name: regression-inference
description: Coefficient confidence intervals (ci_lower / ci_upper at conf_level) and the coefficient covariance and correlation on a regression — the per-family critical value, the opt-in vcov flag, where each engine's and modifier's matrix comes from, how it relates to the standard errors, weighted fits, the refusals, and what R's vcov does differently. Topical design; per-model params in atomic op-reg-* skills.
type: guide
kind: design
applies_to: process, compose, predict
covers: [ci_lower, ci_upper, conf_level, confidence interval, df, vcov, Vcov, Correlation, covariance, coefficient correlation, standard errors, std_errors, regressions]

Regression coefficient intervals and covariance

Confidence intervals

Every fit with standard errors carries ci_lower / ci_upper (keyed like coefficients), conf_level and, for a t interval, df: bound = β ∓ q·SE. Set "conf_level": 0.9 (flat, 0<c<1, default 0.95).

Present exactly where std_errors is: lasso / elastic net only on kept predictors (plug-in, approximate), selection keyed by selected_features. An undefined bound (df ≤ 0) is null. t on df, so it differs from a pasted ±1.96·SE in about the third digit on small samples.

Covariance

Set "vcov": true on a regression slot (flat spec field, NOT under params) to get the coefficient covariance and its correlation. Off by default: Vcov / Correlation are then absent and the response is byte-identical.

{"regressions": [{"type": "REG_OLS", "target": "mpg", "predictors": ["wt", "hp"], "vcov": true}]}

Shape

Response.Regressions[i].Vcov and .Correlation are square-symmetric tables in the full encoding, keyed (intercept) first, then the predictors in spec order. Correlation is cov2cor (unit diagonal); a zero or undefined variance gives null in that row and column. Under selection the keys are (intercept) + the retained predictors in selected_features order; selected-out predictors are absent.

Where the matrix comes from

| Resample jackknife | (n−1)/n·Σ(β₋ᵢ−β̄)(β₋ᵢ−β̄)ᵀ over the leave-one-out refits | | Resample bootstrap | sample covariance (B−1 divisor) of the successful replicates; seeded = deterministic | | Selection | the engine’s matrix from the final refit on the selected predictors |

For OLS, ridge, GLM and both modifiers, √diag(Vcov) equals StdErrors exactly; selection + resample follows the resample SEs over the selected set. The Bayesian diagonal is StdErrors² · a_N/(a_N−1), not equal: StdErrors there is the scale of the marginal t, not the posterior variance.

Weighted fits (frequency or probability) use the N* basis the weighted SE uses; see weighting.

Robust standard errors

"covariance": {"type": "hc3"} on REG_OLS swaps the estimator behind std_errors, p_values, the intervals and vcov (√diag still = StdErrors): hc0–hc3, cluster (CR1, OLS df = clusters − 1) or hac (Newey–West over order_by). Absent = classical, byte-identical. Matches R sandwich + lmtest::coeftest. REG_GLM takes it too: the sandwich over the IRLS scores (R sandwich on glm), still Wald z. REG_BAYES_LINEAR: Müller’s sandwich posterior REPLACES the posterior covariance (mean kept; normal intervals, no df).

Refusals

PROCESSING_REGRESSION_VCOV_UNSUPPORTED, details.reason penalty: l1 / elasticnet has no sampling covariance of its data-dependent active set, never approximated — refused under resample or selection too. A robust covariance is refused beside any penalty, resample or selection.

Predict and runtime apply the same rule, so predict reports it before any fit. pulse_errors_lookup carries the recovery.

Against R

• vcov(lm) and vcov(ridge) agree to rounding.
• vcov(glm) uses weights one IRLS iteration behind its coefficients; Pulse inverts at the converged fixed point, matching R refined to that point to 1e-11 (raw vcov(glm) only to about 1e-6).
• Gamma GLM: Pulse fixes dispersion at 1, R estimates it, so the covariances differ by that factor. No oracle case covers gamma.

Response shaping

vcov, correlation, ci_lower, ci_upper, df, conf_level ride the standard preset (the MCP default) and are absent from minimal. Exclude with regressions[*].<key> paths in return.exclude; return.precision rounds them like any figure. See response-shaping.

See

• regression-modeling — choosing a model and modifiers.
• op-reg-ols — the vcov Param for OLS and ridge.
• op-reg-glm — the vcov Param for GLM.
• op-reg-bayes-linear — the vcov Param for the Bayesian linear model.
• weighting — the N* basis.
• response-shaping — presets and exclude paths.
• matrix-results — the same matrix encodings.