A New View of Statistics

© 1997 Will G Hopkins

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Generalizing to a Population:

 More Than One Dependent Variable

These are called multivariate models. In other words, things like:

jump  sprint <= sex 

You read this as: what is the effect of sex on a person's ability to jump and sprint? Sure, sex might have an effect on each of these separately, but let's put them together and look at the overall effect on both. This would be an example of multivariate analysis of variance (MANOVA). The test statistics are unusual (e.g. Hotelling t2, Wilks' lambda).

Multivariate models are easy in principle, but in practice it's hard to interpret the outcome statistics. I advise you to analyze your dependent variables separately, or do a dimension reduction first, then analyze each dimension separately.

Multivariate models have been adapted for analysis of repeated measures. For example, replace sprint in the above model with a second measurement of jump, and you could write:

jump1  jump2 <= sex,

where jump1 and jump2 are jump heights on the first and second occasion. I deal with this approach to repeated measures later.
  Models With Variables of Uncertain Status

There's only one kind of model here, but it goes by various names: path analysis, structural equation modeling (SEM, not to be confused with standard error of the mean), and linear structural relationships (LISREL).

I've never used this kind of modeling, so this section will be brief and untrustworthy. As far as I can see, the technique can be applied only in cross-sectional studies with hundreds of subjects. It represents an attempt to establish a chain of cause and effect between variables. The stats program does it by looking at all the correlations between all the variables, then creating the best chain, like so:

numeric <= numeric <= numeric <= numeric...

The program produces correlations for each link, and a correlation between the variable on the far right (the cause) and the far left (the effect). Validity of measurement for each variable, where known, can be taken into account.

There is now a huge literature on this topic, which in my opinion goes way beyond what is justified for cross-sectional data. Let's face it, cross-sectional studies can only ever provide suggestive evidence; in the end you need longitudinal studies to nail cause and effect. That's where repeated-measures models come to the fore. Read on.

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