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IntroductionSynthesis of evidence order generic viagracialis price 10mg provided by randomised controlled trials (RCTs) is commonly used to develop clinical guidelines and make reimbursement decision for pharmacological interventions. While the order generic viagracialis price 10mg dose of a drug is of central importance, meta-analyses that examine their efficacy and safety often focus on comparing only agents or classes of drugs, ignoring potential variability due to different doses. As different dose schedules may result in considerable heterogeneity in efficacy and safety, one common approach is to restrict the database at certain dose range (e.g., the therapeutic dose), discard all studies outside that range and then examine the role of dose in a subgroup analysis for the lowest and the highest dose categories.1 This approach fails, however, to synthesise the whole relevant evidence. Alternatively, researchers might opt to order generic viagracialis price 10mg perform many meta-analyses, each restricted to studies that examine a particular drug-dose combination.

This will inevitably result in many underpowered meta-analyses.In this paper, we present a recently developed evidence synthesis method of a doseâeffect meta-analysis (DE-MA) approach that offers a middle ground order generic viagracialis price 10mg between âlumpingâ all doses together into a single meta-analysis and âsplittingâ them to many dose-specific meta-analyses. In DE-MA, we model the changes in the drug effect along the range of all studied dosages. There are order generic viagracialis price 10mg two common approaches to conduct DE-MA. Two-stage and one-stage models.

In the two-stage model, the doseâeffect curve is estimated within each study and then synthesised across studies.2 3 These two steps are performed simultaneously in the one-stage model.4We first provide the statistical explanations of the two models, and then illustrate the models by using a collection of RCTs examining the efficacy of selective serotonin reuptake inhibitors (SSRI) order generic viagracialis price 10mg antidepressants.5The analysis is implemented in R6 and is made available along with dataset and the results on GitHub (https://github.com/htx-r/Dose-effect-MA-EBMH-article-).MethodsIn this section, we describe the two-stage DE-MA model with summarised data. Then we present briefly order generic viagracialis price 10mg the one-stage model. Finally, we discuss other issues related to this topic, namely. Statistical testing of doseâeffect coefficients and how to assess heterogeneity order generic viagracialis price 10mg and make predictions.

The models which are illustrated here to conduct DE-MA have been implemented in various software packages, for example, the drmeta command (in Stata7) and the dosresmeta package8 (in R).6Doseâeffect shape within a studyLet us consider the case of an RCT where several doses are examined (one dose per arm) denoted by where the index j enumerates the dose levels starting with zero. The outcome order generic viagracialis price 10mg is measured in each arm on an additive scale (e.g., a mean, a log-odds). The doseâeffect model within a study associates the change in the order generic viagracialis price 10mg outcome (ie, the treatment effect) to the change in the dose. Let us assume a trial like the one presented in table 1 that has a placebo arm, a dichotomous outcome and the changes in the outcome are measured using the odds ratio (logOR) of each dose level j relevant to a reference dose.

Using the placebo arm as a reference (at dose , and assuming a linear association between logOR and dose, the doseâeffect model isView this table:Table 1 We present the data of Feighner et al study on the observed five dose levels order generic viagracialis price 10mg. The data consist of the observed dose, the number of responses, the total number of participants, the odds ratio (OR) and its 95% CI, log transformations of OR and its standard error (SE) The estimated coefficient Î² shows how much an increase in the dose will impact on the change in logOR.Typically, the referent dose is assigned to the zero or the minimal dose to make interpretation easier. The doses are centred around the referent dose so the order generic viagracialis price 10mg relationship quantifies the change in relative effects. However, this centralisation induces correlation between the logORs in each study (as they are all order generic viagracialis price 10mg estimated relative to the outcome of the.

Such correlations should be estimated and accounted for using the Longnecker and Greenland method.2 9In practice, multiple changes in the doseâeffect shape are expected so that the linear model is not often a realistic assumption. More flexible models are needed to account for those changes10 such as order generic viagracialis price 10mg restricted cubic spline (RCS). RCS is a piecewise function. The dose spectrum is split into intervals (using some changepoints, called knots) and in each interval a cubic polynomial is fitted.11 Restrictions in the estimation of the polynomial coefficients are then imposed to ensure that they are order generic viagracialis price 10mg connected and forming a smooth function which is linear in the two tails.

The location and the number of those knots determine the shape of the RCS order generic viagracialis price 10mg. The locations indicate intervals where changes in the shape might occur, and the number reflects how many such changes are anticipated. In general, setting k knots creates a order generic viagracialis price 10mg RCS model with regression coefficients. For identifiability, the minimum number of knots is three and the doseâeffect shape is.

This function is a combination of linear and non linear transformations.11Of note, a two-stage approach requires that the study examines at least three dose-level data including the referent level and that enables estimating the two regression coefficients in the linear and spline (nonlinear, ) order generic viagracialis price 10mg parts of the equation.Any type of function could be used in the doseâeffect association. For study indicator i, the general order generic viagracialis price 10mg form of the doseâeffect model can be written. The term refers to the p doseâeffect parameter and f denotes the doseâeffect shape.Synthesis of doseâeffect shapes across studiesConsider that we have fit the RCS model in k studies and we have obtained k sets of estimates ( ). Each pair order generic viagracialis price 10mg of coefficients represents the shape of the doseâeffect within each study.

Now, we synthesise the shapes across studies by combining their coefficients. We may set a common underlying coefficient for all studies, for example, and (common-effect model) order generic viagracialis price 10mg. Alternatively, the underlying study-specific coefficients can be assigned a two-dimensional normal distribution with mean and a varianceâcovariance matrix to reflect the heterogeneity order generic viagracialis price 10mg across the studies (random-effects model). In the general case, the doseâeffect shape f involving p coefficients which are similarly synthesised using a multivariate normal distribution.What we describe above is the two-stage approach.

The doseâeffect curves order generic viagracialis price 10mg are estimated within each study and then synthesised across studies in two separate steps. This requires each study to report non-referent doses at least as many as the number of the doseâeffect coefficients. Otherwise, the coefficients will be non-identifiable and order generic viagracialis price 10mg the study should be excluded from the analysis. For example, to estimate order generic viagracialis price 10mg a doseâeffect quadratic shape or a RCS with three knots, two coefficients need to be estimated and hence each study needs to report at least two logORs (which means at least three dose levels).

Studies that report less dose levels, shall be excluded from the synthesis.In the one-stage approach, within and across study estimation of the shape are performed simultaneously.4 This allows for borrowing information across studies and the study-specific coefficients can be estimated even if the study itself does not report the required number of doses. This means order generic viagracialis price 10mg that, with the one-stage approach, we can include in the synthesis studies that report only one logOR (two dose levels) even if we want to estimate RCS.There are different ways to present the results from the DE-MAs. The doseâeffect shape as a function of any dose can be presented in graphical or tabular form by plugging-in the dose values and the estimated coefficients in order generic viagracialis price 10mg the assumed function (see figures 1 and 2). Another useful presentation of the results could be to show absolute estimates of the outcome, such as estimates of probability for efficacy at any given dose, see figure 3.

This can be done in order generic viagracialis price 10mg two simple steps. First, we estimate the absolute probability of the response at the reference dose (e.g., zero) and then we combine this with the estimated relative treatment effect at each dose (e.g., with the estimated logOR) to obtain the absolute outcome (e.g., the probability to respond at an active dose level).The estimated doseâeffect curves of citalopram in Feighner et al study. The fluoxetine-equivalent doses are presented order generic viagracialis price 10mg versus the odds ratio with two different doseâeffect shapes. The linear model in grey (dashed) and the restricted cubic spline (with knots at 20.0, 23.6 and order generic viagracialis price 10mg 44.4) in red (solid).

The 95% confidence bands are shaded around each curve." data-icon-position data-hide-link-title="0">Figure 1 The estimated doseâeffect curves of citalopram in Feighner et al study. The fluoxetine-equivalent doses order generic viagracialis price 10mg are presented versus the odds ratio with two different doseâeffect shapes. The linear model in grey (dashed) and the restricted cubic spline (with knots at 20.0, 23.6 and 44.4) in red (solid). The 95% confidence bands are shaded around each curve.Dose-effect curves for selective serotonin reuptake inhibitors order generic viagracialis price 10mg.

These curves are estimated using the restricted cubic spline function where knots are order generic viagracialis price 10mg set at doses 20.0, 23.6 and 44.4 mg/day. For data synthesis, we apply a one-stage (grey, solid) and two-stage (red, dashed) approaches.The 95% confidence bands are shaded around each curve. SSRI, selective serotonin reuptake inhibitor." data-icon-position data-hide-link-title="0">Figure 2 Dose-effect curves for selective serotonin reuptake order generic viagracialis price 10mg inhibitors. These curves are estimated using the restricted cubic spline function where knots are set at doses 20.0, 23.6 and 44.4 mg/day.

For data synthesis, we apply a one-stage (grey, solid) and two-stage (red, dashed) approaches.The 95% confidence bands are shaded around each curve order generic viagracialis price 10mg. SSRI, selective serotonin order generic viagracialis price 10mg reuptake inhibitor.The synthesised doseâeffect curves across studies of SSRI. The fluoxetine-equivalent doses are presented versus the predicted absolute effect. The doseâeffect order generic viagracialis price 10mg function is the restricted cubic spline (with knots at 20.0, 23.6 and 44.4).

The solid line represents the mean absolute effect and the shaded area is its 95% confidence bands. The dashed (horizontal) line represents the placebo absolute effect at 37.7% order generic viagracialis price 10mg. SSRI, selective serotonin reuptake inhibitor." data-icon-position data-hide-link-title="0">Figure order generic viagracialis price 10mg 3 The synthesised doseâeffect curves across studies of SSRI. The fluoxetine-equivalent doses are presented versus the predicted absolute effect.

The doseâeffect function is the order generic viagracialis price 10mg restricted cubic spline (with knots at 20.0, 23.6 and 44.4). The solid line represents the mean absolute effect and the shaded area is its 95% confidence bands. The dashed (horizontal) line represents the placebo absolute effect order generic viagracialis price 10mg at 37.7%. SSRI, selective serotonin reuptake inhibitor.HeterogeneityHeterogeneity in the study-specific coefficients introduces heterogeneity in the order generic viagracialis price 10mg relative treatment effects, which is what we will call heterogeneity from now on.

It is a function of the dose and can be measured by the variance partition coefficient (VPC).4 The VPC is a study-specific and dose-specific which shows the percentage of heterogeneity out of the total variability specific to the study. VPC can order generic viagracialis price 10mg be computed for each non-referent dose in each study. An average of the study-specific VPCs by dose level could be seen as a dose-specific I2. It is useful to plot the study-specific VPCs (as %) against the order generic viagracialis price 10mg dose levels to gauge the level of heterogeneity.ResultsWe illustrate the models by re-analysing a dataset about the role of dose in the efficacy of SSRIs.

Drug-specific doses are converted into fluoxetine-equivalents (mg/day) using a validated formula.5 The outcome is response to treatment defined as 50% reduction in order generic viagracialis price 10mg symptoms. The data include 60 RCTs, which recruited 15 174 participants in 145 different dose arms (see online supplemental appendix figure 1, 2 and table 1).Supplemental materialDoseâeffect model within a studyTo exemplify the process, we consider the study by Feighner et al.13 Table 1 presents the data at the five examined dose arms. The four logORs are estimated as the odds of each order generic viagracialis price 10mg non-referent category (10, 20, 40, 60 mg/day) relative to the odds in the referent dose (Placebo, 0 mg/day). The study-specific estimated logORs and their SEs can be used to fit a linear doseâeffect model.A log linear trend is then estimated based on the aggregate data presented by Feighner et al (figure 1).13 The Greenland and Longnecker method is used to back estimate the covariance of these four empirical logORs used as dependent variable of the linear doseâeffect model.The linear doseâeffect coefficient is estimated at 0.0156 (95% CI 0.0083 to 0.0230) on the log scale.

The OR at dose 10 to be which means OR increases by for a 10-unit increase in dose.Biologically, it is quite unrealistic to assume a constant effect of order generic viagracialis price 10mg fluoxetine-equivalents on the relative odds of the outcome. We expect order generic viagracialis price 10mg the shape to increase up to a dose level and then flatten out. The exact value of the dose, at which the doseâeffect model is levelling out, is unknown. And it would be good to specify a doseâeffect model that is able order generic viagracialis price 10mg to capture this plausible mechanism.For this reason, we use a RCS function, rather than a linear function, for fluoxetine-equivalents.

RCSs are generated using three knots at 20, 23.6 and 44.4 dose levels which represent the 10%, 50% and 90% percentiles, of the observed non-zero dose distribution. A Wald-test indicates order generic viagracialis price 10mg large incompatibility between this study and the hypothesis of a linear function ( , p =0.033). Figure 1 indicates a large positive doseâeffect up to 30 mg/day of fluoxetine-equivalents and no increase in the effect beyond that value.The fact that the shape is estimated from just a single study results in a large uncertainty around the RCS curve.Synthesis of doseâeffect shapes across studiesWe first synthesise the doseâeffect coefficients from all order generic viagracialis price 10mg studies assuming a random-effects two-stage model. For RCS in the two-stage model, only 17 studies can be synthesised (those with at least three dose levels).

The results order generic viagracialis price 10mg are depicted in figure 2. The estimated linear coefficient at 0.0186 (95% CI 0.0118 to 0.0253) and the spline coefficient is â0.0628 (95% CI order generic viagracialis price 10mg â0.0876 to â0.0379).The random-effects one-stage model can include all 60 studies. The estimated linear and spline coefficients are very close to those from the two-stage model ( 0.0189 (95% CI 0.0146 to 0.0232) and â0.0621 (95% CI â0.0814 to â0.0428)) which is also shown in the agreement of the two shapes in figure 2. The important difference between the results from the two approaches is that the confidence bands are tighter from the one-stage due to including double as many studies as the two-stage approach does.In order generic viagracialis price 10mg figure 3, we show the probability of response as a function of the dose as estimated from the meta-analysis.

After meta-analysing all placebo arms, the probability of response to placebo is estimated at 37.7% (dashed line in figure 3). Then, increase of the dose up to 30 mg/day of fluoxetine-equivalent results in 50% probability order generic viagracialis price 10mg to respond. Beyond 40 mg/day, the probability of response flattens out.For the two-stage and the one-stage models, the statistical hypothesis can be rejected with estimated p-values less than 0.001 for both the linear and order generic viagracialis price 10mg spline coefficients. This can be seen as a statistical evidence that the linear model hypothesis is rejected, and the RCS is preferable with both the linear and the spline part.

The hypothesis of no dose-effect association is not also accepted order generic viagracialis price 10mg (p-value<0.001).Figure 4 shows the variance partition component along with the observed doses. At dose 20 mg/day, the total variability that is attributed solely to heterogeneity ranges between 4% and 40%, which is considered to be moderate. Overall, the majority of VPC values does not exceed 60%.The variance partition component of each observed dose (non-referent doses in each study) presented in circles order generic viagracialis price 10mg. Each circle represents a order generic viagracialis price 10mg study.

The fitted line is LOWESS curve." data-icon-position data-hide-link-title="0">Figure 4 The variance partition component of each observed dose (non-referent doses in each study) presented in circles. Each circle order generic viagracialis price 10mg represents a study. The fitted line is LOWESS curve.DiscussionResearchers can conduct a DE-MA by following two steps. The first step is to estimate a doseâeffect curve within each study order generic viagracialis price 10mg.

The second step is to synthesise those curves across order generic viagracialis price 10mg studies. These two steps can be performed either separately (two-stage model)2 3 or simultaneously (one-stage model).4 In this article, we detail these two models, alongside considerations for statistical testing of the doseâeffect parameters, estimation of heterogeneity and presentation of the results. We use the presented models to re-analyse RCT data comparing various SSRIs in terms of response .We describe the models for a dichotomous outcome and the effect size we used as odds ratio order generic viagracialis price 10mg. However, the model can be adapted easily to other measures like risk ratio and hazard ratio.

Likewise, the model can be employed with other data types such as continuous outcome with (standardised) mean order generic viagracialis price 10mg differences.14Recently, two extensions of the presented models have been introduced in the literature. The one-stage and two-stage models have been extended to a Bayesian setting15 to take advantage order generic viagracialis price 10mg of its great flexibility. One of these advantages is to implement the exact binomial distribution for binary data, instead of the approximate normal distribution for the relative treatment effect in the frequentist settings. The assumption of a normal distribution can be hard to meet when the sample size is small as shown in recent simulations.15 The doseâeffect model has been also extended to network meta-analysis which allows for modelling the doseâeffect relationship simultaneously to more than two agents.16 17Researchers should be careful when they report the findings of DE-MA and follow the existing order generic viagracialis price 10mg reporting guidelines.

Xu et al proposed a checklist with 33 reporting items for such analysis.18 The majority of these items (27) come from the Preferred Reporting Items for Systematic Reviews and Meta-Analyses statement after some modifications.19 The other six items are added from Meta-analyses Of Observational Studies in Epidemiology checklist to cover key considerations of observational studies.20 They used the proposed checklist to assess quality of reporting in the published DE-MAs. They found that while reporting in the introduction and results was on average good, further improvements are required order generic viagracialis price 10mg in reporting methods. Xu and colleagues also studied the association between reporting quality and study order generic viagracialis price 10mg characteristics. They observed that studies including more authors or methodologist have a better reporting quality.

They conclude that while the quality of reporting has improved over the years, further refinement in the reporting checklists is required.The main challenge in DE-MA is how to define order generic viagracialis price 10mg the doseâeffect shape. The shape selection can be guided by previous studies (such as dose-finding studies), clinical experience and biological plausibility informed by pharmacodynamic and pharmacokinetic studies. Additional evidence could be provided order generic viagracialis price 10mg by considering the goodness of fitness measures of various shapes21 or via graphical inspection of the data. Yet, the order generic viagracialis price 10mg RCS model has sufficient flexibility to capture different shapes.

In our case study, using only three knots was sufficient to capture the expected drug behaviour SSRIs while requires only three dose levels to be reported in at least one study. This makes RCS an attractive choice for the majority of analyses.18 However, the number and location of knots should be chosen carefully based on the anticipated drug behaviour and the clinical knowledge.Researchers may encounter order generic viagracialis price 10mg additional challenges if observational studies are synthesised instead of RCTs as it was the case in this paper. First, defining the dependent and independent variables in observational studies could be difficult. For example, if we want to evaluate the association between the alcohol consumption and the use of tobacco, the shape will depend on whether order generic viagracialis price 10mg alcohol is set as a dependent or independent variable.

Second, categorisation of non-pharmacological order generic viagracialis price 10mg exposures (such as environmental exposure, diet and so on), which are often the focus of observational studies, is often difficult. There might be open-ended categories to which assignment of a specific dose is not obvious (e.g., smoking two packages per day and above) and exposure categories might be differently defined across studies.22 23 These challenges could induce additional uncertainty in the analysis. In such cases, sensitivity analysis is recommended to investigate the robustness of the DE-MA results.In conclusion, the DE-MA enables clinicians to understand how the effect of a drug changes as a function of its dose. Such analysis should be conducted in practice using the one-stage model that incorporates evidence from all available studies..

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