Role of effect size variance in Precision
The third
element determining precision is the dispersion of the effect size index.
For t-tests, dispersion is indexed by the standard deviation of the group
means. If we will be reporting precision using the metric of the original
scores, then precision will vary as a function of the SD. (If we will
be reporting precision using a standard index, then the SD is assumed
to be 1.0 and so the SD of the original metric is irrelevant.) For tests
of proportions the variance of the index is a function of the proportions.
Variance is highest for proportions near .50 and lower for proportions
near 0.0 or 1.0. As a practical matter, variance is fairly stable until
proportions fall below .10 or above .90). For tests of correlations the
variance of the index is a function of the correlation. Variance is highest
when the correlation is zero.
Role of effect size in Precision
Effect size,
which is a primary factor in computation of power, has little (if any)
impact in determining precision. In the running example we would report
a 20 point effect with a 95% confidence interval of plus/minus some 13
points. A 30 point effect would similarly be reported with a 95% confidence
interval of plus/minus some 13 points.
While effect
size plays no direct role in precision, it may be related to precision
indirectly. Specifically, for procedures that work with mean differences,
the effect size is a function of the mean difference and also the SD within
groups. The former has no impact on precision; the latter affects both
effect size and precision (a smaller SD yields higher power and better
precision in the raw metric). For procedures that work with proportions
or correlations the absolute value of the proportion or correlation affects
the index's variance, which in turn may have an impact on precision.
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