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Network-meta analysis made easy: detection of inconsistency using factorial analysis-of-variance models
- Hans-Peter Piepho^{1}Email author
https://doi.org/10.1186/1471-2288-14-61
© Piepho; licensee BioMed Central Ltd. 2014
Received: 8 February 2014
Accepted: 9 April 2014
Published: 10 May 2014
Abstract
Background
Network meta-analysis can be used to combine results from several randomized trials involving more than two treatments. Potential inconsistency among different types of trial (designs) differing in the set of treatments tested is a major challenge, and application of procedures for detecting and locating inconsistency in trial networks is a key step in the conduct of such analyses.
Methods
Network meta-analysis can be very conveniently performed using factorial analysis-of-variance methods. Inconsistency can be scrutinized by inspecting the design × treatment interaction. This approach is in many ways simpler to implement than the more common approach of using treatment-versus-control contrasts.
Results
We show that standard regression diagnostics available in common linear mixed model packages can be used to detect and locate inconsistency in trial networks. Moreover, a suitable definition of factors and effects allows devising significance tests for inconsistency.
Conclusion
Factorial analysis of variance provides a convenient framework for conducting network meta-analysis, including diagnostic checks for inconsistency.
Keywords
- Analysis of variance
- Baseline contrast
- Heterogeneity
- Inconsistency
- Linear mixed model
- Network meta-analysis
- Pairwise treatment contrast
- PRESS residual
- Studentized residual
Background
Results from several randomized trials can be combined by meta-analysis methods. In the simplest case, all trials comprise the same set of treatments, typically only two, i.e., a new treatment and a control or baseline treatment. When trials differ in design, i.e., in the sets of treatments tested, joint analysis may be done by what has come to be called network meta-analysis (NMA). Such analyses combine different sources of pairwise treatment comparisons across trials, i.e., direct comparisons from trials that jointly test both treatments of interest and indirect comparisons from trials that only test one of the two treatments, but are connected through other treatments via the trial network. A key assumption of many methods for NMA is consistency of treatment effect estimates across designs, defined by the set of treatments tested. In particular, consistency implies agreement between direct and indirect evidence on a treatment contrast. Several methods have been proposed for detecting inconsistency in trial networks [1–4].
Most methods for analysis of NMA operate on pairwise contrasts of treatments with a baseline treatment or control, henceforth denoted as baseline contrasts. Some methods for detecting inconsistency in meta-analysis networks based on baseline contrasts are relatively complex on account of the fact that baseline treatments may vary among trials and sources of inconsistency have to be traced through loops of the network [3–5]. It has been shown by Piepho et al. [6] that NMA can be greatly simplified by modelling treatment means rather than treatment contrasts using factorial analysis-of-variance (ANOVA) models, and that such analyses can produce identical or essentially the same results as analyses using baseline contrasts. The present paper will therefore focus on the ANOVA approach and illustrate its versatility. Specifically, we will explore ways to detect inconsistency using standard procedures for linear models available in most statistical packages. The methods will be illustrated using the diabetes example published by Senn et al. [7]. This example has also been used by Krahn et al. [1] to illustrate their proposed methods for detection of inconsistency using a baseline contrast parameterization, so our results can be compared directly to that paper in order to appreciate the degree of agreement between both model formulations and the resulting tests and diagnostic checks for inconsistency. The presentation assumes that the reader has access to a mixed model package using restricted maximum likelihood (REML) to estimate variance components and is familiar with the essentials of the underlying theory [8]. Program code in SAS for all analyses presented is given in Additional file 1.
Methods
In this section, we describe the basic models we are using. In the Results section, minor extensions and associated statistics derived from the various models, such as influence diagnostics, are introduced as needed.
Description of factors used for representing factorial ANOVA models for NMA
Factor symbol | Factor description |
---|---|
G | Group of trials, trial type, design |
S | Study, trial |
T | Treatment |
where × is an operator for crossing two factors or model terms, S is a factor identifying the individual trial, and T denotes the treatment factor. Effects in (1b) can be equated with those in (1a) as follows: S ≡ β _{ i }, T ≡ γ _{ j }, and S.T ≡ u _{ ij }.
where / is a nesting operator [9]. Note that the nesting relation G/S in (2b) is resolved as G/S = G + G.S. This structure is then fully crossed with T, as indicated by the crossing operator × on the left-hand side of eq. (2b). Effects in (2a) and (2b) can be equated as follows: G ≡ α _{ h }, G.S ≡ β _{ hi }, T ≡ γ _{ j }, G.T ≡ v _{ hj }, and G.S.T ≡ u _{ hij }.
where y is the observed treatment mean in a trial, η is the linear predictor, modelled, e.g., using (1) or (2), and e is the random normal error associated with the observed mean. The errors are modelled to have zero mean and variance var(e) equal to the observed squared standard error of a mean, assumed to be a known constant when fitting (3). This analysis is easily performed using linear mixed model software with weighting facility by defining the inverse of var(e) as weight and fixing the residual variance at unity [13]. The approach is fully efficient, because the variance-covariance matrix of the vector of means y is diagonal with elements equal to var(e) [14].
Following Krahn et al. [1], we will initially consider analyses by models (1) and (2) when all effects in the linear predictor (eq. 1b or 2b) are taken as fixed. Subsequently, we will consider analyses that model heterogeneity [i.e., the interaction effects S.T and G.S.T in eqs. (1b) and (2b), respectively] as random, which is common practice (see, e.g., [6] and [15]). One may argue that if heterogeneity is detected, then the effect for heterogeneity may be used as an error term for testing inconsistency because heterogeneity effects are nested within the effects for inconsistency. This leads to an analysis with random interaction effects S.T or G.S.T. Conversely, one may insist that heterogeneity be modelled as a fixed effect. Then if heterogeneity is detected, it may be concluded that there is no further basis for testing inconsistency because of the nested structure of effects for heterogeneity in relation to inconsistency. In this situation, one may try to find subsets of trials that do not display heterogeneity and analyse these subsets separately [2]. This philosophy is in agreement with that put forward by Nelder [16], who argued that testing main effects in a two-way fixed-effects ANOVA is justified only when the interaction is deemed to be absent and the model is reduced accordingly. Here, we will present results for both approaches (interactions for heterogeneity fixed or random) and compare the results. Our favoured approach is to model heterogeneity as random when performing checks and tests for inconsistency as well as when comparing treatment means.
Results
Ten treatment groups of the diabetes example of Senn et al.[7]
Four-letter abbreviation of treatment | Treatment |
---|---|
acar | Acarbose |
benf | Benfluorex |
metf | Metformin |
migl | Miglitol |
piog | Pioglitazone |
plac | Placebo |
rosi | Rosiglitazone |
sita | Sitagliptin |
SUal | Sulfonylurea alone |
vild | Vildagliptin |
Fitting models (1) and (2)
Wald-type chi-squared tests for heterogeneity ( u _{ ij } )
Design | Wald statistic | Number of studies | Degrees of freedom | p-value |
---|---|---|---|---|
benf:plac | 4.38 | 2 | 1 | 0.0363 |
metf:plac | 42.16 | 3 | 2 | <0.0001 |
migl:plac | 6.45 | 3 | 2 | 0.0398 |
rosi:plac | 21.27 | 6 | 5 | 0.0007 |
rosi:metf | 0.19 | 2 | 1 | 0.6655 |
An overall test for heterogeneity is obtained by fitting model (2). The trial × treatment interaction (G.S.T) yields a chi-squared statistic of 74.45 on 11 d.f. (p < 0.0001). This chi-squared statistic for overall heterogeneity is equal to the sum of the chi-squared statistics for heterogeneity for the five designs in Table 3. When dropping the effect G.T from the model, the Wald-test for the effect G.S.T becomes a joint test for inconsistency and heterogeneity. The chi-squared statistic for this test equals 96.98 on 18 d.f. (p < 0.0001), and it is equal to Generalized Cochran’s Q [1]. Further note that the model T + S + T.S produces the same overall Q of 96.98. At this point, we can conclude that there is significant heterogeneity.
All chi-squared statistics presented so far are identical to those in Table 3 of Krahn et al. [1], who used a model based on baseline contrasts. We also obtain their chi-squared statistic for inconsistency, when we fit G.S.T as fixed and test the effect G.T (chi-squared = 22.53, d.f. = 7, p = 0.0021). But we favor a mixed model analysis with random trial × treatment interaction (G.S.T), because we consider it the major error term for testing the design × treatment interaction (G.T), which assesses inconsistency. At the same time, the trial effect needs to be modelled as fixed in order to maintain equivalence with the baseline contrast approach [6, 7]. When we take the interaction effect for heterogeneity (G.S.T) as random, assuming a constant variance for this effect, the chi-squared statistic for inconsistency (G.T) drops to 2.27. The REML estimate of the variance for heterogeneity is ${\widehat{\mathit{\sigma}}}_{\mathit{u}}^{2}=0.06932$. Note that this estimate corresponds to half the variance for heterogeneity with the baseline contrast approach [6, 15] (usually denoted as τ ^{2}). Since the test for inconsistency now involves an estimated variance component, we use the Kenward-Roger method for approximating the denominator d.f. of a Wald-type F-statistic [17]. We find F = 0.32 on 7 numerator and 11 denominator d.f. and p = 0.9268. By this analysis, there is no significant inconsistency, which is in contrast to the analysis with fixed effects for G.S.T. Note that this analysis treats the residual variances of the individual trials as known constants, although they are, in fact, estimated when analysing individual trials. The added uncertainty associated with these variance estimates could be accounted for by using the Kenward-Roger method in a single-stage analysis modelling individual patient data [14], but differences compared to the two-stage analysis employed here are expected to be small so long as the sample sizes per treatment and trial are large enough, as is usually the case.
A very simple further check for inconsistency is to fit both G.T and G.S.T as random. The best linear unbiased predictors (BLUPs) of the G.T effects give a direct indication which treatment × design combinations contribute most to the inconsistency. With the diabetes example, the variance component for G.T is estimated to be zero, so the BLUPs for all G.T effects are zero, which is in agreement with the non-significant Wald-test for inconsistency.
Locating inconsistency by detachment of individual designs
Locating inconsistency in the network may be based on a detachment of an individual design from the others by a suitable model formulation that allows testing the contribution of that individual design to inconsistency in the network as well as the inconsistency that remains after detaching that design. Krahn et al. [1] showed how to code a detachment model for baseline contrasts. Here, we show how to implement this approach based on a straightforward extension of the factorial model (2).
Definition of detachment factors for testing inconsistency [D k .T; k ∈ (1,…,15)]
Design (factor G) | Design no. (k) | Factor for detachment | Level of factor for the fifteen designs | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |||
acar:plac | 1 | D1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
acar:SUal | 2 | D2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
benf:plac^{§} | 3 | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
metf:plac | 4 | D4 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
metf:acar:plac | 5 | D5 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
metf:SUal | 6 | D6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
migl:plac^{§} | 7 | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
piog:plac | 8 | D8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
piog:metf | 9 | D9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
piog:rosi | 10 | D10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
rosi:plac | 11 | D11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
rosi:metf | 12 | D12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
rosi:SUal | 13 | D13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
sita:plac^{§} | 14 | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
vild:plac^{§} | 15 | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
Wald-type chi-squared tests for inconsistency using detachment factors [D k .T; k ∈ (1,…,15)]
Design | Design no. ( k) | Number of studies | Degrees of freedom for D k.T | Effect D k.G.S.T fixed | Effect D k.G.S.T random | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
D k.T | D k.G.T | D k.T | D k.G.T | ||||||||
Wald statistic | p-value | Wald statistic | p-value | Wald statistic | p-value^{§} | Wald statistic | p-value^{§} | ||||
acar:plac | 1 | 1 | 1 | 0.09 | 0.7699 | 22.45 | 0.0010 | 0.02 | 0.8889 | 2.25 | 0.8782 |
acar:SUal | 2 | 1 | 1 | 0.01 | 0.9091 | 22.52 | 0.0010 | 0.01 | 0.9430 | 2.26 | 0.8765 |
metf:plac | 4 | 3 | 1 | 0.46 | 0.4976 | 22.07 | 0.0012 | 0.04 | 0.8379 | 2.22 | 0.8814 |
metf:acar:plac | 5 | 1 | 2 | 0.15 | 0.9297 | 22.39 | 0.0004 | 0.07 | 0.9634 | 2.18 | 0.8129 |
metf:SUal | 6 | 1 | 1 | 15.02 | 0.0001 | 7.52 | 0.2758 | 1.63 | 0.2343 | 0.92 | 0.9835 |
piog:plac | 8 | 1 | 1 | 5.28 | 0.0215 | 17.25 | 0.0084 | 0.43 | 0.5299 | 1.96 | 0.9062 |
piog:metf | 9 | 1 | 1 | 5.40 | 0.0201 | 17.13 | 0.0088 | 0.43 | 0.5318 | 1.94 | 0.9081 |
piog:rosi | 10 | 1 | 1 | 0.05 | 0.8280 | 22.49 | 0.0010 | 0.01 | 0.9065 | 2.27 | 0.8751 |
rosi:plac | 11 | 6 | 1 | 6.24 | 0.0125 | 16.30 | 0.0122 | 0.74 | 0.4112 | 1.87 | 0.9168 |
rosi:metf | 12 | 2 | 1 | 0.01 | 0.9199 | 22.52 | 0.0010 | 0.01 | 0.9276 | 2.25 | 0.8795 |
rosi:SUal | 13 | 1 | 1 | 15.76 | <0.0001 | 6.77 | 0.3424 | 1.79 | 0.2146 | 0.66 | 0.9930 |
Using influence diagnostics for design × treatment means
In order to detect influential or outlying observations in the network, we use a two-stage approach. In the first stage, we compute design × treatment means using model (2). In the second stage, we fit an additive two-way model of the form G + T to the design × treatment means. From this analysis, we can obtain residual and influence diagnostics [18, 19] by standard procedures with most linear mixed model packages. The key idea is that observations contributing substantially to inconsistency will display strong G.T interaction effects, which in turn will be captured by the residuals of the additive model G + T.
Three different options are considered for handling the effect for heterogeneity (G.S.T) in the first-stage analysis based on model (2): (i) taking it fixed, (ii) taking it random and (iii) dropping it. It turns out that with options (ii) and (iii), the treatment means of designs 3, 4, 7, 11 and 12 are correlated, meaning that weighting by the inverse of the squared standard errors is only an approximate method (note that the designs in question are precisely the ones represented by several trials). An exact analysis requires carrying the full variance-covariance matrix of design × treatment means forward and specifying this as the residual variance-covariance matrix of the model fitted at the second stage [14]. This is easily done in SAS using the REPEATED statement with the option TYPE = LIN(1). Note that option (iii) is in line with common practice when the baseline contrast formulation is used [1] and heterogeneity is deemed absent. But heterogeneity was found to be significant for the diabetes data, so one may argue that this effect should be in the model for checking consistency. If the effect is in the model and taken as fixed (option i), effectively all trials are given the same weight, whereas when the effect is dropped (option iii), each trial is weighted according to the variances of the means, which explains the differences in results. Both analyses are not fully satisfactory because heterogeneity is not appropriately taken into account. Taking heterogeneity as random (option ii) is common practice in meta-analysis [6, 15], and this is also our preferred approach over option (i) for the reasons stated at the end of the Methods section.
Studentized residuals and PRESS residuals
Design | Observation | Treatment | Model for heterogeneity | |||||
---|---|---|---|---|---|---|---|---|
G.S.T fixed | G.S.T random | G.S.T dropped | ||||||
PRESS residual | Studentized residual | PRESS residual | Studentized residual | PRESS residual | Studentized residual | |||
1 | 1 | acar | 0.0545 | 0.2443 | 0.0785 | 0.1453 | 0.0642 | 0.2925 |
2 | plac | -0.0545 | -0.2443 | -0.0785 | -0.1453 | -0.0642 | -0.2925 | |
2 | 3 | acar | -0.0234 | -0.1022 | 0.0619 | 0.1056 | -0.0259 | -0.1142 |
4 | SUal | 0.0234 | 0.1022 | -0.0619 | -0.1056 | 0.0259 | 0.1142 | |
3 | 5 | benf | ||||||
6 | plac | . | . | . | . | . | . | |
4 | 7 | metf | 0.0547 | 0.3026 | -0.0781 | -0.2282 | -0.0814 | -0.6783 |
8 | plac | -0.0547 | -0.3026 | 0.0781 | 0.2282 | 0.0814 | 0.6783 | |
5 | 9 | acar | -0.0894 | -0.2408 | -0.1507 | -0.2601 | -0.1137 | -0.3070 |
10 | metf | -0.0276 | -0.0930 | 0.0036 | 0.0075 | 0.0060 | 0.0205 | |
11 | plac | 0.1359 | 0.3615 | 0.1193 | 0.2273 | 0.1057 | 0.2833 | |
6 | 12 | metf | 0.6807 | 3.6726 | 0.6095 | 1.1614 | 0.6910 | 3.8755 |
13 | SUal | -0.6807 | -3.6726 | -0.6095 | -1.1614 | -0.6910 | -3.8755 | |
7 | 14 | migl | . | . | . | . | . | . |
15 | plac | . | . | . | . | . | . | |
8 | 16 | piog | -0.4337 | -2.5934 | -0.2802 | -0.5585 | -0.3638 | -2.2987 |
17 | plac | 0.4337 | 2.5934 | 0.2802 | 0.5585 | 0.3638 | 2.2987 | |
9 | 18 | metf | -0.4719 | -2.9147 | -0.2927 | -0.5779 | -0.3467 | -2.3246 |
19 | piog | 0.4719 | 2.9147 | 0.2927 | 0.5779 | 0.3467 | 2.3246 | |
10 | 20 | piog | -0.1074 | -0.5173 | -0.0073 | -0.0141 | -0.0445 | -0.2173 |
21 | rosi | 0.1074 | 0.5173 | 0.0073 | 0.0141 | 0.0445 | 0.2173 | |
11 | 22 | plac | -0.2802 | -1.9593 | -0.2100 | -0.6391 | -0.3181 | -2.4974 |
23 | rosi | 0.2802 | 1.9593 | 0.2100 | 0.6391 | 0.3181 | 2.4974 | |
12 | 24 | metf | -0.1105 | -0.5920 | -0.0616 | -0.1610 | -0.0179 | -0.1005 |
25 | rosi | 0.1105 | 0.5920 | 0.0616 | 0.1610 | 0.0179 | 0.1005 | |
13 | 26 | rosi | -0.7077 | -3.7022 | -0.6733 | -1.2693 | -0.7424 | -3.9701 |
27 | SUal | 0.7077 | 3.7022 | 0.6733 | 1.2693 | 0.7424 | 3.9701 | |
14 | 28 | plac | . | . | . | . | . | . |
29 | sita | . | . | . | . | . | . | |
15 | 30 | plac | . | . | . | . | . | . |
31 | vild | . | . | . | . | . | . |
Presenting multiple comparisons of treatment means
Adjusted means for the ten treatments
Treatment | Adjusted mean | Letter grouping | ||
---|---|---|---|---|
rosi | 0.212 | c | ||
piog | 0.317 | b | c | |
metf | 0.318 | b | c | |
migl | 0.496 | b | c | |
acar | 0.605 | b | c | |
benf | 0.709 | a | c | |
vild | 0.746 | a | c | |
sita | 0.876 | a | c | |
SUal | 1.029 | a | b | |
plac | 1.446 | a |
Pairwise differences of the ten treatment means
benf | metf | migl | piog | plac | rosi | sita | SUal | vild | |
---|---|---|---|---|---|---|---|---|---|
acar | -0.1045 (0.3659) | 0.2870 (0.2504) | 0.1085 (0.3280) | 0.2880 (0.3054) | -0.8414 (0.2384) | 0.3924 (0.2526) | -0.2714 (0.4165) | -0.4238 (0.2568) | -0.1414 (0.4159) |
benf | 0.3915 (0.3153) | 0.2130 (0.3575) | 0.3925 (0.3492) | -0.7369 (0.2776) | 0.4968 (0.3038) | -0.1669 (0.4401) | -0.3194 (0.3622) | -0.0369 (0.4395) | |
metf | -0.1785 (0.2703) | 0.0010 (0.2176) | -1.1284 (0.1494) | 0.1053 (0.1600) | -0.5584 (0.3727) | -0.7109 (0.2272) | -0.4284 (0.3721) | ||
migl | 0.1795 (0.3093) | -0.9499 (0.2253) | 0.2839 (0.2569) | -0.3799 (0.4091) | -0.5324 (0.3238) | -0.2499 (0.4085) | |||
piog | -1.1294 (0.2119) | 0.1043 (0.2163) | -0.5594 (0.4019) | -0.7119 (0.2914) | -0.4294 (0.4013) | ||||
plac | 1.2337 (0.1235) | 0.5700 (0.3414) | 0.4175 (0.2326) | 0.7000 (0.3408) | |||||
rosi | -0.6637 (0.3631) | -0.8162 (0.2290) | -0.5337 (0.3624) | ||||||
sita | -0.1525 (0.4132) | 0.1300 (0.4824) | |||||||
SUal | 0.2825 (0.4126) |
Discussion and conclusions
This paper has illustrated how a factorial ANOVA approach can be used to perform NMA and to locate inconsistency in a given network. It was shown in Piepho et al. [6] that this analysis is either fully equivalent (summary measures, normal response in case of individual patient data) or very similar (individual patient data with non-normal responses and non-identity link functions in a GL(M)M framework) to the more commonly used approach to meta-analysis based on baseline contrasts. We think that the ANOVA approach has some practical advantages. Interpretation of results is facilitated by the focus on t treatment means rather than on t(t - 1)/2 pairwise contrasts. Standard procedures for multiple comparison of treatment means further aid the communication of results. Also, the approach may be appealing to those familiar with ANOVA of factorial experiments. It has been demonstrated that standard diagnostic procedures for linear models can be used to identify influential designs in the network and to detect sources of inconsistency. The results obtained for the diabetes example agree very closely with those obtained using recently proposed procedures based on a baseline-contrast approach [1]. We hope that this paper will help to popularize the ANOVA approach as a viable and easy-to-use approach to NMA.
Declarations
Authors’ Affiliations
References
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Pre-publication history
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