Statistical Methods for Psychology

spaced values of the independent variable. The example will be simpler statistically if the

units on the abscissa are evenly spaced.) The group means using my composite measure on

the abscissa are plotted in Figure 12.3, where you can see that the rated attractiveness does

increase with increasing levels of Composite.

The overall analysis of variance really asked if a horizontal straight line through Y 5 2.917

(the grand mean) would fit the data adequately. The Fled to rejection of that null hypothesis

because several means were far from 2.917. Our next question asks whether a nonhorizontal

straight line provides a good fit to the data. A glance at Figure 12.3 would suggest that this is

the case. We will then follow that question by asking whether systematic residual (nonerror)

variance remains in the data after fitting a linear function, and, if so, whether this residual vari-

ance can be explained by a quadratic function.

To run a trend analysis, we will return to the material we discussed under the headings

of linear and orthogonal contrasts. (Don’t be confused by the use of the word linearin the

last sentence. We will use the same approach when it comes to fitting a quadratic function.

Linear in this sense simply means that we will form a linear combination of coefficients

and means, where nothing is raised to a power.)

In Section 12.3 we defined a linear contrast as

The only difference between what we are doing here and what we did earlier will be in

the coefficients we use. In the case in which there are equal numbers of subjects in the

groups and the values on the abscissa are equally spaced, the coefficients for linear, quad-

ratic, and higher-order functions (polynomial trend coefficients) are easily tabled and are

found in Appendix Polynomial. From Appendix Polynomial we find that for five groups

the linear and quadratic coefficients are

Linear: 22 21012

Quadratic:2 21 22 212

We will not be using the cubic and quartic coefficients shown in the appendix, but their

use will be evident from what follows. Notice that like any set of orthogonal linear coeffi-

cients, the requirements that 5 0 and are met. The coefficients do not form a

“standard set,” because the sum of the absolute values of the coefficients does not equal 2.

That is not a problem here.

As you should recall from Section 12.3, we calculate a sum of squares for the contrast as

SScontrast=

nc^2

aa

2

j

gaj gaibj= 0

c=a 1 X 11 a 2 X 21 a 3 X 3 1 Á 1 akXk=aajXj

12.10 Trend Analysis 405

1 2 3 4 5

Composite

Mean

3.4

3.2

3.0

2.8

2.6

Figure 12.3 Scatterplot of mean versus composite group

polynomial trend
coefficients