Méta régression

Méta régression
(MR)
Marc Sznajder, Patricia Samb
(décors: Roger Hart
Costumes: Donald Cardwell )
Staff DIHSP janvier 2009
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Références

de Koning L, Merchant AT, Pogue J, Anand SS. Eur Heart J.
2007;28:850-6. Waist circumference and waist-to-hip ratio as
predictors of cardiovascular events: meta-regression analysis of
prospective studies.
Department of Clinical Epidemiology and Biostatistics, McMaster University, Hamilton,
Ontario, Canada L8L 2X2.

Thompson SG, Higgins JP. Stat Med. 2002;21:1559-73. How should
meta-regression analyses be undertaken and interpreted?
MRC Biostatistics Unit, Institute of Public Health, Robinson Way, Cambridge CB2 2SR,
UK. [email protected]
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Description of MR

extension to meta-analysis, (and a generalization of
subgroup analyses)

examines the relationship between one or more
study-level characteristics and the sizes of effect
observed in the studies (continuous covariates).

can be used to investigate heterogeneity of effects
across studies.
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Points majeurs des MR
Prendre en compte l’hétérogénéité
statistique liée à la diversité clinique et
méthodologique des études incluses (idem
méta-analyses)
 Lier la taille de l’effet observé à une ou
plusieurs caractéristiques des études
incluses

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MR

Use of trial-level covariates (≠ regression analyses
with individual data) (unité d’analyse: l’essai, donc si peu
d’essais: pas valide)

MR should be weighted to take account of both:
within-trial variances of treatment effects
and the residual between-trial heterogeneity
(that is, heterogeneity not explained by the covariates in
the regression)
This corresponds to random effects metaregression.
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Points majeurs des MR (suite)
Diagramme:
 visualise la précision de l’effet estimé de
chaque traitement

 l’unité
d’analyse est l’essai
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Log relative risk of stroke in 13 trials of aspirin versus placebo,
according to aspirin dose, together with a summary random effects
meta-regression. The area of each circle is inversely proportional to
the variance of the log relative risk estimate.
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Points majeurs des MR (suite)

‘residual heterogeneity’ must be
acknowledged in the statistical analysis.

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‘random effects’ rather than ‘fixed effect’
meta-regression

(If residual heterogeneity exists, a random effects analysis
appropriately yields wider confidence intervals for the regression
coefficients than a fixed effect analysis)

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Points majeurs des MR (suite)

The MR should clearly be weighted
(so that the more precise studies have more influence in th
analysis)
weight for each trial: inverse of the
sum of the within-trial variance and the
residual between-trial variance
(= random effects analysis).

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Points majeurs des MR (suite)
It is appropriate to use meta-regression
to explore sources of heterogeneity
even if an initial overall test for
heterogeneity is non-signicant
(test with low power).

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Points majeurs des MR (suite)

Estimating the residual between-trial variance is
somewhat problematic.
The estimate is usually imprecise because it is
based on rather a limited number of trials.
 One way to allow for the imprecision is to adopt
a Bayesian approach, using, for example,
non-informative priors (help !!)
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Points majeurs des MR (suite)

Logiciel permettant de réaliser des métarégressions à effet aléatoire:
Stata
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Point « mineur » des MR (suite)

relationship described by a MR: observational
association across trials.

Although the original studies may be randomized trials,
the meta-regression is across trials and does not have
the benefit of randomization to underpin a causal
interpretation

risque de biais de confusion
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Autres points « mineurs » des MR
(suite)
Si faible nombre d’essais dans la MR et nombre élevé de
covariables analysées (analyses multiples et post hoc):


risque élevé de conclusions faussement positives
D’où l’importance de pré-spécifier dans le protocole
les variables à analyser et d’en limiter le nombre;
Cependant pas tjs facile d’identifier à l’avance les
variables pertinentes (CDSR)
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Weighted Least Squares (WLS) Regression

Homoscedasticity: the variance of residual error
should be constant for all values of the independent(s).

Violation of homoscedasticity occurs when the
magnitude of the dependent is correlated with the
variance of the independent
(cf figure suivante).

One possible cause of this might be a skewed rather
than normally distributed dependent variable
Wolberg, John (2006). Data analysis using the method of least squares:
Extracting the most information from experiments. NY: Springer.
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EX: The variance in Preference increases for higher values of Age.
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Weighted Least Squares (WLS) Regression
(suite)
WLS regression compensates for violation of
the homoscedasticity assumption by weighting
cases differentially:
cases whose value on the dependent variable
corresponds to large variances on the
independent variable(s) count less (and
inversely) in estimating the regression
coefficients.
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Weighted Least Squares (WLS) Regression
(suite)
Cases are weighted by the reciprocal of their estimated
point variance
That is, cases with greater weights contribute more to the
fit of the regression line.
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Weighted Least Squares (WLS) Regression
(suite)



Weighted predicted/residual plots can be
used to assess the goodness of fit of the
weighted model.
Weighted predicted is plotted on the x axis and
weighted residual on the y axis.
When there is good fit, the residuals will no
longer form a funnel shape but instead be
uniformly distributed around the 0.0 line of the y
axis.
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The funnel shape has disappeared.
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Because of the reduction in heteroscedasticity,
standard errors will be small but estimates will be very similar
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Waist circumference and waist-to-hip ratio as predictors of
cardiovascular events: meta-regression analysis of prospective studies
(2007)

AIMS: The objectives of this study were to determine the association of
waist circumference (WC) and waist-to-hip ratio (WHR) with the risk of
incident cardiovascular disease (CVD) events and to determine whether the
strength of association of WC and WHR with CVD risk is different.

METHODS AND RESULTS: This meta-regression analysis used a search
strategy of keywords and MeSH terms to identify prospective cohort
studies and randomized clinical trials of CVD risk and abdominal obesity
from the Medline, Embase, and Cochrane databases. Fifteen articles (n =
258 114 participants, 4355 CVD events) reporting CVD risk by categorical
and continuous measures of WC and WHR were included. For a 1 cm
increase in WC, the relative risk (RR) of a CVD event increased by 2%
(95% CI: 1-3%) overall after adjusting for age, cohort year, or treatment.
For a 0.01 U increase in WHR, the RR increased by 5% (95% CI: 4-7%).
These results were consistent in men and women. Overall risk estimates
comparing the extreme quantiles of each measure suggested that WHR
was more strongly associated with CVD than that for WC (WHR: RR = 1.95,
95% CI: 1.55-2.44; WC: RR = 1.63, 95% CI: 1.31-2.04), although this
difference was not significant. The strength of association for each
measure was similar in men and women.

CONCLUSION: WHR and WC are significantly associated with the risk of
incident CVD events. These simple measures of abdominal obesity should
be incorporated into CVD risk assessments.
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Waist circumference and waist-to-hip ratio as predictors of
cardiovascular events: meta-regression analysis of prospective studies
(2007)
Méthodo:

Weighted-least-squares (WLS) regression in studies that
reported risk estimates by quantiles of WC or WHR.

Outcome: natural logarithm of CVD risk in each quantile

Beta-coefficients represented the change in log CVD risk for a 1 U
increase in WC or WHR

Inverse (quasi)-variance of risk estimates as regression weights
in order to include the reference category in the regression
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Waist circumference and waist-to-hip ratio as predictors of
cardiovascular events: meta-regression analysis of prospective studies
(2007) (suite)
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Méthodo (suite):
Heterogeneity in beta-coefficients was explored using a random
effects meta-regression model (‘metareg’ module, Stata ver 8.2).
We included predictors for mean age, mean follow-up, and the type
of data (categorical or continuous) used to derive beta-coefficients,
with beta-coefficients as the outcome
Beta-coefficients were weighted by their inverse variances and
pooled using the DerSimonian and Laird random effects model to
allow for differences between studies (‘meta’ module, Stata ver
8.2).
Cochrane's Q was used to assess heterogeneity among the betacoefficients.
Pooled beta-coefficients with 95% confidence intervals were
exponentiated and plotted to assess the statistical significance of
the estimates. Risk estimates for WC were evaluated for a 1 cm
increase, and estimates for WHR were evaluated for a 0.01 U
increase.
We calculated the predicted changes in WC and WHR for an
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equivalent increase in CVD risk to give WHR a meaningful
interpretation.
Funnel plots of moderately and maximally adjusted beta-coefficients.
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Pooled exponentiated beta-coefficients and 95%
confidence intervals plotted by sex and level of adjustment.
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