CHAPTER 17 More of a Good Thing: Multiple Regression 235

• ^Y

a

bX (This is the straight-line model from Chapter 16, where X is the

predictor variable, Y is the outcome, and a and b are parameters.)

• ^Y

a

bX

cX

dX

2

3 (In this multiple regression model, variables can

be squared or cubed. But as long as they’re multiplied by a coefficient —

which is a slope from the model — and the products are added together,

the function is still considered linear in the parameters.)

• ^Y

a

bX

cZ

dXZ (This multiple regression model is special because of

the XZ term, which can be written as X

Z

*

, and is called an interaction. It is

where you multiple two predictors together to create a new interaction

term in the model.)

In textbooks and published articles, you may see regression models written in

various ways:»

» ^{A collection of predictor variables may be designated by a subscripted}

variable and the corresponding coefficients by another subscripted variable,

like this: Y

b

b X

b X

0

1

1

2

2^.»

» ^{In practical research work, the variables are often given meaningful names,}

like Age, Gender, Height, Weight, Glucose, and so on.»

» ^{Linear models may be represented in a shorthand notation that shows only}

the variables, and not the parameters, like this: Y = X + Z + X * Z instead of Y = a

+ bX + cZ + dX * Z or Y = 0 + X + Z + X * Z to specify that the model has no

intercept. And sometimes you’ll see a “~” instead of the “=”. If you do, read the

“~” as “is a function of,” or “is predicted by.”

Being aware of how the calculations work

Fitting a linear multiple regression model essentially involves creating a set of

simultaneous equations, one for each parameter in the model. The equations

involve the parameters from the model and the sums of various products of the

dependent and independent variables. This is also true of the simultaneous

equations for the straight-line regression in Chapter 16, which involve estimating

the slope and intercept of the straight line and the sums of X Y X

, ,

2, and XY. Your

statistical software solves these simultaneous equations to obtain the parameter

values, just as is done in straight-line regression, except now, there are more

equations to solve. In multiple as in straight-line regression, you can also get the

information you need to estimate the standard errors (SEs) of the parameters.