• determining whether the t in (6.23) or (6.24)

is distributed as a Student™s t random variable

Figure 6.11: The reject rate percentage of the

under the null hypothesis. When n X and n Y

one-sample t test when the observations are auto-

are small (n X + n Y < 30 if t is computed

correlated (see text). The ˜equivalent sample size™

with (6.23); n X < 30 and n Y < 30 if t

n is given by (6.26) (thin curve) and is estimated

is computed with (6.24)), the distribution of

with (6.26) (thick curve).

t deviates markedly from any t distribution

[363]. Thus the t test can not be used with

small equivalent sample sizes. An alternative We conducted a Monte Carlo experiment (cf.

is described in [6.6.9]. [6.3.1]) with a one-sample t test to examine

how well it works when the equivalent sample

While the discussion above has focused on the

size n is determined by (6.26). Time series of

difference of means test, the same considerations

length n = 15, 30, 60 . . . were generated by an

apply to one-sample t tests such as the paired

auto-regressive process of ¬rst order with ± =

difference test (cf. [6.6.6]).

0.6 (see Chapter 10). Such processes have auto-

correlation functions of the form ρ(k) = ± |k| . If

6.6.8 The De¬nition and Estimation of the

we insert the equivalent sample size, as de¬ned

Equivalent Sample Size. Let us assume that the

by (6.26), into (6.22) and use a signi¬cance level

data are given with constant time steps δ, such that

of 5%, we observe fewer rejections of the true

the ith sample xi is taken at time t = iδ. Then the

null hypothesis ˜H0 : µ = 0™ (Figure 6.11) than

variance of the sample mean is

expected. The deviation from the nominal 5%

Var X = σ 2 /n X , (6.25) level is considerable when n is less than 30. This

happens because the distribution of t is not well

where

approximated by the distribution of t (n ) under H0 .

nX

nX = , (6.26) Estimates of n can be obtained either from

n X ’1

1+ 1’ ρ X (k)

k

physical reasoning or by means of a statistical

k=1 nX

estimator. Estimates based on physical reasoning

(see Section 17.1) and ρ X (k) is the auto-

should state lower bounds for n because optimistic

correlation function

estimates will result in t -values that are frequently

1 too large and, consequently, cause more frequent

ρ X (k) = 2 Cov(Xi , Xi+k )

σ rejection of the null hypothesis when it is true than

indicated by the signi¬cance level.

(see Section 11.1). We will drop the subscript

˜X ™ for notational convenience in the rest of this Statistical estimators of n use estimates of the

auto-correlation function ρ(k) in combination with

subsection.

6: The Statistical Test of a Hypothesis

116

where x and y are sample means and S 2 is the

(6.26). Various reasonable estimators of n [363, p

454] result in t tests that tend to reject H0 more pooled sample variance (6.17). Compute the

pooled sample lag-1 correlation coef¬cient ±

frequently than speci¬ed by the signi¬cance level.

Estimation is discussed further in [17.1.3]. using

The Monte Carlo experiment described above

nX nY

+

was repeated using this best estimator n in place i=2 xi xi’1 i=2 yi yi’1

±= , (6.30)

of the known equivalent sample size n (6.26). (n X + n Y ’ 2)S 2

p

Figure 6.11 shows that the test now rejects the true

null hypothesis more frequently than speci¬ed by

where xi = xi ’ µ X and yi = yi ’ µY .

the signi¬cance level of 5%. We therefore suggest

Use Appendix H to determine the critical

that the t test not be used with equivalent sample

value of t that is appropriate for a sample of

sizes smaller than 30. Instead, we advise the use

size n X + n Y , which has a lag-1 correlation

of ˜Table-Look-Up test,™ described next, in such

coef¬cient ±.

predicaments.

6.6.10 The Hotelling T 2 test. The multivariate

6.6.9 The ˜Table-Look-Up Test.™ The ˜Table-

version of the t test, which is used to test the null

Look-Up test™ [454] is a small sample alternative

hypothesis

to the conventional t test that avoids the dif¬culties

of estimating an equivalent sample size while

H0 : µ X = µ Y , (6.31)

remaining as ef¬cient as the optimal asymptotic

test when equivalent sample sizes are large.12 is called the Hotelling T 2 test. The assumptions

The Table-Look-Up test procedure is as follows. implicit in this parametric test are identical to

those required for the t test except that they

• The paired difference (or one sample) case: to

apply to vector, rather than scalar, realizations

test ˜H0 : µ = µ0 ™ using a sample of size n X

of an experiment. It is necessary to make the

compute

sampling assumption that the realizations of the

(x ’ µ0 ) m-dimensional random vectors X and Y occur

t= √, (6.27) independently of each other. It is also necessary

SX / n X

to make similar distributional assumptions: that

all observations in a sample come from the

2

where x is the sample mean and S X is the

same distribution and that those distributions are

sample variance.

multivariate normal. In addition, we also assume

Compute the sample lag-1 correlation coef¬- that both X and Y have the same covariance matrix

cient ± X using Σ, so that X ∼ N (µ X , Σ) and Y ∼ N (µY , Σ).

The covariance matrix Σ = (σi j ) is generally

nX

i=1 xi xi’1

±X = not known and must be estimated from the data in

(6.28)

(n x ’ 1)S X

2

a manner analogous to (6.17):

nX nY

k xik x jk +

where xi = xi ’ µ X . Use Appendix H k yik y jk

σij = , (6.32)

to determine the critical value for t that is n X + nY ’ 2

appropriate for a sample of size n with lag-1

where x jk = x jk ’ µ X j and y jk = y jk ’ µY j .

correlation coef¬cient ± X .

The optimal test statistic is given by

• The two sample case (assuming σ X = σY

n X + nY ’ m ’ 1 1

and that lag-1 correlation ± X = ±Y ): to test 1

T2 = +

˜H0 : µ y = µx ™ using X and Y samples of m(n X + n Y ’ 2) n X nY

size n X and n Y respectively, compute ’1

— (µ X ’ µY )T Σ (µ X ’ µY ). (6.33)

x’y

t= , (6.29) This statistic measures the distance in m space

S p n X + nY

1 1

between the sample mean vectors µ X and µY in

dimensionless units. Note the similarity to the t

12 The Table-Look-Up test assumes that the sample(s) comes