Responder analysis of continuous trial outcomes

Converts continuous outcomes (mean change, SD and sample size per arm, across studies) into responder proportions and a range of between-arm effect measures: the risk difference (RD), risk ratio (RR), odds ratio (OR) and number needed to treat (NNT), under a parametric model for the change scores. Responders are defined by a minimal important difference (MID) threshold (the cut-point / "dichotomization" approach of Anzures-Cabrera, Sarpatwari and Higgins, 2011). For a threshold-free alternative see responder_cles().

Usage

responder_analysis(
  data,
  mid,
  direction = c("higher", "lower"),
  method = c("individual", "weighted", "unweighted", "median", "smd"),
  se_method = c("binomial", "delta"),
  pooling = c("fixed", "random"),
  control = c("matched", "median"),
  tau_method = c("DL", "REML"),
  dist = c("normal", "lognormal", "t"),
  df = NULL,
  mid_sd = 0,
  ci_type = c("wald", "logit"),
  ci_method = c("wald", "hksj"),
  conf_level = 0.95
)

Arguments

data: A data frame with one row per study and columns study, change_e, sd_e, n_e, change_c, sd_c, n_c. See sample_responder_data.
mid: Single finite number: the minimal important difference threshold.
direction: "higher" (a larger change indicates response) or "lower".
method: Methods to compute: any of "individual", "weighted", "unweighted", "median", "smd". Defaults to the first four.
se_method: Per-study SE model for "individual": "binomial" (default) or "delta". The "binomial" variance p(1 - p) / n is a pseudo-binomial approximation: p is a probability implied by the estimated mean and SD, not a proportion of observed dichotomized patients, so it does not include the uncertainty in the reported mean and SD. "delta" propagates that uncertainty and is generally preferable for summary-statistic inputs; "binomial" is the default only for continuity with earlier results.
pooling: "fixed" (default) or "random" effects, for the "individual" and "smd" methods.
control: Baseline-risk rule for the summary methods (median, unweighted, weighted): "matched" (default) pools the control arm the same way as the experimental arm; "median" always takes the control responder proportion from the median control arm (the Sofi-Mahmudi 2024 baseline), which yields point estimates only. Ignored by "individual" and "smd".
tau_method: Between-study variance estimator when pooling = "random": "DL" (DerSimonian-Laird, default) or "REML" (needs the metafor package; falls back to DL with a warning if unavailable).
dist: Change-score distribution: "normal" (default), "lognormal" or "t".
df: Degrees of freedom when dist = "t".
mid_sd: Optional standard deviation of the MID threshold; when > 0 its uncertainty is propagated into the effect-measure variances.
ci_type: "wald" (default) or "logit" (keeps proportion and risk-difference intervals within valid bounds via the logit transform and Newcombe's MOVER method).
ci_method: Random-effects interval method: "wald" (Normal, default) or "hksj" (Hartung-Knapp-Sidik-Jonkman, a t-based interval that is better calibrated when the number of studies is small).
conf_level: Confidence level (default 0.95).

Value

A data frame with one row per requested method and columns: method, pooling, k, p_e, p_c, rd/rd_lb/rd_ub, rr/rr_lb/rr_ub, or/or_lb/or_ub, nnt/nnt_lb/nnt_ub, var_rd, and the heterogeneity statistics tau2, i2, q, q_p, pi_lb, pi_ub (for the pooled methods). Proportions, risk differences and CLES are on the proportion scale; multiply by 100 for percentages.

Methods

individual: Dichotomize each study, then pool the per-study effect measures (fixed or random effects). The most defensible option; the per-study SE follows se_method.
weighted: Pool the mean change by inverse variance and the SD by the within-study pooled SD, dichotomize the pooled summaries, and obtain variances by the delta method.
unweighted, median: Dichotomize the arithmetic mean / median of the study means and SDs. Summaries with no variance model: intervals are NA.
smd: Pool the standardized mean difference (Hedges' g), bridge to an odds ratio via the logistic link (lnOR = (pi / sqrt(3)) g), and combine with the weighted-pooled control responder rate to recover risks. The second approach of the reference; not included by default.

For the summary methods (median, unweighted, weighted) the control proportion is, by default, pooled the same way as the experimental arm (control = "matched"). Set control = "median" to instead take the baseline risk from the median control arm for every summary method, as in the Sofi-Mahmudi (2024) simulation study; the experimental arm is still pooled by the chosen method. Because the median control arm carries no sampling-variance model, control = "median" reports point estimates only (no intervals) for the summary methods. The individual and smd methods pool per-study contrasts and ignore control.

References

Sofi-Mahmudi A (2024). Identifying an optimal strategy for converting pain as a continuous outcome to a responder analysis. Master's thesis, McMaster University. https://hdl.handle.net/11375/30210

Thorlund K, Walter SD, Johnston BC, Furukawa TA, Guyatt GH (2011). Pooling health-related quality of life outcomes in meta-analysis: a tutorial and review of methods for enhancing interpretability. Research Synthesis Methods, 2(3), 188 to 203. doi:10.1002/jrsm.46

Anzures-Cabrera J, Sarpatwari A, Higgins JPT (2011). Expressing findings from meta-analyses of continuous outcomes in terms of risks. Statistics in Medicine, 30(25), 2867 to 2880. doi:10.1002/sim.4298

Examples

responder_analysis(sample_responder_data, mid = 1)
#>       method pooling k       p_e       p_c        rd     rd_lb     rd_ub
#> 1 individual   fixed 3        NA        NA 0.2554475 0.1869705 0.3239244
#> 2   weighted    <NA> 3 0.4742782 0.2205372 0.2537410 0.1985865 0.3088955
#> 3 unweighted    <NA> 3 0.4767051 0.2279613 0.2487438        NA        NA
#> 4     median    <NA> 3 0.4869694 0.2150781 0.2718912        NA        NA
#>         rr    rr_lb    rr_ub       or    or_lb    or_ub      nnt   nnt_lb
#> 1 2.148809 1.712779 2.695841 3.198098 2.296864 4.452953 3.914699 3.087140
#> 2 2.150558 1.790679 2.582764 3.188531 2.442773 4.161963 3.941027 3.237341
#> 3 2.091167       NA       NA 3.085185       NA       NA 4.020201       NA
#> 4 2.264151       NA       NA 3.464085       NA       NA 3.677941       NA
#>     nnt_ub       var_rd tau2 i2      q       q_p pi_lb pi_ub
#> 1 5.348437 0.0012206534    0  0 1.6054 0.4481173    NA    NA
#> 2 5.035589 0.0007918911   NA NA     NA        NA    NA    NA
#> 3       NA           NA   NA NA     NA        NA    NA    NA
#> 4       NA           NA   NA NA     NA        NA    NA    NA

# Random-effects individual method with relative measures:
responder_analysis(sample_responder_data, mid = 1,
  method = "individual", pooling = "random")
#>       method pooling k p_e p_c        rd     rd_lb     rd_ub       rr    rr_lb
#> 1 individual  random 3  NA  NA 0.2554475 0.1869705 0.3239244 2.148809 1.712779
#>      rr_ub       or    or_lb    or_ub      nnt  nnt_lb   nnt_ub      var_rd
#> 1 2.695841 3.198098 2.296864 4.452953 3.914699 3.08714 5.348437 0.001220653
#>   tau2 i2      q       q_p    pi_lb     pi_ub
#> 1    0  0 1.6054 0.4481173 -0.18848 0.6993749