By David J. Bartholomew
The classical statistical challenge quite often contains a chance distribution which depends upon a couple of unknown parameters. the shape of the distribution could be identified, in part or thoroughly, and inferences need to be made at the foundation of a pattern of observations drawn from the distribution; usually, yet now not inevitably, a random pattern. This short bargains with difficulties the place a few of the pattern individuals are both unobserved or hypothetical, the latter class being brought as a method of higher explaining the information. occasionally we're drawn to some of these variable themselves and infrequently within the parameters of the distribution. Many difficulties that may be forged into this kind are taken care of. those contain: lacking information, combos, latent variables, time sequence and social size difficulties. even supposing all should be accommodated inside a Bayesian framework, such a lot are most sensible taken care of from first principles.
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In R. E. Millsap & A. ), The Sage handbook of quantitative methods in psychology. London: Sage Publications Ltd. Chapter 5 A General Latent Variable Model Abstract This chapter shows that, starting from the two-way analysis of variance with random effects, it is possible to arrive at a general latent variable model. It does this, first by enlarging the family of conditional distributions considered and secondly, and more fundamentally, by allowing the random effect associated with the rows of the two-way table to be linear in a set of (unobserved) latent variables.
The normal model expressed as a linear equation in Eq. 4) of the last chapter can be cast into this form by writing the conditional distribution of x as xij jy _ Nðl þ ai þ k1i y1 þ k2i y2 þ Á Á Á kqi yq ; r2i : ð6:2Þ There are good practical reasons in favour of using a standard normal distribution for the prior distribution for each of the elements of y and assuming them to be mutually independent. This is because the normal distribution commonly arises in measurement work, partly since many naturally occurring quantities used as measures have distributions close to normal and partly because it spaces individuals along the scale of measurement in a way which accords with our intuition.
In Spearman’s terminology, l þ ki y represents the general ability of the individual and ei the ‘specific factor’. The use of the term ‘factor’ here is out of line with much modern usage, but it accounts for the fact that Spearman called his model the ‘two-factor model’ whereas it would now be called a ‘one-factor model’. Also it is obviously a special case of the general factor model considered in Chap. 5. Spearman used the symbol g instead of y in Eq. 3) to designate what he called the general factor.