DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT Individual fit, heterogeneity, and missing data in multi-groupSEMMichael C. NealeVirginia Institute for Psychiatric and Behavioral Genetics, Virginia CommonwealthUniversity, Box 980126, Richmond, VA 23298-0126September 21, 2005AbstractAnalysis of raw data allows great flexibility in structural equation model-ing. First, data in which some observations are missing at random or miss-ing completely at random are automatically handled within this approach.Second, it is possible to fit finite mixture distribution models to raw data.Third, models for continuous moderator variables, including hierarchical lin-ear models, become easy to specify. Fourth, measures of individual fit thatare readily available may be used to detect outliers in the population, orpopulation heterogeneity. The contribution to the likelihood fit function isone such measure, but it depends on the amount of missing data for thecase in question. Two Q statistics for measuring individual fit as a z-score,which are independent of the amount of missing data, are compared for thebivariate case.IntroductionFor many years, structural equation models have been fitted to summary statistics,primarily covariances, but sometimes to the means as well. More recently, programs such asMx and Amos have shown the advantages of fitting models to raw data, and it is extensionsof this method that form the main focus of this chapter. A frequently asked question is“How does one fit structural equation models to the raw data by maximum likelihood?” Insome ways, this is a strange question because it is a simpler question to answer than onethat asks about the origin of the formula used for fitting models to covariance matrices.Therefore, this chapter begins with an elementary introduction to maximum likelihood(ML), including the concepts of individual fit and the multivariate normal distribution.The second section discusses various alternative measures of individual fit, suitable for usewhen some of the data are missing. In the third section I consider moderator variables as apotential source of non-normality. If these moderators have been measured, it is possible toThe author is grateful for support from PHS grants MH-40828, HL48148, MH45268, MH49492, MH41953,AA09095, RR08123, MH01458DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT INDIVIDUAL FIT 2explicitly model their effects. For the case of binary moderators, the model may be specifiedas a two-group structural equation model, but continuous moderators require an extensionto ML analysis of raw data such that there is a different model for every subject in thesample. Finally, although individual fit statistics can be useful for the detection of outliers ormixture distributions, and to judge the value of adding moderating variables, heterogeneitymay always not be directly related to an observed moderator variable. Formal methodsfor detecting ‘latent’ heterogeneity require the application of finite mixture distributions,which are described in the fourth section.Fitting Models by Maximum LikelihoodBasic principlesLikelihood is a simple concept based in probability theory. The relationship of like-lihood to probability is so close that it can be confusing for the novice. To take a trivialexample, suppose a coin is tossed 100 times, and that 53 times it shows heads. The proba-bility of this outcome given that the coin has a probability p = .5 of heads is easy to calculatefrom the binomial distribution, which is:(h + t)!(h! × t!)× (p)h×(1 −p)t. (1)When this formula is applied to our example, where h = 53, t = 47, and p = .5 we obtain100!(53! × 47!)× .553× (1 − .5)47= .07.This probability statement is the usual way of viewing the outcome, or results of an exper-iment. If we ran a number of such experiments we would obtain a series of probabilities forthe particular sequence of number of heads observed in the experiments. In contrast, withlikelihood, we are interested in whether the coin is really unbiased, and wish to examine oneparticular set of results as a function of p, which would not be .5 for a biased coin. Figure 1shows the likelihood of obtaining 53 heads plotted for values of p from zero to one. Theoriginal probability we calculate under the assumption that p = .5 appears as .07 in thefigure; this is also the likelihood for p = .5. We notice that the curve has a maximum (the‘maximum likelihood estimate’ or MLE) at a value somewhat greater than .5; in fact thisis at .53 corresponding to the 53/100. Elementary calculus can be used to show that theMLE of a proportion is the observed proportion, but this simple relation does not hold forall maximum likelihood estimates. Often, the calculus and algebra involved is very compli-cated and it may defy analytic solution, so that we resort to using numerical optimizationto find MLE’s. This optimization approach is used in all the structural equation modelingprograms AMOS, EQS, CALIS, LISREL and Mx.Likelihood theory concerns itself with comparisons between the height of the curve— the likelihood – at various values of parameters such as p. In Figure 1 the likelihoodat the maximum is Lˆp=.53= .0797332 which can be compared with the likelihood underthe hypothesis that coin is unbiased, Lp=.5= 0.0665905, via a likelihood ratio test. Twicethe difference between the logarithm of these likelihoods is asymptotically distributed as χ2DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT INDIVIDUAL FIT 30.30.40.50.60.7p00.020.040.060.080.1Likelihood0.490.50.510.520.530.540.55p-6-5.8-5.6-5.4-5.2-52 Log-likelihoodDifference in fit=.36Figure 1. (Upper) Likelihood curve based on the binomial distribution for the outcome of 53heads from 100 tosses. The likelihood varies as a function of the parameter p, the probabilityof obtaining heads. (Lower) Twice the log-Likelihood showing the difference (χ2) between themaximum likelihood estimate of .53 and the population value of point five.DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT DRAFT INDIVIDUAL FIT 4with one degree of freedom (fixing p at the prior chosen value of .5 instead of allowing it tovary as a free parameter gives one degree of freedom). In this case we have2 × (log 0.0665905 − log .0797332) = 0.360248which is considerably below the value of
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