Residuals for brma Objects

Computes residuals (observed minus fitted values) from a fitted brma object, with options for standardization.

Usage

# S3 method for class 'brma'
residuals(
  object,
  type = "outcome",
  unit = "estimate",
  conditioning_depth = "marginal",
  bias_adjusted = FALSE,
  ...
)

Arguments

object

a fitted brma object

type

the type of residuals to compute. Options are:

"outcome" (default): Raw residuals (observed - fitted). Available for all model types.
"pearson": Pearson (semi-standardized) residuals, dividing raw residuals by the marginal standard error $\sqrt{v_i + \tau^2}$. Only available for normal outcome models without selection (weightfunction) bias adjustment.
"rstandard": Internally standardized residuals, dividing raw residuals by their standard errors computed using the hat matrix. Only available for normal outcome models without selection (weightfunction) bias adjustment.
"LOO-PIT" (alias: "rstudent"): Leave-one-out probability integral transform (PIT) residuals computed via Pareto smoothed importance sampling. The LOO-CDF value for each observation is computed and transformed to standard normal quantiles via $\Phi^{-1}(u_i)$. Under a correctly specified model, these residuals should follow a standard normal distribution. This is the recommended standardized residual for Bayesian models as it properly accounts for estimation uncertainty and leverage. Available for all model types. Note: This requires that the loo has been computed previously (see add_loo() function).

unit

output unit. Only "estimate" is implemented currently..

conditioning_depth

conditioning depth for non-LOO residuals. "marginal" uses fixed effects only, "cluster" conditions on cluster-level random effects, and "estimate" conditions on the full estimate-level fitted value. LOO-PIT residuals always use the estimate-unit LOO target.

bias_adjusted

whether residuals should be computed from bias-adjusted fitted values. Defaults to FALSE, which means residuals are computed as the difference between observed values and raw (biased) predictions including PET/PEESE terms. Set to TRUE to compute residuals from bias-corrected fitted values. Applies to outcome and Pearson residuals. Note that bias-adjusted residuals are not residuals in the traditional sense.

...

additional arguments.

Value

A numeric vector of residual means, one per estimate.

Details

Raw residuals (type = "outcome") are computed as: $$e_i = y_i - \hat{\mu}_i$$ where $y_i$ is the observed effect size and $\hat{\mu}_i$ is the fitted value (prediction from the fixed effects).

Pearson residuals (type = "pearson") divide raw residuals by the marginal standard error: $$r_i^{Pearson} = \frac{e_i}{\sqrt{v_i + \tau^2}}$$ where $v_i$ is the sampling variance and $\tau^2$ is the relevant heterogeneity component. Only available for normal outcome models without selection (weightfunction) bias adjustment.

Standardized residuals (type = "rstandard") use the hat matrix to compute residual standard errors that account for the uncertainty in estimated coefficients: $$z_i = \frac{e_i}{\sqrt{[(I-H)M(I-H)']_{ii}}}$$ where $H$ is the hat matrix and $M$ is the marginal variance-covariance matrix. For models without moderators, this simplifies to the Pearson formula. Only available for normal outcome models without selection (weightfunction) bias adjustment.

LOO-PIT residuals (type = "LOO-PIT") are the Bayesian equivalent of studentized deleted residuals (Vehtari et al. 2017) . They are computed via leave-one-out probability integral transformation: $$r_i = \Phi^{-1}(u_i)$$ where $u_i = \sum_s w_{is} F(y_i | \theta^{(s)})$ is the LOO-weighted CDF value, $w_{is}$ are the normalized PSIS weights, and $F$ is the cumulative distribution function of the estimate-unit predictive distribution used by LOO. Under a correctly specified model, LOO-PIT residuals should follow a standard normal distribution. Unlike traditional standardized residuals, LOO-PIT residuals properly account for estimation uncertainty and leverage without requiring a hat matrix. This is the recommended method for standardized residuals in Bayesian meta-analysis.

For meta-regression models, fitted values incorporate moderator effects. For models without moderators, all fitted values equal the pooled effect.

For GLMM models (binomial or Poisson), observed effect sizes and their sampling variances are computed from the raw frequency data using the same formulas as metafor::escalc with the default zero-cell adjustment (adding 0.5 to all cells when any cell is zero). GLMM residuals and LOO-PIT values are therefore approximate effect-size-scale diagnostics, not exact PIT diagnostics for the raw count likelihood.

The residuals are computed separately for each posterior sample, naturally propagating uncertainty in model parameters to the residuals.

References

Vehtari A, Gelman A, Gabry J (2017). “Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC.” Statistics and computing, 27(5), 1413–1432. doi:10.1007/s11222-016-9696-4 .

Examples

if (FALSE) { # \dontrun{
if (requireNamespace("metadat", quietly = TRUE)) {
  data(dat.lehmann2018, package = "metadat")

  fit <- bPET(
    yi      = yi,
    vi      = vi,
    data    = dat.lehmann2018,
    measure = "SMD",
    seed    = 1,
    silent  = TRUE
  )

  # raw residuals (default)
  residuals(fit)

  # Pearson and internally standardized residuals
  residuals(fit, type = "pearson")
  residuals(fit, type = "rstandard")

  # LOO-PIT residuals require stored LOO
  fit <- add_loo(fit)
  residuals(fit, type = "LOO-PIT")

  # residuals from bias-adjusted predictions
  residuals(fit, bias_adjusted = TRUE)

  # check Pareto k diagnostics
  plot(loo(fit))
}
} # }