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In this approach, the classic nonlinear exponential model (Equation 18.21) is now in the form of a latent growth curve with structured loadings (as in Browne & duToit, 1991; McArdle & Hamagami, 1996). Individual trajectories start at different initial levels, but then rise or fall in exponential fashion towards some asymptotic values. In this approach, the group curve is based on the latent means and is not based on an averaging of exponential functions (cf. Keats, 1983; Tucker, 1966). This common factor approach allows us to use current computing techniques to examine the empirical t of this nonlinear model. A related approach has been used with a form of the von Bertalanffy model, dyn /dt = (



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The first aspect of health maintenance that may be influenced by psychosocial factors is health preservation. Health preservation processes encompass risk behaviors, preventative health behaviors, susceptibility to stress-related symptoms, and resistance to the onset of disease. Together, these processes represent efforts to preserve a healthy nondiseased state including those that reduce the likelihood of developing disease among healthy individuals who are at risk for the development of a specific disease (i.e., primary prevention).





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these two models, see the special issue of Multivariate Behavioral Research, 1990, Volume 25, Issue 1 (also see 1996, Volume 31, Issue 4 for discussion of indeterminacy per se) In understanding this debate, it is important to note that all procedures for CA and ECFA are the same except for one: CA starts with 10 in the main diagonal of the correlation matrix and CFA starts with a communality estimate (thus taking into account the existence of variance unique to the single variable) This is the only mathematical difference between the two Everything else is the same (which is why they are both special cases of the general factor analytic model) The Case for Common Factor Analysis The rationale for CFA comes from Equation 61 and assumptions about data Including the uniqueness term in the equation makes it a CFA.

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The uniqueness term includes all of the variable s variance not associated with the factors, part of which is random error So, the CFA rationale goes, CFA should be used whenever at least some reliabilities are less than 10 that is, whenever some variables contain any random error Of course, this argument runs, who can show, or assume, that all their variables are without random error Where is the evidence for such variables in the social sciences And if we know the variables have error, is it not rational to build that into our mathematical models Dropping the uniqueness term also means that the factors and only the factors underlie the scores for each variable Hence in the population, the communality is to be 10 This is the justi cation for using 10 in the main diagonal of the correlation matrix.

This means that the multiple correlation of the factors with each of the variables is also 10 Unfortunately, the derivative is that the variables, being sets of linear combinations of a fewer number of factors, will form a nonGramian correlation matrix Such a matrix has an in nite number of solutions and so cannot be factored at all Therefore, CA is a self-contradictory model (The only reason that CA works is that the model is wrong for the data no two of the variables being analyzed have a multiple correlation of 10 with the same factors, so none truly t the model) Although component advocates raise the problem of estimating communalities and factor scores, such estimates are consistent and easily made The variations on factor scores are variations among scores that generally correlate 9 or better in simulation and plasmode studies.

Fig. 8.2 Longer horns are an advantage to a ghting beetle (a), but short-horned males make better runners (b). (a) White bars, shorter-horned winner; blue bars, longer-horned winner. (b) White bars, shorter-horned outperformed longer-horned; purple bars, longer-horned outperformed shorter-horned; lilac bars, no difference in performance between morphs. (From Moczek, A.P. & Emlen, D.J. (2000) Male horn dimorphism in the scarab beetle, Onthophagus tarsus: do alternative reproductive tactics favour alternative phenotypes Animal Behaviour, 59, 459 66. Reproduced with permission of Elsevier.)

This is much better than in other areas For example, major ability tests often correlate 7 to 8, yet are seen as interchangeable Also the correlation between CA factor scores from one study to the next is much less than 10 and is probably no greater then that from one CFA to another, so where is the added precision.

which follows directly from the de nition of convex functions. Suppose that the theorem is true for distributions with k 1 mass points. Then writing pi = pi /(1 pk ) for i = 1, 2, . . . , k 1, we have

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