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 H

OMNIBUS BROWN DECLARATION

© Jeremy Foster, Emma Barkus, Christian Yavorsky 2006
First published 2006
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6 • UNDERSTANDING AND USING ADVANCED STATISTICS

can be done in a number of ways including transforming the scores, using a
more stringent significance level (perhaps 0.01 rather than 0.05), applying a
non-parametric procedure.

Statistical significance
Probability testing is at the centre of statistical analysis and is essentially concerned with deciding how probable it is that the results observed could have been
due to chance or error variation in the scores. To make the explanation simpler,
we shall take the case of testing to see whether there is a difference between two
groups of respondents. Suppose we have measured the amount of concern people
have with their body image using a questionnaire in which a high score indicates
a high level of concern, and done this for a group of women and for a group of
men. The null hypothesis states that there is no difference between the scores of
the two groups. The research (or alternative) hypothesis states that there is a
difference between the scores of the two groups. The research hypothesis may
predict the direction of the outcome (e.g. women will have a higher score than
the men) in which case it is a directional or one-tailed hypothesis. Or the research
hypothesis may just predict a difference between the two groups, without specifying which direction that difference will take (e.g. women and men will score
differently from one another) in which case it is referred to as a non-directional
or two-tailed hypothesis.
In our example, we want to know how likely it is that the difference in the
mean scores of the women and men was the result of chance variation in the
scores. You will probably recognise this as a situation in which you would turn
to the t-test, and may remember that in the t-test you calculate the difference
between the means of the two groups and express it as a ratio of the standard
error of the difference which is calculated from the variance in the scores. If this
ratio is greater than a certain amount, which you can find from the table for t,
you can conclude that the difference between the means is unlikely to have
arisen from the chance variability in the data and that there is a ‘significant’
difference between the means.
It is conventional to accept that ‘unlikely’ means having a 5% (0.05) probability or less. So if the probability of the difference arising by chance is 0.05 or
less, you conclude it did not arise by chance. There are occasions when one uses
a more stringent probability or significance level and only accepts the difference
as significant if the probability of its arising by chance is 1% (0.01) or less. Much
more rarely, one may accept a less stringent probability level such as 10% (0.1).
In considering the level of significance which you are going to use, there are
two types of errors which need to be borne in mind. A Type I error occurs when a
researcher accepts the research hypothesis and incorrectly rejects the null hypothesis. A Type II error occurs when the null hypothesis is accepted and the research

Basic Features of Statistical Analysis and the GLM • 7

hypothesis is incorrectly rejected. When you use the 5% (0.05) significance
level, you have a 5% chance of making a Type I error. You can reduce this by
using a more stringent level such as 1% (0.01), but this increases the probability
of making a Type II error.
When a number of significance tests are applied to a set of data it is generally
considered necessary to apply some method of correcting for multiple testing. (If
you carried out 100 t-tests, 5% of them are expected to come out ‘significant’
just by chance. So multiple significance testing can lead to accepting outcomes as
significant when they are not, a Type I error.) To prevent the occurrence of a
Type I error some researchers simply set the significance level needed to be
reached at the 1% level, but this does seem rather arbitrary. A more precise
correction for multiple testing is the Bonferroni correction in which the 0.05
probability level is divided by the number of times the same test is being used
on the data set. For example, if four t-tests are being calculated on the same data
set then 0.05 would be divided by 4 which would give a probability level of
0.0125 which would have to be met to achieve statistical significance.

RECAPITULATION OF ANALYSIS OF VARIANCE (ANOVA)
ANOVAs, like t-tests, examine the differences between group means. However,
an ANOVA has greater flexibility than a t-test since the researcher is able to use
more than two groups in the analysis. Additionally ANOVAs are able to consider
the effect of more than one independent variable and to reveal whether the effect
of one of them is influenced by the other: whether they interact.
Variance summarises the degree to which each score differs from the mean,
and as implied by the name ANOVAs consider the amount of variance in a data
set. Variance potentially arises from three sources: individual differences, error
and the effect of the independent variable. The sources of variance in a set of data
are expressed in the sum of squares value in an ANOVA. In a between-groups
analysis of variance, the between-groups sum of squares represents the amount
of variance accounted for by the effect of the independent variable with the estimate of error removed. The sums of squares are divided by the corresponding
degrees of freedom to produce the mean square between groups and the mean
square within groups. The ratio between these two figures produces a value for
F. The further away from one the F value is, the lower the probability that the
differences between the groups arose by chance and the higher the level of
significance of the effect of the independent variable.
Repeated measures ANOVAs occur when the participants complete the same
set of tasks under different conditions or in longitudinal studies where the same
tasks are completed by the same participants on more than one occasion. There
are additional requirements which need to be met in order to ensure that the