Should You Always Center a Predictor on the Mean?

Centering predictor variables is one of those simple but extremely useful practices that is easily overlooked.

It's almost too simple.

Centering simply means subtracting a constant from every value of a variable.? What it does is redefine the 0 point for that predictor to be whatever value you subtracted.? It shifts the scale over, but retains the units.

The effect is that the slope between that predictor and the response variable doesn't change at all.? But the interpretation of the intercept does.

The intercept is just the mean of the response when all predictors = 0.? So when 0 is out of the range of data, that value? is meaningless.? But when you center X so that a value within the dataset becomes 0, the intercept becomes the mean of Y at the value you centered on.

What's the point?? Who cares about interpreting the intercept?

It's true.? In many models, you're not really interested in the intercept.? In those models, there isn't really a point, so don't worry about it.

But, and there's always a but, in many models interpreting the intercept becomes really, really important.? So whether and where you center becomes important too.

A few examples include models with a dummy-coded predictor, models with a polynomial (curvature) term, and random slope models.

Let's look more closely at one of these examples.

In models with a dummy-coded predictor, the intercept is the mean of Y for the reference category—the category numbered 0.? If there’s also a continuous predictor in the model, X2, that intercept is the mean of Y for the reference category only when X2=0.

If 0 is a meaningful value for X2 and within the data set, then there’s no reason to center.? But if neither is true, centering will help you interpret the intercept.

For example, let’s say you’re doing a study on language development in infants.? X1, the dummy-coded categorical predictor, is whether the child is bilingual (X1=1) or monolingual (X1=0).? X2 is the age in months when the child spoke their first word, and Y is the number of words in their vocabulary for their primary language at 24 months.

If we don’t center X2, the intercept in this model will be the mean number of words in the vocabulary of monolingual children who uttered their first word at birth (X2=0).

And since infants never speak at birth, it's meaningless.

A better approach is to center age at some value that is actually in the range of the data. One option, often a good one, is to use the mean age of first spoken word of all children in the data set.

This would make the intercept the mean number of words in the vocabulary of monolingual children for those children who uttered their first word at the mean age that all children uttered their first word.

One problem is that the mean age at which infants utter their first word may differ from one sample to another. This means you're not always evaluating that mean that the exact same age.? It's not comparable across samples.

So another option is to choose a meaningful value of age that is within the values in the data set. One example may be at 12 months.

Under this option the interpretation of the intercept is the mean number of words in the vocabulary of monolingual children for those children who uttered their first word at 12 months.

The exact value you center on doesn't matter as long it's meaningful, holds the same meaning across samples,? and within the range of data.? You may find that choosing the lowest value or the highest value of age is the best option. It's up to you to decide the age at which it’s most meaningful to interpret the intercept.

Originally published at https://www.theanalysisfactor.com/center-on-the-mean/. Updated on March 1, 2024.

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