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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.
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Crippa and Orsini BMC Medical Research Methodology (2016) 16:91
DOI 10.1186/s12874-016-0189-0
Dose-response meta-analysis of
differences in means
Alessio Crippa1* andNicolaOrsini
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
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].
1Department of Public Health Sciences, Karolinska Institutet, Stockholm,
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
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
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.
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
pi,j=1, ...,Ji,i=1, ...,I
where s2
j=0nij 1sd2
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 ¯
,j= j,i=1, ...,I
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)ddVar d
Yi0sdi0ni00– 0
xij ¯
Yij sdij nij dij Var dijd
ij Var d
YiJisdiJiniJidiJiVar diJid
iJiVar d
aY is the continuous outcome
bNumber of patients
cMean difference
dStandardized mean difference
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In case the outcome is measured on different scales the
effect sizes can be based on standardized mean differences
ij =¯
Yij ¯
,j=1, ...,Ji,i=1, ...,I(4)
Var d
,j=1, ...,Ji,
i=1, ...,I
Cov d
,j= j,i=1, ...,I
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
i,i=1, ...,I
iis the covariance matrix of the residual error term,
with Var dijalong the diagonal and Cov dij,dijoff-
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
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
Emax Edi|xi=θ1ixθ3i
Logistic E di|xi=θ1i/{1+exp [2ixi)]3i}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
p3 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]
with two transformations [23] defined as
wherethe‘+’notation,withu+=uif u0andu+=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
θjand the
corresponding (co)variance matrix
Vj, by minimizing
that generally requires numerical optimization algo-
rithms. If fis a linear combination of the parameters
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Crippa and Orsini BMC Medical Research Methodology (2016) 16:91 Page 4 of 10
θi, as in the case of regression splines and fractional
polynomials, the close solution can be written as
Vi=Var ˆ
where Xiindicates the Ji×pdesign matrix in the i-th study.
The estimated study-specific dose-response coefficients
θiand the accompanying (co)variance matrices
combined by means of multivariate meta-analysis
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,
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
with an approximate (1α/2)% confidence interval (CI),
that in case fis a linear combination of θcan be expressed
2diag fX,
θTCov ˆ
where zα/2is the α/2-th quantile of a standard normal
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
x(0,xmax]E[ ˆ
E[ ˆ
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
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.
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
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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Crippa and Orsini BMC Medical Research Methodology (2016) 16:91 Page 5 of 10
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
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
3.94 7.72
3.94 3.94 7.52
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
θ1=(1.215, 5.738)T
V1=Var ˆ
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
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
θ=(0.937, 1.156)T
Cov ˆ
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
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
θ2Var ˆ
θ1Cov ˆ
θ2Var ˆ
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
30.841.69 0.054 0.12 0.38
40.470.54 0.021 0.043 0.15
50.781.25 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
21.540.05 0.35 0.017 0.00089
31.400.04 0.18 0.0055 0.00018
40.830.02 0.14 0.0047 0.00017
51.080.02 0.51 0.016 0.00051
Piecewise linear 1 0.55 0.065
2 0.59 0.031
30.841.69 0.054 0.12 0.38
40.470.54 0.021 0.043 0.15
50.781.25 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 (x20)+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
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0 5 10 15 20 25 30
Aripiprazole (mg/day)
Mean Difference
Cutler 2006
0 5 10 15 20 25 30
Aripiprazole (mg/day)
Mean Difference
McEvoy 2007
0 5 10 15 20 25 30
Aripiprazole (mg/day)
Mean Difference
Kane 2002
0 5 10 15 20 25 30
Aripiprazole (mg/day)
Mean Difference
Potkin 2003
0 5 10 15 20 25 30
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
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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.
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
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
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.
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.
PANSS, positive and negative syndrome scale
We are grateful to Dr. Stefan Leucht and John M. Davis for providing the data
and raising the methodological question under study.
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] (
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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... Two types of analyses were carried out in this meta-analysis. First, we performed a random-effects dose-response meta-analysis to estimate the change in blood lipids for each 10 g/d increments in olive oil consumption in each primary trial according to the method introduced by Crippa and Orsini (25) . This method requires the dose of olive oil consumption (g/d) in each study arm, the number of participants in intervention and control groups, and the reported mean and SD of change in TC, LD-C, HDL-C, and triglycerides. ...
... We used visual inspection of funnel plots for testing publication bias when more than 10 trials were available for the analyses. Second, we performed a random-effects dose-response meta-analysis to clarify the shape of the dose-response effects of olive oil intake on blood lipids (25) . We used STATA version 16.0 for conducting our statistical analyses. ...
We performed a systematic review and dose-response meta-analysis of randomized trials on the effects of olive oil consumption on blood lipids in adults. A systematic search was performed in PubMed, Scopus, and Web of Science databases until May 2021. Randomized controlled trials (RCT) evaluating the effect of olive oil intake on serum total cholesterol (TC), triglyceride, low- (LDL-C) and high-density lipoprotein cholesterol (HDL-C) in adults were included. The mean difference (MD) and 95%CI were calculated for each 10 g/d increment in olive oil intake using a random effects model. A total of 34 RCTs with 1730 participants were included. Each 10 g/d increase in olive oil consumption had minimal effects on blood lipids including TC (MD: 0.79 mg/dL; 95%CI: -0.08, 1.66; I ² =57%; n=31, GRADE=low certainty), LDL-C (MD: 0.04 mg/dL, %95CI: -1.01, 0.94; I ² =80%; n=31, GRADE=very low certainty), HDL-C (MD: 0.22 mg/dL; %95CI: -0.01, 0.45; I ² =38%; n=33, GRADE=low certainty), and triglycerides (MD: 0.39 mg/dL; 95%CI: -0.33, 1.11; I ² =7%; n=32, GRADE=low certainty). Levels of TC increased slightly with the increase in olive oil consumption up to 30 g/d (MD 30g/d : 2.76 mg/dL, 95%CI: 0.01, 5.51), and then appeared to plateau with a slight downward curve. A trivial nonlinear dose-dependent increment was seen for HDL-C, with the greatest increment at 20 g/d (MD 20g/d : 1.03 mg/dL, 95%CI: -1.23, 3.29). Based on existing evidence, olive oil consumption had trivial effects on levels of serum lipids in adults. More large-scale randomized trials are needed to present more reliable results.
... The dose units for dairy consumption were unified to 200 ml or 200 g according to the average size of several portion sizes (19)(20)(21). The present study aimed to pool three or more sets of continuous or categorical data using dose-response meta-analyses proposed by Crippa et al. (22,23). When the studies included in the meta-analysis all had more than two exposure groups, the two-stage method was used; when any of the included studies had fewer than three exposure groups, the one-stage method was used (23). ...
... Furthermore, compared with the previous systematic reviews, we not only conducted meta-analyses, but also discussed the dose-response relationships between dairy consumption and birth outcomes (16,17). Particularly, in addition to dose-response meta-analysis for categorical variables, the method of the dose-response metaanalysis of differences in means was also applied (22). Moreover, the present study used the one-stage approach that no longer excluded studies with fewer than three exposure groups, and thus more relevant studies were included for aggregating data (23). ...
Full-text available
Purpose This study aimed to systematically review current evidence and quantitatively evaluate the associations between milk or dairy consumption during pregnancy and birth outcomes. Methods This systematic review had been reported in accordance with the guidelines of Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) statement. A supplementary literature search in PubMed, Web of Science, Cochrane Library, and Embase was conducted on 30 March 2021. Studies that assessed the association of maternal consumption of milk or dairy with birth-related outcomes were identified. The dose-response meta-analyses of continuous data and categorical data were applied. One-stage approach and two-stage approach were used where appropriate. Results In total, 42 studies were eligible for the present systematic review, and 18 of them were included in the outcome-specific meta-analyses. The dose-response meta-analysis [Number of studies ( N ) = 9] predicted a maximum mean change in birthweight of 63.38 g [95% Confidence Interval (CI) = 0.08, 126.67] at 5.00 servings per day. Intake of dairy products had the greatest protective effect on small for gestational age at a maximum of 7.2 servings per day [Relative risk (RR) = 0.69, 95% CI = 0.56, 0.85] ( N = 7). The risk of large for gestational age was predicted to be maximum at 7.20 servings per day of dairy consumption, with the RR and 95% CI of 1.30 (1.15, 1.46; N = 4). In addition, the relationship between dairy consumption and low birth weight (RR = 0.70, 95% CI = 0.33, 1.50; N = 5) and pre-mature birth (RR = 1.13, 95% CI = 0.87, 1.47; N = 5) was not significant, respectively. Conclusions Maternal consumption of dairy during pregnancy has a potential effect on fetal growth. Further well-designed studies are warranted to clarify the specific roles of individual dairy products. Systematic Review Registration identifier: PROSPERO 2020 CRD42020150608
... When the highest or lowest category was open-ended, we assumed that the open-ended interval length was the same as the adjacent interval when estimating the midpoint [18].). In our data set, as the reference dose of exposures varied from study to study, the data first had to be centered [19]. Taking the average of the lowest dose of each study as the initial value of the exposure dose level, a restricted cubic spline model with knots was used to fit the potential nonlinear dose-response relationship. ...
Full-text available
Aim At present, the relationship between serum homocysteine (Hcy), fibrinogen (FIB), lipoprotein-a (LPa), and PAD is uncertain, and there has been no meta-analysis to establish the dose–response relationship between their exposure levels and PAD. Methods and results Relevant literature published in PubMed, Embase, and Web of Science was retrieved. The robust error meta-regression method was used to assess the linear and non-linear dose–response relationship between exposure level and PAD risk. A total of 68 articles, involving 565,209 participants, were included. Combined with continuous variables, the serum Hcy, FIB, and LPa levels of PAD patients were significantly higher than those of healthy individuals. The odds ratios (ORs) of PAD for individuals with high Hcy, FIB, and LPa levels compared with those with low levels were 1.47, 1.14, and 1.76, respectively. The study also showed that circulating Hcy, FIB, and LPa were significantly elevated in patients with PAD compared with controls. The level of Hcy and the risk of PAD presented a U-shaped distribution. The nonlinear dose–response model showed that each 1 μmol/L increase in serum Hcy increased the risk of PAD by 7%. Similarly, for each 10 mg/dL FIB and 10 mg/dL LPa increases, the risk of PAD increased by 3% and 6%, respectively. Conclusions This meta-analysis provided evidence that elevated Hcy, PIB, and LPa levels may increase the risk of PAD, and the risk of PAD increases with the increase in serum exposure within a certain range. By controlling Hcy level, the incidence of PAD may be reduced to control the PAD growing epidemic. Trial registration number: PROSPERO (CRD42021250501),
... Then, the specific effect sizes were pooled by a random-effects model. Finally, a nonlinear dose-response metaanalysis was applied to determine the shape of the effect of different doses of DHA/EPA on albumin, and on the 21 Statistical analyses were performed using STATA software version 16. A two-tailed P value < .05 was considered significant. ...
Context Low serum albumin and pre-albumin concentrations are associated with edema, infection, thrombosis, heart failure, and mortality. Objective This comprehensive systematic review and meta-analysis of clinical trials was conducted to summarize the available findings on the impact of omega-3 supplementation on albumin, pre-albumin, and the C-reactive protein/albumin ratio in hospitalized patients. Data sources PubMed, Web of Science, Scopus, and Google Scholar databases were searched from January 1990 to October 2021. Data Extraction Extracted data from 50 randomized controlled trials (RCTs) with a total number of 3196 participants were analyzed using the random-effects model. The dose-dependent effect was also evaluated. Data Analysis Oral omega-3 supplementation significantly increased serum albumin concentrations in patients with cancer (weighted mean difference [WMD]: 0.19; 95% CI: 0.05, 0.33, P= 0.006), patients on dialysis (WMD: 0.14; 95% CI: 0.01, 0.28, P= 0.042), and those with hypoalbuminemia (WMD: 0.38; 95% CI: 0.03, 0.72, P = 0.033); however, there was no significant effect among patients with gastrointestinal or hepatologic diseases. Moreover, each 1000 mg/day increase in oral omega-3 supplementation resulted in elevated serum albumin levels in cancer patients (WMD: 0.15; 95% CI: 0.07, 0.24, P < 0.001). In addition, a favorable effect of oral omega-3 supplementation on pre-albumin levels was observed among patients with cancer (WMD: 33.87; 95% CI: 12.34, 55.39, P = 0.002). A similar significant effect of parenteral omega-3 supplementation on pre-albumin concentrations was seen among those with gastrointestinal and hepatologic diseases as well (WMD: 23.30; 95% CI: 13.58, 33.03, P < 0.001). No significant effect of oral omega-3 supplementation on the CRP/albumin ratio was found. Conclusions Overall, omega-3 fatty acids supplementation resulted in a favorable change in serum albumin and pre-albumin concentrations in hospitalized patients. Systematic Review Registration PROSPERO registration no. CRD42021285704.
... ). Continuous variable (e.g., body weight) comparison was performed by ES of standard mean difference (SMD) [Equation 2](Viechtbauer 2010;Crippa and Orsini 2016;G. Ma and Chen 2020;G. ...
Abnormal fetal growth increases risks of childhood health complications. Vitamin A supplementation (VAS) is highly accessible, but literature inconsistency regarding effects of maternal VAS on fetal and childhood growth outcomes exists, deterring pregnant women from VAS during pregnancy. This meta-analysis aimed to analyze effects of vitamin A only or vitamin A + co-intervention during pregnancy in healthy mothers (MH) or with complications (MC, night blindness and HIV positive) on perinatal growth outcomes, also assess VAS dose impacts. The Cochrane Library, PubMed, ScienceDirect, Scopus, Embase and Web of Science databases were searched from inception to July 15, 2021. We covered subgroup analyses, including VAS in MH or MC within randomized controlled trial (RCT) or observational studies (OS). Fifty-five studies were included in this meta-analysis (426,098 pregnancies). Vitamin A decreased risk of preterm birth by 9% in MH-RCT (P < 0.001), by 62% in MH-OS (P = 0.029), by 10% in MC-RCT (P = 0.089); decreased LBW by 24% in MC-RCT (P = 0.032); increased neonatal weight in MC-RCT (SMD 0.96; P = 0.051). Besides, vitamin A + co-intervention decreased risks of preterm by 18% in MH-OS (P = 0.021); LBW by 25% in MH-OS (P < 0.001); by 32% in MC-RCT (P = 0.006); decreased neonatal defects by 33% in MH-OS (P = 0.064); decreased anemia by 25% in MH-OS (P = 0.0003); increased neonatal weight in MH-OS (SMD 0.51; P = 0.014); and increased neonatal length in MH-OS (SMD 1.83; P = 0.013). Meta-regression of VAS dose with individual outcomes was not significant, and no side effects were observed for VAS doses up to 4000 mcg (RAE/d). Regardless of maternal health conditions, VAS during pregnancy can safely and effectively improve fetal development and neonatal health even in mothers without VAD.
Context: Previous meta-analyses have suggested that the effects of coenzyme Q10 (CoQ10) on lipid profiles remain debatable. Additionally, no meta-analysis has explored the optimal intake of CoQ10 for attenuating lipid profiles in adults. Objective: This study conducted a meta-analysis to determine the effects of CoQ10 on lipid profiles and assess their dose-response relationships in adults. Methods: Databases (Web of Science, PubMed/Medline, Embase, and the Cochrane Library) were systematically searched until August 10, 2022. The random effects model was used to calculate the mean differences (MDs) and 95% CI for changes in circulating lipid profiles. The novel single-stage restricted cubic spline regression model was applied to explore nonlinear dose-response relationships. Results: Fifty randomized controlled trials with a total of 2794 participants were included in the qualitative synthesis. The pooled analysis revealed that CoQ10 supplementation significantly reduced total cholesterol (TC) (MD -5.53 mg/dL; 95% CI -8.40, -2.66; I2 = 70%), low-density lipoprotein cholesterol (LDL-C) (MD -3.03 mg/dL; 95% CI -5.25, -0.81; I2 = 54%), and triglycerides (TGs) (MD -9.06 mg/dL; 95% CI -14.04, -4.08; I2 = 65%) and increased high-density lipoprotein cholesterol (HDL-C) (MD 0.83 mg/dL; 95% CI 0.01, 1.65; I2 = 82%). The dose-response analysis showed an inverse J-shaped nonlinear pattern between CoQ10 supplementation and TC in which 400-500 mg/day CoQ10 largely reduced TC (χ2 = 48.54, P < .01). Conclusion: CoQ10 supplementation decreased the TC, LDL-C, and TG levels, and increased HDL-C levels in adults, and the dosage of 400 to 500 mg/day achieved the greatest effect on TC.
Context: There is still controversy over the effect of vitamin D3 supplementation on bone health. Objective: The effects of vitamin D3 supplementation on bone mineral density (BMD) and markers of bone turnover, as well as the dose-response relationship between vitamin D3 and bone health in adults, were evaluated. Data sources: The PubMed, Scopus, Cochrane, Web of Science, and AGRIS databases were searched for articles published through April 30, 2022. Thirty-nine of the 6409 records identified met the inclusion criteria. Data extraction: Data were extracted from articles by 2 authors, and data extraction was cross-checked independently. A random-effects model was used to estimate the pooled effect size and the associated 95%CI for the effect of vitamin D3 for each outcome. A one-stage random-effects dose-response model was used to estimate the dose-response relationship between vitamin D3 supplementation and BMD. Data analysis: Results of meta-analysis showed a beneficial effect of vitamin D3 at the lumbar spine (standardized mean difference [SMD] = 0.06; 95%CI, 0.01-0.12) and femoral neck (SMD = 0.25; 95%CI, 0.09-0.41). Dose-response analysis revealed a linear relationship between vitamin D3 supplementation doses and BMD at the femoral neck, lumbar spine, and total hip sites. No significant effect of vitamin D3 supplementation on whole-body or total hip BMD was observed (P > 0.05). Vitamin D3 supplementation significantly decreased BMD at both proximal and distal forearm (SMD = -0.16; 95%CI, -0.26 to -0.06). The variables of ethnicity, age, baseline 25-hydroxyvitamin D (25[OH]D), menopause status, vitamin D3 dosing frequency, and bone health status (P interaction = 0.02) altered the effect of vitamin D3 supplementation on BMD. Additionally, a nonlinear relationship between vitamin D3 supplement doses and markers of bone turnover was found. Conclusion: A protective effect of vitamin D3 supplementation on BMD of the lumbar spine, femoral neck, and total hip is implicated. Systematic review registration: PROSPERO registration number CRD42017054132.
Background There are controversial findings regarding the effect of vinegar on blood pressure based on the evidence accumulated so far. Methods A systematic search was conducted through PubMed, Scopus, and ISI Web of Science up to April 2022. We estimated the change in blood pressure for each 30 ml/d increments in vinegar consumption in each trial and then, calculated the mean difference (MD) and 95%CI using a fixed-effects model. A dose-response meta-analysis of differences in means provided us with the estimation of the dose-dependent effect. The certainty of evidence was rated by the GRADE tool. Results Each 30 ml/d increment in vinegar consumption reduced SBP by -3.25 mmHg (95%CI: -5.54, -0.96; I² = 67.5%, GRADE=low). Levels of SBP decreased linearly and slightly (Pnonlinearity = 0.69, Pdose-response = 0.02) up to vinegar consumption of 30 ml/d (MD30ml/d: -3.36, 95%CI: -5.77, -0.94). Each 30 ml/d increment in vinegar consumption reduced DBP by -3.33 mmHg (95%CI: -4.16, -2.49; I² = 57.1%, GRADE=low). Levels of DBP decreased linearly and slightly (Pnonlinearity = 0.47, Pdose-response = 0.004) up to vinegar consumption of 30 ml/d (MD30ml/d: -2.61, 95%CI: -4.15, -1.06) Conclusions According to the findings, vinegar significantly reduces systolic and diastolic blood pressure and may be considered an adjunct to hypertension treatment. Thus, clinicians could incorporate vinegar consumption as part of their dietary advice for patients.
Characterising the change in the risk of a health‐related outcome according to the levels of an exposure is increasingly popular in systematic reviews. We provide an introduction to the statistical methods currently used to perform linear and non‐linear dose‐response analysis based on aggregated data from multiple studies. We explain how dose‐response associations are estimated within a study and summarised across studies. A re‐analysis of observational studies about coffee consumption and mortality is used to illustrate how to move beyond a linear dose‐response trend, and the complexity of making inference on relative risks when using restricted cubic splines.
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Objective: To develop an evidence base for recommendations on the use of atypical antipsychotics for patients with schizophrenia. Design: Systematic overview and meta-regression analyses of randomised controlled trials, as a basis for formal development of guidelines. Subjects: 12 649 patients in 52 randomised trials comparing atypical antipsychotics (amisulpride, clozapine, olanzapine, quetiapine, risperidone, and sertindole) with conventional antipsychotics (usually haloperidol or chlorpromazine) or alternative atypical antipsychotics. Main outcome measures: Overall symptom scores. Rate of drop out (as a proxy for tolerability) and of side effects, notably extrapyramidal side effects. Results: For both symptom reduction and drop out, there was substantial heterogeneity between the results of trials, including those evaluating the same atypical antipsychotic and comparator drugs. Meta-regression suggested that dose of conventional antipsychotic explained the heterogeneity. When the dose was </=12 mg/day of haloperidol (or equivalent), atypical antipsychotics had no benefits in terms of efficacy or overall tolerability, but they still caused fewer extrapyramidal side effects. Conclusions: There is no clear evidence that atypical antipsychotics are more effective or are better tolerated than conventional antipsychotics. Conventional antipsychotics should usually be used in the initial treatment of an episode of schizophrenia unless the patient has previously not responded to these drugs or has unacceptable extrapyramidal side effects.
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An increasing number of quantitative reviews of epidemiological data includes a doseresponse analysis. Aims of this paper are to describe the main aspects of the methodology and to illustrate the novel R package dosresmeta developed for multivariate dose-response meta-analysis of summarized data. Specific topics covered are reconstructing covariances of correlated outcomes; pooling of study-specific trends; flexible modeling of the exposure; testing hypothesis; assessing statistical heterogeneity; and presenting in either a graphical or tabular way the overall dose-response association.
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It is a major goal of clinical pharmacology to understand the dose-effect relationship in therapeutics. Much progress towards this goal has been made in the last 2 decades through the development of pharmacokinetics as a discipline. The study of pharmacokinetics seeks to explain the time course of drug concentration in the body. Recognition of the crucial concepts of clearance and volume of distribution has provided an important link to the physiological determinants of drug disposition. Mathematical models of absorption, distribution, metabolism and elimination have been extensively applied, and generally their predictions agree remarkably well with actual observations. However, the time course of drug concentration cannot in itself predict the time course or magnitude of drug effect. When drug concentrations at the effect site have reached equilibrium and the response is constant, the concentration-effect relationship is known as pharmacodynamics. Mathematical models of pharmacodynamics have been used widely by pharmacologists to describe drug effects on isolated tissues. The crucial concepts of pharmacodynamics are potency — reflecting the sensitivity of the organ or tissue to a drug, and efficacy — describing the maximum response. These concepts have been embodied in a simple mathematical expression, the Emax model, which provides a practical tool for predicting drug response analogous to the compartmental model in pharmacokinetics for predicting drug concentration. The application of pharmacodynamics to the study of drug action in vivo requires the linking of pharmacokinetics and pharmacodynamics to predict firstly the dose-concentration, and then the concentration-effect relationship. This may be done directly by equating the concentration predicted by a pharmacokinetic model to the effect site concentration, but this simplistic approach is often not appropriate for various reasons, including delay in drug equilibrium with the receptor site, use of indirect measures of drug action, the presence of active metabolites, or homeostatic responses, thus often necessitating the use of more complex models. The relative pharmacodynamic bioavailability of different preparations of the same drug may be determined from the time course of a drug effect. Bioavailability determined in this way may differ markedly from bioavailability defined by measurements of drug concentration if active metabolites are formed or if effects are produced in the non-linear region of the concentration-effect relationship. The influence of changes in the extent of plasma protein binding may be important in the interpretation of drug concentration measurements since it is generally held that only the unbound fraction is pharmacologically active. Clear examples of this phenomenon are few, but this reflects the general paucity of adequate observations rather than casting doubt on the usual assumption. The design of rational dosing regimens for clinical therapeutics cannot be performed with a knowledge of pharmacokinelics alone. The time course of drug effect may be essentially independent of concentration when a dose produces near maximal effects throughout the dosing interval. If effects are between 20 and 80% of maximum, the response will decrease linearly even though concentrations are declining exponentially. Finally, at relatively small degrees of effect, the time course of drug effect and concentration will be in parallel. The usual ‘rule of thumb’ of dosing every half-life is a conservative strategy for limiting wide fluctuations in drug effect, but demands more from the patient in terms of dosing frequency than may be necessary to achieve consistent drug action. On the other hand, if therapeutic success is dependent more on cumulative response than moment to moment activity, the use of extended dosing intervals may markedly reduce the effectiveness of the same average dose. Considerations of these factors can be incorporated into a dosing scheme by combined application of the principles of pharmacokinelics and pharmacodynamics.
Low‐dimensional parametric models are well understood, straightforward to communicate to other workers, have very smooth curves and may easily be checked for consistency with background scientific knowledge or understanding. They should therefore be ideal tools with which to represent smooth relationships between a continuous predictor and an outcome variable in medicine and epidemiology. Unfortunately, a seriously restricted set of such models is used routinely in practical data analysis – typically, linear, quadratic or occasionally cubic polynomials, or sometimes a power or logarithmic transformation of a covariate. Since their flexibility is limited, it is not surprising that the fit of such models is often poor. Royston and Altman's recent work on fractional polynomials has extended the range of available functions. It is clearly crucial that the chosen final model fits the data well. Achieving a good fit with minimal restriction on the functional form has been the motivation behind the major recent research effort on non‐parametric curve‐fitting techniques. Here I propose that one such model, a (possibly over‐fitted) cubic smoothing spline, may be used to define a suitable reference curve against which the fit of a parametric model may be checked. I suggest a significance test for the purpose and examine its type I error and power in a small simulation study. Several families of parametric models, including some with sigmoid curves, are considered. Their suitability in fitting regression relationships found in several real data sets is investigated. With all the example data sets, a simple parametric model can be found which fits the data approximately as well as a cubic smoothing spline, but without the latter's tendency towards artefacts in the fitted curve. Copyright © 2000 John Wiley & Sons, Ltd.
This paper presents a command, glst, for trend estimation across different exposure levels for either single or multiple summarized case-control, incidence-rate, and cumulative incidence data. This approach is based on constructing an approximate covariance estimate for the log relative risks and estimating a corrected linear trend using generalized least squares. For trend analysis of multiple studies, glst can estimate fixed- and random-effects metaregression models. Copyright 2006 by StataCorp LP.
This article reports the results of a meta-analysis based on dose–response studies conducted by a large pharmaceutical company between 1998–2009. Data collection targeted efficacy endpoints from all compounds with evidence of clinical efficacy during the time period. Safety data were not extracted. The goal of the meta-analysis was to identify consistent quantitative patterns in dose–response across different compounds and diseases. The article presents summaries of the study designs, including the number of studies conducted for each compound, dosing range, the number of doses evaluated, and the number of patients per dose. The Emax model, ubiquitous in pharmacology research, was fit for each compound. It described the data well, except for a single compound, which had nonmonotone dose–response. Compound-specific estimates and Bayesian hierarchical modeling showed that dose–response curves for most compounds can be approximated by Emax models with “Hill” parameters close to 1.0. Summaries of the potency estimates show pharmacometric predictions of potency made before the first dose ranging study within a (1/10, 10) multiple of the final estimates for 90% of compounds. The results of the meta-analysis, when combined with compound-specific information, provide an empirical basis for designing and analyzing new dose finding studies using parametric Emax models and Bayesian estimation with empirically derived prior distributions.
Dose-finding experiments define the safe dosage of a drug in development, in terms of the quantity given to a patient. Statistical methods play a crucial role in identifying optimal dosage. Used appropriately, these methods provide reliable results and reduce trial duration and costs. In practice, however, dose-finding is often done poorly, with widely used conventional methods frequently being unreliable, leading to inaccurate results. However, there have been many advances in recent years, with new statistical techniques being developed and it is important that these new techniques are utilized correctly. Statistical Methods for Dose-Finding Experiments reviews the main statistical approaches for dose-finding in phase I/II clinical trials and presents practical guidance on their correct use. Includes an introductory section, summarizing the essential concepts in dose-finding. Contains a section on algorithm-based approaches, such as the traditional 3+3 design, and a section on model-based approaches, such as the continual reassessment method. Explains fundamental issues, such as how to stop trials early and how to cope with delayed or ordinal outcomes. Discusses in detail the main websites and software used to implement the methods. Features numerous worked examples making use of real data. Statistical Methods for Dose-Finding Experiments is an important collaboration from the leading experts in the area. Primarily aimed at statisticians and clinicians working in clinical trials and medical research, there is also much to benefit graduate students of biostatistics.
If you have ever wondered when visiting the pharmacy how the dosage of your prescription is determined this book will answer your questions. Dosing information on drug labels is based on discussion between the pharmaceutical manufacturer and the drug regulatory agency, and the label is a summary of results obtained from many scientific experiments. The book introduces the drug development process, the design and the analysis of clinical trials. Many of the discussions are based on applications of statistical methods in the design and analysis of dose response studies. Important procedural steps from a pharmaceutical industry perspective are also examined.
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