Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Precision-weighted regression of study effect estimates on study characteristics

MR(y_i, v_i, x_i) = (beta_0, beta_1, tau2)

Maps the effect estimate y_i of each study in a meta-analysis, its within-study variance v_i and the study's value x_i of a characteristic such as dose, mean age or baseline risk to a regression line for the expected effect, together with the between-study variance tau2 that the characteristic leaves unexplained. Studies are weighted by the precision of their estimates, and the random-effects form adds the residual between-study variance to each study's own variance. Ratio measures enter as natural logarithms. In health technology assessment the fitted line gives the relative effect expected for the decision population instead of the average over the trials, and the slope is an association between studies, not a randomised comparison.

  • Random-effects meta-regression model with one study-level covariate

    y_i = beta_0 + beta_1 * x_i + u_i + e_i

    Writes the effect estimate of study i as a point on a regression line in the covariate x_i plus two error terms, the study's deviation u_i from the line, with u_i ~ N(0, tau2), and its sampling error e_i, with e_i ~ N(0, v_i) and v_i treated as known. The slope beta_1 is the change in the expected effect per unit increase in x_i, and tau2 is the residual between-study variance not accounted for by the covariate. A fixed-effect meta-regression omits u_i, so sampling error is the only random variation. Further covariates enter as further terms, with a categorical characteristic of J groups coded as J minus 1 dummy variables.

  • Random-effects precision weight of a study in a meta-regression

    w_i = 1 / (v_i + tau2)

    Weight given to study i when the regression coefficients are estimated by weighted least squares. The weight is the inverse of the study's within-study variance plus the residual between-study variance, so larger studies have more influence on the slope, but less than in a fixed-effect meta-regression, where tau2 is zero and the weight is 1/v_i.

  • Proportion of between-study variance accounted for by meta-regression covariates

    R2_het = (tau2_RE - tau2_MR) / tau2_RE

    Compares the between-study variance of a random-effects meta-analysis without covariates with the residual between-study variance of the meta-regression, and expresses the fall as a share of the first. It describes heterogeneity among true effects only, not the sampling error in the observed estimates.

  • Treatment effect at a chosen covariate value from a centred meta-regression

    theta_z = d + b * (z - x_bar)

    TSD 3 centres the covariate at its mean, so the fitted effect d applies at the mean covariate value x_bar, and the effect at any other value z is recovered by un-centring. A decision model uses it to take the relative effect for its own population, for example its baseline risk, instead of the average over the trials.

Meta-Regression — Functions & Formulae | HealthEconomics.wiki