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Rheumatology Advance Access published online on May 3, 2007
Rheumatology, doi:10.1093/rheumatology/kem072
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© The Author 2007. Published by Oxford University Press on behalf of the British Society for Rheumatology. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

The efficacy of inhibiting tumour necrosis factor {alpha} and interleukin 1 in patients with rheumatoid arthritis: a meta-analysis and adjusted indirect comparisons

R. Nixon1,2, N. Bansback2,3 and A. Brennan2

1MRC Biostatistics Unit, Cambridge, UK, 2School of Health and Related Research, University of Sheffield, UK and 3Centre for Health Evaluation and Outcome Sciences, St Paul's Hospital, Vancouver, Canada

Correspondence to: Richard Nixon, MRC Biostatistics Unit, Institute of Public Health, University Forvie Site, Robinson Way, Cambridge CB2 2SR, UK. E-mail: richard.nixon{at}mrc-bsu.cam.ac.uk


   Abstract
Top
 Abstract
Introduction
 Methods
 Results
 Discussion
 Appendix 1
 References
 
Objective. New treatments that inhibit the cytokines tumour necrosis factor {alpha} (TNF{alpha}) and interleukin 1 (IL-1) in the treatment of rheumatoid arthritis have proven clinical effect against placebo and methotrexate (MTX) in several clinical trials in early and late-stage disease and different severity groups. Since there are no head-to-head randomized controlled trials directly comparing the currently available treatments, etanercept, adalimumab, infliximab or anakinra, we perform a meta-analysis that adjusts for differences between study characteristics, and allows indirect comparisons between treatments.

Methods. Thirteen trials of cytokine antagonists were included from a systematic review of the literature. They reported the primary outcome of American College of Rheumatology (ACR) response criteria at 6 months or beyond. Meta-analytical methods are used to quantify relative treatment effects, using the log odds ratio of an ACR20 or ACR50 response at 6 months, whilst adjusting for study-level variables.

Results. In each of the trials, cytokine treatment was efficacious in comparison with placebo or MTX. For each treatment, the inclusion of MTX in combination improved the response. After adjustment for study-level variables, we found TNF{alpha} antagonists to be more efficacious compared with anakinra (P < 0.05). Indirect comparisons between the three TNF{alpha} antagonists indicated no difference in efficacy. Sensitivity analysis using a different statistical model structure confirmed these results.

Conclusion. When the outcome of interest is the probability of an ACR20 or ACR50 response at 6 months we found: (i) treatment with the IL-1 antagonist anakinra is better than placebo; (ii) for each treatment, the use of combination MTX improves the probability of response; (iii) treatment with any of the TNF{alpha} antagonists is better than with the IL-1 antagonist anakinra; and (iv) all drugs in the TNF{alpha} antagonist class are no different from each other.

KEY WORDS: Rheumatoid arthritis, Biologic agents, Meta-analysis


   Introduction
Top
 Abstract
 Introduction
Methods
 Results
 Discussion
 Appendix 1
 References
 
Rheumatoid arthritis (RA) is a chronic disease, with a prevalence in the UK of ~1% and an annual incidence of 3 per 10 000 adults [1]. It is the most common form of inflammatory arthritis, and has a substantial societal effect in terms of cost, disability and lost productivity [2]. Pro-inflammatory cytokines such as tumour necrosis factor {alpha} (TNF{alpha}) and interleukin 1 (IL-1) are found in increased levels in synovial fluid of patients with active RA. The discovery that inhibiting these cytokines can reduce the manifestations of RA, improving function and retarding radiological measures of disease progression, has led to the development of a number of new therapies [3, 4].

Of the presently available cytokine antagonists, only anakinra targets the IL-1 receptors, blocking IL-1 signalling [3]. It is administered once daily, subcutaneously, normally at a dose of 100 mg. The other three available treatments all seek to neutralize TNF{alpha}, but have different properties. Etanercept is a construct of soluble TNF{alpha} receptors in comparison with monoclonal antibodies, infliximab being chimeric while adalimumab is fully human [5, 6]. Etanercept and adalimumab are both administered subcutaneously, 25 mg twice weekly and 40 mg every other week, respectively. Infliximab is administered intravenously every 8 weeks, with the dose dependent on the patient's weight.

The introduction of these cytokine antagonists has increased RA expenditure considerably, and is estimated to triple the size of the RA market to $8.4 billion in the next 5 yrs [7]. There are currently no head-to-head randomized controlled trials (RCTs) between cytokine antagonists. Instead, each treatment has been the subject of placebo-controlled randomized studies leaving physicians and patients to make crude indirect comparisons when deciding between cytokine therapies. In the meantime, the prospects of a head-to-head study are remote, as there is little incentive for the manufacturers of each treatment to risk market share, and the large sample size required for statistical power would make costs too high for most public agencies.

Without a head-to-head study to determine comparative efficacy, it remains conceivable that many patients are receiving suboptimal treatment. There are plausible reasons to suspect differences between inhibiting the different cytokines IL-1 and TNF{alpha} [8]. Furthermore, questions over the comparative efficacy of the different mode of actions used to inhibit TNF{alpha} have been raised having shown distinctly different results in other inflammatory diseases such as Crohn's disease, sarcoidosis and Wegener's vasculitis. [9, 10]. Traditional meta-analysis techniques [11] can be used to pool results but methodological challenges arise for cytokine antagonists because there are four different treatments, different comparators and significant heterogeneity between trials [12].

Our study uses meta-analytical techniques known as mixed treatment comparison [13] and meta-regression [14] to compare the efficacy of these cytokine treatments by adjusting for important study-level covariates.


   Methods
Top
 Abstract
 Introduction
 Methods
Results
 Discussion
 Appendix 1
 References
 
Identification of RCTs
We reviewed the medical literature published after the introduction of cytokine antagonists in the late 1990s. We searched from January 1990 to January 2005 using MEDLINE, EMBASE, the Cochrane Library, Database of Reviews of Effectiveness (DARE) and the Scientific Citation Index. The MeSH search used in MEDLINE, EMBASE and the NHS DARE consisted of three steps, each containing any possible MeSH relevant to the target condition (RA), study drug (cytokine antagonist or biological or TNF{alpha} or IL-1 or etanercept, adalimumab, infliximab or anakinra) and study method (RCT). All MeSHs were exploded. The steps were then combined to produce relevant citations. We also searched the Scientific Citation Index and Cochrane Library with the keywords ‘rheumatoid arthritis’. Proceedings from the American College of Rheumatology and European Congress of Rheumatology meetings were searched electronically for the years 2001–2004. Food and Drug Administration (FDA) submissions for new drug applications were searched to provide additional information on trial results. The reference lists of identified publications were reviewed to identify any additional studies and/or citations.

We chose to focus on the primary outcome of RA trials: the American College of Rheumatology (ACR) response criteria. The ACR20 is defined as a reduction by 20% or more in the number of tender and swollen joints plus 20% improvement in at least three of the following five measures: pain, patient global assessment, physician global assessment, self-assessed physical disability (measured by the Health Assessment Questionnaire or HAQ [15]) or levels of acute-phase reactant [16]. The ACR50 (improvement of 50% or more) is often deemed to be more clinically relevant [17]. Trials compare treatment with placebo with or without methotrexate (MTX). MTX is the most commonly used disease-modifying anti-rheumatic drug (DMARD) [18]. Both the short-term efficacy and the toxic effects of new drugs for RA are usually evaluated in clinical trials of 6–12 months’ duration [19].

Two researchers (N.B. and A.B.) independently applied the inclusion criteria for the analysis (Fig. 1). Discrepancies between researchers were infrequent and were resolved by discussion. The following inclusion criteria were used: (i) RCTs comparing cytokine antagonists with placebo or MTX; (ii) patients with a clinical diagnosis of RA; (iii) a trial time horizon of at least 6 months; and (iv) sufficient data provided to determine the odds ratios for the ACR20 and ACR50. Primary results were extracted in duplicate, along with all reported study and patient characteristics. All data extracted comply with intention to treat. Where only graphical figures of ACR results were reported, estimates were made using a digitized graph reader.


Figure 1
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  		FIG. 1. Process of study selection for randomized controlled trials in meta-analysis.

 
Following the systematic review, 13 RCTs were included in the final analysis comprising 6694 subjects, of whom 4694 received either TNF{alpha} or IL-1 treatment (Table 1). Three trials study anakinra (n = 392) [20–22]. The largest study of anakinra [23] is excluded since ACR results were not reported. Four trials study etanercept (n = 1637) [24–27] and two trials study infliximab (n = 1432) [28, 29]. Four trials study adalimumab (n = 2233) [30–33]. The Safety Trial of Adalimumab in RA study [34] is excluded because the comparator is a combination of different disease-modifying drugs rather than placebo ± MTX.


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  		TABLE 1. Patient characteristics and results from trials in meta-analysis

 
In each trial, one cytokine antagonist is tested, usually in several doses, against a control arm of placebo or MTX. Treatments were given as monotherapy or in combination with MTX except for infliximab, which normally requires combination MTX. Doses used and frequency of treatment differ in each treatment arm. Sample size ranges from 89 up to 1004. Published ACR20 and ACR50 outcome measures exist at 6 months except for two studies where we have used 12-month data in place of 6-month data [28, 33]. Baseline characteristics differ substantially. For example, mean disease duration at baseline ranges enormously from 1 to 13 yrs, the mean number of previously used DMARDs ranges from 0 to 4, and mean HAQ disability score from 1.3 to 1.9. This heterogeneity in baseline characteristics clearly impacts ACR response results. For example, four trials included patients who had not previously tried MTX treatment and used placebo plus MTX as the control arm [24, 25, 28, 33], and their results show consistently lower odds of response than trials where patients had all previously used MTX. Indeed two of these trials [25, 33] show the control arm with higher results than the cytokine antagonists monotherapy arm. In principle all of the covariates in Table 1 are candidates for the statistical modelling to explain heterogeneity between trials.

Statistical methods
Mixed-treatment comparison evidence synthesis [13] extends traditional meta-analysis to allow indirect comparisons, instead of all studies comparing the same treatment with the same comparator. We use this method to perform an analysis to estimate the odds ratio of an ACR20 or ACR50 response at 6 months if treated with a cytokine antagonist agent compared with placebo or MTX, using the 13 RCTs identified. This mixed-treatment analysis allows the cytokine antagonist to be used with or without MTX as background therapy, and also the placebo to be administered with or without MTX. It is common to combine binary results in a meta-analysis on the odds ratio scale, as ratios are more heterogeneous than absolute measure across studies and odds ratios are mathematically convenient. Our model also examines how these odds ratios vary with important prognostic factors. We use meta-regression to explain the differences between studies by regressing the effect sizes from each study onto study-level characteristics [14]. The effect sizes, adjusted for study-level characteristics, will not be identical, as the regression will not completely explain the heterogeneity, and so a random-effects distribution is placed on the adjusted effects sizes. A detailed description of statistical modelling is available [35], and a brief mathematical description of the models is given in Appendix 1.

To select appropriate study-level covariates for adjustment we performed a series of exploratory analyses. With relatively few studies, multiple analyses using all study-level covariables will have a high probability of finding a spurious explanatory variable and there are insufficient degrees of freedom to sensibly model many covariables. Our exploratory analysis suggested that mean disease duration and mean HAQ disability index at baseline were suitable covariates and these were selected for the model because they are known to have prognostic value in determining the effect of treatment [36, 37]. The regression in the modelling is performed on the log odds ratio scale, so a linear relationship between the log odds ratio and explanatory study-level covariates is assumed. Figure 2 illustrates the relationship between disease duration and response rates, clearly showing that the log odds ratio of a successful ACR50 response is higher for trials with patients with higher mean disease duration at baseline.


Figure 2
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  		FIG. 2. Illustration of the effect of mean disease duration at baseline on trial results. Observed log odds ratio of ACR50 at six months given treatment compared with the control arm, plotted against mean disease duration at baseline. (The linear regression weighted by the inverse of the variance of the log odds ratio estimate is also shown. The area of each circle is inversely proportional to the variance of the log odds estimate.)

 
Two statistical models are considered. The primary model takes account of the fact that the more than one treatment arm can come from an individual study (e.g. two different doses of an individual drug), essentially assuming a random effect across studies. The validation model assumes that each treatment arm is independent of the study; a random effect is placed across each treatment arm irrespective of study. The validation model also assumes MTX treatment effects are the same for each arm. The key difference between these models is that the primary model acknowledges the structure of the arms within studies in its exchangeability assumption, and allows MTX and cytokine antagonist effects to be correlated. The validation model is used as a sensitivity analysis. Both models were fitted using WinBUGS [38], which employs Markov chain Monte Carlo (MCMC) simulation [39].


   Results
Top
 Abstract
 Introduction
 Methods
 Results
Discussion
 Appendix 1
 References
 
Model parameters
The parameter values and associated odds ratios for the primary model are shown in Table 2. The parameter values for all four cytokine treatments were positive indicating all four treatments were associated with an increased odds of an ACR20 or ACR50 response. The inclusion of MTX in combination with each cytokine antagonist, and increased mean disease duration at baseline were both associated with improved odds of response. Mean HAQ disability index at baseline has a negative coefficient, indicating higher disability implies lower odds of response. However, the mean disability at baseline is itself correlated with mean disease duration at baseline (patients with longer disease duration are on average more disabled) and the effect of the negative coefficient is to dampen down the positive relationship between mean disease duration and response rate.


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  		TABLE 2. Odds ratio estimates for achieving an ACR response for the different treatments (compared with placebo), at the mean baseline disease duration and HAQ. Also, the parameter estimates that are used to find these odds ratios, and the effect of MTX use, disease duration and HAQ on results

 
Model validity
The predicted odds ratio of an ACR response for a given cytokine antagonist for each treatment arm in a given study can be estimated from the primary model. Details of how to do this are given in the notes to Table 2. Figure 3 shows the observed log odds ratios (black) alongside the predicted log odds ratios (white) for each study arm. In general, the predicted results match the observed relatively closely. Because the predicted values are fitted from a random effects model, the predictions shrink slightly towards the overall mean for each drug. Some trials report higher response rates than expected even after accounting for the covariate adjustment (e.g. 40 mg adalimumab weekly in the van de Putte study) whilst others report lower response rates than expected (e.g. 75 mg/day anakinra in the Bresnihan 1998 study). The study-level covariables, mean disease duration and mean disability index at baseline, explain 67% and 91% of the between-study heterogeneity in the ACR20 and ACR50 log odds response rates, respectively.


Figure 3
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  		FIG. 3. Validity of prediction. Forrest plot of the observed log odds ratios for each trial arm from the raw data vs predicted values from the model.

 
Comparative efficacy of cytokine antagonists
To compare each treatment, we computed the expected odds ratio of ACR responses for each of the 13 studies with an average baseline disease duration of 8 yrs and average baseline HAQ equal to 1.6 (average values for all 13 studies). Figure 4 shows results for each treatment as monotherapy, each treatment in combination with MTX and the TNF{alpha} antagonists together as a class. The results are found from the parameter estimates given in Table 2. Anakinra monotherapy is found to be efficacious in comparison with placebo with an odds ratio of 1.70 for an ACR20 response, and an odds ratio of 2.13 for an ACR50 response. Etanercept, infliximab and adalimumab monotherapy have higher odds ratios in comparison with placebo (3.58, 3.47 and 3.19, respectively for an ACR20 response; 4.21, 4.14 and 3.97, respectively for an ACR50 response). The use of MTX in combination with any of the cytokine antagonists was found to increase the efficacy of each treatment further. Analysed as a class, the TNF antagonists + MTX were more efficacious than anakinra + MTX (odds ratios of 6.35 vs 3.20 for an ACR20, and 8.53 vs 4.56 for ACR50). The efficacy of the three TNF{alpha} antagonists was almost identical with overlapping confidence intervals, indicating no statistical differences between etanercept, infliximab and adalimumab.


Figure 4
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  		FIG. 4. Comparative efficacy of cytokine antagonists. Odds ratios of an ACR20 and ACR50 response at 6 months for each treatment, and TNF{alpha} antagonists as a class, given both as a monotherapy as in combination with MTX. The odds ratios are compared with placebo and have been adjusted for mean disease duration and mean HAQ disability index at baseline.

 
The results shown earlier are further confirmed in a series of analyses of the relative effect of the cytokines against each other shown in Fig. 5. For example, the odds ratio of an ACR50 response at 6 months if treated with infliximab + MTX compared with etanercept + MTX is 0.98, where an odds ratio of 1 is defined as equivalence. The confidence intervals all contain 1 suggesting there is no statistical difference between the treatments. This relative analysis again, demonstrated that the TNF{alpha} antagonists class was more efficacious compared with anakinra. The odds ratios of 1.96 (CI 1.03–4.01) for ACR20 and 1.93 (CI 1.05–3.50) for ACR50 indicate the effect was statistically significant (P < 0.05).


Figure 5
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  		FIG. 5. Relative efficacy of cytokine antagonists. Odds ratios of an ACR20 and ACR50 response at 6 months for all cytokine antagonist treatment pairs and TNF{propto} antagonists as a class compared with anakinra.

 
Sensitivity analysis
As a sensitivity analysis, odds ratios were also calculated using the validation model. The results (Table 3) show very similar finding and confirm the results of the primary model under a different set of statistical assumptions.


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  		TABLE 3. Odds ratios of an ACR20 and ACR50 response at 6 months for (i) anakinra compared with placebo for average baseline disease duration and HAQ, (ii) TNF{alpha} antagonists compared with anakinra and (iii) each pair of TNF{alpha} antagonists

 

   Discussion
Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
Appendix 1
 References
 
The development of cytokine antagonists has transformed the management of RA. As of early 2005, 13 randomized studies have reported ACR20/50 outcomes at 6 months or beyond for TNF{alpha} and IL-1 inhibitors in comparison with placebo or MTX with or without combination MTX. Comparing the efficacy between therapies robustly has previously been impossible since there are no head-to-head studies, and existing trials have substantial differences in study design and populations. We extend meta-analytical techniques to allow the trials to be pooled and treatments compared indirectly, whilst identifying and adjusting for important study-level variables. In summary, the analyses suggest that when the outcome of interest is the odds ratio of an ACR20 or ACR50 response at 6 months: (i) treatment with the IL-1 antagonist anakinra is better than placebo; (ii) for each treatment, the use of combination MTX improves the probability of response; (iii) treatment with a TNF{alpha} antagonist is better than with the IL-1 antagonist anakinra; and (iv) TNF{alpha} antagonists as a class are no different from each other.

Our results supersede the limited literature on the comparison of cytokine antagonists in RA. In an analysis of four studies of patients with severe RA (n = 1053), Hochberg et al. [40] suggested that the TNF{alpha} antagonists in combination with MTX had similar efficacy. Whilst we have come to a similar conclusion, we have used meta-regression to extend the evidence base to patients with early and less severe RA and increase the sample size to nearly 7000 subjects. A recent commentary postulated that anakinra might not be effective at all [41], supported by a clinical study that found no statistical difference when comparing etanercept in combination and without anakinra, along with some controversy over the missing results from the largest study of anakinra. In contrast, our results suggest that anakinra is effective in comparison with placebo; though this could be revised if missing results were reported.

As in previous studies [36, 42], we found mean disease duration at baseline to be the most important study covariate. However, there is one seeming inconsistency. Analysing trials of conventional DMARDs vs placebo, Anderson et al. found that, the probability of response decreases in patients with longer disease duration, whereas we found that the odds ratio of response increases with the mean disease duration of the trial recipients. In fact, these two results are consistent because, whilst the absolute efficacy of the cytokine antagonists does decrease slowly with disease duration, the absolute efficacy of MTX decreases much more rapidly with disease duration, and thus the ‘relative’ efficacy of the cytokine antagonists vs MTX increases with disease duration.

It is possible that the disease duration covariate has explanatory power because it is correlated with other important variables that might not be measured in each trial. For example, we included the mean number of previous DMARDs as a covariate and found no strong relationship with response rates. However, the mean number of previous DMARDs masks a series of more complex differences between trials whereby some include patients that have never attempted existing DMARDs, others, patients who have only partially responded to DMARDs and others, only patients that have failed a specified number of previous treatments. Without detailed unpublished information about each study, we were unable to quantify these different types of patients and we suspect that the disease duration covariate may indirectly account for some of this between-study variation.

Our statistical methods extend those previously used to compare multiple treatments [13]. Each treatment is used in several arms in each trial rather than just once, and ideas from meta-regression are used to explain heterogeneity. This method can be used for other diseases and treatments and is applicable when (i) data are available on different interventions used to treat the same condition, (ii) a single outcome measure is used in all studies and (iii) heterogeneity between studies might be explained by information available on study-level covariates.

A limitation of our analysis is that we restrict our outcome measure to the ACR response criteria, which have several established limitations [43]. Our analysis does not account for cytokine antagonists effects retarding joint deterioration measured radiographically, nor do we examine safety profile differences; both important considerations when choosing between treatments. We were not able to use our techniques to analyse these alternative outcomes because radiographic progression is less frequently reported with different measures in different studies. Similarly, adverse events are difficult to synthesize because there can be substantial inconsistency in the definitions and selection of adverse events published across studies.

The other main limitations to the statistical analysis concern model assumptions and data availability. First, our primary model assumes that the treatment effects are the same for each arm within a study where a particular treatment is used. This is reasonable for MTX, as the same dose is frequently used. However, each arm of a study may use a different dose of each cytokine antagonist. The model could be improved if some dose-response relationship were to be included in the model, but it is not clear how say 25 mg of etanercept could be compared with 20 mg of adalimumab, nor even how 20 mg of adalimumab administered every week should be compared with 40 mg every other week. Secondly, information on the MTX–cytokine antagonist treatment interaction effect is only supplied in the two studies where MTX is not always used in the treatment arms within a study [25, 33], and so is imprecisely estimated. Thirdly, the study-level parameters, which assess the effect of disease duration and HAQ on the treatment effect, are assumed the same for all treatments. These could be different for each cytokine antagonist.

The relationships between variables shown here are generated at the clinical study level and are not necessarily generalizable to the individual patient. In principle, the relationship of 6-month ACR with disease duration or with baseline HAQ of patients across trials may not be the same as these relationships for individual patients within trials [14], because the relationships might be due to other study-level variables that have not been measured. In practice in this case, other studies have shown that disease duration and baseline HAQ are strong predictors of efficacy [36, 37]. This is a generic issue with meta-regression at a study level and individual data is needed to quantify relationships at an individual patient level.

A head-to-head randomized study of cytokine antagonists is unlikely. Although direct comparisons top the hierarchy of evidence [44], when they are absent, the approach of mixed-treatment comparison using indirect randomized data is the best tool for comparing treatments and making recommendations. Policy makers, clinicians and patients must make decisions and our study suggests that they can consider etanercept, infliximab and adalimumab as equally likely to be effective in terms of their odds of ACR response. Decision-makers may wish to take account of other factors beyond this including evidence on radiographic progression, adverse events and extending out to quality of life measures and economic costs. Such analyses require further evidence synthesis and sometimes complex modelling and uncertainty analysis [45].

The results of this study have implications for both current and future service provision in many countries. For instance in the US, prior to implementation of the Medicare Modernization Act, patients with public insurance were 30% more likely to receive infliximab even though it can cost up to $7000 or more a year when infusion costs are included [46]. Our analysis served as input to a Congressionally mandated evaluation of the Medicare Replacement Drug Demonstration (Section 641 of the Medicare Drug Improvement and Modernization Act of 2003) which extended Medicare coverage to include etanercept and adalimumab [47]. With new products expected to emerge and direct head-to-head trials remaining unlikely, uncertainty over the optimal treatment for RA could well increase. The results of our analysis can be updated as new trial data are published and can be incorporated into analyses ranging from exploratory drug development through to long-term cost-effectiveness analyses.

In summary, we have undertaken a systematic review and meta-analysis of cytokine antagonists for RA using mixed-treatment comparison and meta-regression to adjusting for study-level heterogeneity. The results suggest that if ACR response criteria are taken as the outcome measure then: treatment with the IL-1 antagonist anakinra is better than placebo; that for each cytokine antagonist, the use of combination MTX improves the probability of response; that treatment with a TNF{alpha} antagonist is better than with the IL-1 antagonist anakinra and that TNF{alpha} antagonists as a class are no different from each other. While statistical analyses will never replace the need for RCTs, we propose this methodology for providing evidence-based comparison between cytokine antagonists to support patient and physician decision.

Each author is funded by their respective institutions. Project grant funding for health economic analyses related to rheumatoid arthritis has been previously received from Wyeth, Interleukin Genetics Inc., Abbott Laboratories, for N.B. and A.B. and from the British Society of Rheumatology and the US Agency for Healthcare Research and Quality for R.N., N.B. and A.B. R.N. secondment to the School of Health and Related Research was part funded by project work with AstraZeneca in a related area.

Formula


   Appendix 1
Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Appendix 1
References
 
For the statistical modelling, our notation is as follows: i = 1 ... 13 denotes the study index; j = 0 ... Ji the arm within the study, where 0 indexes the control group and Ji is the number of treatment regimen being tested in study i; ki denotes the cytokine antagonist used in study i, where 1 indicates anakinra, 2 etanercept, 3 infliximab and 4 adalimumab. nij denotes the number of patients in arm j of study i; rij the number of patients achieving ACR20 (or ACR50, hereafter referred to as a generic ACR). mij is an indicator variable for treatment with MTX, it is 1 if MTX is given in arm j of study i and 0 otherwise. Two study-level covariates are also included: x1i is the average disease duration and x2i the average baseline HAQ for each study. These two covariates are recentred about their means across studies to aid model fitting.

Assume each patient in arm j of study i independently has a probability pij of achieving ACR


Formula 1

(1)

{alpha}i is the log odds of ACR in the control arm of study i, these are fixed effects, and are treated as nuisance parameters. {theta}ij is the log odds ratio of ACR for study i treatment arm j given treatment with cytokine antagonist ki compared with placebo; and ß the log odds ratio for treatment with MTX. These are assumed to act independently of each other, so the log odds ratio for the relevant cytokine antagonist mono-therapy vs placebo is the same as for the relevant cytokine antagonist combination therapy with MTX vs MTX. The average baseline disease duration and average baseline HAQ of patients are included as meta-regression covariables This is written


Formula 2

(2)

For model 2, we assume each of the log odds ratios has been sampled from a bivariate normal distribution. This is assuming all the treatment effects are exchangeable between studies for both treatment with a cytokine antagonist and with MTX.


Formula 3

(3)

{sigma}2 is the heterogeneity between studies, and µß and µki the overall log odds ratio of ACR given treatment with MTX and cytokine antagonist k, respectively. The between-study variance of the treatment effects is {sigma}2 for both treatment with a cytokine antagonist and with MTX and the correlation is fixed at 1/2{sigma}2. This assumes that the heterogeneities between study treatment effects on the log odds ratio scale are invariant with respect to the choice of comparison [13]. For example, the variance of the log odds ratio of treatment with a cytokine antagonist compared with placebo is the same as for treatment with MTX compared with placebo. This is also the same variance of the log odds ratio of treatment with a cytokine antagonist compared with MTX. In the validation model, only the cytokine antagonist treatment effects are assumed exchangeable (between trial arms, not studies) and the affect of MTX on treatment is a fixed effect.

The model also allows for multiple treatment arms of the same drug within a study, and assumes that treatment effects are the same for each arm within a study where the same treatment is used.

The model assumes that the {gamma} parameters are the same for all treatments, and ßi and {theta}i have been adjusted for study-level covariates. Thus the {theta}i is interpreted as the log odds ratio for the relevant cytokine antagonist at the mean value of the study-level covariates.


   References
Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Appendix 1
 References
 

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Submitted 8 September 2006; revised version accepted 23 February 2007.
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