Choosing statistical tests

M E D I C I N E
REVIEW ARTICLE
Choosing Statistical Tests
Part 12 of a Series on Evaluation of Scientific Publications
Jean-Baptist du Prel, Bernd Röhrig, Gerhard Hommel, Maria Blettner
SUMMARY
Background: The interpretation of scientific articles often
requires an understanding of the methods of inferential
statistics. This article informs the reader about frequently
used statistical tests and their correct application.
Methods: The most commonly used statistical tests were
identified through a selective literature search on the
methodology of medical research publications. These tests
are discussed in this article, along with a selection of
other standard methods of inferential statistics.
Results and conclusions: Readers who are acquainted not
just with descriptive methods, but also with Pearson’s
chi-square test, Fisher’s exact test, and Student’s t test
will be able to interpret a large proportion of medical
research articles. Criteria are presented for choosing the
proper statistical test to be used out of the most
frequently applied tests. An algorithm and a table are
provided to facilitate the selection of the appropriate test.
Cite this as: Dtsch Arztebl Int 2010; 107(19): 343–8
DOI: 10.3238/arztebl.2010.0343
Medical knowledge is increasingly based on em-
pirical studies and the results of these are
usually presented and analyzed with statistical
methods. It is therefore an advantage for any physician
if he/she is familiar with the frequently used statistical
tests, as this is the only way he or she can evaluate the
statistical methods in scientific publications and thus
correctly interpret their findings. The present article
will therefore discuss frequently used statistical tests
for different scales of measurement and types of
samples. Advice will be presented for selecting statisti-
cal tests—on the basis of very simple cases.
Statistical tests used frequently in medical
studies
In order to assess which statistical tests are most often
used in medical publications, 1828 publications were
taken from six medical journals in general medicine,
obstetrics and gynecology, or emergency medicine. The
result showed that a reader who is familiar with de-
scriptive statistics, Pearson’s chi-square test, Fisher’s
exact test and the t-test, should be capable of correctly
interpreting the statistics in at least 70% of the articles
(1). This confirmed earlier studies on frequently used
statistical tests in medical scientific literature (2, 3).
There have however been changes over time in the
spectrum of the tests used. A survey of the analytical
statistical procedures used in publications of the journal
Pediatrics in the first six months of 2005 found that the
proportion of inferential methods had increased from
48 to 89% between 1982 and 2005 (4). There was also a
trend towards more complex test procedures. Never-
theless, here too, the most frequent tests were the t-test,
the chi-square test, and Fisher’s exact test. This article
will accordingly discuss these tests and their proper ap-
plication, together with other important statistical tests.
If the reader is familiar with this limited number of
tests, he/she will be capable of interpreting a large pro-
portion of medical publications. Information about the
rarer statistical tests can be found in the corresponding
articles, in advanced literature (5–7), or by consulting
an experienced statistician.
What is the purpose of statistical tests?
Clinical studies [for example, [5, 8]) often compare the
efficacy of a new preparation in a study group with the
efficacy of an established preparation, or a placebo, in a
control group. Aside from a pure description (9), we
Institut für Epidemiologie, Universität Ulm: Dr. med. du Prel
Medizinischer Dienst der Krankenversicherung Rheinland-Pfalz (MDK), Referat
Rehabilitation/Biometrie: Dr. rer. nat. Röhrig
Institut für Medizinische Biometrie, Epidemiologie und Informatik (IMBEI) Uni-
versitätsmedizin Mainz: Prof. Dr. rer. nat. Hommel, Prof. Dr. rer. nat. Blettner
Deutsches Ärzteblatt International |Dtsch Arztebl Int 2010; 107(19): 343–8 343

M E D I C I N E
would like to know whether the observed differences
between the treatment groups are just random or are
really present. This is because differences can be due to
chance variability (scatter) in a parameter, such as the
success of the treatment in the study group.
Definition
If a scientific question is to be examined by comparing
two or more groups, one can perform a statistical test.
This means that a null hypothesis must be formulated,
which can in principle be rejected. Moreover, a suitable
test parameter must be identified (10, 11).
For example, a clinical study might investigate
whether an antihypertensive drug works better than
placebo. The test variable may then be the reduction in
diastolic blood pressure, calculated from the mean dif-
ference in blood pressure between the active treatment
and placebo groups. The null hypothesis is then: “There
is no difference between the active treatment and the
placebo with respect to antihypertensive activity”
(effect = 0).
A statistical test then calculates the probability of ob-
taining the observed data (or even more extreme data),
if the null hypothesis is correct. A small p-value means
that this probability is slight. The null hypothesis is re-
jected if the p-value is less than a level of significance
which has been defined in advance. A test variable (test
statistic) is calculated from the observed data and this
forms the basis of the statistical test. In our case, this
might be the difference in mean blood pressure after six
months. If specific assumptions are made about the dis-
tribution of the data (for example, normal distribution),
the theoretical (expected) distribution of the test
variable can be calculated.
The value of the test variable calculated from the
observations is then compared with the distribution
expected if the null hypothesis were correct (5). If this
value is greater or less than a specific limit, it is
unlikely that the null hypothesis is correct and the null
hypothesis is accordingly rejected. The result is then
“statistically significant at the level α”. The statistical
test is thus a decision whether the observed value can
be explained by chance, or whether it is greater than
chance (statistically significant). The terms “level of
significance” and the principle of the interpretation of
p-values have already been discussed (10, 11). The
underlying steps in a statistical test are shown once
again in the Box.
It is possible to be mistaken, either in the rejection or
in the retention of the null hypothesis. The reason for
this is that the values exhibit scatter, as, for example,
not all patients react equally to a drug. An “error of the
first type” is the mistaken rejection of the null hypo-
thesis; the maximal probability of this error is the level
of significance α. This is often chosen to be 5% (10,
11). An “error of the second type” is the mistaken reten-
tion of the null hypothesis; the probability of this is ß,
which is the same as 1 minus the power of the study.
The power of the study is specified before the study
starts and depends on the sample size, as well as other
factors. A power of 80% is often selected (10, 11).
Important steps in the decision
for a statistical test
The decision for a statistical test is based on the scien-
tific question to be answered, the data structure, and the
study design. Before the data are recorded and the sta-
tistical test is selected, the question to be answered and
the null hypothesis must be formulated. The test and the
level of significance must be specified in the study
protocol before the study is performed. It must be
decided whether the test should be one-tailed or two-
tailed. If the test is two-tailed, this means that the direc-
tion of the expected difference is unclear. One does not
know whether there is a difference between the new
drug and placebo with respect to efficacy. It is unclear
in which direction the difference may be. (The new
drug might even work less well than the placebo). A
one-tailed test should only be performed when there is
clear evidence that the intervention should only act in
one direction.
The outcome variable (endpoint) is defined at the
same time the question to be answered is formulated.
Two criteria are decisive for the selection of the statisti-
cal test:
● The scale of measurement of the test variable
(continuous, binary, categorical)
● The type of study design (paired or unpaired).
Scales of measurement: continuous, categorical, or binary
The different scales of measurement have already been
discussed in the articles on study design and descriptive
statistics, under the selection of suitable measures and
methods of illustration (9, 12).
For example, in the comparison of two antihyperten-
sives, the endpoint can be the antihypertensive activity
in the two treatment groups. The reduction in blood
pressure is a continuous endpoint. It is also necessary to
BOX
Steps in a statistical test
● Statement of the question to be answered by the study
● Formulation of the null and alternative hypotheses
● Decision for a suitable statistical test
● Specification of the level of significance (for example,
0.05)
● Performance of the statistical test analysis: calculation
of the p-value
● Statistical decision: for example
– p<0.05 leads to rejection of the null hypothesis and
acceptance of the alternative hypothesis
– p≥0.05 leads to retention of the null hypothesis
● Interpretation of the test result
344 Deutsches Ärzteblatt International |Dtsch Arztebl Int 2010; 107(19): 343–8

M E D I C I N E
distinguish whether a continuous endpoint is (approxi-
mately) normally distributed or not.
If however one only considers whether the diastolic
blood pressure falls under 90 mm Hg or not, the end-
point is then categorical. It is even binary, as there are
only two possibilities. If there is a meaningful sequence
in the categorical endpoints, this can be described as an
“ordinal endpoint.”
Paired and unpaired study designs
A statistical test is used to compare the results of the
endpoint under different test conditions (such as treat-
ments). There are often two therapies.
If results can be obtained for each patient under all
experimental conditions, the study design is paired (de-
pendent). For example, two times of measurement may
be compared, or the two groups may be paired with
respect to other characteristics.
Typical examples of pairs are studies performed on
one eye or on one arm of the same person. Typical
paired designs include comparisons before and after
treatment. “Matched pairs,” for example in case-control
studies, are a special case. This involves selecting per-
sons from one group with the same specified character-
istics as persons in another group. The data are then no
longer independent and should be treated as if they
were paired observations from one group (5).
With an unpaired or independent study design, re-
sults for each patient are only available under a single
set of conditions. The results of two (or more) groups
are then compared. There may be differences in the
sizes of the groups.
Common statistical tests
The most important statistical tests are listed in the
Table. A distinction is always made between “categori-
cal or continuous” and “paired or unpaired.” If the end-
point is continuous, normal and non-normal distribu-
tions are distinguished (Table).
Group comparison of two categorical endpoints
The group comparison for two categorical endpoints is
illustrated here with the simplest case of a 2 x 2 table
(four field table) (Figure 1). However, the procedure is
similar for the group comparison of categorical end-
points with multiple values (Table).
● Unpaired samples:
If the frequency of success in two treatment
groups is to be compared, Fisher’s exact test is the
correct statistical test, particularly with small
samples (7). For large samples (about n >60), the
chi-square test can also be used (Table).
● Paired samples:
One example of the use of this test would be an
intervention within a group at two anatomical
sites, such as the implantation of two different
sorts of IOL lenses in the right and left eyes, with
the endpoint “Operation successful: yes or no.”
The samples to be compared are paired. In such a
case, one has to perform the McNemar test (7).
TABLE
Frequently used statistical tests (modified from [3])
Statistical Test
Fisher’s exact test
Chi-square test
McNemar test
Student’s t-test
Analysis of variance
Wilcoxon’s rank sum test (also
known as the unpaired Wilcoxon
rank sum test or the Mann-Whit-
ney U test)
Kruskal-Wallis test
Friedman test
Log rank test
Pearson correlation test
Spearman correlation test
Description
Suitable for binary data in unpaired samples: the
2 x 2 table is used to compare treatment effects
or the frequencies of side effects in two treat-
ment groups
Similar to Fisher’s exact test (albeit less precise).
Can also compare more than two groups or more
than two categories of the outcome variable. Pre-
conditions: sample size >ca. 60. Expected
number in each field ≥5.
Preconditions similar to those for Fisher’s exact
test, but for paired samples
Test for continuous data. Investigates whether
the expected values for two groups are the
same, assuming that the data are normally dis-
tributed. The test can be used for paired or un-
paired groups.
Test preconditions as for the unpaired t-test, for
comparison of more than two groups. The
methods of analysis of variance are also used to
compare more than two paired groups.
Test for ordinal or continuous data. In contrast to
Student’s t-test, does not require the data to be
normally distributed. This test too can be used
for paired or unpaired data.
Test preconditions as for the unpaired Wilcoxon
rank sum test for comparing more than two
groups
Comparison of more than two paired samples, at
least ordinally scaled data
Test of survival time analysis to compare two or
more independent groups
Tests whether two continuous normally dis-
tributed variables exhibit linear correlation
Tests whether there is a monotonous relationship
between two continuous, or at least ordinal, vari-
ables
FIGURE 1 Test selection for
group comparison
with two categorical
endpoints;
*1
Preconditions:
sample size
>ca. 60. Expected
number in each
field ≥5

M E D I C I N E
Continuous and at least ordinally scaled variables
Figure 2 shows a decision algorithm for test selection.
Normally distributed variables—parametric
tests: So-called parametric tests can be used if the end-
point is normally distributed.
Where subjects in both groups are independent of
each other (persons in first group are different
from those in second group), and the parameters
are normally distributed and continuous, the un-
paired t-test is used. If a comparison is to be made
of a normally distributed continuous parameter in
more than two independent (unpaired) groups,
analysis of variance (ANOVA) can be used. One
example would be a study with three or more
treatment arms. ANOVA is a generalization of the
unpaired t-test. ANOVA only informs you
whether the groups differ, but does not say which
groups. This requires methods of multiple testing
(11).
● Paired samples:
The paired t-test is used for normally distributed
continuous parameters in two paired groups. If a
normally distributed continuous parameter is
compared in more than two paired groups,
methods based on analysis of variance are also
suitable. The factor describes the paired
groups—for example, more than two points of
measurement in the use of a therapy.
Non-normally distributed variables—non-
parametric tests: If the parameter of interest is not
normally distributed, but at least ordinally scaled, non-
parametric statistical tests are used. One of these tests
(the “rank test”) is not directly based on the observed
values, but on the resulting rank numbers. This necessi-
tates putting the values in order of size and giving them
a running number. The test variable is then calculated
from these rank numbers. If the necessary precondi-
tions are fulfilled, parametric tests are more powerful
than non-parametric tests. However, the power of para-
metric tests may sink drastically if the conditions are
not fulfilled.
The Mann-Whitney U test (also known as the
Wilcoxon rank sum test) can be used for the com-
parison of a non-normally distributed, but at least
ordinally scaled, parameter in two unpaired
samples (5). If more than two unpaired samples
are to be compared, the Kruskal-Wallis test can be
used as a generalization of the Mann-Whitney U
test (13).
● Paired samples:
The Wilcoxon signed rank test can be used for the
comparison of two paired samples of non-
normally distributed, but at least ordinally scaled,
parameters (13). Alternatively, the sign test should
be used when the two values are only distin-
guished on a binary scale—for example, improve-
ment versus deterioration (7). If more than two
paired samples are being compared, the Friedman
test can be used as a generalization of the sign
test.
Other test procedures
Survival time analysis
If the point of interest is not the endpoint itself, but the
time till it is reached, survival time analysis is the most
suitable procedure. This compares two or more groups
with respect to the time when an endpoint is reached
(within the period of observation) (13). One example is
the comparison of the survival time of two groups of
cancer patients given different therapies. The endpoint
here is death, although it could just as well be the
occurrence of metastases. In contrast to the previous
tests, it almost never happens that all subjects reach the
endpoint in survival time analysis, as the period of ob-
servation is limited. For this reason, the data are also
described as (right) censored, as it is still unclear when
all subjects will reach the endpoint when the study
ends. The log rank test is the usual statistical test for the
FIGURE 2Algorithm for test
selection for group
comparison of
a continuous
endpoint

M E D I C I N E
comparison of the survival functions between two
groups. A formula is used to calculate the test variable
from the observed and the expected numbers of events.
This value can be compared with the known distribu-
tion which would have been expected if the null hy-
pothesis were correct—the chi-square distribution in
this case. A p-value can thus be calculated. A rule can
then be given for deciding for or against the null
hypothesis.
Correlation analysis
Correlation analysis examines the strength of the corre-
lation between two test variables, for example, the
strength of the correlation between the body weight of a
neonate and its body length. The selection of a suitable
measure of association depends on the scale of
measurement and the distribution of the two par-
ameters. The parametric variant (Pearson correlation
coefficient) exclusively tests for a linear correlation be-
tween continuous parameters. On the other hand, the
non-parametric variant—the Spearman correlation co-
efficient—solely tests for monotonous relationships for
at least ordinally scaled parameters. The advantages of
the latter are its robustness to outliers and skew dis-
tributions. Correlation coefficients measure the
strength of association and can have values between –1
and +1. The closer they are to 1, the stronger is the as-
sociation. A test variable and a statistical test can be
constructed from the correlation coefficient. The null
hypothesis to be tested is then that there is no linear (or
monotonous) correlation.
Discussion
The null hypotheses for these statistical tests described
in this article are that the groups are equal. These com-
monly used tests are also known as “inequality tests”.
There are however other types of test. “Trend tests”
examine whether there is a tendency for increasing or
decreasing values in at least three groups. There are
also “superiority tests”, “non-inferiority tests,” and
“equivalence tests.” For example, a superiority test
examines whether an expensive new drug is better than
the conventional standard medication by a specific and
medically relevant difference. A non-inferiority test
might examine whether a cheaper new medicine is not
much worse than a conventional medicine. The accept-
able level of activity is specified before the start of the
study on the basis of expert medical knowledge. An
equivalence test is intended to show that a medication
has approximately the same activity as a conventional
standard medication. The advantages of the new medi-
cation might be simpler administration, fewer side
effects, or a lower price.
The methods of regression analysis and the related
statistical tests will be discussed in more detail in the
course of this series on the evaluation of scientific pub-
lications.
The present selection of statistical tests is obviously
incomplete. Our intention has been to make it clear that
the selection of a suitable test procedure is based on
criteria such as the scale of measurement of the end-
point and its underlying distribution. We would like to
recommend Altman’s book (5) to the interested reader
as a practical guide. Bortz et al. (7) present a compre-
hensive overview of non-parametric tests (in German).
The selection of the statistical test before the study
begins ensures that the study results do not influence
the test selection. Moreover, the necessary sample size
depends on the test procedure selected. Problems in
planning sample size will be discussed in more detail
later in this series.
Finally, the point must be made that a statistical test
is not necessary for every study. Statistical testing can
be dispensed with in purely descriptive studies (12) or
when the interrelationships are based on scientific
plausibility or logical arguments. Statistical tests are
also usually not helpful when investigating the quality
of a diagnostic test procedure or rater agreement (for
example, in the form of a Bland-Altman diagram) (14).
Because of the probability of error, statistical tests
should be used “as often as necessary, but as little as
possible.” The risk of purely chance results is
especially high with multiple testing (11).
Conflict of interest statement
The authors declare that no conflict of interest exists according to the
guidelines of the International Committee of Medical Journal Editors.
Manuscript received on 14 October 2009, revised version accepted on
22 February 2010.
Translated from the original German by Rodney A. Yeates, M.A., Ph.D.
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Corresponding author
Prof. Dr. rer. nat. Maria Blettner
Institut für Medizinische Biometrie,
Epidemiologie und Informatik (IMBEI)
Universitätsmedizin Mainz
Obere Zahlbacher Str. 69
55131 Mainz, Germany

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