01.12.2012 | Research article | Ausgabe 1/2012 Open Access

# Meta-regression models to address heterogeneity and inconsistency in network meta-analysis of survival outcomes

- Zeitschrift:
- BMC Medical Research Methodology > Ausgabe 1/2012

## Electronic supplementary material

## Competing interests

## Authors’ contributions

## Background

## Methods

### Multidimensional network meta-analysis models for survival data

_{ BC }) can be obtained from the estimates of the effect of B versus A (d

_{ AB }) and the effect of C versus A (d

_{ AC }): d

_{ BC }= d

_{ AC }- d

_{ AB }; as such, transitivity holds [2, 3, 6]. For an arm-based model with the outcome of treatment x as a function of time, f

_{ x }(t) where x = A, B, or C and t represents time, this consistency assumption translates into:

_{ jkt }reflect the underlying hazard rate in trial j for intervention k at time point t. μ

_{ jbt }are the study j specific hazard rates at time t with comparator treatment b. δ

_{ jbk }reflects the study specific constant log HRs of treatment k relative to comparator treatment b and are drawn from a normal distribution with the pooled estimates expressed in terms of the overall reference treatment A: d

_{ bk }=d

_{ Ak }− d

_{ Ab }with d

_{ AA }= 0. Variance σ

^{2}reflects the heterogeneity across studies. As an alternative to a non-parametric baseline hazard function, the development of the hazard rate over time can be described by parametric functions, such as Weibull or Gompertz.

#### Two-dimensional treatment effects without covariates (Model 1)

_{ jkt }again reflects the underlying hazard rate in trial j for intervention k at time point t and is now described as a function of time t with p={-2,-1,-0.5,0,0.5,1,2,3} and t

^{0}=ln(t) with treatment and study specific scale and shape parameters β

_{0jk }and β

_{1jk }. (In the example presented below details on the likelihood and data structure are provided). If β

_{1jk }equals 0, a constant log hazard function is obtained, reflecting exponentially distributed survival times. If β

_{1jk }≠ 0 and p = 1 a linear log hazard function is obtained which corresponds to a Gompertz survival function [8]. If β

_{1jk }≠ 0 and p = 0 a Weibull hazard function is obtained. The vectors $\left(\begin{array}{c}\hfill {\mu}_{0jb}\hfill \\ \hfill {\mu}_{1jb}\hfill \end{array}\right)$ are trial specific and reflect the true underlying scale and shape parameters of the comparator treatment b. δ

_{0jbk }is the study specific difference in the scale parameter β

_{0}of the log hazard curve for treatment k relative to comparator treatment b. δ

_{0jbk }are drawn from a normal distribution with the pooled estimates expressed in terms of the overall reference treatment A: d

_{0Ak }− d

_{0Ab }with d

_{0AA }= 0. The parameters d

_{0Ak }correspond to the treatment effect of k relative to overall reference treatment A with a proportional hazard model. The pooled difference in the shape parameter β

_{1}of the log hazard curve for treatment k relative to comparator treatment b is expressed as d

_{1Ak }− d

_{1Ab }with d

_{1AA }= 0. d

_{1Ak }reflects the change in the log HR over time. For a proportional hazards model d

_{1Ak }equals 0. By incorporating d

_{1Ak }in addition to d

_{0Ak }a multidimensional treatment effect is used. For additional flexibility, the first order fractional polynomial model can be generalized to a 2

^{nd}order fractional polynomial model, representing 3-dimensional treatment effects [8].

_{ 0 }

^{2}reflects the heterogeneity in the difference in the scale parameters across studies. A random effects model with only a heterogeneity parameter for d

_{0Ak }implies that the between study variance of the log hazard ratios remains constant over time.

#### Two-dimensional treatment effects with treatment specific covariate interactions (Model 2)

_{ xbk }reflects the impact of study level covariate X

_{ j }on the difference in the scale parameters of the hazard functions with treatment k relative to control treatment b. Now d

_{0bk }is the difference in scale treatment k relative to b when the covariate value equals zero. Since β

_{ xbk }= β

_{ xAk }− β

_{ xAb }with β

_{ xAA }= 0, k − 1 different and independent regression coefficients for β

_{ xAk }will be estimated with the model. As an alternative to independent treatment-by-covariate interactions, one can also assume exchangeable interaction effects [10].

#### Two-dimensional treatment effects with constant covariate interaction (Model 3)

_{ j }on the scale parameter of each treatment k relative to A is the same for all treatments. This assumption can be defended when treatments indirectly compared are all from the same class and there is no (biological) reason to assume that a patient characteristic, or any other contextual aspect of the study, modifies treatment effects differently for the different drugs compared. Furthermore, the assumption of constant treatment-by-covariate interaction can also be useful when evaluating the impact of study (design) characteristics (or bias) on treatment effects. [10, 14]. The corresponding network meta-analysis model will be:

_{ x }X

_{ j }will cancel out for the comparison of treatment k relative to b when b ≠ A

_{0Ak }with k = B, C, corresponding to the log-HR of treatment B and C relative to A at time t = 0 and covariate value X = 0. The slope of the AB and AC planes as a function of ln(time) represents parameter d

_{1Ak }, the impact of time on the log HR of treatment k relative to A. The slope of the AB and AC planes as a function of the covariate value represents β

_{ xAk }, which is the impact of covariate X on the treatment effect parameter d

_{0Ak }(the scale). Figure 1A represents Model 1 where it is assumed that the covariate is not an effect modifier of d

_{0Ak }and therefore β

_{ xAk }= 0. In Figure 1B the effect of the covariate for the AB comparison is different from the AC comparison (Model 2). Figure 1C reflects Model 3 with the same effect of the covariate for the AB and the AC comparison.

_{0Ak }can be easily extended to fit trials with 3 or more treatment arms by decomposing a multivariate normal distribution as a series of conditional univariate distributions [9, 13]:

#### Higher dimensional models with heterogeneity and covariate effects acting on multiple treatment effect parameters

_{ j }on the pooled treatment effects in terms of scale, δ

_{0jbk }, and shape δ

_{1jbk }and δ

_{2jbk }, with treatment k relative to control treatment b.

_{0}, σ

_{1}, σ

_{2}representing the heterogeneity in treatments effects δ

_{0jbk }, δ

_{1jbk }and δ

_{2jbk }respectively. ρ

_{01}, ρ

_{02}and ρ

_{12}are the correlations between these parameters. Although such a general model is very flexible to explore heterogeneity and inconsistency, identifiability is expected to be a challenge.

### Illustrative example

_{1}, t

_{2}], (t

_{2}, t

_{3}], …, (t

_{ q }, t

_{ q+1}] with t

_{1}=0. For each time interval m =1,2,3,…,.q extracted survival proportions were used to calculate the patients at risk at the beginning of that interval and incident number of deaths. (A more specific explanation is provided in the Additional file 1 of this paper.) A binomial likelihood distribution of the incident events for every interval can be described according to:

_{ jkt }is the observed number of events in the m

^{th}interval ending at time point t

_{ m+1}for treatment k in study j. n

_{ jkt }is the number of subjects at risk just before the start of that interval adjusted for the subjects censored in the interval. p

_{ jkt }is the corresponding underlying event probability. When the time intervals are relatively short, the hazard rate h

_{ jkt }at time point t for treatment k in study j can be assumed to be constant for any time point within the corresponding m

^{th}time interval. The hazard rate corresponding to p

_{ jkt }for the m

^{th}interval can be standardized by the unit of time used for the analysis (e.g. months) according to: h

_{ jkt }= − ln(1 − p

_{ jkt })/Δt

_{ jkt }where Δt

_{ jkt }is the length of the interval. For the model estimation we assign this underlying hazard to time point t

_{ m+1}.

_{ xAk }∼ normal(0, 10

^{4}) will be replaced with β

_{ x }∼ normal(0, 10

^{4}). With the fixed effects model, it is not necessary to define a prior distribution for σ

_{0}.

## Results

### Illustrative example

Model | Dbar | Dhat | pD | DIC |
---|---|---|---|---|

Random effects Weibull model without covariates | 1462.4 | 1432.3 | 30.1 | 1492.5 |

Fixed effects Weibull model without covariates | 1468.9 | 1443.0 | 25.9 | 1494.8 |

Models with study data as covariate | ||||

Random effects Weibull model with treatment specific covariate interactions | 1460.5 | 1428.5 | 32.0 | 1492.5 |

Fixed effects Weibull model with treatment specific covariate interactions | 1464.7 | 1436.7 | 28.0 | 1492.7 |

Random effects Weibull model with constant treatment covariate interactions | 1459.7 | 1428.7 | 31.0 | 1490.7 |

Fixed effects Weibull model with constant treatment covariate interactions | 1466.7 | 1439.9 | 26.8 | 1493.5 |

_{0Ak }). The treatment-by-covariate interaction for non-DTIC vs. DTC (−0.02) was different than the interaction term obtained with a model with a constant interaction term (0.05), which implies that the assumption of a constant covariate interaction can be challenged.

Fixed effects model | Random effects model (model 1) | Random effects model with treatment specific covariate interaction* (model 2) | Random effects model with constant treatment-by-covariate interaction* (model 3) | |||||
---|---|---|---|---|---|---|---|---|

Median of posterior distribution | 95% Credible Interval | Median of posterior distribution | 95% Credible Interval | Median of posterior distribution | 95% Credible Interval | Median of posterior distribution | 95% Credible Interval | |

Pooled estimate for difference in scale β
_{
0
}
| ||||||||

DTIC + IFN vs. DTIC (d
_{
0AB
}) | −0.16 | (−0.63; 0.33) | −0.22 | (−0.76; 0.23) | −0.04 | (−0.54; 0.47) | −0.18 | (−0.58; 0.39) |

DTIC + non-IFN vs. DTIC (d
_{
0AC
}) | −0.07 | (−0.46; 0.34) | −0.19 | (−0.65; 0.30) | −0.16 | (−0.66; 0.32) | −0.10 | (−0.51; 0.35) |

non-DTIC vs. DTIC (d
_{
0AD
}) | −0.27 | (−0.63; 0.13) | −0.30 | (−0.88; 0.24) | −0.17 | (−1.40; 1.11) | −0.43 | (−1.12; 0.06) |

Pooled estimate for difference in shape β
_{
1
}
| ||||||||

DTIC + IFN vs. DTIC (d
_{
1AB
}) | 0.13 | (−0.08; 0.33) | 0.14 | (−0.04; 0.34) | 0.12 | (−0.07; 0.30) | 0.16 | (−0.04; 0.30) |

DTIC + non-IFN vs. DTIC (d
_{
1AC
}) | −0.06 | (−0.23; 0.11) | −0.02 | (−0.19; 0.15) | −0.04 | (−0.21; 0.13) | −0.05 | (−0.19; 0.10) |

non-DTIC vs. DTIC (d
_{
1AD
}) | 0.04 | (−0.17; 0.23) | 0.06 | (−0.15; 0.27) | 0.09 | (−0.11; 0.29) | 0.03 | (−0.16; 0.21) |

Estimate for covariate effect (βx) | ||||||||

DTIC + IFN vs. DTIC (βx
_{
AB
}) | 0.06 | (−0.02; 0.16) | 0.05 | (−0.01; 0.12) | ||||

DTIC + non-IFN vs. DTIC (βx
_{
AC
}) | 0.06 | (−0.05; 0.18) | 0.05 | (−0.01; 0.12) | ||||

non-DTIC vs. DTIC (βx
_{
AD
}) | −0.02 | (−0.21; 0.19) | 0.05 | (−0.01; 0.12) | ||||

Between study variance (heterogeneity) in scale | 0.21 | (0.02; 0.56) | 0.19 | (0.01; 0.52) | 0.18 | (0.01; 0.55) |

_{0Ak }, and d

_{1Ak }, with k = B,C,D, corresponding to respectively DTIC + IFN, DTIC + non-IFN and non-DTIC) the resultant HRs as a function of time were obtained according to ln(HR

_{ Ak }) = d

_{0Ak }+ d

_{1Ak }· ln(t). Figure 4A reflects the HRs over time (along with 95% credible intervals) with a random effects model without adjustment for differences in study date across studies and comparisons. Figure 4B shows the HR over time after adjustment for differences in study date using a random effect model with treatment specific covariate interactions (model 2). The HRs over time are presented for the average study date of all studies. Figure 4C shows the HRs over time after adjustment for study date using a random effect model with a constant treatment-covariate interaction (model 3). Comparing Figure 4A with Figure 4B and 4c illustrates the effect of ignoring the variation in study date across the different comparisons.

Random effects model without covariate interaction (model 1) | Random effects model with treatment specific covariate interaction (model 2) | Random effects model with constant treatment-by-covariate interaction (model 3) | ||||
---|---|---|---|---|---|---|

Median of posterior distribution | 95% Credible Interval | Median of posterior distribution | 95% Credible Interval | Median of posterior distribution | 95% Credible Interval | |

DTIC + IFN vs. DTIC | −1.12 | (−4.20; 3.49) | −2.46 | (−5.72; 1.91) | −1.90 | (−5.10; 2.22) |

DTIC + non-IFN vs. DTIC | 3.63 | (−1.72; 10.75) | 3.77 | (−1.04; 10.72) | 3.21 | (−2.39; 9.65) |

non-DTIC vs. DTIC | 2.66 | (−2.38; 13.49) | 1.16 | (−7.51; 25.76) | 6.60 | (−0.78; 21.39) |

DTIC + non-IFN vs. DTIC + IFN | 4.71 | (−1.38; 11.37) | 6.26 | (0.26; 13.29) | 5.13 | (−1.15; 11.50) |

non-DTIC vs. DTIC + IFN | 3.81 | (−2.40; 14.11) | 3.46 | (−6.00; 27.88) | 8.48 | (−0.65; 24.00) |

non-DTIC vs. DTIC + non-IFN | −0.84 | (−9.15; 10.3) | −2.79 | (−13.90; 22.17) | 3.27 | (−6.72; 20.74) |