## Statistics.du.se

D-level Essay in Statistics 2009
** **

How to Analyze Change from Baseline:

Absolute or Percentage Change？
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Richard Stridbeck
How to Analyze Change from Baseline:
Absolute or Percentage Change?
In medical studies, it is common to have measurements before and after some medical interventions. How to measure
the change from baseline is a common question met by researchers. Two of the methods often used are absolute changeand percentage change. In this essay, from statistical point of view, we will discuss the comparison of the statistical powerbetween absolute change and percentage change. What's more, a rule of thumb for calculation of the standard deviation ofabsolute change is checked in both theoretical and practical way. Simulation is also used to prove both the irrationality of theconclusion that percentage change is statistical ine¢ cient and the nonexistence of the rule of thumb for percentage change.

Some recommendations about how to measure change are put forward associated with the research work we have done.

Key Words: Absolute Change, Percentage Change, Baseline, Follow-up, Statistical Power, Rule of Thumb.

change to evaluate the change of weight. Neovius (2007) alsochose absolute change as the change measurement in their
obesity research, while Kim (2009) chose percentage changeto measure the fat lost in di¤erent part of an obese man'sbody in a weight loss program. In a cystic …brosis clinical
In medical studies, a common way to measure treatment study, Lavange (2007) used percentage change as well. We
e¤ect is to compare the outcome of interest before treat-
see that, both of the two methods have been used in di¤erent
ment with that after treatment. The measurements before
kinds of clinical studies.

and after treatment are known as the baseline (B) and the
The properties of absolute change and percentage change
follow-up (F ), respectively. How to measure the change from
have been discussed by Tönqvist (1985). From his point of
baseline is a common question met by researchers. There
view, one of the advantages of percentage change is that per-
are many methods that can be used as the measure of di¤er-
centage change is independent of the unit of measurement.

ence. Two of them, which are used in a lot of clinical studies,
For instance, a man who weighs 100 Kg lost 10% of weight
are absolute change (C = B
F ) and percentage change
after a treatment, i.e. 10 Kg. Equivalently, he lost 22.05
F ) B). In di¤erent books and articles, absolute
pounds (1Kg = 2:2046 Pounds). 10 Kg and 22.05 pounds are
change may also be called change, while percentage change is
essentially the same weight, but the absolute change scores
also called relative change.

are di¤erent. However, no matter what the unit of measure-
There is a simple example that will show us the di¤erence
ment is, the percentage change is 10% all the time. More
between absolute change and percentage change more clearly:
details about the advantage of absolute change can be found
Two obese men A and B participate in a weight loss program.

in the article of Tönqvist (1985). Although there are many
Their weights at the beginning of the program are 150 Kg and
advantages for the two change measurement methods, Tön-
100 Kg, respectively. When they …nish the program, the man
qvist (1985) did not give any recommendations about how to
A who weighs 150 Kg lost 15 Kg, while another man lost 10
make a choice between absolute change and percentage change
Kg. From the example, we see that, the man A lost 5 Kg more
based on these properties.

than that the man B, but the percent of weight they lost are10% in both cases. We want to know, which change measure-
In other literatures, several suggestions about which
ment is best to show the treatment e¤ect of the weight loss
method to choose are mentioned. Vickers (2001) suggested
avoiding using percentage change. That is because he com-
In di¤erent clinical studies, either absolute change or per-
pared the statistical power of di¤erent methods by doing
centage change may be chosen. In the study of healthy dieting
a simulation and concluded that percentage change from
and weight control, Waleekhachonloet (2007) used absolute
baseline is statistically ine¢ cient. Kaiser (1989) also gave
How to Analyze Change from Baseline: Absolute or Percentage Change?
some recommendations for making a choice between absolute
2. Comparison of the statistical
change and percentage change. He suggested using the change
power of absolute change and
measurement that has less correlation with baseline scores. Atest statistic developed by Kaiser (1989) was also derived, i.e.

percentage change
the ratio of the maximum likelihood of absolute change tothat of percentage change. The absolute change is recom-mended if the value of the test statistic is larger than one,
In clinical research, it is common to test whether there is a
treatment e¤ect after a medical intervention. In order to
while percentage change is preferred when it is less than one.

test the treatment e¤ect, it's necessary to choose a suitable
That is, a simple rule helping researchers to make a choice
measurement of the di¤erence between baseline and follow-up
Actually, the primary consideration for choosing a change
From a statistical point of view, an important criterion for
measurement method is di¤erent from di¤erent points of view.

a good statistical method is high statistical power. Therefore,
From a clinical point of view, we prefer to use a change mea-
from the two common change measurement methods, absolute
surement that may show the health-improvement for the pa-
change and percentage change, the one with a higher statis-
tients in a more observable way. For example, in study of
tical power will be preferred.

asthma, the primary outcome variable is often FEV (ForcedExpiratory Volume L/s).

The e¢ ciency of a treatment is
2.1 Statistical Power
evaluated by calculating the percentage change in FEV frombaseline. In hypertension studies, it is common to use the ab-
According to the hypothesis testing theory, statistical power
solute change in blood pressure instead of percentage change.

is the probability that a test reject the false null hypothesis.

From statistical point of view, we prefer the method which
The de…nition of statistical power can be expressed as
has the highest statistical power as Vickers (2001) did.

equation (1).

Another issue concerned by researchers is the standard
deviation of the treatment e¤ect (change scores). For two
Statistical power = P (reject H0 j H0 is False)
medical interventions that may lead to the same expected
change, the e¤ect of the intervention that has a smaller stan-
0 is the null hypothesis.

In a t-test, equation (1) can be rewritten as equation (2).

dard deviation seems more stable and e¤ective. And clini-cians may always prefer that medical intervention. Since itis not practical for researchers to get all the interested exper-
Statistical power
P (reject H0 j H0 is False)
imental datasets which recorded the details of baseline and
follow-up scores for each patient, there is a rule of thumb
21 . It may help calculating the stan-
dard deviation of change scores from the standard deviation
where t =2 is the t-value under the signi…cant level
of baseline scores.

two-side t-test, and pt is the p-value of the t-test.

The aim of this essay is to show, the statistical e¢ ciency
From expression (2), we may see that the larger the ex-
of percentage change under some conditions in contrast with
pected absolute value of the t-statistic is, the higher the sta-
Vickers' (2001) conclusion, and the rationality of the rule of
tistical power will be.2 Equivalently, the smaller the expected
thumb. The essay is organized as follows. The second section
p-value is, the higher the statistical power will be. Therefore,
is the comparison of the statistical power of absolute change
from di¤erent measurement methods, we will choose the one
and percentage change by constructing a test statistic under
that has a larger expected absolute value of t-statistic or a
certain distribution assumption. In the third section, the ra-
smaller expected p-value.

tionality of rule of thumb is discussed in a theoretical way.

Simulations of some of the issues discussed in section two and
2.2 A Clinical Example of Blood Pressure
three are carried out in the fourth section. The …fth section
is an empirical investigation of the usefulness of the rule ofthumb by using some real datasets. Finally, in the discussion
To easily interpret the di¤erence of two measurement meth-
section, we discuss the results got from the previous sections,
ods, an example from a clinical trial is shown in Table 1. In
and give some suggestions.

the table, there are the records of the supine systolic bloodpressures (in mmHg) for 5 patients before and after takingthe drug captopril.

Let (Bj; Fj) denote a baseline/follow-up pair of scores for
patient j in the treatment group, j = 1; 2;
1 Personal communication Prof. Johan Bring, E-mail: johan.bring@statisticon.se2 However, it must be emphasized that, in this essay, "statistical power" actually means something slightly di¤erent from this.

Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
can get absolute change Cj = Bj
Fj and percentage change absolute change and percentage change are asymptotic nor-
Bj for patient j by calculating from the mally distributed, i.e.

baseline Bj and follow-up Fj scores immediately.

In this example, j is the patients' ID number, and here
n = 5. In columns 2 and 3, there are baseline and follow-
up scores for each patient. Absolute change and percentage
where C and C are the mean and the standard deviation
change that calculated from baseline and follow-up scores are
P are the mean and the standard deviation of
shown in column 4 and 5, respectively.

P , respectively.

In this case, t-test can be used for both absolute change
Table 1. Supine systolic blood pressure (in mmHg) for 5
and percentage change. For absolute change, the null hypoth-
patients with moderate essential hypertension, immediately
esis of t-test is H0 : C = 0. From C
before and after taking the drug captopril3
the t-statistic for an absolute change t-test is
C is an estimate of the standard deviation of ab-
solute change.

Similarly, in the percentage case, the null hypothesis of
t-statistic for a percentage change t-test is
From Table 1, we see that there is a decreasing e¤ect for
the blood pressure of each patient after taking the drug cap-
topril. Absolute change and percentage change show the de-
crease in di¤erent ways. From a statistical point of view, we
where bP is an estimate of the standard deviation of per-
should compare the statistical power for the two methods.

centage change.

We have mentioned the relation between statistical power
and the absolute value of t-statistic. We know that, when
2.3 Comparison of Statistical Power
the signi…cant level
is …xed, if the expected absolute value
We have mentioned that Vickers (2001) compared the statis-
of the t-statistic of absolute change is larger, the statistical
tical power of di¤erent methods by doing a simulation. How-
power of that will be higher. The opposite is also true, i.e.

ever, his conclusion just based on an ideal simulation proce-dure, and he did not compare the statistical power theoreti-cally. Kaiser (1989) developed a test statistic which comparedthe maximum likelihood of the two methods. It has nothing
to do with statistical power. But Kaiser (1989) gave an idea
that it is easier to do comparison by constructing a ratio test
Statistical P ower of Absolute Change
For comparison of the statistical power of the two meth-
Statistical P ower of P ercentage Change
ods, we construct a ratio test statistic by using the test sta-tistic or p-value of the treatment e¤ect test. Before that, we
where E (jtCj) and E (jtP j) are the expected absolute value
need to know the distributions of absolute change and per-
of the t-statistic of absolute change and percentage change,
centage change. That is because, for di¤erent distributions of
absolute change or percentage change, di¤erent test methods
So, when R > 1, absolute change has higher statisti-
will be used. In order to construct a ratio test statistic, we
cal power than percentage change, and we choose absolute
should know the test statistics used in both numerator and
change. If R < 1, the percentage change with the higher
denominator of the ratio test statistic.

statistical power is preferred.

In the case of small sample size, it is common to assume
that one of the distributions of absolute change and percent-
2.3.1 When t-test is Suitable for both Absolute
age change is normal. In some speci…c situation, both ab-
Change and Percentage Change
solute change and percentage change may be normally dis-
When the sample size n of the clinical experiment is large,
tributed. Even though for a dataset that is not normally dis-
according to the Central Limit Theorem, both the mean of
tributed, if the distribution is close to normal distribution or
3 Hand, DJ, Daly, F, Lunn, AD, McConway, KJ and Ostrowski, E (1994): A Handbook of Small Data Sets. London: Chapman and Hall.

Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
the distribution is symmetric without extreme observations,
statistical power than percentage change, and we choose ab-
t-test may also be used. In that situation, the test statistic R
solute change. For R0 > 1, the percentage change is preferred.

is also applicable.

In this section, another ratio test statistic R0 for non-
If we simulate some datasets, by using the ratio test sta-
normal distribution situation is discussed. This is the supple-
tistic R, we can compare the statistical power of the absolute
ment of the normal distribution case. In the following simula-
change and percentage change of the datasets that we simu-
tion part, we will concentrate more on the normal distribution
lated. In contrast with Vickers' (2001) claim, some datasets
case shown in subsection 2.3.1, and the details in subsection
with R < 1 will be shown, which re‡ects that percentage
2.3.2 will not be discussed any more.

change has higher statistical power than absolute change un-der some conditions. In the simulation section, we will talkmore about the comparison of the statistical power of the twomethods.

3. Rule of Thumb for the Standard
2.3.2 When the t-test is not suitable for At Least One
Deviation of Change Scores
For the cases when the assumptions for the t-test are notsatis…ed for at least one of the tests, another test should be
The standard deviation of the treatment e¤ect is an im-
portant parameter that is of interest in the planning of
considered. Wilcoxon rank sum test4 is an alternative method
studies. The standard deviation of the change scores is the
proposed by Wilcoxon (1945). Bonate (2000) mentioned that
focus in the second section of this essay. For the case when
it is the non-parametric counterpart to the paired samples
t-test may be used instead of a non-parametric test, in or-
t-test and should be used when normal assumptions are vio-
der to calculate the ratio test statistic R, we should work out
lated. He suggested that the Wilcoxon rank sum test is always
both the mean and the standard deviation of absolute change
a better choice when the distribution of the data is unknown
and percentage change …rst. It is easy to get these values in
or uncertain. Therefore, when the paired samples t-test does
case the datasets of the experiment which give the baseline
not work, we choose Wilcoxon rank sum test instead.

and follow-up scores for each patient are known. However, in
Since we can not use t-statistic to construct the ratio test
clinical research, it is not always possible and practical to get
statistic any more, we may choose to use the expected p-value
the scores for each patient, especially in the planning phase of
of the treatment e¤ect test.

a study. If we require some datasets to support our research
Similarly to (5), according to what is mentioned in equa-
work, we may …nd some experiment datasets interesting for
tion (2), we may construct another ratio test statistic R0 by
our research from experiments that someone else has done.

taking the ratio of expected p-value, i.e.

One of the good ways to …nd the datasets is searching from
published clinical articles. Most of the time, we may …nd some
examples in these articles which show us the summary of the
baseline scores, the follow-up scores, and their standard devi-
ations. And we may get relevant datasets from these tables.

Statistical P ower of Absolute Change
However, the scores for each patient in these clinical research
Statistical P ower of P ercentage Change
articles are seldom published. In this case, how can we know
the standard deviation of the change scores?
C ) and E (pP ) are the expected p-value of ab-
solute change and percentage change, respectively.

There is a rule of thumb which describes the relationship
For the cases that t-test still works, E (p
between the standard deviation of the change scores and that
of the baseline scores.

pected p-value of the t-test for absolute change and percentage
change, respectively. When Wilcoxon rank sum test is used
instead of t-test, E (p
W ilcoxon) and E (pP ) =
E (pP W ilcoxon). E (pC W ilcoxon) and E (pP W ilcoxon) arethe expected p-value of the Wilcoxon rank sum test for ab-
3.1 Theoretical Derivation of the Rule of
solute change and percentage change, respectively. Therefore,
Thumb for Absolute Change
we may get three di¤erent alternative forms for the ratio test
The general expression for the rule of thumb of absolute
statistic R0.

We have talked about that the smaller the expected p-
value is, the higher the statistical power will be. When the sig-
is …xed, if R0 < 1, absolute change has higher
4 Wilcoxon rank sum test is a non-parametric test for assessing whether two independent samples of observations come from the same distribution.

Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
where k is a constant that should be determined; SD (C)
If we assume that SD (B) = 1, then we get SD (C) =
and SD (B) are the standard deviation of absolute change
2r. The smooth curve in Figure 1 shows the relation-
and baseline scores, respectively.

ship between the standard deviation of the absolute change
The relationship between the standard deviation of ab-
SD (C) and the correlation coe¢ cient r. We see that SD (C)
solute change scores and that of the baseline scores can be
decreases from 1:4 to 0 as the correlation coe¢ cient increases
derived from properties of the variance of C,
from 0 to 1. When r 6 0:8, SD (C) roughly has a linear de-crease. After that, when the correlation coe¢ cient r tends to1, the ratio decreases quickly to 0.

where r = cov (B; F )
V ar (B) V ar (F ) is the correla-
tion coe¢ cient between baseline and follow-up scores.

We assume that V ar (F ) = mV ar (B), where m is the
ratio of the variance of follow-up scores to that of baseline
scores. In a speci…c case, m is a constant which may be cal-culated from the known dataset. Then, equation (9) can be
V ar (B) V ar (F )(10)
V ar (B) mV ar (B)
Correlation Coefficient r
Equivalently, from equation (10), we get the relation equa-
Figure 1. Relation curve between the standard deviation of
absolute change SD (C) and the correlation coe¢ cient r
when SD (B) = 1.

Thus, in equation (8), k = 1
shows that the rule of thumb is determined by the correlation
Therefore, when r
0:75, the empirical form of the rule
coe¢ cient r and the ratio m.

2 holds. If the correlation
When the baseline and follow-up scores have the same
coe¢ cient r changes to another value, the form of the rule
variance, m = 1, then we get k = 1
of thumb will be also changed. When r tends to 1, a little
pression of the rule of thumb becomes
change in r may result in a signi…cant change in the standarddeviation of absolute change.

3.3 Rule of Thumb for Percentage Change
Therefore, the standard deviation of the absolute change
is connected to the standard deviation of the baseline scores
Earlier we stated that P = (B
via the correlation coe¢ cient r.

centage change is the ratio of absolute change to the baselinescore. Then we get
3.2 The Relation between the Standard Devi-ation of Absolute Change and the Correlation
The empirical form of the rule of thumb is shown in expression
Since percentage change is a ratio of two variables, its dis-
tribution is uncertain. It is hard to derive the expression of
(12) shows the relation between the standard deviation of the
the standard deviation of percentage change from the stan-
absolute change and that of the baseline scores. So, how does
dard deviation of baseline scores as we did in equation (10).

the standard deviation of absolute change depend on the cor-
We have discussed the rule of thumb for SD (C), and we
relation coe¢ cient?
see that the standard deviation of percentage change depends
Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
not only on the absolute change C but also on the baseline
the patients in the treatment group, the …nal follow-up scores
score B. If we …x the value of C, B may also keep on changing
F all have an absolute decrease of 5 units from F 0 after the
from one sample to another. As a result, it seems there is no
medical intervention, while there is no change of the follow-up
stable relationship between the standard deviation of percent-
scores for patients in the control group, i.e.

age change SD (P ) and the baseline score B. A rule of thumbfor percentage change may, therefore, not be stated. This con-
clusion will be proved in the following simulation part.

He changed the correlation coe¢ cient r, and got di¤erent
simulation results under di¤erent correlation coe¢ cient. Us-ing these simulation results, he calculated statistical power for
each method and made the statistical ine¢ ciency conclusion.

In the following subsections, we will do simulations based
on Vickers' (2001) method. But some change and improve-
In section 2, we discussed the comparison of the statistical ment will be done to his code.

power of absolute change and percentage change, by con-
structing a ratio test statistic based on normal distribution.

In the third section, we discussed the rule of thumb for ab-
4.1.1 A Case that Percentage Change Has Higher Sta-
solute change and percentage change theoretically.

This section will do some simulations to show the prob-
In Vickers'(2001) simulation method, a …xed absolute change
lems that we have discussed in a practical way. The …rst
from the simulated follow-up scores F 0 to the …nal follow-up
thing we want to prove is, in contrast with Vickers' (2001)
scores F was set to each patient in the treatment group. If we
conclusion, that percentage change can be statistically e¢ -
change the …xed absolute change to a …xed percentage change,
cient under some conditions. The second thing that will be
maybe we will get something di¤erent. To be more random-
proved is the di¢ culty of de…ning a rule of thumb for percent-
ized, just like what may happen in practice, we use a random
percentage change instead of …xed percentage change. Thepercentage changes P are simulated from a normal distribu-
Percentage tion. We should notice that the changes we did to Vickers'
Change under Some Conditions
(2001) simulation will result in the change of the correlationcoe¢ cient between the baseline and follow-up scores. The cor-
Vickers (2001) suggested avoiding using percentage change,
relation coe¢ cient of the baseline and follow-up scores in the
because of his conclusion that percentage change from base-
simulation result is not the r that we used in equation (13) any
line is statistically ine¢ cient. He made that conclusion based
more, even though the real value of the correlation coe¢ cient
on the comparison of statistical power calculated from his
may be very close to the value we used in the simulation.

simulation results.

However, since the correlation coe¢ cient will not a¤ect the
Vickers (2001) did the simulation in the following way.

comparison of the statistical power for the two methods, we
First, he simulated 100 pairs of baseline and follow up scores
will give the value of used in each simulation procedure, but
for 100 patients. The baseline scores B are simulated from
do not talk more about it. What's more, in the following sim-
a normal distribution, i.e. B
N (50; 10). In order to get
ulation, we just concentrate on the patients in the treatment
100 scores B, he simulated 100 B0 …rst, B0
he got B from the equation B = B0 + 50. He also simu-
We have developed a ratio test statistic R in section 2, and
lated another 100 scores Y , Y
N (0; 10), which are de…ned
we will use it to do the comparison between absolute change
as the post-treatment scores of the control group. Then the
and percentage change. The simulation can be divided into
follow-up scores F 0 are simulated from B0 and Y by using
two steps. In step 1, we simulate 100 pairs of baseline/follow-
the equation (13). We should note that F 0 is not the …nal
up scores. In the second step, the test statistic R is calculated
follow-up scores.

based on the scores we simulated in step 1.

From equation (5), we know that, in order to simulate
a dataset such that R < 1, we should let the percentage
N (0; 10), we obtain that
change have a large mean and small standard deviation. So,
N (50; 10). Finally, Vickers (2001) simulated 100 g from
in this case, we simulate P from the normal distribution
Binomial (1; 0:5) for each patient. These patients who got
N (0:5; 0:01). We set r = 0:75 and simulate B from the dis-
g = 1 were put into the treatment group, and the other pa-
tribution N (200; 20). According to Vickers (2001) simulation
tients were put in the control group. So, there are nearly
method, we obtain a dataset of scores. Figure 2 shows a part
50 patients in both treatment group and control group. For
of the simulation results of baseline and follow-up scores in
Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
step 1. From Figure 2, we see that there is a nearly 50%
In order to show a more general result, we repeat the pro-
decrease from the baseline score for each patient.

cedure in both step 1 and step 2 100 times, and check thedistribution of R. As shown in Figure 3, the solid line on theleft is the distribution of R based on the datasets we simu-lated. We see that, when P
N (0:5; 0:01), the value of R is
much less than 1. As a result, in this case, percentage changehas a higher statistical power.

In this simulation, we set the percentage change P nor-
mally distributed with a large mean and small standard de-viation. However, it is unreasonable to have such a small
standard deviation in practice. If we increase the standarddeviation of P , what kind of result will come to us?
Figure 3 shows the distribution of R under di¤erent stan-
dard deviation of P . We see that, the value of the test sta-
tistic R increases as the standard deviation of P increases.

Although R increases, it is still less than 1. In this case, weprefer percentage change to absolute change.

4.1.2 A Case that Absolute Change Has A LittleHigher Statistical Power
B N(200,20), P N(0.5,0.01)
In the last section, we simulated a case where percentagechange had a higher statistical power, which is in contrast
Figure 2. Change from Baseline Scores to Follow-up Scores
with Vickers' (2001) conclusion. If we consider more about
the simulation method, we should notice that we used per-centage change to do simulation in that case. It may be afactor which a¤ects the simulation results such that percent-age change has a higher statistical power.

B N(200,20), P N(0.5,SD(P))
Figure 3. Distribution of R (r = 0:75).

B N(200,20), C N(100,5)
Since the test statistic R is the ratio of two expected val-
Figure 4. Change from Baseline Scores to Follow-up Scores
ues, in order to calculate the expected value, we repeat the
score simulation procedure in step 1 100 times, and then weget 100 datasets of scores. In the second step, using the 100datasets, we work out the value of R, and check if it is less
Now, we just change P
N (0:5; 0:01) to C
and keep other conditions the same. Part of the baseline and
Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
follow-up scores are shown in Figure 4. It seems similar withthe scores in Figure 2. This is because we set the expectedabsolute change to 100, which is 50% of baseline scores. So,in a similar absolute change case, what kind of result we willget?
B N(200,SD(B)), C N(Mean(C),10)
Figure 6. Distribution of R (r = 0:75).

We have done 3 simulations based on a modi…cation of
Vickers'(2001) method so far. The …rst one shows that per-centage change has higher statistical power. The other two
B N(200,20), P N(100,SD(C))
show that percentage has nearly the same statistical power
Figure 5. Distribution of R (r = 0:75).

with absolute change.

All of them proved that percent-
age change can be statistical e¢ cient under some conditions.

Therefore, Vickers (2001) conclusion is not correct.

Figure 5 shows the distribution of R under di¤erent stan-
4.2 Nonexistence of Rule of Thumb for Per-
dard deviation of C. Comparing with the distributions in
Figure 3, we …nd it has a di¤erent kind of change when thestandard deviation of C changes. The distributions in Figure
We have discussed the rule of thumb for percentage change
3 mainly perform a location di¤erence, while the distributions
theoretically in section 3. In this section we will simulate
in Figure 5 have di¤erent kurtosis and spread.

another dataset to check if the rule of thumb for percentage
Even though the expected value of R in Figure 5 is larger
change exists. The simulation will show how the standard de-
than 1, it is really close to 1. In this case, it seems both
viation of percentage change SD (P ) depends on the baseline
absolute change and percentage change can be used. The dif-
ference between the statistical powers of the two methods is
This simulation is also based on Vickers' (2001) sim-
ulation method.

In this case, the baseline scores follows
N (50; 10), and the percentage change has a distribution
N (0:1; 0:02). Following the simulation steps, we will
4.1.3 Another Case that Percentage Change Has Lit-
get a score dataset, and the standard deviations of absolute
tle Di¤erence with Absolute Change
change and percentage change can be calculated.

After we get the baseline and follow-up scores, we make a
In this case, we reduce the standard deviation of the baseline
simple transformation that both baseline and follow-up scores
scores to 10 and compare the results with that of the previous
decrease 5 units, i.e.

From Figure 6, we …nd that the expected value of the ratio
test statistic is much more close to 1. If we also reduce themean of C, then R is completely less than 1. This is another
After transformation, we get a new dataset of baseline and
case that shows, under some conditions, the statistical powers
follow-up scores, and calculate the standard deviations of ab-
of the two methods are nearly the same.

solute change and percentage change of new scores. Repeat
Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
the simulation procedure 100 times, each time we may get
standard deviation of absolute change does not change. Ac-
4 standard deviations, SD (C), SD
tually, we can prove that in a theoretical way.

Then, we calculate the mean of 100 simulation results for the
4 standard deviations, respectively. When the correlation co-e¢ cient changes, we get the relation curve between the stan-
After the transformation, C does not change. Therefore,
dard deviation of change scores and the correlation coe¢ cient
the standard deviation of absolute change will not change,
both before and after transformation.

From …gure 7, we also see that, the standard deviations of
percentage change under di¤erent correlation coe¢ cients be-
come larger after transformation. The smaller the correlationcoe¢ cient is, the larger the change of the standard deviation
of percentage change will be.

We have mentioned that SD (P ) = SD (C
case, C does not change, but B becomes smaller. As a result,the standard deviation of percentage change becomes larger.

This re‡ects that the standard deviation of percentage change
Before T ransformation
depends on the baseline scores. Therefore, it's di¢ cult to have
After T ransformation
a rule of thumb for the standard deviation of the percentagechange based on only the standard deviation of the baseline
Correlation Coefficient r
B N(50,10), P N(0.1,0.02)
5. Demonstration of the Rule of
Thumb for Absolute Change
We have discussed the rule of thumb for absolute change
theoretically in the third section of this essay. A gen-
eral expression of the rule of thumb is given in equation (12).

We know that, when r
0:75, the empirical form of the rule of
thumb in expression (7) holds. In this section, we will concen-
trate on the demonstration of the rule of thumb for absolute
change. We collect some real datasets to check whether the
Before T ransformationAfter T ransformation
rule of thumb works well.

The real datasets are searched from examples of clinical
research articles and medical literatures. As shown in Table
2, we have collected two kinds of datasets. The datasets of
the …rst 10 cases contain the scores for each patient, while thedata sets of the last 5 cases just contain some data summary,
e.g. mean, standard deviation, etc., of the baseline scores and
Correlation Coefficient r
absolute change scores.

B N(50,10), P N(0.1,0.02)
If we know the scores for each patient, we can calculate
Figure 7. Relation curve between the standard deviation of
not only the standard deviations but also the correlation co-
absolute change (above) or percentage change (below) and
e¢ cient between the baseline scores and the follow-up scores.

the correlation coe¢ cient r before and after transformation.

So, for the …rst 10 cases, we also get the value for the general
From …gure 7, we observe that, after transformation, the
form of the rule of thumb.

Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
Table 2. Comparison of the real standard deviation and the value got from the rule of thumb for absolute change
Comparing the values of SD (C) and SD (B)
In this case, SD (C) = 2:68 and
are obvious di¤erences between the two values for these real
2rSD (B) = 2:66, the two values are nearly the same,
datasets. In some cases, the di¤erence between SD (C) and
which re‡ect that the rule of thumb are also a¤ected by the
2 is very large.

For the cases that has a correlation coe¢ cient between 0.6
If we also take m into account, we will get the real value of
and 0.9, which is close to 0.75, the di¤erence between SD (C)
the standard deviation of absolute change. Actually, we have
2 may be acceptable. For example, in case 1,
proved that in equation (11) of section 3.

r = 0:63, it is close to 0.75. In this case, SD (C) = 8:99 and
When we know nothing about the correlation between the
2 = 8:09 , it seems that the two values are close
baseline and follow-up scores, just like the last …ve cases in
to each other. However, in case 5, when r = 0:17, the value
Table 2, the rule of thumb may not be suitable. In this case,
of SD (C) is nearly three times of the value of SD (B)
we should be more careful.

This is not acceptable. These facts show that the rule of
From the analysis based on real datasets in this section,
thumb in expression (7) is valid when r
0:75 or when r is
we learned that when the ratio m = V ar (F )
close to that value.

to 1 and the correlation coe¢ cient is nearly 0.75, the rule of
If we take the correlation coe¢ cient r into account, by
2 will be practical. If these con-
comparing the values of SD (C) and
ditions are not satis…ed, it is not a good rule to follow. If we
…nd that the values of
2rSD (B) are closer to SD (C)
ignore the two conditions and insist on using the rule, as we
2, especially when m
know from Table 2, it may result in a big mistake.

5 Douglas G.Alman (1991): Practical Statistics for Medical Research. London: Chapman and Hall. Page 4756 Douglas G.Alman (1991): Practical Statistics for Medical Research. London: Chapman and Hall. Page 4757 Pagano M, Gauvreau K (2000): Principles of Biostatistics, Second Edition, Duxbury. Table B.158 Pagano M, Gauvreau K (2000): Principles of Biostatistics, Second Edition, Duxbury. Table B.159 Bradstreet, T.E. (1994) "Favorite Data Sets from Early Phases of Drug Research - Part 3." Proceedings of the Section on Statistical Education
of the American Statistical Association. <http://www.math.iup.edu/ tshort/Bradstreet/part3/part3-table3.html> 2009-06-08
1 0 Hand, DJ, Daly, F, Lunn, AD, McConway, KJ and Ostrowski, E (1994): A Handbook of Small Data Sets. London: Chapman and Hall.

1 1 Ryan, Joiner, Cryer (1985): Minitab Handbook, Second Edition. PWS-KENT Publishing Company. Page 318, Pulse Data1 2 Bonate P (2000): Analysis of Pretest-Posttest Design. Boca Raton: Chapman and Hall/CRC. Table 3.11 3 Bonate P (2000): Analysis of Pretest-Posttest Design. Boca Raton: Chapman and Hall/CRC. Table 3.41 4 Bonate P (2000): Analysis of Pretest-Posttest Design. Boca Raton: Chapman and Hall/CRC. Table 9.11 5 Waleekhachonloet O, Limwattananon C, Limwattananon S, Gross C (2007): Group behavior therapy versus individual behavior therapy for
healthy dieting and weight control management in overweight and obese women living in rural community. Obesity Research & Clinical Practice,1: 223-232. Table 3
1 6 Neovius M, Rössner S (2007): Results from a randomized controlled trial comparing two low-calorie diet formulae. Obesity Research & Clinical
Practice, 1: 165-171. Table 1 & Table 2
1 7 Neovius M, Rössner S (2007): Results from a randomized controlled trial comparing two low-calorie diet formulae. Obesity Research & Clinical
Practice, 1: 165-171. Table 1 & Table 2
1 8 Neovius M, Rössner S (2007): Results from a randomized controlled trial comparing two low-calorie diet formulae. Obesity Research & Clinical
Practice, 1: 165-171. Table 1 & Table 2
1 9 Neovius M, Rössner S (2007): Results from a randomized controlled trial comparing two low-calorie diet formulae. Obesity Research & Clinical
Practice, 1: 165-171. Table 1 & Table 2
Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
in more detail. It has higher statistical power than the twomethods we talked about. But for the people who are not
6. Discussion and Conclusions
statisticians, this method can not be understood as easily asabsolute change or percentage change.

In this essay we compared the use of absolute change and From a clinical point of view, clinicians may prefer to
percentage change. According to the de…nition of statis-
choose the method that will show the health-improvement
tical power, we developed a ratio test statistic R under cer-
more obviously. Some researchers may choose the method
tain distribution assumption, which can help us decide which
that may be understood by most people that are interested
method will be used, absolute change or percentage change.

in his research. Sometimes, we may make a choice just based
When R > 1, absolute change has a higher statistical power.

on some empirical information. Most of the time, the choice
In that case, we prefer absolute change to percentage change.

depends more on the research work and the researcher's own
If R < 1, we choose percentage change.

Based on Vickers'(2001) simulation method, with the help
of the ratio test statistic, we did some simulations to com-pare the statistical power of the two methods. In contrastwith Vickers' (2001) conclusion that the percentage change
is statistical ine¢ cient, we simulated some datasets in whichpercentage change has higher statistical power, or has nearlythe same statistical power with absolute change. In this way,
Bonate P. (2000): Analysis of Pretest-Posttest Design.

we showed that percentage can be statistical e¢ cient under
Boca Raton: Chapman and Hall/CRC.

some conditions.

Kaiser L. (1989): Adjusting for baseline: change or
Another issue often concerned by researchers is the stan-
percentage change? Statistics in Medicine, 10: 1183-
dard deviation of change scores. There is a rule of thumb
that may help us get the standard deviation of change scoresquickly from the standard deviation of the baseline scores.

Kim M. et al. (2009): Comparison of epicardial, ab-
The general form of the rule of thumb for absolute change can
dominal and regional fat compartments in response to
be derived in a theoretical way. From the derivation in section
weight loss.

Nutr Metab Cardiovasc Dis, 1:7.

3, we know that, when the ratio m = V ar (F )
0:75, the empirical form of the rule of thumb
2 holds. We also checked these con-
avange L., Engels J. and Accurso F. (2007): An-
alyzing percent change in cystic …brosis clinical trials.

ditions in a practical way by collecting some real data to
The 21st Annual North American Cystic Fibrosis Con-
compare the real value and the value got from the rule of
ference, Anaheim, California.

thumb. And we got the same conclusion. So, if we knownothing about the correlation between baseline and follow-up
Neovius M. and Rössner S. (2007): Results from a
scores, we should be more careful to use the rule.

randomized controlled trial comparing two low-calorie
For percentage change, a rule of thumb does not exist.

diet formulae. Obesity Research & Clinical Practice, 1:
That is because the standard deviation of percentage change
depends on the baseline scores, and it is very hazardous tostate a rule. We also proved this by doing a simulation with
Törnqvist L., Vartia P. and Vartia Y. (1985): How
a simple transformation. The simulation result showed how
should relative changes be measured? American Statis-
the standard deviation of percentage change depends on the
tician, 39: 43-46.

baseline scores.

Vickers A. (2001): The use of percentage change from
In this essay, we didn't give any rules to make a choice
baseline as an outcome in a controlled trial is statis-
between absolute change and percentage change. We devel-
tically ine¢ cient: a simulation study.

oped a ratio test statistic which may be helpful, but it is not a
Research Methodology, 1:6.

good rule to tell us how to make a choice. That is because, inthe ratio test statistic, expected value should be used. For a
speci…c dataset in practice, we can not calculate the expected
Limwattananon S. and Gross C. (2007):
value. That is the limitation of the test statistic.

behavior therapy versus individual behavior therapy
Actually, there is not a most optimal method to tell us
for healthy dieting and weight control management in
which method to choose. From a statistical point of view,
overweight and obese women living in rural community.

we would like to choose the method with higher statistical
Obesity Research & Clinical Practice, 1: 223-232.

power. Beside the two change measurement methods from
Wilcoxon F. (1945): Individual comparisons by rank-
baseline, there are also some other methods. One of them
ing methods, Biometrics Bull., 1, 80.

is analysis of covariance, which is mentioned by both Vick-ers (2001) and Kaiser (1989). Bonate (2000) discussed thisZhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
#correlation coefficient
#percentage change
method="stack",main="Change from Baseline to Follow-up")for(i in (1:length(b))){
}mtext(side=1,line=3, "B N(200,20), P N(0.5,0.01)")
2 0 Responsible Programmer: Ling Zhang.

Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
#correlation coefficient
f<-b*r+y*(1-r 2) 0.5+200h<-rnorm(n,0.5,s)
#percentage change
#absolute value of tc, i.e.

#absolute value of tc, i.e.

#expected value of jtcj
#expected value of jtpj
#value of the ratio test statistic R
Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
#correlation coefficient
f<-b*r+y*(1-r 2) 0.5+200h<-rnorm(n,100,5)
method="stack",main="Change from Baseline to Follow-up")
for(i in (1:length(b))){
}mtext(side=1,line=3, "B N(200,20), C N(100,5)")
#correlation coefficient
f<-b*r+y*(1-r 2) 0.5+200h<-rnorm(n,100,s)
#absolute value of tc, i.e.

#absolute value of tp, i.e.

#expected value of jtcj
#expected value of jtpj
#value of the ratio test statistic R
Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
for(muc in c(100,50)){
#correlation coefficient
f<-b*r+y*(1-r 2) 0.5+200h<-rnorm(n,muc,10)
#absolute value of tc, i.e.

#absolute value of tp, i.e.

#expected value of jtcj
#expected value of jtpj
#value of the ratio test statistic R
Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
xlab="B N(200,SD(B)), C N(Mean(C),10)",ylab="Density",main="Distribution of R")
#correlation coefficient
#percentage change
#follow-up score before transformation
#follow-up score after transformation
#baseline score before transformation
#baseline score after transformation
Zhang, L. and Han, K. (2009)
How to Analyze Change from Baseline: Absolute or Percentage Change?
mtext(side=1,line=4, "B N(50,10), P N(0.1,0.02)")for(i in 1:21){
mtext(side=1,line=4, "B N(50,10), P N(0.1,0.02)")
Zhang, L. and Han, K. (2009)

Source: http://www.statistics.du.se/essays/D09_Zhang%20Ling%20&%20Han%20Kun.pdf

The Egyptian Cabinet Information and Decision Support Center Center for Future Studies Sustainable Cities in Egypt Learning from Experience: Potentials and Preconditions for New Cities in Desert Areas Dr. Nisreen El-Lahham Dr. Waleed Hussen September 2009 Sustainable Cities in Egypt

Review Ocular side-effects of urological Nikolaos A. Kostakopoulos, Vasileios G. Argyropoulos Department of Urology, IASO General Hospital, Athens, Greece Corresponding author: Nikolaos A. Kostakopoulos Email: 'kostako@doctor.com' Η αυξανόμενη γήρανση του πληθυσμού έχει σαν αποτέλεσμα συχνότερη εμφάνιση συμπτωμάτων καλοήθους υπερπλασίας του προστάτη, ασταθούς κύστεως και στυτικής δυσλειτουργίας. Αυτό έχει σαν επακόλουθο ευρεία συνταγογράφηση. φαρμάκων, όπως αναστολέων της φωσφοδιεστεράσης, αντιμουσκαρινικών παραγόντων και α-αναστολέων που προκαλούν μερικές φορές σοβαρές παρενέργειες από τους οφθαλμούς. Σ΄αυτή την ανασκόπηση θα αναφερθούν περιληπτικά οι διάφορες δυνητικές οφθαλμικές παρενέργειες, η συχνότητά τους, η φυσική τους ιστορία και η σημασία τους για τον κλινικό γιατρό.