Quantifying, displaying and accounting for heterogeneity in the meta-analysis of RCTs using standard and generalised Qstatistics
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Abstract
Background
Clinical researchers have often preferred to use a fixed effects model for the primary interpretation of a meta-analysis. Heterogeneity is usually assessed via the well known Q and I^{2} statistics, along with the random effects estimate they imply. In recent years, alternative methods for quantifying heterogeneity have been proposed, that are based on a 'generalised' Q statistic.
Methods
We review 18 IPD meta-analyses of RCTs into treatments for cancer, in order to quantify the amount of heterogeneity present and also to discuss practical methods for explaining heterogeneity.
Results
Differing results were obtained when the standard Q and I^{2} statistics were used to test for the presence of heterogeneity. The two meta-analyses with the largest amount of heterogeneity were investigated further, and on inspection the straightforward application of a random effects model was not deemed appropriate. Compared to the standard Q statistic, the generalised Q statistic provided a more accurate platform for estimating the amount of heterogeneity in the 18 meta-analyses.
Conclusions
Explaining heterogeneity via the pre-specification of trial subgroups, graphical diagnostic tools and sensitivity analyses produced a more desirable outcome than an automatic application of the random effects model. Generalised Q statistic methods for quantifying and adjusting for heterogeneity should be incorporated as standard into statistical software. Software is provided to help achieve this aim.
Keywords
Random Effect Model Reference Interval Funnel Plot Asymmetry Clinical Trial Unit Fixed Effect EstimateList of Abbreviations
- IPD
Individual Patient Data
- FE
fixed effect
- DL
DerSimonian and Laird
- RE
Random effects
- PM
Paule-Mandel.
Background
Meta-analysis provides a way of quantitatively synthesising the results of medical studies or trials that target a particular research question. As shown in a 2005 review of the clinical research literature [1], it is still most common to meta-analyse results across clinical studies using the inverse variance approach, to yield a 'fixed' or 'common' effect estimate. By obtaining individual patient data (IPD) from all trials in a meta-analysis, some aspects of clinical heterogeneity can be minimised through data cleaning [2]. However, regardless of whether the meta-analysis is based on IPD or aggregate data, substantial statistical heterogeneity between studies may still remain.
Cochran's Q statistic has long been used to assess statistical heterogeneity in meta-analysis. When Q is larger than its expected value E[Q] under the null hypothesis of no heterogeneity, the difference Q - E[Q] can be used to furnish the most popular estimate of the heterogeneity parameter, using the DerSimonian and Laird method [3]. Higgins and Thompson's I^{2} statistic [4, 5] is also a simple function of Q and quantifies the proportion of total variation that is between trial heterogeneity. Unlike Q, I^{2} is designed to be independent of the number of trials constituting the meta-analysis and independent of the outcome's scale, so it can easily be compared across meta-analyses. It is now reported as standard, with or without Cochran's Q.
The presence of significant and substantial heterogeneity demands some form of action. Ideally, after exploration of the data, heterogeneity can be explained by variation in the constituent trial's characteristics. If this is not possible then some may feel a meta-analysis inappropriate altogether, whereas some would opt for fitting a random effects model to the data instead. There is no accepted rule for deciding on when a move from a fixed to a random effects model is the right course of action [6]. Clearly, all other things being equal, the larger the magnitude of the heterogeneity the stronger the case for a shift. However, as the amount of heterogeneity increases, so too does the potential impact of moving from one model to the other. Thus, with increasingly diverging interpretations, it is sometimes very difficult to make a satisfactory decision on which model to choose, or indeed whether to pool the trials in a meta-analysis at all.
In Methods we review the standard approach to meta-analysis and heterogeneity quantification based on the Q statistic. We then introduce a similar approach based on a 'generalised Q' statistic that has recently been proposed. In Results we analyse the summary data from 18 separate IPD meta-analyses to see whether the original conclusions could have been sensitive to the choice of fixed or random effects model. A more in-depth analysis is then conducted on the two meta-analyses with the largest observed heterogeneity. The 18 meta-analysis are then used to illustrate the relative performance of the standard and generalised Q statistics in measuring the extent of heterogeneity present. Finally, in Discussion and Conclusions we review the issues raised and offer recommendations for the future quantification and reporting of heterogeneity in meta-analysis.
The data
The summary statistics for 18 meta-analyses carried out by the MAG.
Meta-analysis | # trials | Q, P-value | Fixed Effect HR (CI) P-value | |
---|---|---|---|---|
cervix 1 [15] | 18 | 44.48, 0.00 | 62 | 1.05 (0.93-1.19) 0.39 |
cervix 2 [17] | 18 | 20.83 0.23 | 18 | 0.76 (0.67-0.85) 0.00 |
cervix 3 [15] | 5 | 9.18, 0.06 | 56 | 0.65 (0.53-0.80) 0.00 |
bladder 1 [14] | 9 | 7.27, 0.51 | 0 | 0.91 (0.83-1.01) 0.08 |
bladder 2 [16] | 6 | 2.25, 0.81 | 0 | 0.75 (0.60-0.96) 0.02 |
nsclc 1 [8] | 17 | 28.98, 0.02 | 45 | 1.04 (0.96-1.12) 0.33 |
nsclc 2 [8] | 7 | 3.63, 0.73 | 0 | 0.98 (0.83-1.14) 0.76 |
nsclc 3 [8] | 25 | 22.32, 0.56 | 0 | 0.90 (0.83-0.97) 0.01 |
nsclc 4 [8] | 11 | 39.63, 0.00 | 75 | 0.84 (0.74-0.95) 0.01 |
ovarian 1 [7] | 19 | 21.92, 0.24 | 18 | 0.98 (0.91-1.06) 0.69 |
ovarian 2 [7] | 11 | 12.83, 0.23 | 22 | 0.93 (0.83-1.05) 0.23 |
ovarian 3 [10] | 9 | 14.78, 0.06 | 46 | 0.88 (0.79-0.98) 0.02 |
ovarian 4 [10] | 9 | 10.35, 0.24 | 23 | 0.91 (0.80-1.05) 0.21 |
ovarian 5 [10] | 12 | 2.57, 1.00 | 0 | 1.02 (0.93-1.12) 0.66 |
port [11] | 9 | 13.06, 0.11 | 39 | 1.21 (1.08-1.34) 0.00 |
sarcoma [9] | 14 | 11.80, 0.54 | 0 | 0.89 (0.76-1.03) 0.12 |
oeso [12] | 6 | 10.37, 0.07 | 52 | 0.89 (0.78-1.01) 0.06 |
glioma [13] | 12 | 13.29, 0.27 | 17 | 0.85 (0.78-0.92) 0.00 |
Methods
The ϵ_{ i }term relates to the precision of study i's estimate, and is assumed to follow a N (0, Open image in new window ) distribution.
where W_{ i }= 1/ Open image in new window is study i's precision.
Heterogeneity quantification using the standard Q-statistic
where Open image in new window , and is referred to as the 'typical' within study variance.
when Q > M - 1.
From a philosophical perspective, fixed effect and random effects estimates target very different quantities. Fixed effect models estimate the weighted mean of the study estimates, whereas random effects models estimate the mean of a distribution from which the study estimates were sampled. However, if model (1) is correct and we are additionally willing to assume that the u_{ i } terms are independent of the ϵ_{ i }terms, then they should both provide estimates of the same parameter θ. Another consequence of this independence assumption is that the individual study estimates Open image in new window should be independent of the ϵ_{ i }terms, and hence we do not expect the magnitude of the effect estimate to be correlated with its precision.
Heterogeneity quantification using a 'generalised' Q-statistic
where Open image in new window and where Open image in new window is also calculated from equation (2) by replacing W_{ i } with Open image in new window . Like the standard Q statistic in equation (3), this also follows a Open image in new window distribution under the null hypothesis of no heterogeneity. Paule and Mandel [23] (PM) and DerSimonian and Kacker [22] propose to estimate τ^{2} by iterating equation (5) until Q(τ^{2}) equals its expected value of M-1; this estimate will be referred to as Open image in new window . DerSimonian and Kacker recommend using Open image in new window since it is still very easy obtain, is guaranteed to have at most one solution and provides a more accurate estimate of τ^{2} that closely mirrors both the REML estimate and the generalized Bayes estimate [24], which are both much harder quantities to obtain computationally.
Viechtbauer [25] suggests that equation (5) can additionally be used to provide an α-level confidence set for Open image in new window , by finding the values of τ^{2} that equate Q(τ^{2}) with the α/2th and 1- α/2th percentiles of the Open image in new window distribution. He showed that this method performed very well in a simulation study that evaluated its coverage properties compared to a range of other methods - such as Biggerstaff and Tweedie [26] and Sidik and Jonkman [27] - primarily because it is based on an exact χ^{2} distribution, rather than a distributional approximation.
for any estimate of the between study variance Open image in new window . From now on we will refer to Inconsistency statistics specifically utilising the DL method as Open image in new window and those specifically utilising the PM method as Open image in new window . The term I^{2} will be reserved for discussing the general concept of Inconsistency.
Reference intervals for Open image in new window and Open image in new window
where Open image in new window and Open image in new window represent the values of τ^{2} equating Q(τ^{2}) to the lower α/2 and upper 1-α/2 percentiles of the relevant χ^{2} distribution.
Results
A standard Q-statistic analysis
One could use the reference intervals around Open image in new window to directly test for the presence of heterogeneity, as apposed to Q; a strategy suggested by Medina et. al. [28]. From Figure 1 we see that only 3 out of the 7 meta-analyses with significant Q statistics produced significant Open image in new window statistics at the 10% level. Since Q and Open image in new window are so closely related it is perhaps surprising to some reviewers that such differing conclusions could arise.
From Figure 1, the two meta-analyses with the most apparent statistical heterogeneity were NSCLC 4 [8] and Cervix 1 [15]. They also exhibit the most marked differences between their fixed and random effects estimates, as highlighted by large deviations from the diagonal - shown in red in Figure 2. These two meta-analyses are now discussed further, in order to demonstrate how we chose to investigate these heterogeneous data sets.
The NSCLC 4 meta-analysis
This meta-analysis compared the effectiveness of supportive care plus chemotherapy versus supportive care alone for patients with advanced non-small cell lung cancer. The fixed effect hazard ratio estimate of 0.84 suggests a substantial and highly significant benefit from the addition of chemotherapy with a p-value for a null effect of 0.005. The random effects model estimate of 0.77 suggested an even more extreme benefit of chemotherapy. However, such was the magnitude of heterogeneity detected - as revealed by an I^{2} of 75% - this estimate is attributed much less certainty, with a p-value of 0.04.
Subgroup analyses for the two examples.
Trial Group | # trials | Q, P-value, Open image in new window (%) | Fixed Effect HR (CI) P-value | Random Effects HR (CI) P-value |
---|---|---|---|---|
NSCLC data | ||||
all | 11 | 39.6 (1.97e-05) 74.8 | 0.84 (0.74-0.95) 5.42e-03 | 0.77 (0.59-0.99) 0.042 |
Cisplatin | 8 | 22.2 (2.34e-03) 68.5 | 0.73 (0.63-0.85) 6.63e-05 | 0.70 (0.53-0.93) 0.014 |
Q_{ int } = 39.62 - (22.20 + 8.72) = 8.70 (p = 0.003) | ||||
all* | 11 | 0.84 (0.61-1.16) 0.21 | ||
Cervix data | ||||
>14 days | 11 | 12.76 (0.24) 22 | 1.25, (1.07,1.46) 0.005 | 1.27 (1.06,1.53) 0.0099 |
≤ 14 days | 7 | 20.74 (0.002) 71 | 0.83, (0.69,1.00) 0.046 | 0.87 (0.60,1.25) 0.44 |
Q_{ int } = 44.48 - (12.76 + 20.74) = 10.98 (p = 9e-04) |
The Cervix 1 meta-analysis
Significant heterogeneity persisted in the results of trials using shorter chemotherapy cycles. The fixed effect result suggested a modest benefit from short cycle chemotherapy, whereas the random effects model suggested less of an effect and a much wider confidence interval overlapping the null effect of 1. However our conclusions were also guided by a sensitivity analysis of the shorter duration trials, excluding the MRC CeCa trial. Figure 4 (right) shows a Baujat plot [32, 33] of the data; on the horizontal axis is the contribution of each study to the overall Q statistic in equation (3), on the vertical axis is the difference between the fixed effect estimate Open image in new window with and without each study, standardised by the total variance of the fixed effect estimate without that study. If the fixed effects model is correct, each point's horizontal component should be approximately Open image in new window distributed. The CeCa trial is way out on its own, whereas the other trials all fall within the 95th percentile of this distribution. Thus the total heterogeneity present is very much a product of this single trial. Furthermore, the CeCa trial's large vertical component shows that its inclusion Significantly alters the fixed effects estimate too. Excluding the CeCa trial gave a fixed-effect result still favouring short cycle chemotherapy (HR = 0.76, 95%CI = 0.62-0.92) and heterogeneity was much reduced. Repeating the sensitivity analysis using a random effects model gave very similar results (HR = 0.75,95%CI = 0.58-0.95).
A generalised-Qanalysis
A simulation study
The point estimates and confidence intervals for Open image in new window differ from the original Open image in new window - in particular the confidence intervals for Open image in new window are noticeably wider. In order to see if this extra width truly reflected the uncertainty in the estimation of Open image in new window , or instead if it was over-conservative, we conducted twelve simulation studies, each one based on the characteristics of a meta-analysis which exhibited some heterogeneity (from 'Glioma' to 'NSCLC 4'). From each one we took the number of studies M, within study variances σ^{2} and the DL heterogeneity estimate Open image in new window . For meta-analysis j, j = 1, ..., 12, we then simulated 10,000 new meta-analyses of M_{ j } study estimates Open image in new window for i = 1, ... M_{ j } . The choice of θ = 0 is clearly unimportant. Since the within study variances and the true τ^{2} values were held fixed, the true value of I^{2} stayed fixed at the original value reported in Table 1, and Open image in new window . We then calculated the proportion of 95% reference intervals for Open image in new window and Open image in new window that contained the true value. Figure 6 (left) shows the results. Higgins and Thompson's Open image in new window reference interval appears to exhibit sub-optimum coverage, which is especially clear when the true value of I^{2} is large. Reference intervals for Open image in new window based on equation (6) appear to well maintain the desired coverage across all 12 simulation scenarios.
Discussion
NSCLC 4 and Cervix 1
As mentioned in Methods, in the presence of heterogeneity we still expect fixed and random effects estimates to be targeting a single quantity. However, in Results the two meta-analyses with the largest heterogeneity also showed that largest empirical differences between Open image in new window and Open image in new window . The NSCLC 4 data was a good example of this, being the meta-analysis with the largest outward heterogeneity, but with also clear funnel plot asymmetry. If we had been ignorant as to the type of chemotherapy used in each study, and therefore had no way of explaining the heterogeneity, we would perhaps have considered applying a random effects model, despite suspecting small study effects. Random effects estimation in this context can start to look considerably less attractive, because Open image in new window gives more (rather than less) relative weight to the smaller studies than Open image in new window since for any study i, W_{ i } ≥ Open image in new window , a fact first highlighted by Greenland [34]. This has lead some to propose bias adjustment procedures to counteract small study effects [35, 36, 37]. Henmi and Copas [38] have recently advocated an interesting compromise; to use the fixed effects point estimate Open image in new window - that is robust to small study effects - but surrounded by a confidence interval derived under the random effects model. As shown in (Table 2), when applied to all studies in the NSCLC 4 meta-analysis this puts a 95% confidence interval of (0.61-1.16) around Open image in new window = 0.84, with an associate p-value of 0.21, bringing the treatment's benefit severely into doubt. Fortunately, we were able to plausibly explain most of the asymmetry present by the differing types of chemotherapy regimens used, providing a much more useful answer with added clinical insight.
For the Cervix meta-data, stratifying the trials by chemotherapy cycle duration helped to partially explain the heterogeneity. Again, in doing so it raised interesting clinical questions about the effective treatment of this cancer. The remaining heterogeneity present in the short cycle chemotherapy trials was removed by excluding an outlying study in a sensitivity analysis, guided by the results of a Baujat plot. Throwing data away is generally frowned upon by statisticians, and more sophisticated methods for incorporating so called 'outliers' have been proposed [39]. However, for small outlying studies this strategy is clearly a convenient and effective option. We could find no explanation for the extreme effect found by the CeCa trial in its design or patient population, but it is perhaps worth noting that, along with the PMG and LGOG trials, its results were never published in a peer reviewed journal. Clearly, one of the advantages of a meta-analysis is to bring together the totality of evidence, including especially trials whose results were not fully disseminated in the past. We do not know if the extreme results observed specifically in the CeCa and LGOG trials influenced their original non-publication, but it is certainly worrying that the overall picture of evidence is far easier to interpret in their absence.
Standard or Generalised Qstatistic?
In Methods and Results we described and demonstrated the use of meta-analytical techniques based on the generalised Q statistic. Are these worth using? As can be seen from Figure 5 (right), whenever Open image in new window is zero so is Open image in new window . For non-zero values Open image in new window is generally greater than Open image in new window , the difference between the two appears to increase as the magnitude of the heterogeneity increases. This suggests that when a substantial amount of heterogeneity is present, Open image in new window may be systematically underestimating it because a one-iteration formula is not sufficient to arrive at an estimate near the truth. This underestimation does not effect in any meaningful way the estimate for θ. Across the 18 meta-analyses, the random effects estimates for Open image in new window based on Open image in new window and Open image in new window were very similar (and are therefore not shown) since the overall mean estimate is fairly insensitive to small changes in τ^{2}[25, 40]. However, the variance of Open image in new window , V_{ RE }, and I^{2} are far more sensitive to changes in τ^{2} and hence accurate estimation is important for these quantities.
Conclusions
In this paper we have restricted our focus to the estimation of the meta-analytical quantities τ^{2}, I^{2} and the overall mean parameter θ, as well as providing confidence intervals for the latter two. We note that this does not reflect the state-of-the-art in what can estimated via a random effects meta-analysis; one can for instance also estimate trial level effect parameters (θ + u_{ i }), predict the likely effects of future studies and test hypotheses relating to these additional parameters [19]. With this in mind, we make the following tentative conclusions.
The actual magnitude of the estimate τ^{2} is often overlooked as a heterogeneity measure [41], and in keeping with modern developments the Dersimonian and Laird estimate is no longer considered to be the best choice [22, 24]. We recommend using the PM estimate for τ^{2} - and by extension the Open image in new window it implies - since it is still very easy to calculate, but shares much of the accuracy and rigor of more complex methods. Van der Tweel and Bollen [42] use the PM method to estimate the overall random effects mean θ_{ RE } and heterogeneity parameter within the context of a sequential meta-analysis, but appear to stick with the original Open image in new window for other aspects of their analysis. We recommend that practitioners additionally make use of the PM estimate in the Inconsistency measure Open image in new window . R code to estimate Open image in new window , θ_{ RE } and Open image in new window (with confidence intervals) is provided below.
An I^{2} of over 75% has traditionally been considered as indicating a high level of inconsistency, I^{2}'s of above 50% as moderate and I^{2}'s of below 25% as low. It is tempting to consider a random effects model when the I^{2} is high. However, the range of the reference intervals shown in Figure 6 (left) highlights the considerable uncertainty around this measure. The recently updated Cochrane handbook [6] now gives overlapping rather than mutually exclusive regions for low, moderate and high heterogeneity, but when the heterogeneity is measured with as much uncertainty as in the Cervix 3 meta-analysis (90% reference intervals for Open image in new window of 0% to 93%) any categorisation feels dubious. Inconsistency intervals based on the Open image in new window statistic will generally be wider than those based on the standard Open image in new window measure but is a more accurate reflection of the uncertainty present. These findings are based on a fairly large simulation study for widely varying τ^{2}, typical within study variance s^{2} and trial number M. Although the simulated data were normally distributed, we do not think the conclusions would have changed if the study effects had been drawn from a more non-standard distribution. By plotting Open image in new window at the lower and upper reference levels, as well at a spread of more central measures such as the mean, median and mode, one can easily and effectively convey this uncertainty to the analyst. For a comprehensive comparison of methods for estimating the heterogeneity parameter τ^{2} see Biggerstaff and Tweedie [26] or Viechtbauer [25].
In the presence of heterogeneity, the naive and automatic application of the random effects model has been widely criticised. It is sensible to conduct a further investigation the data [34, 43, 44], but this may not lead to the identification of any explanatory factors. If unexplained heterogeneity also leads to large differences between the fixed and random effects estimates, there is the obvious prospect that conflicting clinical interpretations could arise. When funnel plot asymmetry is the predominant cause of this, I^{2} statistics have a less meaningful interpretation. For this reason Rücker et. al [37] have recently proposed an alternative 'G' statistic, that expresses the inconsistency between studies after this asymmetry has been accounted for (through a bias correction for small study effects). As demonstrated on the NSCLC meta-analysis, the Henmi-Copas method combining a fixed effects estimate with a 'random effects' confidence interval provides an alternative way of dealing with funnel plot asymmetry without making an explicit bias correction. Both the approaches of Rücker et. al. and Henmi and Copas appear to offer sensible and practical solutions to this problem, and merit further investigation.
R code
This code calculates point estimates and α-level confidence intervals for Open image in new window , Open image in new window and Open image in new window , given the estimated effect sizes y within study standard errors s and desired type I error Alpha. This code is based on the algorithm suggested by DerSimonian and Kacker [22].
PM = function(y = y, s = s, Alpha = 0.1){
K = length(y) ; df = k -1 ; sig = qnorm(1-Alpha/2)
low = qchisq((Alpha/2), df) ; up = qchisq(1-(Alpha/2), df)
med = qchisq(0.5, df) ; mn = df ; mode = df-1
Quant = c(low, mode, mn, med, up) ; L = length(Quant)
Tausq = NULL ; Isq = NULL
CI = matrix(nrow = L, ncol = 2) ;MU = NULL
v = 1/s^2 ; sum.v = sum(v) ; typS = sum(v*(k-1))/(sum.v^2 - sum(v^2))
for(j in 1:L){
tausq = 0 ; F = 1 ;TAUsq = NULL
while(F>0){
TAUsq = c(TAUsq, tausq)
w = 1/(s^2+tausq) ; sum.w = sum(w) ; w2 = w^2
yW = sum(y*w)/sum.w ; Q1 = sum(w*(y-yW)^2)
Q2 = sum(w2*(y-yW)^2) ; F = Q1-Quant[j]
Ftau = max(F,0) ; delta = F/Q2
tausq = tausq + delta
}
MU[j] = yW ; V = 1/sum(w)
Tausq[j] = max(tausq,0) ; Isq[j] = Tausq[j]/(Tausq[j]+typS)
CI[j,] = yW + sig*c(-1,1) *sqrt(V)
}
return(list(tausq = Tausq, muhat = MU, Isq = Isq, CI = CI, quant = Quant))
}
Authors' information
JB is a biostatistician working within the London and Cambridge MRC hubs for trials methodology research. JFT is the head of the Meta-analysis group at the MRC Clinical Trials Unit (CTU). AJC is a senior statistician within the MRC CTU and also a senior lecturer in medical statistics at University College, London. SB is a systematic reviewer at the CTU, working within the Meta-analysis group.
Notes
Acknowledgements
None declared.
Supplementary material
References
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