Black Sea Journal ofEngineering and Science 2(1): 16-22 (2019)
BSJ Eng. Sci. / Aysel YENİPINAR et al. 16
Black Sea Journal of Engineering and Science
Open Access Journal
e-ISSN: 2619-8991
Research Article
Volume 2 - Issue 1: 16-22 / January 2019
DETERMINING SAMPLE SIZE IN LOGISTIC
REGRESSION WITH G-POWER
Aysel YENİPINAR
1
*, Şeyma KOÇ
1
, Demet ÇANGA
2
, Fahrettin KAYA
3
1
Kahramanmaraş Sütçü Imam University, Faculty of Agriculture, Department of Animal Science, 46040,
Onikişubat, Kahramanmaraş, Turkey
2
Osmaniye Korkut Ata University, Department of Food Processing, 80050, Bahçe, Osmaniye, Turkey
3
Kahramanmaraş Sütçü Imam University, Andırın Vocational School, Computer Technology, Department of
Animal Science, 46410, Onikişubat, Kahramanmaraş, Turkey
Received: October 11, 2018; Accepted: November 23, 2018; Published: January 01, 2019
Abstract
There are several methods used to determine the sample size. Investigator; because of the insufficient precious
resources such as time, labor, money, tools and equipment, it works by pulling the sample with a suitable sampling
method from the population it is examining. According to the statistics obtained from the sample, he will make
comments about the population and make decisions. The correctness of the decisions made is closely related to the size
of the sample. For this reason, the problem of determining sample size is one of the first and important problems of an
investigator. A small sample of information causes loss of information and misjudgments. A very large sample is
contrary to the purpose of sampling and resources are wasted. The calculation of the sample size can now be done very
easily via free programs.
Keywords: Sample size determination, G-Power, Logistic regression
*Corresponding author: Kahramanmaraş Sütçü Imam University, Faculty of Agriculture, Department of Animal Science, 46040,
Onikişubat, Kahramanmaraş, Turkey
E mail: ayselyenipinar46@gmail.com (A. YENIPINAR)
Aysel YENİPINAR
https://orcid.org/0000-0001-2345-6789
Şeyma KOÇ
https://orcid.org/0000-0001-5708-9905
Demet ÇANGA
https://orcid.org/0000-0003-3319-7084
Fahrettin KAYA
https://orcid.org/0000-0003-1666-4859
Cite as: Yenipinar A, Koc S, Canga D, Kaya F. 2019. Determining sample size in logistic regression with G-power. BSJ Eng Sci, 2(1): 16-
22.
1. Introduction
The general problem of many researchers is the problem
of determining sample size in their studies. In addition to
the fact that logistic regression analysis is a difficult
method of analysis, there is no common acceptance in
determining sample size. Different views and formulas
were developed by the authors to determine the sample
size in logistic regression analysis. Hsieh et al. (1998)
proposes to compare averages or compare ratios to
calculate a simple sample size for linear and logistic
regression. Moineddin et al. (2007) conducted a
simulation study of sample size for multi-level logistic
regression models. In the study, it was determined that
the sample size of the individual and group level,
multilevel logistic regression models parameters and
BSPublishers
Black Sea Journal of Engineering and Science
BSJ Eng. Sci. / Aysel YENİPINAR et al. 17
variance components to evaluate the effect on the
accuracy of estimates. Demidenko (2007) derived the
general Wald-based power and sample size formulas for
Logistic regression in the sample size determination
study for the revised logistic regression and then applied
them to binary exposure and agitator to obtain closed
form expression. Herrera and Gomez (2008) conducted a
Monte Carlo simulation study. They examined the
different sample sizes of reference and focal groups in
their studies and determined the effect of logistic
regression on power and type I error. In their logistic
regression analysis, Nemes et al. (2009) reported that the
odds ratio was more predicted in small to medium sample
size studies, whereas small sample size was systematic
bias and deviation from zero.
Different package programs have been produced to
determine sample size with the help of the findings and
opinions of the researchers in logistic regression analysis.
G* Power is one of the programs that calculations of the
sample size. It is a program of 17 Mb sizes, easy to install
and use. G*Power (Erdfelder et al., 1996) was designed as
a general stand-alone power analysis program for
statistical tests commonly used in social and behavioral
research. However, it can also be used for science and
medicine.
The purpose of this article is to introduce the researchers
to the calculation of the sample size problem for logistic
regression analysis. The results of logistic regression
analysis can be interpreted easily for medical data as it is
easy to interpret for binary variables (yes-no, not- none
or limit values up- down, etc.). Cost, labor and time are
very important for the researcher when calculating
sample size. These factors are becoming more important,
especially because of the high cost in the field of medicine
and the difficulty in the application of some medical
procedures according to the ethical rules. In logistic
regression analysis, G-power can be used as an
alternative to complex formulas and simulation
applications in determining sample size and it is a
program that everyone can easily apply to solve the
problem of determining sample size.
2. Material and Method
2.1. Material
In this study, a numerical example is used to understand
the use of G power. In our example, 45% male X = 1, 55%
female and X = 0.70% of the girls and 80% of the boys
were taken as normal birth weight. These numerical
sample values were obtained randomly from the births
of Kahramanmaraş Maternity Hospital between January-
June 2018.Probability of error (significance level) α =
0.05 and the power of the test was taken as 1-β =
0.95.Impact magnitude (calculated in one way or look at
previous work).
2.2. Method
G* Power is a major extension of, and improvement over,
the previous versions. It runs on widely used computer
platforms (Windows XP, Windows Vista, and Mac OS X
10.4) and covers many different statistical tests of the t,
F, and chi2 test families. In addition, it includes power
analyses for z tests and some exact tests. G*Power
provides improved effect size calculators and graphic
options, supports both distribution-based and design-
based input modes, and offers all types of power
analyses in which users might be interested.
G*Power (Figure 1 shows the main window of the
program) covers statistical power analyses for many
different statistical tests of the F test, t test, χ2-test and z
test families and some exact tests. G*Power provides
effect size calculators and graphics options. G*Power
supports both a distribution-based and a design-based
input mode. It contains also a calculator that supports
many central and noncentral probability distributions.
G*Power is free software and available for Mac OS X and
Windows XP/Vista/7/8.
Figure 1. The main window of G*Power
Types of analysis; G*Power offers five different types of
statistical power analysis:
1. A priori (sample size N is computed as a function of
power level 1−β, significance level α, and the
detected population effect size)
2. Compromise (both α and 1−β are computed as
functions of effect size, N, and an error probability
ratio q = β/α)
3. Criterion (α and the associated decision criterion
Black Sea Journal of Engineering and Science
BSJ Eng. Sci. / Aysel YENİPINAR et al. 18
are computed as a function of 1−β, the effect size,
and N)
4. Post-hoc (1−β is computed as a function of α, the
population effect size, and N) 5. Sensitivity
(population effect size is computed as a function of
α, 1−β, and N).
Program handling; Perform a Power Analysis Using
G*Power typically involves the following three steps:
1. Select the statistical test appropriate for your
problem.
2. Choose one of the five types of power analysis is
available
3. Provide the input parameters required for the
analysis and click "Calculate".
Plot parameters; In order to help you explore the
parameter space relevant to your power analysis, one
parameter (α, power (1−β), effect size, or sample size)
can be plotted as a function of another parameter.
A logistic regression model describes the relationship
between a binary response variable Y (with Y = 0 and Y =
1 denoting non-occurance and occurance of an event,
respectively) and one or more independent variables
(covariates or predictors) Xi. The variables Xi are
themselves random variables with probability density
function f X(x) (or probability distribution f X(x) for
discrete X).
In a simple logistic regression with one covariate X the
assumption is that the probability of an event P = Pr (Y =
1) depends on X in the following way:
)exp(1
)exp(
10
10
xββ
xββ
P
(1)
For β
1
≠ 0 and continuous X this formula describes a
smooth S-shaped transition of the probability for Y = 1
from 0 to 1 (β
1
> 0) or from 1 to 0 (β
1
< 0) with increasing
x. This transition gets steeper with increasing β
1
.
Rearranging the formula leads to: log(P/(1−P)) = β
0
+
β
1
X. This shows that the logarithm of the odds P/(1−P),
also called a logit, on the left side of the equation is linear
in X. Here, β
1
is the slope of this linear relationship. The
interesting question is whether covariate X
i
is related to
Y or not. Thus, in a simple logistic regression model, the
null and alternative hypothesis for a two-sided test are:
H
0
: β
1
= 0
H
1
: β
1
≠0 (2)
The procedures implemented in G*Power for this case
estimates the power of the Wald test. The standard
normally distributed test statistic of the Wald test is:
z=
=
(3)
where
1
ˆ
β
is the maximum likelihood estimator for
parameter β
1
and var(
1
ˆ
β
) the variance of this estimate.
2.2.1. Effect size index
In the simple logistic model the effect of X on Y is given
by the size of the parameter β
1
. Let p
1
denote the
probability of an event under H
0
, that is
exp(β
0
) = p
1
/(1−p
1
) (4)
and p
2
the probability of an event under H
1
at X=1, that is
exp(β
0
+ β
1
) = p
2
/(1−p
2
). (5)
Then
exp(β
0
+ β
1
)/exp(β
0
)
= exp(β
1
) = [p
2
/(1− p
2
)]/[p
1
/(1− p
1
)]
= odds ratio OR, (6)
which implies
β
1
= log[OR] (Şahin, 1999) . (7)
Given the probability p
1
(input field Pr(Y=1|X=1) H
0
) the
effect size is specified either directly by p
2
(input field
Pr(Y=1|X=1) H
1
) or optionally by the odds ratio (OR)
(input field Odds ratio). Setting p
2
= p
1
or equivalently OR
= 1 implies β
1
= 0 and thus an effect size of zero. An effect
size of zero must not be used in a priori analyses. Besides
these values the following additional inputs are needed.
(Faul et al., 2009).
R
2
other X:
In models with more than one covariate, the influence of
the other covariates X
2
,...,X
p
on the power of the test can
be taken into account by using a correction factor. This
factor depends on the proportion R
2
= ρ
2
1,2,3...p
of the
variance of X
1
explained by the regression relationship
with X
2
,...,X
p
. If N is the sample size considering X
1
alone,
then the sample size in a setting with additional
covariates is:
N| = N/(1−R
2
) (8)
This correction for the influence of other covariates has
been proposed by Hsieh et al. (1998). R
2
must lie in the
interval [0,1].
X distribution:
1 Bimonial
P(k) = (
)π
k
(1−π)
N−k
, (9)
where k is the number of successes ( X = 1) in N trials of a
Bernoulli process with probability of success π, 0 < π < 1
Black Sea Journal of Engineering and Science
BSJ Eng. Sci. / Aysel YENİPINAR et al. 19
2 Exponential
f(x) = (1/λ)e
−1/λ
, (10)
exponential distribution with parameter λ > 0.
3 Lognormal
f(x) = 1/(xσ√2π)exp[−(lnx − µ)
2
/(2σ
2
) (11)
lognormal distribution with parameters µ and σ> 0.
4 Normal
f(x) = 1/(σ√2π)exp[−(x − µ)
2
/(2σ
2
) (12)
normal distribution with parameters µ and σ> 0.
5 Poisson
(P(X = k) = (λk/k!)e
−λ
, (13)
Poisson distribution with parameter λ > 0.
6 Uniform
(f(x) = 1/(b−a) for a ≤ x ≤ b, f(x) = 0 (14)
otherwise, continuous uniform distribution in the interval
[a,b], a < b).
7 Manual (Allows to manually specify the variance of
β
ˆ
under H
0
and H
1
)
G*Power provides two different types of procedure to
calculate power: An enumeration procedure and large
sample approximations. The Manual mode is only
available in the large sample procedures.
The odds ratio is found when H1 and H0 values are
entered and calculated.When it transfers to the other
window it looks like these transfer. Type 1 error is 0.05
and power is 0.95. Take R as 0, and enter the X
distribution binominal.
Pr(Y=1| X=1)= H
1
=0.7 factor + probability of occurrence
Pr(Y=1| X=1)= H
0
=0.3 factor - probability of occurrence
And after calculating chose calculate and transfer to
main window.
R
2
other =X In the presence of other variables, the
variance of the main locator is '0' if there is no other
locator. If there is correlation, it is squared according to
Low / medium / high correlation. For example R = 0.20
then R
2
= 0.04
X parm μ: Factor + events available in reference studies.
If factor + and factor - equals 0.50 must be entered. In
this example + factor % 45 so that it must be used %45.
2.2.2. Options
In Input mode, you can choose between two input modes
for the effect size: The effect size may be given by either
specifying the two probabilities p
1
, p
2
defined above, or
instead by specifying p1 and the odds ratio OR.
Procedure G*Power provides two different types of
procedure to estimate power. An "enumeration
procedure" proposed by Lyles et al. (2007) and large
sample approximations. The enumeration procedure
seems to provide reasonable accurate results over a wide
range of situations, but it can be rather slow and may
need large amounts of memory. The large sample
approximations are much faster. Results of Monte-Carlo
simulations indicate that the accuracy of the procedures
proposed by Demidenko (2007) and Hsieh et al. (1998)
are comparable to that of the enumeration procedure for
N > 200. The procedure base on the work of Demidenko
(2007) is more general and slightly more accurate than
that proposed by Hsieh et al. (1998). It is therefore
recommended to use the procedure recommended by
Demidenko (2007) as a standard procedure. The
enumeration procedure of Lyles et al. (2007) may be
used to validate the results (if the sample size is not too
large). It must also be used, if one wants to compute the
power for likelihood ratio tests.
The enumeration procedure provides power analyses for
the Wald-test and the Likelihood ratio test. In logistic
regression analysis, Wald test can be used to test
whether the coefficients are important. The distribution
of the test statistic of the Wald test approaches the
standard normal distribution. For each variable, the Z
test is performed using standard errors in the list. The
Wald test is meaningful if the sample volume is large.
(Buse, 1982).
The highest probability estimator of the slope parameter
is compared to the estimated value of the standard error.
β
1
=0, while the distribution of the test statistic is suitable
for standard normal distribution. The test statistic of this
test;
W=
(15)
is obtained by the formula (Buse, 1982).
The general idea is to construct an exemplary data set
with weights that represent response probabilities given
the assumed values of the parameters of the X
distribution. Then a fit procedure for the generalized
linear model is used to estimate the variance of the
regression weights (for Wald tests) or the likelihood
ratio under H
0
and H
1
(for likelyhood ratio tests). The
size of the exemplary data set increases with N and the
enumeration procedure may thus be rather slow (and
may need large amounts of computer memory) for large
sample sizes. The procedure is especially slow for
analysis types other then "post hoc", which internally call
the power routine several times. By specifying a
threshold sample size N you can restrict the use of the
enumeration procedure to sample sizes < N. For sample
sizes ≥ N the large sample approximation selected in the
option dialog is used. Note: If a computation takes too
long you can abort it by pressing the ESC key. 2. G*Power
provides two different large sample approximations for a
Wald-type test. Both rely on the asymptotic normal
Black Sea Journal of Engineering and Science
BSJ Eng. Sci. / Aysel YENİPINAR et al. 20
distribution of the maximum likelihood estimator for
parameter β
1
and are related to the method described by
Whittemore (1981). The accuracy of these
approximation increases with sample size, but the
deviation from the true power may be quite noticeable
for small and moderate sample sizes. This is especially
true for X-distributions that are not symmetric about the
mean, i.e. the lognormal, exponential, and Poisson
distribution, and the Binomial distribution with π ≠ 1/2.
The approach of Hsieh et al. (1998) is restricted to
binary covariates and covariates with standard normal
distribution. The approach based on Demidenko (2007)
is more general and usually more accurate and is
recommended as standard procedure. For this test, a
variance correction option can be selected that
compensates for variance distortions that may occur in
skewed X distributions (see implementation notes). If
the Hsieh procedure is selected, the program
automatically switches to the procedure of Demidenko if
a distribution other than the standard normal or the
binomial distribution is selected (Faul at al., 2007).
3. Results and Discussion
Select the statistical test appropriate for your problem In
Step1, the statistical test is chosen using the distribution
based or the design-based approach.
Distribution-based approach to test selection First select
the family of the test statistic (i.e., exact, F, t, χ2, or z test)
using the Test family menu in the main window. The
Statistical test menu adapts accordingly, showing a list of
all tests available for the test family.
Is there a relationship between birth weight and gender?
Is it a predictor for gender X addiction?
In this example, 45% male X = 1, 55% female and X = 0.
70% of the girls and 80% of the boys were taken as
normal birth weight. Probability of error (significance
level) α = 0.05 and the power of the test was taken as 1-β
= 0.95.Impact magnitude (calculated in one way or look
at previous work).
Figure 2. Choice of Logistics regression from the
windows
In order to select the logistic regression menu from the
statical test, the test family Z test selected first.
Figure 3. Choice of options menu from the windows
On the pre-pop-up screen there is a section called option,
from which can calculate the sample size from odds ratio
or two probabilities from two different locations.
Especially with variance correction, the sample size
increases slightly. Because other features are selectively
closed, no changes are made.
Figure 4. It can be checked with variance correction
option
Define the following values;
Probability of error (significance level) α = 0.05
The power of the test was taken as 1-β = 0.95.
Impact magnitude (calculated in one way or look at
previous work).
Black Sea Journal of Engineering and Science
BSJ Eng. Sci. / Aysel YENİPINAR et al. 21
Figure 5. Probability of error and power values
Figure 6. Choose Determine and write next window H
0
and H
1
values
Enter the X distribution binominal.
Figure 7. Write next window σ,1-β and x- parm values.
Specified and calculated σ, 1-β and x- parm values are
written in the pop-up window. After than calculate.
3. Conclusion
In this study, how to calculate the sample size with g-
power in logistic regression analysis is explained. As a
result, a sample size of 68 in numerical sample and the
probability of correctly rejecting the H0 hypothesis
indicating that there is no relation between the main
predictor variable and result variable is 95%.
The limit of the work is more than one argument. The
limitation of sample size calculation with G * power is
that there are multiple independent variables in multiple
logistic regression analysis and no separate sample size
can be calculated for each. So experts advise to calculate
the power for the most striking variable.
Conflict of interest
The authors declare that there is no conflict of interest.
Acknowledgements
This research was presented as an oral presentation at
the International Congress on Domestic Animal Breeding
Genetics and Husbandry (ICABGEH-2018) held on 26-28
September 2018 in Antalya.
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