Statistical Methods for Psychology

where and are taken to be heterogeneous variances. As noted above, the expression

that I have just denoted as is notnecessarily distributed as t on n 11 n 22 2 df. If we knew

what the sampling distribution of actually looked like, there would be no problem. We

would just evaluate against that sampling distribution. Fortunately, although there is no

universal agreement, we know at least the approximate distribution of.

The Sampling Distribution of t‘

One of the first attempts to find the exact sampling distribution of was begun by Behrens

and extended by Fisher, and the general problem of heterogeneity of variance has come to

be known as the Behrens–Fisher problem.Based on this work, the Behrens–Fisher distri-

bution of was derived and is presented in a table in Fisher and Yates (1953). However,

because this table covers only a few degrees of freedom, it is not particularly useful for

most purposes.

An alternative solution was developed apparently independently by Welch (1938) and by

Satterthwaite (1946). The Welch–Satterthwaite solutionis particularly important because

we will refer back to it when we discuss the analysis of variance. Using this method, is

viewed as a legitimate member of the t distribution, but for an unknown number of degrees of

freedom. The problem then becomes one of solving for the appropriate df, denoted df:

The degrees of freedom (df) are then taken to the nearest integer.^13 The advantage of this

approach is that df is bounded by the smaller of n 12 1 and n 22 1 at one extreme and

n 11 n 2 – 2 dfat the other. More specifically,

In this book we will rely primarily on the Welch–Satterthwaite approximation. It has

the distinct advantage of applying easily to problems that arise in the analysis of variance,

and it is not noticeably more awkward than the other solutions.

Testing for Heterogeneity of Variance

How do we know whether we even have heterogeneity of variance to begin with? Since we

obviously do not know and (if we did, we would not be solving for t), we must in

some way test their difference by using our two sample variances ( and ).

A number of solutions have been put forth for testing for heterogeneity of variance. One

of the simpler ones was advocated by Levene (1960), who suggested replacing each value of X

either by its absolute deviation from the group mean—dij= ƒXij 2 Xjƒ—or by its squared

s^21 s^22

s^21 s^22

Min(n 12 1, n 22 1)...df¿.

¿

¿

df¿=

a

s^21

n 1

1

s^22

n 2

b

2

a

s^21

n 1

b

2

n 121

1

a

s^22

n 2

b

2

n 221

¿

t¿

t¿

t¿

t¿

t¿

t¿

t¿

s^21 s^22

214 Chapter 7 Hypothesis Tests Applied to Means

(^13) Welch (1947) later suggested that it might be more accurate to write
df¿=G
a
s^21
n 1
1
s^22
n 2
b
2
a
s^21
n 1
b
2
n 1111
a
s^22
n 2
b
2
n 211
W 22
Behrens–Fisher
problem
Welch–
Satterthwaite
solution