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Generalizing to a Population:
SIMPLE MODELS AND TESTS continued

Linear Regression
Let's use the same example that I used to introduce the concept of statistical models. As you can see, data for two variables like weight and height scream out to have a straight line drawn through them. The straight line will allow us to predict any person's weight from a knowledge of that person's height. Obviously, the prediction won't be perfect, so we will also be able to say how strong the linear relationship is between weight and height, or how well the straight line fits the data (the goodness of fit).

Here's how we represent the model:
model: numeric <= numeric
example: weight <= height

You normally think about a straight line as Y = mX + c, where m is the slope and c is the intercept. The way I would write this relationship, using the above notation, is simply Y <= X. We don't have to worry about showing the constants, but the stats program worries about them. They're the parameters in the model.

The Slope
The most interesting parameter in a linear model is usually the slope. If the slope is zero, the line is flat, so there's no relationship between the variables. In the example, the slope is about 0.75 kg per cm (an increase in weight of 0.75 kg for each cm increase in height). We can also calculate the slope in two ways that don't have those ugly units (kg per cm).

One way is to calculate the percent change in weight per percent change in height. It's unusual, but sometimes it's the best way, especially for variables that need log transformation. The slope expressed as % per % comes directly out of the analysis of log-transformed variables.

The other way to remove the units is to normalize the two variables by dividing their values by their standard deviations, then fit the straight line. The resulting slope is known as a standardized regression coefficient. It represents the change in weight, expressed as a fraction of the standard deviation, per standard deviation change in height. You can also generate it by multiplying the slope (in kg per cm) by the ratio of the standard deviations for height over the standard deviation for weight. In a simple linear regression, the value of the standardized regression coefficient is exactly the same as the correlation coefficient, and you can interpret its magnitude in the same way. In the example, the value is about 0.9, or a difference of 0.9 standard deviations in weight per change of one standard deviation in height. That's a really strong relationship!

Goodness of Fit
The stats program works out values for the slope and intercept (the parameters) that give the best fit. I'll explain how after I've dealt with all four simple models. Meanwhile, we want a measure of how good the fit is. The correlation coefficient is one such measure. Another way to represent the fit is to square the correlation coefficient, multiply it by 100, then call the result the percent of variance explained, or percent R2. For example, the R2 represents the proportion of variation in weight that can be attributed to height, when there is a linear relationship between weight and height. A correlation of 0.9 is equivalent to an R2 of 0.81 or 81%. I'll explain more about goodness of fit in a few pages' time.

The p value or the confidence interval for the correlation coefficient tell us how good the fit is likely to be in the population. The program can also give confidence intervals or p values for the slope and intercept. The correlation coefficient can be considered as a test statistic for whether the line fits the data at all. But stats programs can also produce another statistic for this purpose, called the F ratio. The values for F are quite different from those for r, but there is a one-to-one relationship between them, and the r and the F have the same p value for a given sample.

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Last updated 10 Dec 00