Effect-size estimation — --spasqr-mode wald

The default --spasqr-mode score is optimized for genome-wide screening: it tests $H_0!: \beta_G = 0$ against $H_1!: \beta_G \neq 0$ at every $\tau$ using a rank-score statistic computed from a single null-model fit per chromosome. Score mode returns calibrated $p$-values and signed $Z$-scores per marker, but not the effect-size estimate $\hat\beta_G$ itself.

For follow-up analyses — characterizing the magnitude of a top hit, plotting per-quantile effect curves, or feeding effects into a downstream meta-analysis — switch to Wald mode with --spasqr-mode wald. Wald mode fits the full model $Q_\tau(Y \mid X, G_j) = X^\top \beta_\tau + G_j\, \beta_{G,\tau}$ once per (marker, $\tau$) by smoothed M-estimation, and returns

  • $\hat\beta_{G,\tau}$ — the per-quantile genetic effect on the transformed $Y$ scale,
  • $\widehat{\mathrm{SE}}(\hat\beta_{G,\tau})$ — sandwich-form standard error using the bandwidth-aware Hessian and the empirical score meat,
  • $Z_{j,\tau} = \hat\beta_{G,\tau} / \widehat{\mathrm{SE}}$ — Wald statistic,
  • $P_{j,\tau}$ — two-sided $p$-value from the standard normal.

When to use Wald mode

Use case Recommended mode
Genome-wide screening (millions of variants) score — much faster, gives calibrated $P_\mathrm{CCT}$.
Effect-size estimation on top hits (≤ a few hundred variants) wald — gives $\hat\beta_G$ + SE per $\tau$.
Generating summary statistics for meta-analysis wald restricted via --extract.
Sensitivity analyses (alternative bandwidth, tau-specific) wald — finer control over per-marker fit.

Wald mode refits the full model per (marker, $\tau$), so total cost scales as $\mathcal O(\text{n_markers} \times \text{n_taus})$ versus score mode’s $\mathcal O(\text{n_markers})$. For a genome-wide scan with 9 tau levels this is roughly 9× slower than score mode — fine for a candidate list, infeasible for the full genome.

Example

Restrict to a set of GWAS hits (one ID per line in hits.txt) and re-run in Wald mode:

./grab --method SPAsqr \
     --spasqr-mode wald \
     --bfile geno \
     --pheno pheno_int.txt --pheno-name Y1 \
     --covar covar.txt --covar-name covar1,covar2 \
     --pred-list grab_predlist.txt \
     --spasqr-taus 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 \
     --extract hits.txt \
     --spasqr-h-scale 10 \
     --threads 8 \
     --out spasqr_wald

This produces spasqr_wald.Y1.SPAsqr with one row per (marker, $\tau$):

CHROM  POS  ID  REF  ALT  MISS_RATE  ALT_FREQ  MAC  HWE_P  TAU  BETA  SE  Z  P

Column meanings:

  • CHROM POS ID REF ALT MISS_RATE ALT_FREQ MAC HWE_P — same as score mode.
  • TAU — the quantile level for this row.
  • BETA — Wald estimate $\hat\beta_{G,\tau}$ on the pheno-transform scale (e.g. INT scale if --pheno-transform int).
  • SE — sandwich-variance standard error.
  • Z — Wald statistic $\hat\beta_{G,\tau} / \widehat{\mathrm{SE}}$.
  • P — two-sided $p$-value, $P = 2 \cdot \Phi(- Z )$.

Interpreting per-quantile effects

For a single hit, you typically have 5–9 rows (one per $\tau$). The effect-quantile curve $\tau \mapsto \hat\beta_{G,\tau}$ tells you where in the conditional distribution of $Y$ the variant acts:

  • Flat curve — mean-only effect; classical OLS / linear GWAS would have caught it. SPASQR still gives the same hit with slightly conservative SE.
  • Monotone in $\tau$ — location-scale effect; the variant shifts both the centre and the tails.
  • U-shaped or non-monotone — heteroskedastic / dispersion effect; the variant changes the spread of $Y$ without shifting the centre. These are the hits where SPASQR typically beats classical mean-based GWAS.

Plot $\hat\beta_{G,\tau}$ against $\tau$ with $\pm 1.96 \cdot$ SE error bars; cross-check the sign pattern against the per-$\tau$ $Z$-scores from score mode.

Bandwidth and solver notes

  • Wald mode defaults to --spasqr-h-scale 10 (narrower bandwidth than score mode’s 3). This reduces smoothing bias in $\hat\beta_G$ at the cost of slower convergence; it is the right default when you care about the estimate, not just the test.
  • The same --spasqr-solver choice applies (qmme default, conquer as alternative). qmme is the recommended solver for both modes.
  • --spasqr-tol controls convergence tolerance for the M-estimation iterations. The default 1e-7 is usually fine; tighten to 1e-9 for very rare variants or extreme tail $\tau$.

When NOT to use Wald mode

  • For genome-wide screening, always use score mode first. Score mode’s $P_\mathrm{CCT}$ is the calibrated GWAS $p$-value; Wald per-$\tau$ Z-scores are not multiplicity-adjusted.
  • For rare-variant testing at extreme $\tau$ (e.g. $\tau \in {0.05, 0.95}$) with sparse data, Wald sandwich SEs can be unstable. Use score mode instead, which leverages SPA in the tails.

Note

  • Wald mode honors the same --pred-list, --sp-grm-plink2, --pheno-transform and --spasqr-taus flags as score mode.
  • The output schema differs: Wald is one row per (marker, $\tau$) with TAU BETA SE Z P columns; score is one row per marker with P_CCT P_tau{val}... Z_tau{val}... columns.
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