index variable

by Christos Giannoulis, PhD

Attributes are often measured using multiple variables with different upper and lower limits. For example, we may have five measures of political orientation, each with a different range of values.

Each variable is measured in a different way. The measures have a different number of categories and the low and high scores on each measure are different.

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I recently gave a free webinar on Principal Component Analysis. We had almost 300 researchers attend and didn’t get through all the questions. This is part of a series of answers to those questions.

If you missed it, you can get the webinar recording here.

Question: Can we use PCA for reducing both predictors and response variables?

In fact, there were a few related but separate questions about using and interpreting the resulting component scores, so I’ll answer them together here.

How could you use the component scores?

A lot of times PCAs are used for further analysis — say, regression. How can we interpret the results of regression?

Let’s say I would like to interpret my regression results in terms of original data, but they are hiding under PCAs. What is the best interpretation that we can do in this case?

Answer:

So yes, the point of PCA is to reduce variables — create an index score variable that is an optimally weighted combination of a group of correlated variables.

And yes, you can use this index variable as either a predictor or response variable.

It is often used as a solution for multicollinearity among predictor variables in a regression model. Rather than include multiple correlated predictors, none of which is significant, if you can combine them using PCA, then use that.

It’s also used as a solution to avoid inflated familywise Type I error caused by running the same analysis on multiple correlated outcome variables. Combine the correlated outcomes using PCA, then use that as the single outcome variable. (This is, incidentally, what MANOVA does).

In both cases, you can no longer interpret the individual variables.

You may want to, but you can’t. (more…)

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One common reason for running Principal Component Analysis (PCA) or Factor Analysis (FA) is variable reduction.

In other words, you may start with a 10-item scale meant to measure something like Anxiety, which is difficult to accurately measure with a single question.

You could use all 10 items as individual variables in an analysis–perhaps as predictors in a regression model.

But you’d end up with a mess.

Not only would you have trouble interpreting all those coefficients, but you’re likely to have multicollinearity problems.

And most importantly, you’re not interested in the effect of each of those individual 10 items on your (more…)

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