Fractional Polynomial Meta-Analysis Models for Repeated Measures of Continuous Outcomes

Methods

Fractional polynomial meta-analysis models for repeated measures of continuous outcomes in non-comparative studies

6MWT and FVC, were modeled using fractional polynomial meta-analysis, which estimate the development of outcomes over time.4 Both first order and second order fractional polynomial models were evaluated regarding their goodness of fit to the data. The random effects first order polynomial models are described in the following equations:

(1)

reflects the ‘underlying’ change from baseline outcome in study j at time point t. Note the absence of a link function which was not necessary given that data analysed with this model were continuous. The outcome was described as a function of time t with p = (-2, -1, -0.5, 0, 0.5, 1, 2) and t0 = ln(t), and fixed shape parameter λ. Variance σ2reflects the heterogeneity in the scale parameters across studies. The absence of an intercept in the model is due to the fact that 6MWT and FVC were both modeled as change from baseline. Negative values of P were also not used for this reason.Simply replacing βj by λ in the linear model leads to the fixed effects model.

If we need more flexibility how the outcomes change over time, we can add a second time related factor and obtain a 2nd order polynomial. The second order random effects polynomial models are described as follows:

(2)

Σis the between study covariance matrix, where σ1 represents the heterogeneity for the first shape parameter, σ2 the heterogeneity for the second shape parameter, and ρ is the correlation between these parameters. This model

For all repeated measures models (fixed and random effects, 1st and 2nd order), model selection was conducted using both the DIC and by evaluating the modeled curves. Verifying the modeled curves was a way to avoid over-fitting the data. Over-fitting can lead to low DIC (good fit), but produce lines that are not biologically plausible.

The models presented here are for fitting non-comparative data – that is considering one treatment at a time (referred to as non-comparative model in the results). These models also extend to comparative models in which curves for two treatment groups are fit simultaneously. Given the sparseness of comparative data, a two-step modeling approach similar to that used for survival analysis was used. Non-comparative fractional models were estimated for ERT evidence and then used as priors in the comparative model (referred to as combined model in the results). Thus the combined analyses first modeled the ERT curves and set these as anchors within the comparative portion of the analysis.

Meta-analysis model for reported survival proportion

For survival data, too few events (i.e., many zero-counts) and very limited repeated measures did not allow for using the repeated measures meta-analysis suggested in the statistical analysis plan (explained below). Nonetheless, an analysis was conducted by converting datainto events over person time and modeling these using Poisson models. Given the available evidence base, in which only one trial contained data on non-ERT outcomes, we conducted single sample meta-analyses in each treatment and compared the resulting effect estimates. Here we present traditional non-comparative meta-analysis.The fixed effects model to pool results of multiple studies can be expressed as follows. First, the likelihood is:

where r is the number of events and E is the person-time at risk. The parameter of interest is the study hazard. λj. From this likelihood, the equation for meta-analysis becomes:

(3)

reflects the logarithm of the ‘underlying’ outcome in study j and λis the average expected outcome with the treatment of interest on a normal scale. σ2 represents the between-study heterogeneity.

Meta-analysis models for repeated cross-sectional proportions in non-comparative studies

Ambulation and ventilator status were dichotomous measures reported as proportions at base line and follow-up (which is not the cumulative incidence). As neither of these outcomes were captured in comparative studies, analyses were conducted independently within the ERT and non-ERT evidence bases.. The models employed for these outcomes were slightly more complex than single-point meta-analyses as they included follow-up time (i.e. studies with 6 months follow-up were differentiated from those with 5 years follow-up. For the random effects meta-analysis model study-specific intercept and slope, and , were fit. These study-specific effects were drawn from a random effects distribution:

(4)

reflects the ‘underlying’ outcome in study j at time t transformed (if necessary) to a normally distributed scale through a link function g();explicitly, , where is the unknown likelihood function parameter generated by the observed data.For these outcomes, given that both were dichotomous, a logit link function was used and the unknown likelihood parameter generated by the observed data was pjt, the proportions of patients in study j at time t who did needed wheelchairs and ventilators, respectively. Although this model only allowed for two outcomes per study (at baseline and at end of study), it did allow for varying end of study time points from one study to the next, as was the case in this evidence base for these outcomes.The random effects distribution was a multivariate normal distribution with means set to the parameters of interest, and .represents the mean transformed outcome at baseline. represents the mean change in transformed outcome over one time unit, in this case the mean change in log-odds over a 1-month period. In this model the uninformative priors for slope and intercept followed a multivariate normal distribution. represents the mean transformed outcome at baseline specific to study j and similarlyrepresents the mean change in transformed outcome over one time unitspecific to study j.The variance parameter was a 2-by-2 matrix that was informed by the variance about each effect parameter, as well as their covariance, which can be expressed as where ρ is the correlation. Thus a hierarchical structure was imposed in the random effects model, in which the overall ‘true’ effect across all study populations was the central tendency that, with some between-study heterogeneity Σ, determined the individual study treatment effects one level lower in the hierarchy.

Equation (7) demonstrates the fixed effects meta-analysis model for non-comparative unrepeated change from baseline in general form:

(5)

Thus the fixed effects model assumes the same slope and intercept for all studies. Where possible both fixed and random effects models were fit and compared using the DIC (see 3.2.7).

Combining comparative and non-comparative studies

When modeling evidence bases that contained both comparative and non-comparative data, a two step approach was used to combine these into a single analysis. In the first step, the outcome measure within the single-arm evidence wasmodeled using non-comparative meta-analysis (either traditional non-comparative meta-analysis or non-comparative fractional polynomial meta-analysis). In the second step, the comparative data were analyzed using comparative modeling; however, rather than using non-informative priors, informative priors informed by the first step were used instead. Thus, data from the comparative and non-comparative studies were combined into a single analysis.

Meta-regression

Most meta-analyses include studies that are clinically and methodologically diverse. Between-study heterogeneity is the true variation in treatment effects or outcomes of interest among different studies caused by systematic differences in known and unknown study design and patient characteristics across studies influencing these..Meta-regression analysis was used to model the development of outcome over time as a function of patient characteristics to explain between study heterogeneity:6 The fractional polynomial models can be extended with study level covariates to explore between-study heterogeneity.

(6)

Bx reflects the impact of study level covariate Xjon the shape parameter of the time-related outcome function. This can easily be extended to the 2nd order fractional polynomial model with two adjustment covariates B1x and B2x both from a multivariate normal prior distribution.

The following covariates were evaluated with the analysis:

• Percent males
• Disease duration
• Age
• Disease severity

Disease duration was defined as the time between disease diagnosis and study initiation; age was defined as age at study initiation; and disease severity was measured as the maximum value between proportion of patients using wheelchairs and proportion of patients using ventilators. Unsurprisingly, the disease severity for all studies included in the 6MWT outcome were defined according to the proportion of patients using ventilators rather than wheelchairs. These data were very well reported. Studies with missing values were removed from the meta-regression adjusted analyses. Only two studies (Restel et al, 2014 and Hartung et al, 2008) had missing values for disease duration, gender and disease severity. One more study (Adreassen et al, 2014) had missing disease duration.

Given that meta-analysis is performed with study level data, covariates that relate to continuously distributed patient characteristics were reduced to dichotomous variables to avoid any issues with the ecological fallacy.6 All variables were measured at the study level (i.e., not arm-specific). Therefore, percentage of males was the percentage of study participants that were male. The cut-off values used to dichotomize the variables are as follows: ≥50% males, >46 years of age, >12 years disease duration, and >35 disease severity score. The values were chosen to be whole numbers (with the exception of age) that were close to the median value. The same cut-offs were used across all outcomes and models.

Likelihood, link functions and prior distributions

To perform the meta-analyses within a Bayesian framework, likelihood distributions need to be defined to relate the data to the parameters of the models. The second provides the likelihood and link functions for the outcome data in the available evidence base.

Likelihood and link functions for outcome data

In order not to influence the observed results by the prior distribution, non-informative prior distributions were used for the model parameter(s). Table 3presents an overview of the prior distributions to be used in the Bayesian analyses.

Table Prior distributions for model parameters used for analysis in a Bayesian framework

Table A1: The search strategy employed for searches in Medline and Embase

mp denotes multi-purpose and implies a search through all fields;.sh. denotes a Medical Subject Heading (MeSH) term;exp denotes explode and implies that a term and a collection of variations on that term are searched for; * is used for truncation; ? is a single space wildcard term.

Table A2: The search strategy employed to search Cochrane Central Register of Controlled Trials

Table A3: Study characteristics

* Sub study of Gungor et al, 2013¥ Thisstudy had eight years of follow-up, but used a non-eligible enzyme replacement (unlicensed) for the first three years. Therefore, only data from the last five years met our inclusion criteria and was used.28

Figure A1: Observed ventilator and ambulation status over by study for ERT and non-ERT arms

Legend:Each colored line represents a study. The size of the circles represent the approximate sample sizes at each time point. Panel A (top left) represents the observed ventilator status among studies including patients on ERT. Panel B (top right) represents the observed ventilator status among studies including patients not on ERT. Panel C (bottom left) represents the observed ambulation status (wheelchair use) over time among studies including patients on ERT. Panel D (bottom right) represents the observed ambulation status (wheelchair use) over time among studies including patients not on ERT

Figure A2:Modeled ventilator and ambulation status over by study for ERT and non-ERT arms with each colored line representing a single study.

Legend: Panel A shows the modeled ambulation status (wheelchair use) over time using a random effects model. Panel B shows the modeled ventilator status over time using a random effects model. The full lines represent the modeled outcomes over time. The dashed lines of corresponding color represent the 95% credible intervals for the modeled outcomes.