This article has Open Peer Review reports available.
Dose-response meta-analysis of differences in means
- Alessio Crippa^{1}Email author and
- Nicola Orsini^{1}
https://doi.org/10.1186/s12874-016-0189-0
© The Author(s) 2016
Received: 27 October 2015
Accepted: 13 July 2016
Published: 2 August 2016
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. E _{max}, 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
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].
Randomized clinical trials often investigate a continuous 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 methodological research focused on developing and improving methods for performing dose-response analysis in a single study [4, 5]. A conclusive result is hardly obtained by a single investigation and there is often the need to synthesize information collected from multiple studies. In such a case meta-analytic methods can be used to define an overall 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 presented 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 implementation [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 summarizing 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 E _{max} model, where the random component accounts for heterogeneity in placebo effect across trials [17]. Heterogeneity, however, may be related to other study characteristics rather than differences in placebo response such as implementation, participants, intervention, and outcome definition. Thomas et al. [18] adopted hierarchical Bayesian models to summarize and describe, independently, the distribution of study-specific model parameters derived from an E _{max} model.
The mentioned strategies assumed pre-specified models 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 E _{max} 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 variance/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
Notation for aggregated data in the i-th study used in dose-response meta-analysis of differences in meas
dose | mean(Y)^{ a } | sd(Y) | n ^{ b } | d^{ c } | Var(d) | d^{∗} ^{ d } | Var(d^{∗}) |
---|---|---|---|---|---|---|---|
0 | \(\bar Y_{i0}\) | s d _{ i0} | n _{ i0} | 0 | – | 0 | – |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
x _{ ij } | \(\bar Y_{ij}\) | s d _{ ij } | n _{ ij } | d _{ ij } | Var(d _{ ij }) | \(d^{*}_{ij}\) | \(\text {Var} \left (d^{*}_{ij} \right)\) |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
\(x_{iJ_{i}}\) | \(\bar Y_{iJ_{i}}\) | \(sd_{iJ_{i}}\) | \(n_{iJ_{i}}\) | \(d_{iJ_{i}}\) | \(\text {Var} \left (d_{iJ_{i}} \right)\) | \(d^{*}_{iJ_{i}}\) | \(\text {Var} \left (d^{*}_{iJ_{i}} \right)\) |
Effect sizes and their variance/covariance
Dose-response analysis
\(\boldsymbol {\hat \Sigma }_{i} \) is the covariance matrix of the residual error term, with Var(d _{ ij }) along the diagonal and \(\text {Cov} \left (d_{ij}, d_{ij^{'}} \right)\) off-diagonal.
Frequently used dose-response models
Model | Equation | No. of parameters |
---|---|---|
Linear | E[d _{ i }|x _{ i }]=θ _{1i } x _{ i } | 1 |
Quadratic | \(\displaystyle \mathrm {E} \left [ \boldsymbol {d}_{i} | \boldsymbol {x}_{i} \right ] = \theta _{1i} \boldsymbol {x}_{i} + \theta _{2i} \boldsymbol {x}_{i}^{2}\) | 2 |
E _{max} | \( \displaystyle \mathrm {E} \left [ \boldsymbol {d}_{i} | \boldsymbol {x}_{i} \right ] = \theta _{1i} \boldsymbol {x}_{i}^{\theta _{3i}}/ \left (\theta _{2i} + \boldsymbol {x}_{i}^{\theta _{3i}} \right)\) | 3 |
Logistic | E[d _{ i }|x _{ i }]=θ _{1i }/{1+exp[(θ _{2i }−x _{ i })]/θ _{3i }} | 3 |
The most common choice in dose findings [2] is the use of the E _{max} model which is expressed in terms of three parameters: the maximum effect (θ _{1i }), the dose to produce half of the maximum effect (θ _{2i }) and the steepness of the curve (θ _{3i }) [20]. As other non-linear models, the E _{max} 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 but may 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.
where the ‘+’ notation, with u _{+}=u if u≥0 and u _{+}=0 otherwise, has been used.
for each combination of p _{1} and p _{2} in the predefined set of values {−2,−1,−0.5,0,0.5,1,2,3}; for p=0, x ^{ p } becomes log(x). The best fitting fractional polynomial is typically chosen based on the Akaike’s Information Criterion [24].
where X _{ i } indicates the J _{ i }×p design matrix in the i-th study.
Meta-analysis
A fixed-effects model assumes no statistical heterogeneity 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 practice, unless it is known that the studies are performed in a similar way and are sampled from the same population [25]. The Cochran’s Q test [26] is typically used to test statistical heterogeneity across studies (H _{0}:Ψ=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 heterogeneity is expected and should be taken into account in the analysis. A random-effects model allows the dose-response coefficients, θ _{ i }, to vary across studies. Statistical heterogeneity is captured by the between-studies variance Ψ while θ represents the mean of the distribution of dose-response coefficients and an estimate, \(\boldsymbol {\hat \theta }\), can be obtained using (restricted) maximum likelihood estimation [15].
where z _{ α/2} is 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 (ED_{ γ }) to produce an almost complete effect, e.g. γ % of the observed maximum predicted response.
where x _{max} 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 \(\boldsymbol {\hat \theta }\), the pooled estimate of the dose-response coefficients. The approach consists in re-sampling the dose-response coefficients θ from its approximate normal distribution and derive the distribution of \(\mathrm {\widehat {ED}}_{\gamma }\) based on the samples. Approximated confidence intervals for \(\mathrm {\widehat {ED}}_{\gamma }\) can 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, and the Cochrane Database of Systematic 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, controlled clinical trials of schizoaffective patients providing dose-response results for at least two non-zero dosages of aripiprazole were eligible.
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 |
Study-specific dose-response coefficients and corresponding covariances for different dose-response models considered in the analysis
Model | id | \(\hat \theta _{1}\) | \(\hat \theta _{2}\) | \(\text {Var} \left (\hat \theta _{1} \right)\) | \(\text {Cov} \left (\hat \theta _{1},\hat \theta _{2} \right)\) | \(\text {Var} \left (\hat \theta _{2} \right)\) |
---|---|---|---|---|---|---|
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 | |
3 | 0.84 | −1.69 | 0.054 | −0.12 | 0.38 | |
4 | 0.47 | −0.54 | 0.021 | −0.043 | 0.15 | |
5 | 0.78 | −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 |
2 | 1.54 | −0.05 | 0.35 | −0.017 | 0.00089 | |
3 | 1.40 | −0.04 | 0.18 | −0.0055 | 0.00018 | |
4 | 0.83 | −0.02 | 0.14 | −0.0047 | 0.00017 | |
5 | 1.08 | −0.02 | 0.51 | −0.016 | 0.00051 | |
Piecewise linear | 1 | 0.55 | 0.065 | |||
2 | 0.59 | 0.031 | ||||
3 | 0.84 | −1.69 | 0.054 | −0.12 | 0.38 | |
4 | 0.47 | −0.54 | 0.021 | −0.043 | 0.15 | |
5 | 0.78 | −1.25 | 0.23 | −0.62 | 1.8 |
A p-value < 0.001 for the multivariate Wald-type test H _{0}:θ=0 provided strong evidence against the null hypothesis of no relation between different doses of aripiprazole and mean change PANSS score. The Q test (Q=3.5, p-value = 0.899) did not detect substantial statistical heterogeneity across studies.
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 x _{max} = 19.32 mg/day. The model suggested a slight decrease in the predicted mean PANSS score for dosages greater than 20 mg/day. The estimated dose to produce 50 % and 80 % of the predicted maximum effect were \(\mathrm {\widehat {ED}}_{50} =\) 5.82 mg/day (95 % CI: 5.10, 8.58) and \(\mathrm {\widehat {ED}}_{80} =\) 10.43 mg/day (95 % CI: 9.02, 16.73).
Sensitivity analysis
To evaluate the sensitivity of the dose-response curve to the choice of the parametric model f adopted, instead, we considered three alternatives: fractional polynomials; quadratic; and E _{max}. Since two studies only had two non-referent doses, the study-specific (sigmoidal) E _{max} models as described in Table 2 cannot be estimated. A common solution is to fix the steepness of the curve θ _{3} to be 1, also referred to as hyperbolic E _{max} [20].
The “best” fractional polynomials (p _{1}=0.5, p _{2}=1) provided overall a similar dose-response curve when compared to the spline model, with slightly higher value for the maximum predicted response (right panel of Fig. 3). The hyperbolic E _{max} 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 θ _{3} to be 1. The dose-response curve described by the quadratic model fall in between the spline and the hyperbolic E _{max} curves.
Discussion
In this paper we proposed a statistical method to combine 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 coefficients (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 modeling 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 studies may have investigated a lower number of dose levels, exclusion of these studies may result in substantial loss of information. In addition, many non-linear models assume a specific behaviour (e.g. monotonicity) requiring 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 quantitative 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 assumptions 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 important 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 summary 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 distribution of the conditional mean effect size. Extension of the introduced methodology to percentiles is not straightforward and may represent an interesting topic of future research.
An additional limitation of aggregated dose-response data is that supplementary information for approximating the covariance terms may not be available. Articles may report directly mean or standardized mean differences and standard errors for non-referent dose groups. Whenever the standard deviation for the outcome variable in the control group (\({s_{i0}^{2}}\)) cannot be obtained, it may be approximated using the pooled standard deviation based on the non-referent dose levels (\(s_{p_{ij}}^{2}\)). Alternatively a specific value may be imputed and a sensitivity analysis can be performed to evaluate how the results of the meta-analysis vary for different values of \({s_{0j}^{2}}\). Further limitations relate to the general application of meta-analysis based on aggregated data. These include restrictions in subgroup analyses, the impossibility of assessing the appropriateness of individual analyses, and to harmonize variable definitions and analyses for reducing the extent of heterogeneity, as well as specific biases such as aggregation (or ecological) bias in meta-regression models. Meta-analysis of individual patient data, however, are often difficult to undertake especially for the availability of individual data, so that usage of aggregated data may represent the only alternative [36]. Specific to aggregated dose-response data, different dose references and exposure range may complicate the analysis. The presented methodology assume that all the selected studies share a common dose-response model. Important departure from this assumption may limit and/or impact the pooling of individual dose-response coefficients. An alternative 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 graphical 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 E _{max} model presented higher drug efficacy for low dosages. Compared to the previous models, the E _{max} 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 overall 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 evaluate 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
Declarations
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.
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Authors’ Affiliations
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