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Crippa and Orsini BMC Medical Research Methodology (2016) 16:91
DOI 10.1186/s12874-016-0189-0
RESEARCH ARTICLE Open Access
Dose-response meta-analysis of
differences in means
Alessio Crippa1* andNicolaOrsini
1
Abstract
Background: Meta-analytical methods are frequently used to combine dose-response findings expressed in terms
of relative risks. However, no methodology has been established when results are summarized in terms of differences
in means of quantitative outcomes.
Methods: We proposed a two-stage approach. A flexible dose-response model is estimated within each study (first
stage) taking into account the covariance of the data points (mean differences, standardized mean differences).
Parameters describing the study-specific curves are then combined using a multivariate random-effects model
(second stage) to address heterogeneity across studies.
Results: The method is fairly general and can accommodate a variety of parametric functions. Compared to
traditional non-linear models (e.g. Emax, logistic), spline models do not assume any pre-specified dose-response curve.
Spline models allow inclusion of studies with a small number of dose levels, and almost any shape, even non
monotonic ones, can be estimated using only two parameters. We illustrated the method using dose-response data
arising from five clinical trials on an antipsychotic drug, aripiprazole, and improvement in symptoms in shizoaffective
patients. Using the Positive and Negative Syndrome Scale (PANSS), pooled results indicated a non-linear association
with the maximum change in mean PANSS score equal to 10.40 (95 % confidence interval 7.48, 13.30) observed for
19.32 mg/day of aripiprazole. No substantial change in PANSS score was observed above this value. An estimated
dose of 10.43 mg/day was found to produce 80 % of the maximum predicted response.
Conclusion: The described approach should be adopted to combine correlated differences in means of quantitative
outcomes arising from multiple studies. Sensitivity analysis can be a useful tool to assess the robustness of the overall
dose-response curve to different modelling strategies. A user-friendly R package has been developed to facilitate
applications by practitioners.
Keywords: Meta-analysis, Dose-response, Mean differences, Random-effects
Background
The identification and characterization of dose-response
relationships is an essential part of the analysis in many
scientific disciplines such as toxicology, pharmacology,
and epidemiology. This is particularly important in the
development and testing of new compounds (e.g. a new
drug, pharmaceutical treatment) where trials at different
stages aim to evaluate the efficacy of increasing levels of
dosage (Phase II-III trials) or to derive a dose-response
curve for selection of optimal doses (Phase IV trials) [1, 2].
*Correspondence: alessio.crippa@ki.se
1Department of Public Health Sciences, Karolinska Institutet, Stockholm,
Sweden
Randomized clinical trials often investigate a continu-
ous outcome variable, such as the efficacy or safety of
a drug, reporting the change from baseline of a medical
score, or the final value of a clinical measurement. The
dose-response results are typically summarized by dose-
specific means and standard deviations [3]. Measures of
effect are expressed in terms of mean or standardized
mean differences using a dose level, usually the placebo
group, as referent [1]. Over the last few years method-
ological research focused on developing and improving
methods for performing dose-response analysis in a sin-
gle study [4, 5]. A conclusive result is hardly obtained by a
single investigation and there is often the need to synthe-
size information collected from multiple studies. In such a
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Crippa and Orsini BMC Medical Research Methodology (2016) 16:91 Page 2 of 10
case meta-analytic methods can be used to define an over-
all relation or to investigate heterogeneity across study
findings.
A method for pooling aggregated dose-response data
where the outcome is a log relative risk was originally pre-
sented by Greenland and Longnecker in 1992 [6]. Since
then, several papers have refined and covered specific
aspects of the methodology such as model specification
[7, 8], modeling strategies [9, 10], and software implemen-
tation [11, 12]. Other methodological articles extended
the approach for continuous outcome but in the case
where individual patient data are available, mainly in the
context of time-series environmental studies [13–15].
Only a few alternatives have been proposed to pool
aggregated dose-response data where the findings are
summarized by differences in means. Davis and Chen
[16] in 2004 described a methodology for summariz-
ing dose-response curves of first and second generation
antipsychotics in schizoaffective patients. The authors
reconstructed drug-specific dose-response curves and
conducted a meta-analysis to compare the effectiveness of
medium vs high dosages. A common alternative to analyze
the drug effect consists of fitting a random-intercept Emax
model, where the random component accounts for het-
erogeneity in placebo effect across trials [17]. Heterogene-
ity, however, may be related to other study characteristics
rather than differences in placebo response such as imple-
mentation, participants, intervention, and outcome defi-
nition. Thomas et al. [18] adopted hierarchical Bayesian
models to summarize and describe, independently, the
distribution of study-specific model parameters derived
from an Emax model.
The mentioned strategies assumed pre-specified mod-
els that do not allow for non-monotonic curves which
may occur in practice [19], as in case of dose-response
data of antipsychotics. In addition, fitting study-specific
sigmoidal curves such as the Emax model requires
that the single studies have assessed at least three
dose levels in order to estimate model parameters.
Discarding studies not providing enough data points
represents a loss of information and may introduce
bias.
The aim of this paper is to formalize and propose a
general and flexible methodology to pool dose-response
relations from aggregated data where the changes in the
distribution of the quantitative outcome are expressed
in terms of differences in means. We first present the
data necessary for a dose-response meta-analysis and
derive formulas for obtaining effect sizes and their vari-
ance/covariance structure. We describe flexible dose-
response models with particular emphasis on regression
splines. The method is then applied to dose-response data
from clinical trials on use of aripiprazole and symptoms
improvement in schizoaffective patients.
Methods
Dose-response data
The notation and data required for a dose-response meta-
analysis for a generic study are displayed in Table 1. We
consider Istudies indexed by i=1, ...,Ireporting the
results of a common treatment at different dose levels
xij,j=1, ...,Ji,wherex0i=0 indicates the control or
placebo group in the i-th study. The study-specific results
typically consist of dose-specific means of an outcome
variable, Yij, that measures the efficacy of the j-th dose
in the i-th study [3]. Additional information about the
number of patients allocated in each treatment, nij ,and
the sample standard deviations of Yij,sdij , is generally
reported or obtained from the study-specific results.
Effect sizes and their variance/covariance
A common way to reduce heterogeneity in placebo
response is to compute the effect size (or treatment effect)
as difference between dose-specific means and placebo
mean. In case all studies measure the outcome on a com-
mon and interpretable scale, the difference can be based
on the absolute scale
dij =¯
Yij −¯
Yi0,j=1, ...,Ji,i=1, ...,I(1)
Assuming common study-specific population standard
deviations, the variance of dij is defined as
Var dij =nij +ni0
nijni0
s2
pi,j=1, ...,Ji,i=1, ...,I
(2)
where s2
pi=Ji
j=0nij −1sd2
ij/Ji
j=0nij −1is the
square of the pooled standard deviation for the i-th study.
Since the study-specific mean differences dij use the same
referent values, ¯
Yi0, they cannot be regarded as indepen-
dent. The covariance term is defined as
Cov dij,dij=Va r ¯
Yj0=s2
i0
ni0
,j= j,i=1, ...,I
(3)
Table 1 Notation for aggregated data in the i-th study used in
dose-response meta-analysis of differences in meas
dose mean(Y)asd(Y) nbdcVar (d)d∗dVar d∗
0¯
Yi0sdi0ni00– 0 –
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
xij ¯
Yij sdij nij dij Var dijd∗
ij Var d∗
ij
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
xiJi¯
YiJisdiJiniJidiJiVar diJid∗
iJiVar d∗
iJi
aY is the continuous outcome
bNumber of patients
cMean difference
dStandardized mean difference
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Crippa and Orsini BMC Medical Research Methodology (2016) 16:91 Page 3 of 10
In case the outcome is measured on different scales the
effect sizes can be based on standardized mean differences
d∗
ij =¯
Yij −¯
Yi0
spi
,j=1, ...,Ji,i=1, ...,I(4)
with
Var d∗
ij=1
nij
+1
ni0
+d∗
ij
2
2Ji
j=0nij
,j=1, ...,Ji,
i=1, ...,I
Cov d∗
ij,d∗
ij=1
ni0
+
d∗
ijd∗
ij
2Ji
j=0nij
,j= j,i=1, ...,I
(5)
Dose-response analysis
The chosen effect sizes and the corresponding
(co)variances are used to estimate the study-specific dose-
response curves. The dose-response curves characterize
the relative efficacy of the dose under investigation using
the placebo effect as referent (i.e. the relative efficacy
for the placebo is zero by definition). The dose-response
models are expressed through the parametric model f,
which specifies how the effect size varies according to the
dose values. The functional relationship fis parametrized
in terms of θi,thep×1 vector of dose-response coef-
ficients. We consider the case of mean differences, dij,
but the same principles apply for standardized mean
differences, d∗
ij. The study-specific curves can be written
as
di=f(xi,θi)+εi,εi∼N0,
ˆ
i,i=1, ...,I
(6)
ˆ
iis the covariance matrix of the residual error term,
with Var dijalong the diagonal and Cov dij,dijoff-
diagonal.
Several alternatives are available to model the dose-
response curve (i.e. for the choice of f). Table 2 shows
the definition of 4 types of models ordered according to
the number of parameters, ranging from 1 for a linear
modelupto3foralogisticmodel.SeeBretzetal.fora
comprehensive description [2].
The most common choice in dose findings [2] is the use
of the Emax model which is expressed in terms of three
Table 2 Frequently used dose-response models
Model Equation No. of parameters
Linear E di|xi=θ1ixi1
Quadratic E di|xi=θ1ixi+θ2ix2
i2
Emax Edi|xi=θ1ixθ3i
i/θ2i+xθ3i
i3
Logistic E di|xi=θ1i/{1+exp [(θ2i−xi)]/θ3i}3
parameters: the maximum effect (θ1i),thedosetopro-
duce half of the maximum effect (θ2i) and the steepness
of the curve (θ3i) [20]. As other non-linear models, the
Emax model assumes a specific shape that does not allow
for non-monotonic curve and its estimation requires at
least three non reference dose levels. Quadratic models
are defined by only p=2 coefficients butmay poorly fit at
extreme dose values [9]. Other non-linear models such as
logistic and sigmoidal models, are commonly defined by
p≥3 coefficients so that study-specific aggregated data
may not be sufficient to estimate the parameters.
We propose the use of regression splines to flexibly
model the dose of interest. Splines represent a family of
smooth functions that can describe a wide range of curves
(i.e. U-shaped, J-shaped, S-shaped, threshold) [21]. The
curves consist of piecewise polynomials over consecu-
tive intervals defined by kknots. Their use may facilitate
curve fitting since many non-linear curves can be exam-
ined by estimating only a small number of coefficients.
For instance, a restricted cubic spline model with three
knots k=(k1,k2,k3)is defined only in terms of p=2
coefficients [22]
E[di|xi]=θ1ix1i+θ2ix2i(7)
with two transformations [23] defined as
x1=x
x2=(x−k1)3
+−k3−k1
k3−k2(x−k2)3
++k2−k1
k3−k2(x−k3)3
+
(k3−k1)2
(8)
wherethe‘+’notation,withu+=uif u≥0andu+=0
otherwise, has been used.
An alternative flexible approach to model the dose-
response association is represented by fractional poly-
nomials. In particular, a dose-response model based on
fractional polynomial of order two can be written as in
Eq. 6 with the two transformations defined as
x1=xp1and x2=xp2if p1= p2
x1=xp1and x2=xp1log(x)if p1=p2(9)
for each combination of p1and p2in the predefined set of
values {−2, −1, −0.5, 0, 0.5, 1, 2, 3};forp=0, xpbecomes
log(x). The best fitting fractional polynomial is typically
chosen based on the Akaike’s Information Criterion [24].
Once the functional relation fhas been selected, gen-
eralized least square estimation can be performed to effi-
ciently estimates the dose-response coefficients
ˆ
θjand the
corresponding (co)variance matrix
ˆ
Vj, by minimizing
di−f(xi,θi)Tˆ
−1
idi−f(xi,θi)(10)
that generally requires numerical optimization algo-
rithms. If fis a linear combination of the parameters
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θi, as in the case of regression splines and fractional
polynomials, the close solution can be written as
ˆ
θi=XT
i
ˆ
−1
iXi−1
XT
i
ˆ
−1
idi
ˆ
Vi=Var ˆ
θi=XT
i
ˆ
−1
iXi−1(11)
where Xiindicates the Ji×pdesign matrix in the i-th study.
Meta-analysis
The estimated study-specific dose-response coefficients
ˆ
θiand the accompanying (co)variance matrices
ˆ
Viare
combined by means of multivariate meta-analysis
ˆ
θi∼Nθ,
ˆ
Vi+(12)
A fixed-effects model assumes no statistical hetero-
geneity among study results, i.e. differences in the dose-
response coefficients are only related to sampling error.
The assumption of homogeneity may not hold in prac-
tice, unless it is known that the studies are performed
in a similar way and are sampled from the same popula-
tion [25]. The Cochran’s Q test [26] is typically used to
test statistical heterogeneity across studies (H0:=0)
[27]. Selected studies, however, will typically differ with
respect to study design and implementation, selection of
participants, and type of analyses. A certain degree of het-
erogeneity is expected and should be taken into account
in the analysis. A random-effects model allows the dose-
response coefficients, θi, to vary across studies. Statistical
heterogeneity is captured by the between-studies variance
while θrepresents the mean of the distribution of dose-
response coefficients and an estimate,
ˆ
θ, can be obtained
using (restricted) maximum likelihood estimation [15].
As a final result, the pooled dose-response curve can be
presented in either a graphical or tabular form by predict-
ing the mean differences of the outcome for a set of xdose
values
E[
ˆ
d|x]=fX,
ˆ
θ(13)
with an approximate (1−α/2)% confidence interval (CI),
that in case fis a linear combination of θcan be expressed
as
E[
ˆ
d|x]∓zα
2diag fX,
ˆ
θTCov ˆ
θfX,
ˆ
θ(14)
where zα/2is the α/2-th quantile of a standard normal
distribution.
Dose findings
Once the pooled dose-response curve has been estimated,
it may be of interest to determine a set of target doses,
i.e. doses associated with prespecified outcome effects.
In development of new compounds it is often important
to select an optimal dose which is almost as effective as
the maximum effective dose but has less undesired side
effects, which often occur at high dosages. Suppose one
wants to determine which is the lowest dose (EDγ)topro-
duce an almost complete effect, e.g. γ%oftheobserved
maximum predicted response.
The EDγcan be determined as
EDγ=argmax
x∈(0,xmax]E[ ˆ
d|x]
E[ ˆ
d|xmax]≥γ(15)
where xmax is the dose corresponding to the maximum
predicted outcome.
An important step when presenting results from dose
findings analysis is to accompany the previous estimates
with a measure of precision, typically confidence intervals.
Pinheiro et al. [3] proposed the use a parametric bootstrap
approach based on the asymptotic normal distribution of
ˆ
θ, the pooled estimate of the dose-response coefficients.
The approach consists in re-sampling the dose-response
coefficients θfrom its approximate normal distribution
and derive the distribution of
EDγbased on the sam-
ples. Approximated confidence intervals for
EDγcan be
constructed using percentiles of the sampling distribution.
Results
To illustrate the methodology we examined the
dose-response relation between aripiprazole, a second-
generation antipsychotic, and symptoms improvement in
schizoaffective patients. We updated the search strategy
presented in a previous review by Davis and Chen [16]
by searching the Medline, International Pharmaceutical
Abstracts,CINAHL,andtheCochraneDatabaseofSys-
tematic Reviews. To reduce the exclusion of unpublished
papers, additional sources including Food and Drug
administration website, data from Cochrane reviews,
poster presentations and conference abstracts were also
searched. All random-assignment, double-blind, con-
trolled clinical trials of schizoaffective patients providing
dose-response results for at least two non-zero dosages of
aripiprazole were eligible.
Five studies [28–32] met the inclusion criteria and were
included in the analysis. All the studies reported mean
changes from baseline as main outcome variable, using
the Positive and Negative Syndrome Scale (PANSS). The
PANSS scale is an ordinal score derived from 30 items
ranging from 1 to 7. Computations of ratios such as per-
centage changes are not directly applicable and may lead
to erroneous results [33, 34]. To address this issue, the the-
oretical minimum (i.e. 30) needs to be subtracted from
the original score [35]. Information about the means, the
number of patients assigned to each treatment, and the
standard deviation was available from the published data.
Because all the studies measured the outcome variable on
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the same scale, we computed PANSS mean differences as
effect sizes. Data are reported in Table 3.
We used the trial by Cutler et al. [28] to illustrate the
steps required for estimating the dose-response curve for
a single study. For example, the difference in mean PANSS
comparing the dose of 2 mg/day relative to 0 mg/day
is d11 =8.23 −5.3 =2.93 mg/day. Its variance is
Var (d11 )=(85 +92)/(85 ×92)×s2
p1= 7.59, where
s2
p1=119, 419/356 =335.4. The covariance of this
difference in means PANSS is 18.312/85 =3.94. The vari-
ance/covariance structure associated with the vector of
differences in means for this trial d1can be presented in a
matrix form
ˆ
1=⎡
⎣
7.59
3.94 7.72
3.94 3.94 7.52 ⎤
⎦
Toestimatethedose-responsecurveweneedfirstto
specify the model f. We characterized the dose-response
relation using a restricted cubic spline model with three
knots located at the 10th, 50th, and 90th percentiles
(0, 10, and 30 mg/day) of the overall dose distribu-
tion (p= 2). The restricted cubic spline dose-response
model is defined as in Eq. 7. Efficient estimates of the
dose-response coefficients and (co)variance matrix were
obtained by generalized least square estimation
ˆ
θ1=(1.215, −5.738)T
ˆ
V1=Var ˆ
θ1=0.49
−3.65 31.64 (16)
We applied the same procedure to the other stud-
ies included in the meta-analysis in order to obtain the
study-specific
ˆ
θiand
ˆ
Vi,i=1, ..., 5 (Table 4). The study-
specific predicted curves are presented in Fig. 1. Under
a random-effects model, restricted maximum likelihood
estimates were
ˆ
θ=(0.937, −1.156)T
Cov ˆ
θ=0.03
−0.05 0.10 (17)
Ap-value <0.001 for the multivariate Wald-type test
H0:θ=0provided strong evidence against the null
hypothesis of no relation between different doses of arip-
iprazole and mean change PANSS score. The Q test (Q=
3.5, p-value = 0.899) did not detect substantial statistical
heterogeneity across studies.
To communicate results of the pooled dose-response
analysis, we can estimate the pooled mean differences in
PANSS scores using 0 mg/day as referent as 0.937x1−
1.156x2, together with the corresponding 95 % confidence
interval for a generic dose xof interest as following
(0.937x1−1.156x2)∓1.960.03x2
1+0.1x2
2−0.1x1x2
Table 3 Aggregated dose-response data of five clinical trials investigating effectiveness of different dosages of aripiprazole in
schizoaffective patients. The continuous outcome is measured on the Positive and Negative Syndrome Scale and summarized by
mean values (mean(Y)) and standard deviations (sd(Y))
ID Author, Year dose mean(Y) sd(Y) n d Var(d)
1 Cutler, 2006 [28] 0 5.300 18.310 85 0.000 0.000
2 8.230 18.320 92 2.930 7.593
5 10.600 18.310 89 5.300 7.715
10 11.300 18.320 94 6.000 7.515
2 McEvoy, 2007 [29] 0 2.330 26.100 107 0.000 0.000
10 15.040 27.600 103 12.710 13.344
15 11.730 26.200 103 9.400 13.344
20 14.440 25.900 97 12.110 13.764
3 Kane, 2002 [30] 0 2.900 24.280 102 0.000 0.000
15 15.500 26.490 99 12.600 12.038
30 11.400 22.900 100 8.500 11.977
4 Potkin, 2003 [31] 0 5.000 21.140 103 0.000 0.000
20 14.500 20.160 98 9.500 8.563
30 13.900 20.880 96 8.900 8.654
5 Study 94202 [32] 0 1.400 25.730 57 0.000 0.000
2 11.000 25.000 51 9.600 25.447
10 11.500 25.200 51 10.100 25.447
30 15.800 28.510 54 14.400 24.701
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Crippa and Orsini BMC Medical Research Methodology (2016) 16:91 Page 6 of 10
Table 4 Study-specific dose-response coefficients and corresponding covariances for different dose-response models considered in
the analysis
Model id ˆ
θ1ˆ
θ2Var ˆ
θ1Cov ˆ
θ1,ˆ
θ2Var ˆ
θ2
Restricted cubic splines 1 0.55 0.55 0.065 0.065 0.065
2 0.59 0.59 0.031 0.031 0.031
30.84−1.69 0.054 −0.12 0.38
40.47−0.54 0.021 −0.043 0.15
50.78−1.25 0.23 −0.62 1.8
Fractional Polynomials 1 17.47 −7.84 2.9e+02 −2.2e+02 1.8e+02
2 29.59 −12.42 2.5e+02 −1.6e+02 1e+02
3 32.12 −13.47 1.4e+02 −70 37
4 18.48 −6.42 1.8e+02 −93 49
5 21.97 −7.00 2.3e+02 −1.1e+02 55
Emax 1 8.13 3.13 27 24 36
2 11.39 0.00 38 35 42
3 10.54 0.00 37 53 1e+02
4 9.20 0.00 60 1.4e+02 3.6e+02
5 13.28 0.94 23 2.7 2.7
Quadratic 1 1.54 −0.09 0.96 −0.086 0.0083
21.54−0.05 0.35 −0.017 0.00089
31.40−0.04 0.18 −0.0055 0.00018
40.83−0.02 0.14 −0.0047 0.00017
51.08−0.02 0.51 −0.016 0.00051
Piecewise linear 1 0.55 0.065
2 0.59 0.031
30.84−1.69 0.054 −0.12 0.38
40.47−0.54 0.021 −0.043 0.15
50.78−1.25 0.23 −0.62 1.8
where x1and x2are defined as in Eq. 8. For instance,
the model-based predicted mean changes in PANSS score
compared to placebo were 4.52 (95 % CI: 2.96, 6.08) for
5 mg/day, 8.08 (95 % CI: 5.43, 10.73) for 10 mg/day, 9.95
(95 % CI: 6.97, 12.94) for 15 mg/day, 10.38 (95 % CI: 7.49,
13.27) for 20 mg/day, 9.84 (95 % CI: 6.86, 12.83) for 25
mg/day, and 8.83 (95 % CI: 5.11, 12.54) for 30 mg/day. The
pooled predicted dose-response curve together with the
confidence intervals and the model mean differences is
provided in Fig. 2.
The results indicated a statistically significant positive
association between increasing doses of aripiprazole and
the mean change in PANSS score with the maximum value
of 10.39 (95 % CI: 7.48, 13.30) observed at xmax = 19.32
mg/day. The model suggested a slight decrease in the pre-
dicted mean PANSS score for dosages greater than 20
mg/day. The estimated dose to produce 50 % and 80 % of
the predicted maximum effect were
ED50 =5.82 mg/day
(95 % CI: 5.10, 8.58) and
ED80 =10.43 mg/day (95 % CI:
9.02, 16.73).
Sensitivity analysis
A sensitivity analysis is often required to evaluate the
robustness of the pooled dose-response curve. In the
spline model, for example, the location of the knots may
affect the shape of the dose-response curve. Therefore
we considered alternative knots locations including differ-
ent combinations of the 10th, 25th, 50th, 75th and 90th
percentiles of the overall dose distribution (0, 0.5, 10,
18.75, and 30 mg/day). A graphical comparison is pre-
sented in the left panel of Fig. 3. The alternative curves
roughly described the same dose-response shape with
no substantial variation, all indicating an increase in the
mean change PANSS score up to 20 mg/day of arip-
iprazole. We can assess whether there is an increasing
trend above 20 mg/day by simply re-defining x2equal
to (x−20)+in Eq. 8; this approach is known as piece-
wise linear model. The rate of change in the PANSS
mean differences was negative and not statistically sig-
nificant (θ1+θ2=-0.284, p= 0.18) after 20 mg/day of
aripiprazole.
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Crippa and Orsini BMC Medical Research Methodology (2016) 16:91 Page 7 of 10
0 5 10 15 20 25 30
0
5
10
15
20
25
Aripiprazole (mg/day)
Mean Difference
Cutler 2006
0 5 10 15 20 25 30
0
5
10
15
20
25
Aripiprazole (mg/day)
Mean Difference
McEvoy 2007
0 5 10 15 20 25 30
0
5
10
15
20
25
Aripiprazole (mg/day)
Mean Difference
Kane 2002
0 5 10 15 20 25 30
0
5
10
15
20
25
Aripiprazole (mg/day)
Mean Difference
Potkin 2003
0 5 10 15 20 25 30
0
5
10
15
20
25
Aripiprazole (mg/day)
Mean Difference
Study 94202
Fig. 1 Study-specific mean differences in Positive and Negative Syndrome Scale score for increasing dosages of aripiprazole. The first author and
year of publication of the subjects included in the original analyses are reported. Black squares indicate the mean differences and whiskers their
95 % confidence interval. Ariprazole dosage was modeled with restricted cubic splines. Solid lines represent the estimated dose-response curves,
dashed lines the corresponding 95 % confidence intervals. The placebo group (dose = 0) served as the referent group
Fig. 2 Pooled dose-response association between aripiprazole and mean change in Positive and Negative Syndrome Scale score (solid line).
Aripiprazole dosage was modeled with restricted cubic splines in a random-effects model. Dash lines represent the 95 % confidence intervals for the
spline model. The placebo group (dose = 0) served as the referent group. Circles indicate observed mean differences in individual studies; size of
bubbles is proportional to precision (inverse of variance) of the mean differences. Right axis represents percentage of the maximum predicted effect
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Crippa and Orsini BMC Medical Research Methodology (2016) 16:91 Page 8 of 10
Fig. 3 Graphical sensitivity analysis for the pooled dose-response curves between aripiprazole and mean change in Positive and Negative Syndrome
Scale score. The placebo group (dose = 0) served as the referent group. Right axis represents percentage of the maximum predicted effect. Left
panel: different location of the three knots in a restricted cubic spline model. Right panel: different models, restricted cubic splines (solid line),
fractional polynomials (dashed line), quadratic polynomial (dotted line), and Emax model (dot-dashed line). Circles indicate observed mean differences
in individual studies; size of bubbles is proportional to precision (inverse of variance) of the mean differences
To evaluate the sensitivity of the dose-response curve
to the choice of the parametric model fadopted, instead,
we considered three alternatives: fractional polynomials;
quadratic; and Emax. Since two studies only had two non-
referent doses, the study-specific (sigmoidal) Emax models
as described in Table 2 cannot be estimated. A common
solution is to fix the steepness of the curve θ3to be 1, also
referred to as hyperbolic Emax [20].
The “best” fractional polynomials (p1=0.5, p2=1)
provided overall a similar dose-response curve when com-
pared to the spline model, with slightly higher value for the
maximum predicted response (right panel of Fig. 3). The
hyperbolic Emax had substantially higher predicted mean
differences for low values of the dose. The non-linear
model assumes a specific hyperbolic dose-response curve
that did not seem to fit the data and may be dependent
from the choice of fixing θ3to be 1. The dose-response
curve described by the quadratic model fall in between the
spline and the hyperbolic Emax curves.
Discussion
In this paper we proposed a statistical method to com-
bine differences in means of quantitative outcomes.
The method consists of dose-response models estimated
within each study (first stage) and an overall curve
obtained by pooling study-specific dose-response coef-
ficients (second stage). The covariance among study-
specific mean differences is taken into account in the first
stage analysis using generalized least square estimators,
while statistical heterogeneity across studies is allowed by
multivariate random-effects model in the second stage.
One major strength of the proposed method is that
it is fairly general and can accommodate different mod-
eling strategies, including non-linear ones described by
Pinheiro et al. [3]. Non-linear models, however, are
defined by at least three or four parameters, and hence
require an equal number of dose levels for each single
study included in the analysis. Given that some stud-
ies may have investigated a lower number of dose lev-
els, exclusion of these studies may result in substantial
loss of information. In addition, many non-linear models
assume a specific behaviour (e.g. monotonicity) requir-
ing a strong a priori information about the dose-response
curve. The choice of the parametric model is critical,
since it highly influences the final results [3]. Indeed, the
selection of the dose-response model should be informed
by subject-matter knowledge as well as understanding of
the research questions at hand. We presented the use of
regression splines as a flexible tool for modeling any quan-
titative exposure. The major advantage is that a variety
of curves, even non monotonic ones, can be estimated
using only two parameters. It is considered to be closed
to non-parametric regression, since no major assump-
tions about the shape of the curve are needed [9]. A
possible alternative is the use of fractional polynomials.
In comparing the two strategies, we did not find impor-
tant differences between the two strategies and concluded
that both are useful tools to characterize a (non-linear)
dose-response curve. Nonetheless a sensitivity analysis
is generally required to evaluate the robustness of the
combined results.
A possible limitation of the proposed methodology is
that it requires information about dose-specific means
and standard deviations. Studies providing other sum-
mary measures, such as dose-specific medians, would
not be included the analysis. The dose-response analysis
presented in Eq. 6 is based on the asymptotic normal dis-
tribution of the conditional mean effect size. Extension of
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Crippa and Orsini BMC Medical Research Methodology (2016) 16:91 Page 9 of 10
the introduced methodology to percentiles is not straight-
forward and may represent an interesting topic of future
research.
An additional limitation of aggregated dose-response
data is that supplementary information for approximat-
ing the covariance terms may not be available. Articles
may report directly mean or standardized mean differ-
ences and standard errors for non-referent dose groups.
Whenever the standard deviation for the outcome vari-
able in the control group (s2
i0) cannot be obtained, it
may be approximated using the pooled standard deviation
based on the non-referent dose levels (s2
pij ). Alternatively
a specific value may be imputed and a sensitivity anal-
ysis can be performed to evaluate how the results of
the meta-analysis vary for different values of s2
0j.Fur-
ther limitations relate to the general application of meta-
analysis based on aggregated data. These include restric-
tions in subgroup analyses, the impossibility of assessing
the appropriateness of individual analyses, and to har-
monize variable definitions and analyses for reducing the
extent of heterogeneity, as well as specific biases such as
aggregation (or ecological) bias in meta-regression mod-
els. Meta-analysis of individual patient data, however, are
often difficult to undertake especially for the availabil-
ity of individual data, so that usage of aggregated data
may represent the only alternative [36]. Specific to aggre-
gated dose-response data, different dose references and
exposure range may complicate the analysis. The pre-
sented methodology assume that all the selected studies
share a common dose-response model. Important depar-
ture from this assumption may limit and/or impact the
pooling of individual dose-response coefficients. An alter-
native methods has been proposed based on a series of
univariate meta-analyses of effect sizes for a pre-specified
grid of dose-levels [37]. Further work is needed to analyze
this possibility and potential advantages. Depending on
the extent of heterogeneity of the dose-response curves,
however, it may not be opportune to pool study-specific
results, and meta-regression or stratified analyses should
be performed [38].
In our application, we considered the effectiveness
of increasing dosages of aripiprazole in shizoaffective
patients. We described the steps needed to obtain the
overall dose-response curve and to present it in a graph-
ical form. We observed a non-linear association with the
maximum efficacy corresponding to aripiprazole 19.32
mg/day. An estimated dose of 10.46 mg/day, however,
may be sufficient to obtain 80 % of the maximum effect,
which may be relevant for avoiding possible undesired
side effects. Sensitivity analysis showed similar results as
compared to fractional polynomials. The Emax model pre-
sented higher drug efficacy for low dosages. Compared to
the previous models, the Emax model did not seem to fit
properly the data at low dosages.
Conclusions
We described an approach to combine differences in
means of a quantitative outcome contrasting different
dose levels relative to a placebo in randomized trials.
The general framework of the proposed methodology can
include a variety of flexible models. Sensitivity analysis
can be a useful tool to assess the stability of the over-
all dose-response curve to different modelling strategies.
Although the method was presented for the analysis of
randomized trials, it may be extended to observational
studies where mean differences are further adjusted for
potential confounders. Future work is needed to evalu-
ate the properties of the statistical model and validity of
the underlying assumptions. A user friendly procedure is
implemented in the dosresmeta R package [39] with
worked examples available on GitHub.
Abbreviations
PANSS, positive and negative syndrome scale
Acknowledgements
We are grateful to Dr. Stefan Leucht and John M. Davis for providing the data
and raising the methodological question under study.
Funding
This work was supported by Karolinska Institutet’s funding for doctoral
students (KID-funding) (AC) and by a Young Scholar Award from the
Karolinska Institutet’s Strategic Program in Epidemiology (SfoEpi) (NO).
Availability of data and materials
The data on the effectiveness of aripiprazole are publicly available and also
contained in dosresmeta R package on github [39] (https://github.com/
alecri/dosresmeta/blob/master/data/ari.rda).
Authors’ contributions
AC developed the methods and prepared a draft. NO provided critical reviews,
corrections and revisions. Both authors read and approved the final version of
the manuscript.
Authors’ information
AC is a PhD student in Epidemiology and Biostatistics. Dr. NO is Associate
Professor of Medical Statistics.
Competing interests
The authors declare that they have no competing interest.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Received: 27 October 2015 Accepted: 13 July 2016
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