COMPOSITE VARIABLES
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Supplemental Digital Content 1
The derivation of power calculation formula for composite score method
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Let
denote the effect size of
linear model E(
, where
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that are standardized by
and
in the
and
with least square estimators
and
. The power
calculation formula used by Dupont and Plummer (1998) is
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(A1)
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where
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denote its (
is the cumulative probability of the distribution with
) percentile with right tail probability
degrees of freedom and
. However,
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data and may not be useful in the study design stage. One may replace
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standard deviation
of
and
The variance
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since
can be shown as
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in
by
can further be shown as
and
. On the other hand,
with covariance
Accordingly, the effect size
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and
of , respectively, where
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in (A1) depends on
.
can be approximated by
, and the following power calculation formula can
be used in the research design:
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where
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percentile at
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The derivation of power calculation formula for Bonferroni correction
is the cumulative probability of the standard normal distribution with (
.
)
COMPOSITE VARIABLES
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Let
denote the correlation matrix between
correlation vector between
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and
Let
, and let
denote the
denote the bivariate
element of the vector
be a column vector of standardized covariates from
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and
. Let
denote the vector of the least square estimator for the linear model
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Given
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to
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method, if we define the effect size for
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finite sample variance of the least square estimator
when the sample size
is large. Similarly as
is
, which is close
defined in the composite score
as
then one can express
as
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by the fact that
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standardization, one can write down the power calculation formula for
. Since a p-value for individual
is not affected by
as
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The power for multiple testing using Bonferroni correction is therefore
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with
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50, and
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. That means,
for each . For example, if
, then
and
statistical power equals 0.0876 or 8.76%.
, with a sample size n =
and
for each
under Bonferroni correction. Accordingly, the
COMPOSITE VARIABLES
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Reference
Dupont, W.D., & Plummer, W.D.Jr. (1998). Power and sample size calculations for studies
involving linear regression. Controlled Clinical Trials, 19(6), 589-601.