Satisfacción del cliente

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Statistical method

The use of a mixed linear model for performing indirect comparisons by means of a network of comparisons was proposed in 2002 by Lumley (21).

The estimated treatment effect based on trial k comparing treatments i and j is written

Y

ijk _with

estimated variance

2

ijk

σ

_.

The model of the effect

Y

ijk _{consists of three components in addition to sampling error:}

1. the meta true effect of treatments i and j which is represented by

μ

i _and

μ

j _{respectively. The}

meta true effect is the value around which the true effects of treatment established in each trial fluctuate. The observed effect is derived from the true effect more or less skewed by sampling error (related to sampling fluctuations).

2. the random effect

η

ik and

η

jk _{which represents the difference between the true effect of}

jk

η

_is

τ

2

. The true treatment effect i in the trial k is therefore

μ η

i

+

ik.

Y

ijk _{is the estimate of}

the difference between the treatments i and j in the trial k, and is therefore an estimate of the difference between the true effects

μ η

i

+

ik _and

μ η

j

+

jk_.

3. a second random effect

ξ

ij _{representing the change in the effect of treatment i when}

compared with j.

ξ

ij _{therefore makes it possible to capture any inconsistency in the network of}

evidence. Its variance

ω

makes it possible to measure the inconsistency of the network. So in conclusion, the model is

(

)

(

)

(

)

( )

(

)

2 2 2

0,

0,

0,

ijk i ik j jk ij ijk ijk ijk ik ij

Y

N

N

N

μ η

μ η

ξ

ε

ε

σ

η

τ

ξ

ω

=

+

−

+

+

+

=

≈

≈

Where

ε

ijk _{stands for random sampling error which at the level of trial k skews estimation of the}

difference between treatment i and treatment j.

Box - Mixed model, hierarchical

In statistics, an effect is the change induced by a factor in the value of a random variable. In a trial, the treatment factor causes a change in the value of the variable used as endpoint. For most of the time, factors are regarded as fixed. The treatment effect is therefore the mean constant change induced by treatment on the random variable used as an endpoint.

Mixed effects models, also called random effects models, are used to take account of several levels of fluctuation/random errors (random effects) and so to consider that the effects may also vary in a random manner.

Qualification of an effect as random means that this effect varies from one statistical unit to another, in a random manner according to a certain distribution.

In a meta-analysis, modelling using a fixed model represents reality by considering that the effect Y

observed in the trial k is equal to the true treatment effect, plus the random fluctuations of sampling

(i.e. the random error occurring at the level of the trial between the observed value and the true value).

So in this model there is only one effect (the true effect of treatment) and this effect is fixed for all trials (hence the name "fixed effects model").

A random meta-analysis model will stipulate that the true effect of treatment varies from one trial to another, fluctuating at random around a value which is a meta-parameter. The effect being investigated at the level of the trial is therefore not fixed but random, hence the name "random effects model". The meta-analysis then tries to estimate the meta-parameter. So the model stipulates heterogeneity of effect of treatment (the effect varies from one trial to another at random). In this random model, we therefore have two levels of variability due to chance:

1. sampling fluctuations (as with the fixed model) describing the difference between the true value of the effect of treatment in a trial and the effect observed

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Example of application

This method was used in a meta-analysis by Psaty et al. of first-line antihypertensives (24). The

objective was to complete and estimate all possible comparisons in the network. For example, there were no trials directly comparing low-dose diuretics with angiotensin receptor blockers. An indirect

comparison performed implicitly by the method made it possible to document the relative efficacy of these two types of treatment (Figure 14 and Figure 15).

Figure 14 - Network of comparisons included

Psaty BM, Lumley T, Furberg CD, Schellenbaum G, Pahor M, Alderman MH, et al. Health outcomes associated with various antihypertensive therapies used as first-line agents. A network meta-analysis. JAMA 2003;289(19):2534-44.

Figure 15 - Results of network meta-analysis

Psaty BM, Lumley T, Furberg CD, Schellenbaum G, Pahor M, Alderman MH, et al. Health outcomes associated with various antihypertensive therapies used as first-line agents. A network meta-analysis. JAMA 2003;289(19):2534-44.

Copyright © 2003 American Medical Association. All rights reserved.

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