Re-express the outcome in within-case residual-SD units

Scale harmonization for raw-data synthesis: fits an ordinary least squares regression of the outcome on the fixed piecewise terms separately for every tier (case, or case within study), then divides that tier's outcome by the tier's residual standard deviation (root mean squared error). The returned <outcome>_std column is in within-case residual-SD units, so studies measured on different scales can enter one multilevel model. This is scale harmonization, not an effect size: the standardized coefficients are not standardized mean differences. scdbayes does not compute the within-case SMD, which is biased under autocorrelation. The design-comparable BC-SMD is estimated in a Bayesian model (bc_smd_bayes()), and studies are pooled with a Bayesian meta-analysis (meta_bayes()); scdhlm/metafor scripts are optional frequentist comparisons.

Usage

standardize_by_case(
  data,
  formula_fixed = NULL,
  outcome_var = "outcome",
  tiers = tier_vars(data)
)

Arguments

data

Data frame with the outcome and either a phase column or the piecewise columns named in formula_fixed.

formula_fixed

Optional fixed-effects formula (formula object or string, with or without a left-hand side, e.g. "~ time_A + level_AB + trend_AB"). NULL uses the default fixed terms for the detected design. Only syntactic column names and is_safe_formula() syntax are accepted; function calls such as I() raise an error, and default terms skip non-syntactic names.

outcome_var

Outcome column name.

tiers

Columns whose combination defines one series; defaults to tier_vars().

Value

data with a numeric column <outcome_var>_std. The attribute "case_sigma" holds one row per tier: the tier columns, tier (label), n (non-missing observations), p (estimated coefficients), sigma (residual SD, NA when not estimable), and status.

Details

Each participant's scores are divided by the root mean squared error of a participant-specific ordinary regression before the multilevel meta-analysis. Standardizing adds error from estimating the scale factor, so analyzing raw data is preferable when all cases share one scale.

Caveat: the per-case OLS residual SD ignores autocorrelation. Within-case standardized effect sizes are biased under autocorrelation, with neither OLS nor multilevel synthesis performing adequately, and raw-data synthesis of data standardized for each case may share the problem. Use the transformation as a sensitivity analysis, not as the primary synthesis metric.

The regression contains only fixed piecewise terms: no random effects, no moderators, no autocorrelation. With formula_fixed = NULL the terms are the package defaults for the design returned by detect_design_type() after coding with process_scd_phases(). A supplied formula keeps its fixed terms and drops grouping terms such as (1 | case) and ar() terms. Columns constant within a tier are aliased and do not count toward the number of coefficients.

A tier is left NA (with a warning) when it has fewer than p + 2 non-missing observations, where p counts the estimated coefficients including the intercept (so at least two residual degrees of freedom remain), or when its residual SD is zero or not finite.

The transformation suits continuous or interval outcomes. For counts, rates, and short ordinal scales the residual SD depends on the mean and the result is approximate; there in particular, keep a count or ordinal likelihood on the natural scale as the primary analysis.

Examples

d <- simulate_scd_data(n_cases = 3, n_phase_a = 8, n_phase_b = 8)
out <- standardize_by_case(d)
attr(out, "case_sigma")
#>   case tier  n p    sigma status
#> 1   C1   C1 16 4 8.037677     ok
#> 2   C2   C2 16 4 7.817246     ok
#> 3   C3   C3 16 4 9.784271     ok