The experimental design of postmortem studies: the effect size and statistical power
The experimental design of postmortem studies: the effect size and statistical power
Joris Meurs 0 1
0 Department of BioAnalytical Chemistry, VU University Amsterdam , De Boelelaan, 1081 HV Amsterdam , The Netherlands
1 & Joris Meurs
Purpose The aim is of this study was to show the poor statistical power of postmortem studies. Further, this study aimed to find an estimate of the effect size for postmortem studies in order to show the importance of this parameter. This can be an aid in performing power analysis to determine a minimal sample size. Methods GPower was used to perform calculations on sample size, effect size, and statistical power. The minimal significance (a) and statistical power (1 - b) were set at 0.05 and 0.80 respectively. Calculations were performed for two groups (Student's t-distribution) and multiple groups (one-way ANOVA; F-distribution). Results In this study, an average effect size of 0.46 was found (n = 22; SD = 0.30). Using this value to calculate the statistical power of another group of postmortem studies (n = 5) revealed that the average statistical power of these studies was poor (1 - b \ 0.80). Conclusion The probability of a type-II error in postmortem studies is considerable. In order to enhance statistical power of postmortem studies, power analysis should be performed in which the effect size found in this study can be used as a guideline.
Postmortem research; Sample size; Experimental design; Significance; Power; Effect size
Prior to conducting research, several considerations have to
be made. For example, the required sample size has to be
]. Commonly, this is done by performing a
so-called power analysis [
]. In a power analysis, the
sample size is calculated by using four parameters:
significance (a), statistical power (1 - b), variance (r2), and
effect size (d) [
]. A description and the effect on the
sample size of each of these parameters is shown in
Table 1. In order to emphasize the effect of a and 1 - b,
the confusion matrix is shown in Fig. 1. Despite a and
1 - b being mostly straightforward values, determining r2
and d is rather difficult [
]. In case two independent means
are present, Cohen set values of d at 0.20, 0.50, and 0.80
which represent a small, medium, or large effect size
]. The effect sizes in case multiple means
(multiple groups) are present have been set at 0.10, 0.25,
and 0.40, which represent a small, medium, or large effect
size respectively. According to Cohen, his set medium
value for d represents ‘‘an effect likely to be visible to the
naked eye’’ [
]. For instance, this can be a change in
decomposition stage of a cadaver. In quantitative research
this visible effect could be, for example, a significant
change in concentration of a certain analyte in a
postmortem sample. Nevertheless, for inexperienced
individuals it still remains unclear what the actual meaning of d is.
The effect size is defined as the absolute difference
between two independent means and the within-sample
standard deviation [
]. In other words, how much does a
certain situation (e.g., a qualitative or quantitative
experiment) differ from reality? Moreover, for calculating d
values the independent means (la; la) and the within-sample
standard deviation (r) have to be estimated . Hence, the
resulting d will be a rather subjective value. To solve this
problem, a pilot study can be performed and a sample
standard deviation can be used for calculating the effect
]. However, pilot studies lack statistical power .
Hence, performing a pilot study is not desirable.
It is observed that in postmortem research the sample
size is variable. For instance, the sample size can be as low
as nine [
] or as high as 57,903 [
]. Low availability of
samples or legal restrictions can be a reason for small
sample sizes. Although, parameters like the statistical
power should still be taken into account despite these
limitations. No discussion on the sample size used or the
statistical power reached is seen in most publications.
Hence, the probability is of false-negative results cannot be
derived from the data that is shown [
]. Therefore, the aim
of this paper is to show how a minimal sample size can be
estimated without a priori knowledge on the standard
deviation to ensure sufficient statistical power.
Furthermore, the poor statistical power of postmortem studies will
Calculation of the sample size in general cases
Two independent means (Student’s t test)
To calculate the sample size (n) in order to compare two
independent means, Eq. 1 has to be solved [
ðza=2 þ zbÞ d
where, z is the corresponding z score for values of a and b
and d is defined as the absolute difference between the
experimental mean (la) and the control mean (lb) (Eq. 2).
d ¼ jla
To calculate the z score, values for a were set at 0.05 and
0.01 respectively. Likewise, values for b were set at 0.20,
0.10, and 0.05 respectively. All obtained values are shown
below in matrix Z. Column 1 and 2 contain the values for
significance levels of 0.05 and 0.01. Values for b decrease
going down the rows.
2 a ¼ 0:05; b ¼ 0:20
Z ¼ 4 a ¼ 0:05; b ¼ 0:10
a ¼ 0:05; b ¼ 0:05
2 7:849 11:6790 3
¼ 4 10:5074 14:8794 5
a ¼ 0:01; b ¼ 0:20 3
a ¼ 0:01; b ¼ 0:10 5
a ¼ 0:01; b ¼ 0:05
According to Cohen, the effect size is considered as
small, medium, or large at values of 0.20, 0.50, and 0.80
]. Since r/d is inversely related to the effect
size, r/d-values of 5, 2, and 1.25 can be considered as
large, medium, and small respectively. Therefore, values
for the ratio r2/d2 were set from 0 to 5. With these values,
the corresponding sample size (n) was calculated
(Fig. 2).To obtain a reasonable estimate for the minimal
sample size, for all combinations of a and b the sample size
was calculated at the maximum ratio of r2/d2. These values
are shown in Table 2 and Fig. 3.
Multiple means (ANOVA)
In case of multiple means, the sample size should be
determined by using ANOVA. The effect size (f) is then
expressed as follows (Eq. 4) [
Accordingly, the total sample size is calculated by using
Eq. 5, in which N is the total sample size and k is the
noncentrality parameter [
]. This noncentrality
parameter is about 1.5 for a = 0.01 when b = 0.20 and
about 1 for a = 0.05 when b = 0.20 .
f ¼ r
N ¼ f 2
For the one-way ANOVA model, Cohen’s values of
0.10, 0.25, and 0.40 were used to calculate the minimal
sample size at significance levels of 0.05 and 0.01
respectively. These results are shown in Table 3 and Fig. 4.
Statistical power and effect size of postmortem studies
In order to show the poor statistical power of postmortem
studies, a number of studies were selected for post hoc
testing on the sample size in order to determine the
achieved power. For calculations GPower was used [
First, the effect size for a number of postmortem studies
(n = 22) was calculated. This data is shown in Table 4.
Significance level and statistical power were set at 0.05 and
0.80 respectively. A mean effect size of 0.46 (SD = 0.30)
k, group size; values are calculated in GPower [
Fig. 4 Influence of f and k on the sample size
This effect size was used to calculate the achieved
statistical power of another group of postmortem studies
(n = 5). A priori, the significance was set at 0.05. The
results are shown in Table 5. Only for the studies of Mao
et al. [
] and Laiho and Penttila¨ [
] was the achieved
statistical power sufficient (i.e., a value greater than 0.80).
In all other cases, the statistical power was less than 0.80,
which means there is a reasonable probability of a type-II
error. Despite these low power values, the risk of
falsenegative results are not discussed. An example of a
falsenegative result is that no significant difference is found in
concentration while in fact there is a significant difference.
In other words, the null hypothesis (H0) has been falsely
N (k = 3)
N (k = 4)
N (k = 5)
N (k = 8)
N (k = 10)
Achieved power was calculated using GPower [
]. Post hoc testing was performed using a one-way
ANOVA model with fixed effects
a Groups were not divided into equal numbers
Discussion and conclusion
Power analysis can be a useful tool in determining the
sample size needed for qualitative and quantitative
postmortem experiments. Examples of postmortem qualitative
and quantitative research are determining the degree of
] and measuring postmortem vitreous
]. However, in order to calculate the sample
size, values have to be set subjectively.
That can be a cause of choosing a random sample size in
postmortem research. Sample size determination and
achieved statistical power are rarely discussed in
postmortem studies. However, it is important to discuss these
parameters in order to establish the reliability of the
This study is the first to demonstrate that postmortem
studies lack statistical power. In order to achieve sufficient
power, Tables 2 and 3 can be used for obtaining a minimal
sample size for common values of significance and
statistical power. However, it should always be checked a
posteriori if the set levels of power and significance are
achieved by performing a post hoc test. Nevertheless,
An effect size has been estimated for postmortem
The statistical power of postmortem studies is poor.
Power analysis should be performed in order to
enhance statistical power of postmortem studies.
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