H

Power of a significance test is defined, and issues related to sample size are discussed. Rather than avoiding significance testing altogether, one must consider its power to detect a given size effect with the sample at hand (i.e., significance testing with small samples is meaningful if one is looking for large effects). Contrary to those who believe that power is relevant only in the context of significance testing, the authors take the position that power remains an issue in meta-analysis as well, and discuss this in some detail.

The notion that physicists do not perform formal significance tests is also addressed. [Not mentioned, however, is the fact that in some areas, such as particle physics, significance testing plays an important role, though the terminology used is somewhat different. In the chapter, attention is focused on those areas of physics where significance testing is less common.] One reason is that physicists are often trying to improve their estimates of physical constants, rather than testing hypotheses. The argument is also made that many of their studies are equivalent to meta-analyses in the social sciences and their statistics like those used in meta-analysis.

The meaning of objectivity and its relevance to hypothesis testing is also covered. A hypothesis must be formed independently of the data used to evaluate it, so that objective judgments about the world can be made as a result of the hypothesis test. It is suggested that nonzero null hypotheses should be used, and the null values modified to reflect new knowledge; this avoids the nil-hypothesis criticism. Finally, the authors argue that hypothesis testing should not be viewed as a true-false decision but rather as something that affects one's degree of belief in the hypothesis. They sum up their position by saying that "significance testing is a procedure contributing to the (provisional) judgment about the objective validity of a substantive proposition".

An appendix is provided which summarizes the differing positions of the developers of significance testing methods, R. A. Fisher vs. J. Neyman and E. Pearson. The appendix allows the reader to assess the criticisms of significance testing in light of what the developers said. Back to publications

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**Keywords:** Three-way exploratory factor analysis; unique axes;
parallel proportional profiles; factor rotation problem; three-way data
preprocessing; three mode principal components; trilinear
decomposition; trilinear model; multidimensional scaling; longitudinal
factor analysis; factor analysis of spectra; interpretation of factors;
'Real' or causal or explanatory factors; L. R. Tucker; R. B. Cattell

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The next major section in the chapter deals with the uniqueness properties of the PARAFAC model (pp. 147-169). "Uniqueness" or "intrinsic axes" means that given certain assumptions, the solution determined from the data by PARAFAC has no alternative, equal-fitting form (i.e., any other rotation would reduce its fit to the data). Why this is important (pp. 147-150; 163-9) and the minimum conditions for uniqueness (pp. 161-2) are explained. The value of empirically confirming a PARAFAC solution by split-half, bootstrapping, and/or jackknifing procedures is also discussed.

The final section compares PARAFAC with other models, most notably Tucker's T3 (pp. 169-182), Corballis' three-way model (pp. 184-5), and Sands and Young's ALSCOMP (pp. 188-190). PARAFAC1 is described as a special case of the T3 model and vice versa, and a diagram on p. 175 shows how Carroll transforms a T3 representation into the corresponding PARAFAC one (two other methods of embedding T3 in PARAFAC are also discussed on pp. 176-178 and pp. 203-7). A family of related models for three-way profile data, arranged from most general to most restricted, is presented on p. 184. PARAFAC3 is listed between T3 and PARAFAC1 and is discussed on pp. 185-6. PARAFAC2 and DEDICOM, not in the table, are also discussed in relation to PARAFAC3 and T3 (p. 187).

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- See also:
- Celebrity-brand congruence analysis.
- "How can I know if it's real?" (next summary)

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**Keywords:** Multidimensional scaling; factor analyses

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- See also:
- Factor analysis of tongue shapes. (next summary)

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- See also:
- Uniqueness proof for a family of models...
- Determination and proof of minimum uniqueness conditions for PARAFAC1. (next summary)

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Although PP was originally formulated as a principle of rotation to be used with classic two-way factor analysis, it is shown to embody a latent three-way factor model, which is here made explicit and generalized from two to N "parallel occasions". As originally formulated, PP rotation was restricted to orthogonal factors. The generalized PP model is demonstrated to give unique "correct" solutions with oblique, non-simple structure, and even non-linear factor structures.

A series of tests, conducted with synthetic data of known factor composition, demonstrate the capabilities of linear and non-linear versions of the mode, provide data on the minimal necessary conditions of uniqueness, and reveal the properties of the analysis procedures when these minimal conditions are not fulfilled. In addition, a mathematical proof is presented for the uniqueness of the solution given certain conditions on the data.

Three-mode PP factor analysis is applied to a three-way set of real data consisting of the fundamental and first three formant frequencies of ll persons saying 8 vowels. A unique solution is extracted, consisting of three factors which are highly meaningful and consistent with prior knowledge and theory concerning vowel quality.

The relationships between the three-mode PP model and Tucker's multi-modal model, McDonald's non-linear model and Carroll and Chang's multi-dimensional scaling model are explored.

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