Data Analysis with Microsoft Excel: Updated for Office 2007

410 Statistical Methods

Two-Way Analysis of Variance

One-way analysis of variance compares several groups corresponding to a

single categorical variable, or factor. A two-way analysis of variance uses

two factors. In agriculture, for example, you might be interested in the ef-

fects of both potassium and nitrogen on the growth of potatoes. In medicine

you might want to study the effects of medication and dose on the duration

of headaches. In education you might want to study the effects of grade level

and gender on the time required to learn a skill. A marketing experiment

might consider the effects of advertising dollars and advertising medium

(television, magazines, and so on) on sales.

Recall that earlier in the chapter you looked at the means model for a

one-way analysis of variance. Two-way analysis of variance can also be ex-

pressed as a means model:

yijk5mij1eijk

where y is the response variable and mij is the mean for the ith level of one

factor and the jth level of the second factor. Within each combination of the

two factors, you might have multiple observations called replicates. Here

eijk is the error for the ith level of the fi rst factor, the jth level of the second

factor, and the kth replicate, following a normal distribution with mean 0

and variance s^2.

The model is more commonly presented as an effects model where

yijk5m1ai 1 bj1abij1eijk

Here y is the response variable, m is the overall mean, ai is the effect of

the ith treatment for the fi rst factor, and bj is the effect of the jth treatment

for the second factor. The term abij represents the interaction between the

two factors, that is, the effect that the two factors have on each other. For

example, in an experiment where the two factors are advertising dollars and

advertising medium, the effect of an increase in sales might be the same

regardless of what advertising medium (radio, newspaper, or television) is

used, or it might vary depending on the medium. When the increase is the

same regardless of the medium, the interaction is 0; otherwise, there is an

interaction between advertising dollars and medium.

A Two-Factor Example

To see how different factors affect the value of a response variable, con-

sider an example of the effects of four different assembly lines (A, B, C,

or D) and two shifts (a.m. or p.m.) on the production of microwave ovens