New Statistical Algorithms for the Analysis of Mass - FU Berlin, FB MI ...

74 CHAPTER 4. (BIO-)MEDICAL APPLICATIONS
meaning to the target audience, or when results from multiple studies of an
area are being combined.
To give an example for a standardized measure let us look at Cohen’s d
(Cohen, 1988) which is the difference between two means divided by the pooled
standard deviation for those means:
d = mean1 − mean2
�
(SD2 1 + SD2 2)/2
where meani and SDi are the mean and standard deviation for group i = 1, 2.
Note that sample size does not play a part in the calculation that d is
heavily influenced by the denominator in the equation. This is incorporated
in (Hedges and Olkin, 1985)’s ˆg:
ˆg =
¯x1 − ¯x2
� (n1−1)·SD 2 1 +(n2−1)·SD 2 2
((n1+n2)−2)
�
�
3
· 1 −
.
4 · (n1 + n2) − 9
Note that both values can also be computed from the value of the t test of
differences between the two groups (Rosenthal and Rosnow, 1991; Rosnow and
Rosenthal, 1996).
Replication & Bias Estimation
“If science is the business of discovering replicable effects, because
statistical significance tests do not evaluate result replicability,
then researchers should use and report some strategies that do
evaluate the replicability of their results.” (Thompson, 1995)
Empirical evidence for result replicability can be gained either external
or internal (Thompson, 1995). While the external replication involves the
creation of a new sample measured at a different time and/or different location,
internal replicability is based on comparing recombinations of sub-samples of
the full sample at hand. An internal replication is performed to estimate the
bias (also called imprecision or random error) of the analytical method. It
helps to understand the precision of some sample statistics, such as median,
variance or percentiles. Methods of measurements are almost always subject
to some random variation by using subsets of (jackknife) or drawing randomly
with replacement from (bootstrapping) the available data (samples). The
following paragraphs introduce these techniques.
Jackknife (Tukey, 1958): The jackknife is a simple method for approximation
of the bias and variance of an estimator. Basically, the jackknife approach
partitions out the impact of a particular subset of the data on an estimate derived
from the total sample. In other words, jackknife tries to control for a
piece of the sample which may be exerting too much influence on your results
due to sampling error. This is done by systematically recomputing the desired
statistical estimate leaving out one observation at a time from the sample
set. The Jackknife is less general than bootstrap (see below) but easier to apply
to complex sampling schemes, such as multi-stage sampling with varying
sampling weights. The following introduces the calculation of the jackknife
estimator: