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can be found in Marcoulides and Schumacker (2001) For work having a LISREL focus, see Cudeck, du Toit, and S rbom (2001) Improved Methods for Small Samples As noted earlier, SEM methodology is often applied when sample size is small: About 18% of the studies we reviewed used samples of fewer than 100 individuals (McCallum & Austin, 2000, p 215) The quality of results that may occur from a given study with small samples will depend on the features of the model of interest (parameter estimates, standard errors, z tests, test statistics, mediational effects) as well as characteristics of the model such as the communality level of the variables (MacCallum, Widaman, Zhang, & Hong, 1999) and the degree and kind of nonnormality that might exist (eg, Boomsma & Hoogland, 2001; Finch, West, & MacKinnon, 1997; Hoogland, 1999; Yuan & Bentler, 1999b).

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Ideally, SEM methods would perform well across all the design features just mentioned, but this is not the case (eg, Bentler & Yuan, 1999) At a minimum, one would like to have enough power to reject alternative models (Hancock, Lawrence, & Nevitt, 2000; MacCallum et al, 1996), but even the estimation of power depends on having a test statistic that can be relied upon under the given circumstances (ie, typically, that correctly can be referred to the hypothesized noncentral chi-square distribution) Because various test statistics that are presumed to have central chi-square distributions do not behave this way under realistic data-gathering conditions (eg, violation of normality in the case of normal maximum likelihood), it is likely that the noncentral distributions used to calculate power also may not suf ce Clearly, better small sample methods are needed.

n g1 : Y1 {1, 2, . . . , 2nR1 }

In this chapter a new and enhanced version of the popular evolving Takagi-Sugeno approach (Angelov & Filev, 2004) is introduced, called eTS . It builds on the strengths of the original eTS; namely, it is also fully recursive (thus suitable for online applications

Given the model characteristics, some methods are better able to cope with small samples than are other methods For example, Bentler and Yuan (1999) studied the normal theorybased likelihood ratio statistic TML , the Satorra-Bentler rescaled statistic TSB , the Yuan and Bentler version TYB of Browne s (1984) residual-based ADF statistic TB , and their F statistic derived from the residual-based ADF statistic They found that the F statistic performed best of all these mentioned at the smallest sample sizes It also performed very well in Yuan and Bentler s (1999a) and Nevitt s (2000) study, and is available in EQS 6 (Bentler, 2001) However, Fouladi (1999) recommended using a Bartlett (1950) correction and applying it to the Satorra-Bentler (1994) scaled and adjusted statistics She found that the Bartlett-corrected adjusted statistic performed best.