Conjoint Experiment

Conjoint Survey Experiments

When to Use

Use this skill when you need to:

• Design conjoint survey tasks (attributes, levels, profiles) with randomized full or partial factorial profile generation
• Estimate Average Marginal Component Effects (AMCEs) via linear probability model (OLS) with standard errors clustered by respondent
• Compute marginal means (unconditional on other attributes) as an alternative AMCE summary
• Estimate interaction AMCEs to test whether attribute effects vary by other profile attributes
• Analyze respondent-level heterogeneity in AMCEs (subgroup by party ID, ideology, demographics)
• Detect inattentive respondents through straightlining, response time, and consistency checks
• Power-analyze a conjoint study (tasks × respondents × attributes)

Conjoint experiments are used in political science to measure candidate evaluation, immigration preferences, trade-off tolerance for policy packages, and partisan asymmetries.

Background

Conjoint design: Respondents see multiple tasks. In each task, they compare two (or more) profiles that vary across predefined attributes. Each attribute has a set of levels. Levels are randomly assigned to profiles independently across attributes (full factorial) or according to a restricted design (partial factorial, to avoid implausible combinations).

AMCE (Average Marginal Component Effect): The AMCE for level l of attribute A is the average difference in choice probability when A = l compared to a baseline level, averaging over all possible randomizations of other attributes. Because profiles are randomized, the estimator is simply OLS:

choice_ij = μ + Σ_A Σ_l β_{A,l} × 1[attribute_A = l] + ε_ij

where i indexes tasks and j respondents. Standard errors clustered by respondent account for within-respondent correlation across tasks.

Marginal means (Leeper, Hobolt & Tilley 2020) measure the expected choice probability when an attribute takes a given level, averaged over all other attributes — without conditioning on a reference level. This avoids interpretational dependence on the baseline choice.

Interaction AMCE: When an attribute's effect varies by another attribute (or a respondent characteristic), the interaction is estimated by including the product term in the OLS model.

Subgroup AMCE: Splitting the sample by party ID (or ideology) and re-estimating AMCE per subgroup tests partisan asymmetry in attribute preferences (common in immigration and candidate evaluation conjoint studies).

Power: For a binary forced-choice conjoint, the minimum detectable effect (MDE) for an AMCE depends on:

• Number of tasks per respondent (T)
• Number of respondents (N)
• Number of levels per attribute (K) — more levels means each level appears less often

Rule of thumb (Bansak et al. 2021): N × T ≥ 5,000 tasks total for MDE ≈ 0.04.

Environment Setup

pip install pandas>=1.5 statsmodels>=0.14 numpy>=1.23 matplotlib>=3.6

No external data downloads required. Conjoint datasets are typically generated during survey programming (Qualtrics, Formr) and exported as CSV. Store file path as:

export CONJOINT_DATA_PATH="/data/conjoint_survey_results.csv"

Core Workflow

import os
import numpy as np
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches
import warnings
warnings.filterwarnings("ignore")


# ---------------------------------------------------------------------------
# 1. Conjoint Profile Generator
# ---------------------------------------------------------------------------

def generate_conjoint_profiles(
    attributes: dict[str, list],
    n_respondents: int,
    n_tasks: int = 5,
    n_profiles: int = 2,
    seed: int = 42,
) -> pd.DataFrame:
    """
    Simulate a conjoint dataset with randomly assigned profile attributes.

    Parameters
    ----------
    attributes : dict
        {attribute_name: [level1, level2, ...]}
        Example: {'age': ['35', '45', '55'], 'gender': ['Man', 'Woman']}
    n_respondents : int
        Number of survey respondents.
    n_tasks : int
        Number of forced-choice tasks per respondent.
    n_profiles : int
        Number of profiles per task (typically 2 for forced-choice).
    seed : int
        Random seed for reproducibility.

    Returns
    -------
    pd.DataFrame with one row per profile:
    respondent_id, task_id, profile_id, chosen (0/1), + one column per attribute.
    """
    rng = np.random.default_rng(seed)
    attr_names = list(attributes.keys())
    rows = []

    for resp_id in range(1, n_respondents + 1):
        for task_id in range(1, n_tasks + 1):
            # Randomly assign levels to each profile
            profile_levels = {}
            for attr, levels in attributes.items():
                profile_levels[attr] = rng.choice(levels, size=n_profiles, replace=True)

            # Simulate choice based on hidden utility (for testing)
            # In real data, this comes from respondent answers
            chosen_profile = rng.integers(0, n_profiles)

            for prof_id in range(n_profiles):
                row = {
                    "respondent_id": resp_id,
                    "task_id": task_id,
                    "profile_id": prof_id + 1,
                    "chosen": int(prof_id == chosen_profile),
                }
                for attr in attr_names:
                    row[attr] = profile_levels[attr][prof_id]
                rows.append(row)

    return pd.DataFrame(rows)


# ---------------------------------------------------------------------------
# 2. AMCE Estimation via OLS
# ---------------------------------------------------------------------------

def estimate_amce(
    df: pd.DataFrame,
    outcome: str = "chosen",
    attributes: list[str] | None = None,
    reference_levels: dict[str, str] | None = None,
    cluster_var: str = "respondent_id",
) -> pd.DataFrame:
    """
    Estimate Average Marginal Component Effects (AMCE) via linear probability model.

    Parameters
    ----------
    df : pd.DataFrame
        Long-format conjoint dataset (one row per profile).
    outcome : str
        Binary choice indicator (0/1).
    attributes : list of str
        Attribute column names. Inferred from reference_levels keys if not provided.
    reference_levels : dict
        {attribute: reference_level} — the level dropped (baseline).
        If None, the first alphabetical level is used as baseline.
    cluster_var : str
        Variable for clustering standard errors (typically respondent_id).

    Returns
    -------
    pd.DataFrame with AMCE estimates: attribute, level, amce, se, ci_lower, ci_upper, p.
    """
    reference_levels = reference_levels or {}
    attributes = attributes or [c for c in df.columns if c not in
                                  [outcome, cluster_var, "task_id", "profile_id"]]

    # Encode dummies for each attribute
    dummy_frames = []
    for attr in attributes:
        ref = reference_levels.get(attr, sorted(df[attr].unique())[0])
        dummies = pd.get_dummies(df[attr], prefix=attr, drop_first=False)
        ref_col = f"{attr}_{ref}"
        if ref_col in dummies.columns:
            dummies = dummies.drop(columns=[ref_col])
        dummy_frames.append(dummies)

    X_df = pd.concat(dummy_frames, axis=1).astype(float)
    X = sm.add_constant(X_df.reset_index(drop=True))
    y = df[outcome].reset_index(drop=True).astype(float)
    groups = df[cluster_var].reset_index(drop=True)

    model = sm.OLS(y, X)
    result = model.fit(cov_type="cluster", cov_kwds={"groups": groups})

    rows = []
    for col in X_df.columns:
        parts = col.split("_", 1)
        attr = parts[0]
        level = parts[1] if len(parts) > 1 else col
        idx = col
        if idx not in result.params.index:
            continue
        coef = result.params[idx]
        se = result.bse[idx]
        rows.append({
            "attribute": attr,
            "level": level,
            "amce": round(coef, 5),
            "se": round(se, 5),
            "ci_lower": round(coef - 1.96 * se, 5),
            "ci_upper": round(coef + 1.96 * se, 5),
            "p": round(result.pvalues[idx], 4),
        })
    return pd.DataFrame(rows)


# ---------------------------------------------------------------------------
# 3. Marginal Means
# ---------------------------------------------------------------------------

def marginal_means(
    df: pd.DataFrame,
    outcome: str = "chosen",
    attributes: list[str] | None = None,
    weight_var: str | None = None,
) -> pd.DataFrame:
    """
    Compute marginal means: weighted average choice probability per attribute level.

    Parameters
    ----------
    df : pd.DataFrame
    outcome : str
    attributes : list of str
    weight_var : str, optional
        Survey weight column.

    Returns
    -------
    pd.DataFrame with attribute, level, marginal_mean, se, n.
    """
    attributes = attributes or [c for c in df.columns if c not in
                                  [outcome, weight_var or "__none__", "respondent_id",
                                   "task_id", "profile_id"]]
    rows = []
    for attr in attributes:
        for level, grp in df.groupby(attr):
            y = grp[outcome].astype(float)
            if weight_var and weight_var in grp.columns:
                w = grp[weight_var].fillna(1.0)
                mm = np.average(y, weights=w)
            else:
                mm = y.mean()
            se = y.sem()
            rows.append({
                "attribute": attr, "level": str(level),
                "marginal_mean": round(mm, 5),
                "se": round(se, 5),
                "n": len(grp),
            })
    return pd.DataFrame(rows)


# ---------------------------------------------------------------------------
# 4. AMCE Coefficient Plot
# ---------------------------------------------------------------------------

def plot_amce(
    amce_df: pd.DataFrame,
    title: str = "AMCE Estimates",
    figsize: tuple = (9, 7),
    save_path: str | None = None,
) -> plt.Figure:
    """
    Horizontal forest plot of AMCE estimates with 95% CI.

    Parameters
    ----------
    amce_df : pd.DataFrame
        Output of estimate_amce().
    title : str
    figsize : tuple
    save_path : str, optional

    Returns
    -------
    matplotlib Figure.
    """
    df_plot = amce_df.copy()
    df_plot["label"] = df_plot["attribute"] + ": " + df_plot["level"]
    df_plot = df_plot.sort_values(["attribute", "amce"], ascending=[True, False])

    colors = plt.cm.tab10.colors
    attr_list = df_plot["attribute"].unique().tolist()
    attr_color = {attr: colors[i % 10] for i, attr in enumerate(attr_list)}

    fig, ax = plt.subplots(figsize=figsize)
    y_pos = range(len(df_plot))
    row_colors = [attr_color[attr] for attr in df_plot["attribute"]]

    ax.errorbar(
        df_plot["amce"], list(y_pos),
        xerr=[df_plot["amce"] - df_plot["ci_lower"], df_plot["ci_upper"] - df_plot["amce"]],
        fmt="o", color="black", elinewidth=1.2, capsize=3, markersize=6, zorder=3
    )
    # Color-code by attribute
    ax.scatter(df_plot["amce"], list(y_pos), color=row_colors, s=50, zorder=4)
    ax.axvline(0, color="red", linestyle="--", linewidth=1, alpha=0.7)
    ax.set_yticks(list(y_pos))
    ax.set_yticklabels(df_plot["label"].tolist(), fontsize=9)
    ax.set_xlabel("AMCE (change in Pr(chosen))")
    ax.set_title(title, fontsize=13)
    ax.grid(True, axis="x", alpha=0.3)

    # Legend
    patches = [mpatches.Patch(color=attr_color[a], label=a) for a in attr_list]
    ax.legend(handles=patches, fontsize=8, loc="lower right")
    plt.tight_layout()
    if save_path:
        fig.savefig(save_path, dpi=150)
    return fig


# ---------------------------------------------------------------------------
# 5. Subgroup AMCE Heterogeneity
# ---------------------------------------------------------------------------

def subgroup_amce(
    df: pd.DataFrame,
    subgroup_var: str,
    subgroup_values: list,
    outcome: str = "chosen",
    attributes: list[str] | None = None,
    reference_levels: dict[str, str] | None = None,
    cluster_var: str = "respondent_id",
) -> dict[str, pd.DataFrame]:
    """
    Estimate AMCE separately for each subgroup value.

    Parameters
    ----------
    df : pd.DataFrame
    subgroup_var : str
        Column for splitting (e.g., 'party_id').
    subgroup_values : list
        Values to include (e.g., ['Democrat', 'Republican']).

    Returns
    -------
    dict {subgroup_value: amce_df}
    """
    results = {}
    for val in subgroup_values:
        sub = df[df[subgroup_var] == val].copy()
        if len(sub) < 50:
            print(f"Warning: subgroup {val} has only {len(sub)} observations.")
            continue
        amce_df = estimate_amce(
            sub, outcome=outcome, attributes=attributes,
            reference_levels=reference_levels, cluster_var=cluster_var
        )
        results[val] = amce_df
    return results

Advanced Usage

Inattentive Respondent Detection

import numpy as np
import pandas as pd


def detect_inattentive_respondents(
    df: pd.DataFrame,
    respondent_col: str = "respondent_id",
    chosen_col: str = "chosen",
    time_col: str | None = None,
    time_threshold_seconds: float = 5.0,
) -> pd.DataFrame:
    """
    Flag respondents for inattention via straightlining detection.

    Straightlining: always choosing the first (or second) profile across all tasks.
    Fast responders: completing tasks faster than a minimum plausible threshold.

    Parameters
    ----------
    df : pd.DataFrame
    respondent_col : str
    chosen_col : str
    time_col : str, optional
        Column for task completion time in seconds.
    time_threshold_seconds : float
        Minimum plausible time per task.

    Returns
    -------
    pd.DataFrame with respondent_id, n_tasks, always_chose_first, fast_responder, flag.
    """
    # Profile 1 corresponds to the first profile shown in each task
    # Assumes profile_id column exists
    results = []
    for resp_id, grp in df.groupby(respondent_col):
        tasks = grp.groupby("task_id")
        n_tasks = len(tasks)
        choices_first = 0
        for tid, task in tasks:
            first_profile = task["profile_id"].min()
            chosen_profile_id = task.loc[task[chosen_col] == 1, "profile_id"]
            if not chosen_profile_id.empty and chosen_profile_id.iloc[0] == first_profile:
                choices_first += 1

        always_first = (choices_first == n_tasks)
        fast_resp = False
        if time_col and time_col in grp.columns:
            avg_time = grp[time_col].mean()
            fast_resp = avg_time < time_threshold_seconds

        results.append({
            "respondent_id": resp_id,
            "n_tasks": n_tasks,
            "pct_chose_first": round(choices_first / n_tasks, 3),
            "always_chose_first": always_first,
            "fast_responder": fast_resp,
            "flag": always_first or fast_resp,
        })
    return pd.DataFrame(results)


def conjoint_power(
    n_respondents: int,
    n_tasks: int,
    effect_size: float = 0.05,
    alpha: float = 0.05,
) -> dict:
    """
    Approximate power for conjoint AMCE detection.

    Uses the normal approximation: power = Φ(|β| / SE - z_{α/2}).
    SE ≈ 1 / sqrt(N * T / K) where K = 2 profiles per task (binary choice).

    Parameters
    ----------
    n_respondents : int
    n_tasks : int
    effect_size : float
        AMCE to detect (e.g., 0.05 = 5 percentage points).
    alpha : float
        Significance level.

    Returns
    -------
    dict with power, total_tasks, se_approx, mde.
    """
    from scipy.stats import norm
    total_tasks = n_respondents * n_tasks
    se_approx = 1 / np.sqrt(total_tasks / 2)
    z_alpha = norm.ppf(1 - alpha / 2)
    z_beta = abs(effect_size) / se_approx - z_alpha
    power = norm.cdf(z_beta)
    mde = z_alpha * se_approx * 2
    return {
        "power": round(power, 4),
        "total_tasks": total_tasks,
        "se_approx": round(se_approx, 5),
        "mde_90pct": round(mde * 1.28 / z_alpha, 5),
    }

Troubleshooting

External Resources

• Hainmueller, J., Hopkins, D. & Yamamoto, T. (2014). Causal inference in conjoint analysis. Political Analysis, 22(1), 1-30.
• Leeper, T.J., Hobolt, S.B. & Tilley, J. (2020). Measuring subgroup preferences in conjoint experiments. Political Analysis, 28(2), 207-221.
• Bansak, K. et al. (2021). Using conjoint experiments to analyze election outcomes. Political Analysis, 29(3), 380-395.
• Abramson, S.F., Kocak, K. & Magazinnik, A. (2022). What do we learn about voter preferences from conjoint experiments? American Journal of Political Science.
• cregg R package (reference implementation): https://github.com/leeper/cregg

Examples

Example 1: Conjoint Dataset Simulation + AMCE OLS

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# Define attributes and levels for a candidate evaluation conjoint
ATTRIBUTES = {
    "age":         ["35", "45", "55", "65"],
    "gender":      ["Man", "Woman"],
    "education":   ["High School", "College", "Graduate"],
    "experience":  ["None", "Local", "State", "Federal"],
    "party":       ["Democrat", "Republican", "Independent"],
    "issue_pos":   ["Progressive", "Moderate", "Conservative"],
}

# Generate dataset (1200 respondents, 5 tasks each)
df_conjoint = generate_conjoint_profiles(
    attributes=ATTRIBUTES,
    n_respondents=1200,
    n_tasks=5,
    n_profiles=2,
    seed=42,
)

# Inject a realistic preference signal: women, Democrats preferred
rng = np.random.default_rng(123)
pref_cols = ["gender", "party", "experience"]
# Re-draw chosen based on utility
for resp_id, resp_grp in df_conjoint.groupby("respondent_id"):
    for task_id, task_grp in resp_grp.groupby("task_id"):
        idx = task_grp.index
        if len(idx) < 2:
            continue
        # Simple utility: woman +0.15, Democrat +0.20, Federal exp +0.15
        utils = []
        for i in idx:
            u = (0.15 * (df_conjoint.loc[i, "gender"] == "Woman") +
                 0.20 * (df_conjoint.loc[i, "party"] == "Democrat") +
                 0.15 * (df_conjoint.loc[i, "experience"] == "Federal") +
                 rng.gumbel(0, 1))
            utils.append(u)
        chosen_idx = idx[np.argmax(utils)]
        df_conjoint.loc[idx, "chosen"] = 0
        df_conjoint.loc[chosen_idx, "chosen"] = 1

print(f"Dataset size: {len(df_conjoint)} rows, {df_conjoint['respondent_id'].nunique()} respondents")

# Estimate AMCE
reference = {
    "age": "35", "gender": "Man", "education": "High School",
    "experience": "None", "party": "Democrat", "issue_pos": "Progressive",
}
amce_results = estimate_amce(df_conjoint, reference_levels=reference,
                               attributes=list(ATTRIBUTES.keys()))
print("=== AMCE Estimates ===")
print(amce_results.to_string(index=False))

Example 2: AMCE Coefficient Plot

import matplotlib.pyplot as plt

# Using amce_results from Example 1
fig = plot_amce(
    amce_results,
    title="Candidate Evaluation Conjoint — AMCE",
    figsize=(10, 9),
    save_path="amce_coefficient_plot.png",
)
plt.show()

# Also compute marginal means
mm = marginal_means(df_conjoint, attributes=list(ATTRIBUTES.keys()))
print("\n=== Marginal Means (top entries) ===")
print(mm.sort_values("marginal_mean", ascending=False).head(10).to_string(index=False))

Example 3: Subgroup Heterogeneity by Party ID

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# Add respondent-level party ID
n_resp = df_conjoint["respondent_id"].nunique()
resp_party = pd.DataFrame({
    "respondent_id": range(1, n_resp + 1),
    "resp_party": np.random.default_rng(77).choice(
        ["Democrat", "Republican", "Independent"],
        n_resp, p=[0.38, 0.36, 0.26]
    ),
})
df_c2 = df_conjoint.merge(resp_party, on="respondent_id")

subgroup_results = subgroup_amce(
    df_c2,
    subgroup_var="resp_party",
    subgroup_values=["Democrat", "Republican"],
    attributes=["gender", "party", "experience", "issue_pos"],
    reference_levels={"gender": "Man", "party": "Democrat",
                      "experience": "None", "issue_pos": "Progressive"},
)

# Compare AMCE on party attribute across subgroups
fig, axes = plt.subplots(1, 2, figsize=(14, 6), sharey=True)
colors = {"Democrat": "#1f77b4", "Republican": "#d62728"}
for ax, (sub_val, amce_sub) in zip(axes, subgroup_results.items()):
    sub_plot = amce_sub[amce_sub["attribute"] == "party"].copy()
    y_pos = range(len(sub_plot))
    ax.errorbar(
        sub_plot["amce"], list(y_pos),
        xerr=[sub_plot["amce"] - sub_plot["ci_lower"],
              sub_plot["ci_upper"] - sub_plot["amce"]],
        fmt="o", color=colors[sub_val], capsize=4, markersize=8, elinewidth=1.5,
    )
    ax.axvline(0, color="gray", linestyle="--", linewidth=0.8)
    ax.set_yticks(list(y_pos))
    ax.set_yticklabels(sub_plot["level"].tolist())
    ax.set_xlabel("AMCE")
    ax.set_title(f"{sub_val} Respondents — Party Attribute AMCE")
    ax.grid(True, axis="x", alpha=0.3)

plt.suptitle("Partisan Heterogeneity in Candidate Party Preferences", fontsize=13)
plt.tight_layout()
plt.savefig("subgroup_amce_party.png", dpi=150)
plt.show()

# Power analysis
power_result = conjoint_power(n_respondents=1200, n_tasks=5, effect_size=0.05)
print("\n=== Power Analysis ===")
for k, v in power_result.items():
    print(f"  {k}: {v}")