In re: High-Tech Employee Antitrust Litigation

Filing 716

Omnibus Declaration of Christina J. Brown in Support of #715 Reply re Joint Motion to Exclude the Expert Testimony of Edward E. Leamer, Ph.D. , #714 Reply to Joint Motion to Strike the Improper Rebuttal Testimony in Dr. Leamer's Reply Expert Report or, in the Alternative, MOTION for Leave to Submit a Reply Report of Dr. Stiroh filed by Apple Inc.. (Attachments: #1 Exhibit A, #2 Exhibit B, #3 Exhibit C, #4 Exhibit D, #5 Exhibit E, #6 Exhibit F, #7 Exhibit G, #8 Exhibit H, #9 Exhibit I, #10 Exhibit J, #11 Exhibit K, #12 Exhibit L, #13 Exhibit M, #14 Exhibit N, #15 Exhibit O, #16 Exhibit P, #17 Exhibit Q)(Related document(s) #715 , #714 ) (Brown, Christina) (Filed on 2/27/2014) Modified text on 2/28/2014 (dhmS, COURT STAFF).

EXHIBIT O

OMNIBUS BROWN DECLARATION

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HYPOTHESIS TESTS

7.6. Power and Size of Tests
The remaining sections of this chapter study two limitations in using the usual computer output to test hypotheses.
First, a test can have little ability to discriminate between the null and alternative
hypotheses. Then the test has low power, meaning there is a low probability of rejecting
the null hypothesis when it is false. Standard computer output does not calculate test
power, but it can be evaluated using asymptotic methods (see this section) or finitesample Monte Carlo methods (see Section 7.7). If a major contribution of an empirical
paper is the rejection or nonrejection of a particular hypothesis, there is no reason for
the paper not to additionally present the power of the test against some meaningful
alternative hypothesis.
Second, the true size of the test may differ substantially from the nominal size of
the test obtained from asymptotic theory. The rule of thumb that sample size N > 30
is sufficient for asymptotic theory to provide a good approximation for inference on a
single variable does not extend to models with regressors. Poor approximation is most
likely in the tails of the approximating distribution, but the tails are used to obtain
critical values of tests at common significance levels such as 5%. In practice the critical
value for a test statistic obtained from large-sample approximation is often smaller
than the correct critical value based on the unknown true distribution. Small-sample
refinements are attempts to get closer to the exact critical value. For linear regression
under normality exact critical values can be obtained, using the t rather than z and the
F rather than χ 2 distribution, but similar results are not exact for nonlinear regression.
Instead, small-sample refinements may be obtained through Monte Carlo methods (see
Section 7.7) or by use of the bootstrap (see Section 7.8 and Chapter 11).
With modern computers it is relatively easy to correct the size and investigate the
power of tests used in an applied study. We present this neglected topic in some
detail.

7.6.1. Test Size and Power
Hypothesis tests lead to either rejection or nonrejection of the null hypothesis. Correct
decisions are made if H0 is rejected when H0 is false or if H0 is not rejected when H0
is true.
There are also two possible incorrect decisions: (1) rejecting H0 when H0 is true,
called a type I error, and (2) nonrejection of H0 when H0 is false, called a type II
error. Ideally the probabilities of both errors will be low, but in practice decreasing
the probability of one type of error comes at the expense of increasing the probability
of the other. The classical hypothesis testing solution is to fix the probability of a type
I error at a particular level, usually 0.05, while leaving the probability of a type II error
unspecified.
Define the size of a test or significance level
α = Pr type I error
= Pr reject H0 |H0 true ,
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(7.43)

7.6. POWER AND SIZE OF TESTS

with common choices of α being 0.01, 0.05, or 0.10. A hypothesis is rejected if the test
statistic falls into a rejection region defined so that the test significance level equals the
specified value of α. A closely related equivalent method computes the p-value of a
test, the marginal significance level at which the null hypothesis is just rejected, and
rejects H0 if the p-value is less than the specified value of α. Both methods require only
knowledge of the distribution of the test statistic under the null hypothesis, presented
in Section 7.2 for the Wald test statistic.
Consideration should also be given to the probability of a type II error. The power
of a test is defined to be
Power = Pr reject H0 |Ha true
= 1 − Pr accept H0 |Ha true
= 1 − Pr Type II error .

(7.44)

Ideally, test power is close to one since then the probability of a type II error is close to
zero. Determining the power requires knowledge of the distribution of the test statistic
under Ha .
Analysis of test power is typically ignored in empirical work, except that test procedures are usually chosen to be ones that are known theoretically to have power that, for
given level α, is high relative to other alternative test statistics. Ideally, the uniformly
most powerful (UMP) test is used. This is the test that has the greatest power, for given
level α, for all alternative hypotheses. UMP tests do exist when testing a simple null
hypothesis against a simple alternative hypothesis. Then the Neyman–Pearson lemma
gives the result that the UMP test is a function of the likelihood ratio. For more general testing situations involving composite hypotheses there is usually no UMP test,
and further restrictions are placed such as UMP one-sided tests. In practice, power
considerations are left to theoretical econometricians who use theory and simulations
applied to various testing procedures to suggest which testing procedures are the most
powerful.
It is nonetheless possible to determine test power in any given application. In the
following we detail how to compute the asymptotic power of the Wald test, which
equals that of the LR and LM tests in the fully parametric case.

7.6.2. Local Alternative Hypotheses
Since power is the probability of rejecting H0 when Ha is true, the computation
of power requires obtaining the distribution of the test statistic under the alternative hypothesis. For a Wald chi-square test at significance level α the power equals
2
Pr[W> χα (h)|Ha ]. Calculation of this probability requires specification of a particular
alternative hypothesis, because Ha : h(θ) = 0 is very broad.
The obvious choice is the fixed alternative h(θ) = δ, where δ is an h × 1 finite
vector of nonzero constants. The quantity δ is sometimes referred to as the hypothesis error, and larger hypothesis errors lead to greater power. For a fixed alternative
the Wald test statistic asymptotically has power one as it rejects the null hypothesis
all the time. To see this note that if h(θ) = δ then the Wald test statistic becomes
247