Determining the Power of the Test

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195

D - G∑ D J

∑

j =1

n -1

(6)

and we reject H0 if t0 > tα

or if t0 < tα

. The confidence level, α, is typically taken to be 0.05

2,n-1

and is the type I error, or the probability of rejecting H0 when H0 is true. If H0 is rejected, we conclude

that the fixed monitor system does not represent the water quality of the stream at the α confidence level.

If H0 is not rejected, we conclude that "we have not found sufficient evidence to reject H0" (Hines and

Montgomery 1980). This may be because the monitor site accurately represents the stream or because the

sample size (that is, the number of comparisons) is so small that not enough data are available to make the

stronger conclusion to reject H0. So, for verification, we need a large enough sample size to minimize the

type II error (that is, the probability of accepting H0 when H0 is false).

Similarly, one-sided hypotheses can be tested as follows:

H0:D ≤ 0, H1:D > 0, reject H0 if t0 > tα,n-1

(7)

H0:D ≥ 0, H1:D < 0, reject H0 if t0 < tα,n-1

(8)

Determining the Power of the Test

The rejection of the null hypothesis is considered a "strong" conclusion because we control

the type I error (choice of α), or the probability of rejecting H0 when H0 is true. On the other

hand, the acceptance of the null hypothesis is considered to be a "weak" conclusion, because we

do not control the type II error (β), or the probability of accepting H0 when H0 is false.

Thus, to determine the meaning of our conclusion when we accept the hypothesis that a

monitor represents the flow, we must determine the type II error. For the monitor location to be

acceptable, the type II error must be acceptably small.

To estimate the type II error, or β, a statistic d is calculated, and with α and n, β can be

determined from operating characteristic charts available in statistics books (Hines and

Montgomery 1980, p 604). Using Equations 5 and 6, we calculate d as follows:

(9)

Once β is found, the probability of correctly accepting H0 is the power, namely P = 1 β.

Because we want only to correctly accept H0, we desire the power to be as close to 1 as

possible. The question then becomes, What's good enough?

Since we typically choose α to be 0.05, it seems reasonable to attempt to hold β to a similar

probability. However, because we have no direct control over β, probabilities less than 0.2 are

probably sufficient. Thus, we consider comparisons with the power greater than 0.8 to be

acceptable.

Water Quality Technical Note AM-03 (January 1998)