coef(fit) # DML theta, clustered SE
sc_average(fit, scale = "probability") # population AME (delta-method)
sc_ame(fit, "attr_level") # debiased structural AME
sc_counterfactual(fit, A = list(...), B = list(...)) # debiased by default
sc_voteshare_contrast(fit, contrast = c(...)) # debiased vote share for a contrast
sc_mrs(fit, "attr_a", "attr_b", estimand = "population") # debiased theta_j/theta_k
sc_importance(fit, vartype = "orthogonal") # debiased shares (opt-in)7 Estimands
The application chapters fit one model and then read several quantities off it: population averages, the distribution of preferences across respondents, and individual-level summaries. This chapter is the reference for those quantities. It lists each estimand, the function that returns it, what it targets, and how its standard errors are computed, so you can look up the right call without re-reading a worked example.
The quantities form a hierarchy by how much the design has to support to recover them. Population averages (the AMCE and related averages) are available from almost any forced-choice design. Distributional quantities (how preferences spread across respondents) need enough tasks per respondent and enough signal in the moderators \(\mathbf Z\) to separate real heterogeneity from noise. Individual-level quantities (one respondent’s preference vector, their personal trade-offs) ask for more still. Acharya, Hainmueller, and Xu (2026) set out which quantities a given design can recover. The design diagnostic at the end of this chapter estimates which tier your own design reaches. The code below is reference syntax (eval: false); see the application chapters for the same calls run on real fits.
7.1 Overview
| Estimand | Function | Targets | Inference | which_beta applies |
|---|---|---|---|---|
| AMCE \(\hat\theta_k\) | coef(), summary(), plot_amce() |
population-average coefficient (logit) | DML, clustered SE | no |
| Average marginal effect | sc_average() |
population AME (logit or probability) | delta-method on \(\hat\theta\) | no |
| Structural AME | sc_ame() |
population AME (probability) | debiased orthogonal score | no |
| Counterfactual vote share | sc_counterfactual(), sc_voteshare_contrast() |
\(\Pr(\text{choice})\) for a profile pair or contrast | debiased orthogonal score | no |
| Population MRS / WTP | sc_mrs(estimand = "population"), sc_wtp(...) |
ratio \(\theta_j/\theta_k\) | debiased, Fieller interval | no |
| Attribute importance | sc_importance(), plot_importance() |
utility-variance share per attribute | plug-in, or debiased (vartype = "orthogonal") |
yes (plug-in) |
| Fraction preferring | sc_fraction_preferring(), plot_fraction() |
share with \(\hat\beta_k > 0\) | plug-in or wild bootstrap | yes |
| Polarization | sc_polarization() |
dispersion of \(\hat\beta_k(\mathbf Z)\) | plug-in or wild bootstrap | yes |
| Direction and intensity | sc_direction_intensity() |
sign vs. magnitude split | plug-in | yes |
| Compensating differential | sc_compensating() |
benefit/cost trade-off | plug-in, clustered SE | yes |
| Heterogeneity test | sc_heterogeneity_test(), plot_hetero() |
\(\mathrm{Var}(\hat\beta_k(\mathbf Z)) > 0\) | one-sided test | yes |
| Per-respondent betas | predict() |
each respondent’s \(\hat\beta(\mathbf Z_i)\) | model-based | yes |
| Individual MRS / WTP | sc_mrs(), sc_wtp() (default) |
respondent-averaged trimmed ratio | clustered SE | yes |
| Optimal profile | sc_optimal_profile() |
utility-maximizing profile | model-based | yes |
| Preference clusters | sc_clusters() |
latent preference types | k-means on \(\hat\beta(\mathbf Z)\) | yes |
| Subgroup AMCE | sc_subgroup(), plot_subgroup() |
\(\hat\beta\) re-averaged over a subset | clustered SE | yes |
The which_beta column records whether the quantity reads the per-respondent coefficients (and so can be computed from either the Stage-2 hybrid view or the raw DNN view; see below). The population-average estimands marked “no” are computed from the Stage-1 cross-fit and do not depend on that choice.
7.2 Population averages
These are the quantities a standard AMCE analysis already reports, plus their structural counterparts. They are population averages, computed from the Stage-1 cross-fitted first stage, and they come with double/debiased machine-learning (DML) inference.
The population-average parameter \(\hat\theta\) (from coef() / summary()) carries an orthogonal-score correction with a respondent-clustered standard error. The same correction, implemented for the paper’s Supplementary Materials (Section C), extends to the other average quantities, so several sc_* functions return debiased confidence intervals rather than plug-in ones:
| Quantity | Debiased option |
|---|---|
sc_counterfactual() |
vartype = "orthogonal" (the default); vartype = "plugin" for the v0.1 plug-in |
sc_voteshare_contrast() |
always debiased; takes a raw contrast vector in attribute-dummy space |
sc_ame() |
always debiased (the structural AME on the probability scale) |
sc_mrs(), sc_wtp() |
estimand = "population" (debiased ratio theta_j/theta_k with a Fieller interval) |
sc_importance() |
vartype = "orthogonal" (opt-in; the plug-in shares stay the default) |
The debiased estimates use the Stage-1 cross-fitted first stage and are population averages, so subgroup and which_beta do not apply to them. Applied to \(\hat\theta\) itself the orthogonal score reproduces the average-parameter estimate and its clustered standard error exactly. For sc_importance(), the debiased shares are a ratio of orthogonal-score numerators and can fall outside \([0, 1]\) when the first stage is noisy, which is why the always-in-\([0, 1]\) plug-in shares remain the default.
7.3 Distributional quantities
These describe how preferences are spread across respondents rather than their average. They read the per-respondent coefficients \(\hat\beta_k(\mathbf Z_i)\), so they accept which_beta. The fixed-\(T\) summaries (fraction preferring, polarization) also accept a respondent-cluster wild bootstrap for honest standard errors.
sc_fraction_preferring(fit) # share with beta_k > 0
sc_fraction_preferring(fit, se_method = "wild_bootstrap", n_boot = 1000)
sc_polarization(fit, se_method = "wild_bootstrap", n_boot = 1000)
sc_direction_intensity(fit) # sign vs. magnitude
sc_compensating(fit, benefit = "attr_a", cost = "attr_b")
sc_heterogeneity_test(fit, adjust = "bh") # Var(beta_k(Z)) > 0
TipA fixed-\(T\) shrinkage caveat
With few tasks per respondent, the Stage-2 MAP update pulls each respondent’s coefficients toward the mean stage, so a hybrid-view fraction or polarization estimate is conservative: the recovered spread is narrower than the truth. Sanity-check distributional claims with which_beta = "dnn" (the unrefined view). If the qualitative conclusion flips between the two, the design is too sparse to support individual-level claims, and you should report only the population averages. The design diagnostic makes this concrete.
7.4 Individual-level quantities
These summarize a single respondent’s recovered preferences, or build on the full per-respondent coefficient matrix. They are model-based: their credibility rests on the first stage having learned \(\hat\beta(\mathbf Z)\) well, which needs both enough tasks per respondent and moderators \(\mathbf Z\) that explain a real share of the heterogeneity.
beta <- predict(fit) # N_resp x p coefficient matrix
predict(fit, newdata = new_Z, type = "beta") # forward pass on new respondents
sc_mrs(fit, "attr_a", "attr_b") # respondent-averaged trimmed ratio
sc_optimal_profile(fit) # utility-maximizing profile
sc_clusters(fit, k = 3L) # latent preference types
sc_subgroup(fit, list(GroupA = mask, GroupB = !mask)) # beta re-averaged over a subsetThe trimmed ratio in sc_mrs() / sc_wtp() (the default respondent-level estimand, as opposed to estimand = "population") is unbounded when the denominator coefficient is near zero, so it trims at \(\{0.01, 0.99\}\) by default; inspect the ridgelines before reading an individual MRS literally.
7.5 Choosing the coefficient view: which_beta
Every distributional and individual-level sc_* quantity accepts which_beta = c("hybrid", "dnn"). The default "hybrid" reads fit$beta_hat (whatever Stage 2 produced). Passing "dnn" reads fit$beta_hat_dnn (the raw Stage-1 view). When stage2 = "none" the two are numerically identical and the helper silently falls back.
## Same fit, two views of the same distribution
sc_fraction_preferring(fit) # MAP-shrunk
sc_fraction_preferring(fit, which_beta = "dnn") # raw DNNWhen the paper text says “the structural model reveals that X% of respondents prefer Y”, it means the MAP-shrunk fraction (the "hybrid" default). The raw DNN fraction is usually wider because unrefined respondent-level coefficients have more sampling noise. The two views are the basis of the shrinkage sanity check described above.
7.6 Survey-weighted estimands
By default, sconjoint targets the unweighted sample of respondents: each respondent contributes equally after averaging over that respondent’s conjoint tasks. If the original survey supplies respondent weights and the target is a weighted population, pass those weights to scfit() with respondent_weights.
fit_w <- scfit(
formula,
data = dat,
respondent = "respondent_id",
task = "task_id",
profile = "profile_id",
respondent_weights = "survey_weight",
seed = 42
)The weight can be a column name in data or a numeric vector of length nrow(data). It must be finite, non-negative, and constant within respondent. The package normalizes the weights across respondents and then uses them in the respondent-level aggregation step:
\[ \hat\theta_w = \sum_{i=1}^{N} a_i\,\bar\psi_i, \qquad a_i=\frac{w_i}{\sum_{j=1}^{N}w_j}, \qquad \bar\psi_i=\frac{1}{T_i}\sum_{t=1}^{T_i}\psi_{it}. \]
The clustered variance uses the same weighted respondent contributions, with the usual finite-cluster factor:
\[ \widehat V_w = \frac{N}{N-1} \sum_{i=1}^{N} a_i^2(\bar\psi_i-\hat\theta_w)(\bar\psi_i-\hat\theta_w)'. \]
Survey weights do not reweight the first-stage learner. The DNN, elastic net, or GRF still learns the conditional preference function \(f(\mathbf Z)\) from the observed respondent-task rows. The weights enter only after the orthogonal scores or respondent-level plug-in quantities have been formed. This is the right operation when the issue is target-population composition rather than a different training loss.
The same weighted aggregation is used by the production-facing quantities: coef() / summary(), sc_average(), sc_counterfactual(), sc_mrs() / sc_wtp(), sc_compensating(), sc_fraction_preferring(), sc_polarization(), sc_importance(), sc_subgroup(), and sc_direction_intensity().
coef(fit_w)
sc_counterfactual(fit_w, A = list(...), B = list(...))
sc_compensating(fit_w, benefit = "co_partisan", cost = "journalists")
sc_fraction_preferring(fit_w, se_method = "wild_bootstrap", n_boot = 1000)For comparison or diagnostics, fit the model once without weights and once with weights. A material difference means the estimand is sensitive to survey composition; a small difference supports reporting the unweighted structural estimate with a weighting robustness check.
7.7 Design diagnostic: which tiers does my design support?
sc_design_diagnostic() estimates per-coefficient \(\hat R^2_{Z,k}\) from the MAP posterior and maps the design to four recovery tiers per the paper’s design-guidance heuristics:
| Tier | Condition | Quantities recoverable |
|---|---|---|
| mean & aggregate | any reasonable design | AMCE, average vote shares |
| distributional | T ≥ 5 and \(\bar R^2_Z\) ≥ 0.35 | preference shares, importance rankings |
| individual-level | T ≥ 8 and \(\bar R^2_Z\) ≥ 0.55 | respondent-specific quantities |
| ratio (MRS / WTP) | T ≥ 10 and \(\bar R^2_Z\) ≥ 0.55 and N ≥ 5000 | individual MRS / WTP |
These tiers line up with the three sections above: “mean & aggregate” with the population averages, and “distributional” and “individual-level” with the two sections of those names. A design must reach a tier before the quantities in the matching section can be trusted.
diag <- sc_design_diagnostic(fit)
print(diag)
## Top / bottom R^2_Z attributes are also reported -- they show which
## dummies are best-pinned by Z and which rely most on T for recovery.Requires stage2 != "none" (it needs the MAP Hessian’s posterior variance). The v0.2.1 estimator is flagged experimental = TRUE in print output because it has not yet been validated against the paper’s 1,872-cell simulation grid; Chapter 5 shows it in action on the Saha-Weeks design.