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
Outcomes included in study / Likelihood / Link functionNormally distributed continuous data / 6MWT and FVC / yjt ~ Normal(θjt, σ2) / g(θjt) = θjt (Identity)
Binary data / Ambulation status, ventilator use / rjt ~ Binomial(θjt, njt) / g(θjt) = logit(θjt)
Rate data / Single endpoint survival / rjt~ Poisson(λjtEjt) / g(λjt) = log(rjt/ ejt)
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
Model parameters / Prior distribution / CommentNuisance parameters / μj ~ Normal(0, 0.0001)
Comparative treatment effect parameters / d ~ Normal(0, 0.0001)
Non-comparative treatment effect parameters / λ ~ Normal(0, 0.0001)
Covariate effect parameters / βi ~ Normal(0, 0.0001)
Heterogeneity parameters / σ ~ Uniform(0, u) / u is dependent upon the measurement scale and link function used.
Table A1: The search strategy employed for searches in Medline and Embase
No. / Terms / Comments1. / exp Glycogen Storage Disease Type II/ / Population:
Late-onset Pompe Disease (LOPD) terms
2. / (glycogen storage disease type II OR glycogen storage disease type 2 OR gsd?II OR gsd II OR gsd?2 OR gsd 2).mp.
3. / (glycogenosis type II ORglycogenosis type 2).mp
4. / (late* OR late-onset OR juvenile* OR adult* OR child* OR non-infantile*).mp.
5. / (pompe OR pompe-disease OR lopd).mp.
6. / (acid alpha-glucosidase deficiency OR gaa deficiency OR (deficiency AND (acid alpha-glucosidase OR gaa))).mp.
7. / (acid maltase deficiency or AMD).mp. not (macular).mp.
8. / 4 AND (5 OR 6 OR 7)
9. / OR/1-3 OR 8 / Population results
10. / exp enzyme replacement therapy/ / Intervention: Alglucosidase alfa terms
11. / (enzyme replacement therapy OR ert OR enzyme therapy).mp.
12. / (alglucosidase alfa OR alglucosidase* OR genzyme OR myozyme OR lumizyme OR rhGAA OR recombinant human GAA).mp.
13. / OR/10-12 / Intervention results
14. / (natural history OR natural course OR course*) / Comparison:
Natural history (prognosis) terms
15. / exp mortality/
16. / incidence.sh. OR follow-up stud*.sh.
17. / (prognos* OR predict*).mp.
18. / OR/14-17 / Prognosis results
19. / 8 AND (13 OR 18) / Final results
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
No. / Terms / Comments1. / exp Glycogen Storage Disease Type II/ / Population:
Late-onset Pompe Disease (LOPD) terms
2. / (glycogen storage disease type II OR glycogen storage disease type 2 OR gsd?II OR gsd II OR gsd?2 OR gsd 2).mp.
3. / (glycogenosis type II ORglycogenosis type 2).mp
4. / (late* OR late-onset OR juvenile* OR adult* OR child* OR non-infantile*).mp.
5. / (pompe OR pompe-disease OR lopd).mp.
6. / (acid alpha-glucosidase deficiency OR gaa deficiency OR (deficiency AND (acid alpha-glucosidase OR gaa))).mp.
7. / (acid maltase deficiency or AMD).mp. not (macular).mp.
8. / 4 AND (5 OR 6 OR 7)
9. / OR/1-3 OR 8 / Population results
10. / exp enzyme replacement therapy/ / Intervention: Alglucosidase Alfa terms
11. / (enzyme replacement therapy OR ert OR enzyme therapy).mp.
12. / (alglucosidase alfa OR alglucosidase* OR genzyme OR myozyme OR lumizyme OR rhGAA OR recombinant human GAA).mp.
13. / OR/10-12 / Intervention results
14. / 8 AND 13 / Final results
Table A3: Study characteristics
Study ID / Title / Sample Size / Design / Duration(Months) / Region
Adreassen et al, 201416 / Effect of enzyme replacement therapy on isokinetic strength for all major muscle groups in four patients with Pompe disease-a long-term follow-up / 4 / Cohort / 72 / Denmark
Angelini et al, 200933 / Progress in enzyme replacement therapy in glycogen storage disease type ii / 11 / Cohort / 18 / Italy
Angelini et al, 201236 / Observational clinical study in juvenile-adult glycogenosis type 2 patients undergoing enzyme replacement therapy for up to 4 years / 74 / Cohort / 36 / Italy
Bembi et al, 201018 / Long-term observational, non-randomized study of enzyme replacement therapy in late-onset glycogenosis type ii / 24 / Cohort / 36 / Italy
De Vries et al, 201220 / Effect of enzyme therapy and prognostic factors in 69 adults with Pompe disease: An open-label single-center study / 71 / Cohort / 18 / Netherlands
Furusawa et al, 201221 / Effects of enzyme replacement therapy on five patients with advanced late-onset glycogen storage disease type ii: A 2-year follow-up study / 5 / Cohort / 24 / Japan
Gungor et al, 201310 / Impact of enzyme replacement therapy on survival in adults with Pompe disease: Results from a prospective international observational study / 283 / Database/Survey / 108 / Multinational
Hagemans et al, 200629* / Course of disability and respiratory function in untreated late-onset Pompe disease / 52 / Cohort / 24 / Netherlands
Merk et al, 200922 / Glycogen storage disease type ii (pompe disease) - influence of enzyme replacement therapy in adults / 4 / Cohort / 6 / Germany
NCS-LSD cohort study15 / Effectiveness of enzyme replacement therapy in adults with late-onset Pompe disease: results from the NCS-LSD cohort study / 62 / Cohort / 48 / UK
Orlikowski et al, 201119 / Recombinant human acid alpha-glucosidase (rhgaa) in adult patients with severe respiratory failure due to Pompe disease / 5 / Cohort / 12 / France
Papadimas et al, 201123 / Adult Pompe disease: Clinical manifestations and outcome of the first Greek patients receiving enzyme replacement therapy / 5 / Cohort / 38 / Greece
Patel et al, 201224 / The impact of antibodies in late-onset pompe disease: A case series and literature review / 3 / Case series / 52 / USA & Germany
Restel et al, 201425 / Enzymatic replacement therapy in patients with late onset Pompe disease-5-year follow up / 4 / Cohort / 60 / Poland
Strothotte/Regnery26 / Enzyme replacement therapy with alglucosidase alfa in 44 patients with late-onset glycogen storage disease type 2: 12-month results of an observational clinical trial / 44 / Cohort / 12 / Germany
Van Capelle et al, 201027 / Effect of enzyme therapy in juvenile patients with Pompe disease: A three-year open-label study / 5 / Cohort / 36 / Netherlands
Van Capelle et al, 2010b28 / Eight years experience with enzyme replacement therapy in two children and one adult with Pompe disease / 3 / Case series / 60¥ / Netherlands
Van Der Beek et al, 200913 / Rate of disease progression during long-term follow-up of patients with late-onset Pompe disease / 16 / Cohort / 1925 / Netherlands
Van Der Beek et al, 201220 / Clinical features and predictors for disease natural progression in adults with Pompe disease: A nationwide prospective observational study / 66 / Cohort / 60 / Netherlands
Van Der Ploeg et al, 201011 / A randomized study of alglucosidase alfa in late-onset Pompe disease / 90 / Randomized Controlled Trial / 19.5 / USA & Europe
Wokke et al, 200814 / Clinical features of late-onset Pompe disease: A prospective cohort study / 61 / Cohort / 12 / USA & Europe
* 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.