A New View of Statistics

© 2000 Will G Hopkins

Go to: Next · Previous · Contents · Search · Home

Generalizing to a Population:
COMPLEX MODELS: More Than One Independent Variable continued

 Multiple Linear Regression
  model: numeric <= numeric1  numeric2... + interactions
  example: weight <= height  age  height*age
The example shows weights and heights of a sample of people aged between 20 and 60. Each person is represented by a number, which is the person's age rounded to the nearest decade (2 = 15-24 years, 3 = 25-34 years, etc.). Look closely at the way the numbers are distributed. What would you conclude about the effect of age on weight, for any given height? Right! People get heavier as they get older.

Multiple linear regression is the model to use when you want to look at data like these, consisting of two or more numeric independent variables (height, age) and a numeric dependent variable (weight). In this first example, the only effect of age is to produce a uniform increase in weight, irrespective of height. It's just as correct to say there is a uniform increase in weight with height, irrespective of age. These interpretations come straight from the model. Or you can look at the graphical interpretation and think about the effect of age as altering the intercept of the weight-height line in a uniform way. But what about when there's an interaction?

Interpreting the Interaction Term
As you can see, the effect of an interaction is to make different slopes for different ages. The slopes change in a nice linear way with increasing age, just as the intercepts did (and still do). In the example, I've given older people a greater weight for a given height than younger people, which is not necessarily realistic. Real data would certainly not show such clear-cut effects of either height or weight, anyway.

It's one thing for me to show you a clear-cut example with colors for the different ages. It's quite another matter for you to interpret real data, without a colored graph. If you get a substantial interaction with your data, I suggest you look at the values of the parameters in the solution. Use them to work out how your outcome variable is affected by a range of values of the independent variables. That's the only way you will sort out what's going on.

By the way, for publication you would not plot them as I have shown here. In fact, generally you don't plot the data for linear regressions, be they simple or multiple, unless the data show interesting non-linear effects.
Paradoxically Insubstantial Effects
On the previous page I pointed out how one independent variable can make another seem insubstantial in an ANCOVA. The same is true here. It's important, so let's take an example.

Suppose you want to predict running-shoe size (dependent variable) from an athlete's height and weight. These two variables are well correlated, but let's assume the correlation is almost perfect. When two variables have an almost perfect correlation, it means they effectively measure the same thing, even if they are in different units. Now let's put them both into the model. Will weight tell you anything extra about shoe size, when height is already in the model? No, because weight isn't measuring anything extra, so it won't be substantial in the model. But hey, height won't be substantial with weight in the model, for the same reason. So you have the bizarre situation where neither effect is substantial, and yet both are obviously substantial! If you didn't know about this phenomenon, you might look at the p values for each effect in the model, see that they are both greater than 0.05, and conclude that there is no significant effect of either height or weight on shoe size.

The trick is to look at the p value for the whole model as well. None of the effects might be significant, but the whole model will be very significant. And you should always look at the main effects individually, as simple linear regressions or correlations, before you go to the multiple model. You'd find they were both substantial/significant.

So in this example, would you use both independent variables to predict shoe size? Not an easy question to answer. I'd look to see just how much bigger the R2 gets with the second independent variable in the model, regardless of its statistical significance. More on this, next.

Now for two important applications of multiple linear regression: stepwise regression, and on the next page, polynomial regression.

 Stepwise Regression
model: numeric <= numeric1  numeric2  numeric3...
  example: competitive speed <= a set of fitness-test variables
No figure is needed for this one. No interactions either, thank goodness! Numeric1, numeric2, and so on are independent variables, and you try to find the best ones for predicting your dependent variable.

An obvious example is where your dependent variable is some measure of competitive performance, like running speed over 1500 m, and your independent variables are the results of all sorts of fitness tests for aerobic power, anaerobic power, and body composition. What's the best way to combine the tests to predict performance? An interesting and possibly useful question, because you can use the answer for talent identification or team selection. (Why not use the 1500-m times for that purpose? Hmmm...) Anyway, in stepwise regression the computer program finds the lab test with the highest correlation (R2) with performance; it then tries each of the remaining variables (fitness tests) in a multiple linear regression until it finds the two variables with the highest R2; then it tries all of them again until it finds the three variables with the highest R2, and so on. The overall R2 gets bigger as you add in more variables. Ideally of course, you hope to explain 100% of the variance.

Now, even random numbers will explain some of the variance, because you never get exactly zero for a correlation with real numbers. So you need an arbitrary point at which to cut off any further variables from entering the analysis. It's done with the p value, and the default value is 0.15. When a variable enters the model with a p value >0.15, the stepwise procedure halts. You'd hardly call a p value of 0.15 significant, but it's OK if you're using stepwise regression as an exploratory tool to identify the potentially important predictors.

The question of what variables you finally include for your prediction equation is not just a matter of the p values, though. You should be looking at the R2 and deciding whether the last few variables in the stepwise analysis add anything worthwhile, regardless of their significance. If the sample size isn't as big as it ought to be, there's a good chance that the last few variables will contribute substantially to the R2, and yet not be statistically significant. You should still use them, but knowing that their real contributions could be quite a bit different.

OK, what is a worthwhile increase in the R2 as each variable enters the model? Take the square root of the total R2 after each variable has entered, then interpret the resulting correlations using the scale of magnitudes. If the correlations are in the moderate-large range, an increase of 0.1 or more is worthwhile. If the correlation is in the very large to almost perfect range, then smaller increases (0.05 or even less) are worthwhile, as I explain later.

Finally, a warning! If two independent variables are highly correlated, only one will end up in the model with a stepwise analysis, even though either can be regarded as predictors. Go back up this page for the reason. And as discussed in the previous paragraph, the decision to keep both in the model depends on the R.

Go to: Next · Previous · Contents · Search · Home
Last updated 10 Dec 00