Explain between-study heterogeneity with study-level covariates

A random-effects pool reports heterogeneity; a meta-regression asks where it comes from. The residual \(au^2\) is the part the moderators fail to explain, and it belongs in the weights, otherwise the standard errors are those of a fixed-effect fit and are too small. The moderators are study-level, so this is ecological regression: a covariate that predicts the effect across studies says nothing about the same covariate within a study.

Usage

Mareg(yi, vi, X)

Arguments

yi

Study effect estimates.

vi

Their sampling variances, strictly positive.

X

Moderator matrix; supply the intercept column yourself.

Value

List with beta, se, tau2, R2, ll, QE, QM, n, p.

Details

Formula: \(y_i = x_i' beta + u_i + e_i\), Var(u) = tau^2, Var(e_i) = v_i, weights 1/(v_i + tau^2). tau^2 is the moment estimator \(max(0, (Q_E - (n - p)) / (tr W - tr((X'WX)^{-1} X'W^2 X)))\) – van Houwelingen, Arends and Stijnen (2002) Section 4

References

van Houwelingen, H. C., Arends, L. R. and Stijnen, T. (2002). Statistics in Medicine 21(4):589-624. doi:10.1002/sim.1040 .

Examples

Mareg(yi = c(1, 2, 3, 4, 5, 6, 7, 8), vi = c(1, 2, 3, 4, 5, 6, 7, 8), X = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $beta
#> [1] 1
#> 
#> $se
#> [1] 0.1666667
#> 
#> $tau2
#> [1] 0
#> 
#> $R2
#> [1] 1
#> 
#> $ll
#> [1] -12.65381
#> 
#> $QE
#> [1] 2.796192e-26
#> 
#> $QM
#> [1] 12.45204
#> 
#> $tau2_null
#> [1] 2.528936
#> 
#> $n
#> [1] 8
#> 
#> $p
#> [1] 1
#> 
#> $method
#> [1] "Random-effects meta-regression"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"