Author: Brent Kaplan
Date: 2026-03-02

# Hurdle Demand Models

#### Brent Kaplan

#### 2026-03-02

## Introduction

The `beezdemand` package includes functionality for fitting two-part mixed effects hurdle demand models using Template Model Builder (TMB). This approach is particularly useful when:

- Zero consumption values are informative (i.e., represent true non-consumption rather than censored data)
- You want to model both the probability of consuming and the intensity of consumption simultaneously
- Individual-level heterogeneity is important for both parts of the model

### When to Use Hurdle Models vs. Standard Models

Scenario | Recommended Approach
--- | ---
Few zeros, zeros are measurement artifacts | `fit_demand_fixed()` or `fit_demand_mixed()`
Many zeros, zeros represent true non-consumption | `fit_demand_hurdle()`
Need to understand factors affecting whether someone consumes | `fit_demand_hurdle()`
Need individual-level estimates of “quitting price” | `fit_demand_hurdle()`

## Model Specification

The hurdle model has two parts:

### Part I: Binary Model (Probability of Zero Consumption)

\[\text{logit}(\pi_{ij}) = \beta_0 + \beta_1 \cdot \log(\text{price} + \epsilon) + a_i\]

Where:

- \(\pi_{ij}\) = probability of zero consumption for subject \(i\) at price \(j\)
- \(\beta_0\) = intercept (baseline log-odds of zero consumption)
- \(\beta_1\) = slope (effect of log-price on log-odds of zero)
- \(\epsilon\) = small constant (default 0.001) for handling zero prices
- \(a_i\) = subject-specific random intercept

### Part II: Continuous Model (Consumption Given Positive)

With 3 random effects: \[\log(Q_{ij}) = (\log Q_0 + b_i) + k \cdot (\exp(-(\alpha + c_i) \cdot \text{price}) - 1) + \varepsilon_{ij}\]

With 2 random effects: \[\log(Q_{ij}) = (\log Q_0 + b_i) + k \cdot (\exp(-\alpha \cdot \text{price}) - 1) + \varepsilon_{ij}\]

Where:

- \(Q_0\) = intensity (consumption at price 0)
- \(k\) = scaling parameter for exponential decay
- \(\alpha\) = elasticity parameter
- \(b_i\) = subject-specific random effect on intensity
- \(c_i\) = subject-specific random effect on elasticity (3-RE model only)

### Random Effects Structure

The random effects follow a multivariate normal distribution:

- 2-RE model: \((a_i, b_i) \sim \text{MVN}(0, \Sigma_{2 \times 2})\)
- 3-RE model: \((a_i, b_i, c_i) \sim \text{MVN}(0, \Sigma_{3 \times 3})\)

## Getting Started

### Data Requirements

Your data should be in long format with columns for:

- Subject identifier
- Price
- Consumption (including zeros)

    library(beezdemand)

    # Example data structure
    knitr::kable(
        head(apt, 10),
        caption = "Example APT data structure (first 10 rows)"
    )

Example APT data structure (first 10 rows)
id | x | y | group
--- | --- | --- | ---
19 | 0.0 | 10 | a
19 | 0.5 | 10 | a
19 | 1.0 | 10 | a
19 | 1.5 | 8 | a
19 | 2.0 | 8 | a
19 | 2.5 | 8 | a
19 | 3.0 | 7 | a
19 | 4.0 | 7 | a
19 | 5.0 | 7 | a
19 | 6.0 | 6 | a

### Basic Model Fitting

    # Fit 2-RE model (simpler, faster)
    fit2 <- fit_demand_hurdle(
        data = apt,
        y_var = "y",
        x_var = "x",
        id_var = "id",
        random_effects = c("zeros", "q0"), # 2 random effects
        verbose = 0
    )

    # Fit 3-RE model (more flexible)
    fit3 <- fit_demand_hurdle(
        data = apt,
        y_var = "y",
        x_var = "x",
        id_var = "id",
        random_effects = c("zeros", "q0", "alpha"), # 3 random effects
        verbose = 1
    )

### Interpreting Output

    # View summary
    summary(fit2)
    #>
    #> Two-Part Mixed Effects Hurdle Demand Model
    #> ============================================
    #>
    #> Call:
    #> fit_demand_hurdle(data = apt, y_var = "y", x_var = "x", id_var = "id",
    #>     random_effects = c("zeros", "q0"), verbose = 0)
    #>
    #> Convergence: Yes
    #> Number of subjects: 10
    #> Number of observations: 160
    #> Random effects: 2 (zeros, q0)
    #>
    #> Fixed Effects:
    #> --------------
    #>              Estimate Std. Error t value
    #> beta0      -392.08454  146.77234  -2.671
    #> beta1       135.79413   50.43547   2.692
    #> log_q0        1.93813    0.11712  16.548
    #> log_k         0.55505    0.09013   6.158
    #> log_alpha    -2.31478    0.16822 -13.760
    #> logsigma_a    5.76791    1.48290   3.890
    #> logsigma_b   -1.04114    0.22837  -4.559
    #> logsigma_e   -1.63184    0.06063 -26.914
    #> rho_ab_raw   -0.19191    0.23755  -0.808
    #>
    #> Variance Components:
    #> --------------------
    #>           Estimate  Std. Error
    #> alpha       0.0988      0.0166
    #> k           1.7420      0.1570
    #> var_a  102316.8751 303451.6066
    #> var_b       0.1246      0.0569
    #> cov_ab    -21.4107     12.6867
    #> var_e       0.0382      0.0046
    #>
    #> Correlations:
    #> -------------
    #>        Estimate Std. Error
    #> rho_ab  -0.1896      0.229
    #>
    #> Model Fit:
    #> ----------
    #>   Log-likelihood: 2.31
    #>   AIC: 13.38
    #>   BIC: 41.06
    #>
    #> Demand Metrics (Group-Level):
    #> -----------------------------
    #>   Pmax (price at max expenditure): 20.0000
    #>   Omax (max expenditure): 30.9810
    #>   Q at Pmax: 1.5491
    #>   Elasticity at Pmax: -0.4772
    #>   Method: numerical_optimize_observed_domain
    #>
    #> Derived Parameters (Individual-Level Summary):
    #> ----------------------------------------------
    #>   Q0 (Intensity):
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>   3.622   5.496   7.585   7.365   8.544  11.623
    #>   Alpha:
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #> 0.09879 0.09879 0.09879 0.09879 0.09879 0.09879
    #>   Breakpoint:
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>   7.591  13.549  16.183  17.986  21.796  34.419
    #>   Pmax:
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>      20      20      20      20      20      20
    #>   Omax:
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>   16.15   24.51   33.83   32.85   38.11   51.85

    # Extract coefficients
    coef(fit2)
    #>         beta0         beta1    logsigma_a    logsigma_b    logsigma_e
    #> -392.08454145  135.79413071    5.76791495   -1.04113637   -1.63183778
    #>    rho_ab_raw            Q0         alpha             k
    #>   -0.19191226    6.94572417    0.09878808    1.74202557

    # Standardized tidy summaries
    tidy(fit2) |> head()
    #> # A tibble: 6 × 9
    #>   term        estimate std.error statistic  p.value component     estimate_scale
    #>   <chr>          <dbl>     <dbl>     <dbl>    <dbl> <chr>         <chr>
    #> 1 beta0      -392.      147.         -2.67 7.55e- 3 zero_probabi… logit
    #> 2 beta1       136.       50.4         2.69 7.09e- 3 zero_probabi… logit
    #> 3 Q0            6.95      0.813       8.54 1.36e-17 consumption   natural
    #> 4 k             1.74      0.157      11.1  1.32e-28 consumption   natural
    #> 5 alpha         0.0988    0.0166      5.94 2.77e- 9 consumption   natural
    #> 6 logsigma_a    5.77      1.48        3.89 1.00e- 4 variance      natural
    #> # ℹ 2 more variables: term_display <chr>, estimate_internal <dbl>
    glance(fit2)
    #> # A tibble: 1 × 9
    #>   model_class   backend  nobs n_subjects n_random_effects converged logLik   AIC
    #>   <chr>         <chr>   <int>      <int>            <int> <lgl>      <dbl> <dbl>
    #> 1 beezdemand_h… TMB       160         10                2 TRUE        2.31  13.4
    #> # ℹ 1 more variable: BIC <dbl>

    # Get subject-specific parameters
    head(get_subject_pars(fit2))
    #>    id       a_i        b_i        Q0      alpha breakpoint Pmax     Omax
    #> 1  19 -88.44338  0.5148913 11.623368 0.09878808  34.419426   20 51.84539
    #> 2  30 -21.33118 -0.6512299  3.621529 0.09878808  20.997058   20 16.15363
    #> 3  38  40.35752 -0.2348862  5.491712 0.09878808  13.330757   20 24.49548
    #> 4  60  31.73723  0.1663613  8.202899 0.09878808  14.204505   20 36.58857
    #> 5  68  31.19353  0.4228981 10.601806 0.09878808  14.261495   20 47.28877
    #> 6 106 116.82473 -0.2317221  5.509116 0.09878808   7.590564   20 24.57311

## Diagnostics

Use `check_demand_model()` and the plotting helpers for quick post-fit checks.

    check_demand_model(fit2)
    #>
    #> Model Diagnostics
    #> ==================================================
    #> Model class: beezdemand_hurdle
    #>
    #> Convergence:
    #>   Status: Converged
    #>
    #> Random Effects:
    #>
    #> Residuals:
    #>   Mean: 5.392e-05
    #>   SD: 0.1896
    #>   Range: [-0.6119, 0.4986]
    #>   Outliers: 2 observations
    #>
    #> --------------------------------------------------
    #> Issues Detected (1):
    #>   1. Detected 2 potential outliers (|resid| > 3)
    #>
    #> Recommendations:
    #>   - Investigate outlying observations
    plot_residuals(fit2)$fitted

    plot_qq(fit2)

## Understanding Results

### Fixed Effects

Parameter | Interpretation
--- | ---
`beta0` | Part I intercept: baseline log-odds of zero consumption
`beta1` | Part I slope: change in log-odds per unit increase in log(price)
`logQ0` | Log of intensity parameter (population average)
`k` | Scaling parameter for demand decay
`alpha` | Elasticity parameter (population average for 2-RE, mean for 3-RE)

### Subject-Specific Parameters

The `subject_pars` data frame contains:

Parameter | Description
--- | ---
`a_i` | Random effect for Part I (zeros probability)
`b_i` | Random effect for Part II (intensity)
`c_i` | Random effect for alpha (3-RE model only)
`Q0` | Individual intensity: \(\exp(\log Q_0 + b_i)\)
`alpha` | Individual elasticity: \(\alpha + c_i\) (or just \(\alpha\) for 2-RE)
`breakpoint` | Price where P(zero) = 0.5: \(\exp(-(\beta_0 + a_i) / \beta_1) - \epsilon\)
`Pmax` | Price at maximum expenditure
`Omax` | Maximum expenditure

## Model Selection: 2-RE vs 3-RE

### When to Use Each

- 2-RE model: When you believe elasticity (\(\alpha\)) is relatively constant across subjects
- 3-RE model: When you believe elasticity varies meaningfully between subjects

### Likelihood Ratio Test

    # Compare nested models
    compare_hurdle_models(fit3, fit2)

    # Unified model comparison (AIC/BIC + LRT when appropriate)
    compare_models(fit3, fit2)

    # Output:
    # Likelihood Ratio Test
    # =====================
    #            Model n_RE    LogLik df     AIC     BIC
    #    Full (3 RE)    3 -1234.56 12 2493.12 2543.21
    # Reduced (2 RE)    2 -1245.78  9 2509.56 2547.89
    #
    # LR statistic: 22.44
    # df: 3
    # p-value: 5.2e-05

A significant p-value suggests the 3-RE model provides a better fit.

## Visualization

    # Population demand curve
    plot(fit2, type = "demand")

Population demand curve from 2-RE hurdle model.

    # Probability of zero consumption
    plot(fit2, type = "probability")

Probability of zero consumption as a function of price.

    # Distribution of individual parameters
    plot(fit2, type = "parameters")
    plot(fit2, type = "parameters", parameters = c("Q0", "alpha", "Pmax"))

    # Individual demand curves
    plot(fit2, type = "individual", subjects = c("1", "2", "3", "4"))

    # Single subject with observed data
    plot_subject(fit2, subject_id = "1", show_data = TRUE, show_population = TRUE)

## Simulation and Validation

### Simulating Data

The `simulate_hurdle_data()` function generates data from the hurdle model:

    # Simulate with default parameters
    sim_data <- simulate_hurdle_data(
        n_subjects = 100,
        seed = 123
    )

    head(sim_data)
    #   id    x         y delta       a_i        b_i
    # 1  1 0.00 12.345678     0 -0.523456  0.1234567
    # 2  1 0.50 11.234567     0 -0.523456  0.1234567
    # ...

    # Custom parameters
    sim_custom <- simulate_hurdle_data(
        n_subjects = 100,
        logQ0 = log(15), # Q0 = 15
        alpha = 0.1, # Lower elasticity
        k = 3, # Higher k (ensures Pmax exists)
        stop_at_zero = FALSE, # All prices for all subjects
        seed = 456
    )

### Monte Carlo Simulation Studies

The `run_hurdle_monte_carlo()` function assesses model performance through simulation.

Note: Monte Carlo simulations are computationally intensive and not run during vignette building. The example below shows typical usage and expected output format.

    # Run Monte Carlo study
    mc_results <- run_hurdle_monte_carlo(
        n_sim = 100, # Number of simulations
        n_subjects = 100, # Subjects per simulation
        n_random_effects = 2, # 2-RE model
        verbose = TRUE,
        seed = 123
    )

    # View summary
    print_mc_summary(mc_results)

    # Monte Carlo Simulation Summary
    # ==============================
    #
    # Simulations: 100 attempted, 95 converged (95.0%)
    #
    #    Parameter True Mean_Est   Bias Rel_Bias% Emp_SE Mean_SE SE_Ratio Coverage_95%  N
    #        beta0 -2.0    -2.01  -0.01      0.5   0.12    0.11     0.92         94.7 95
    #        beta1  1.0     1.02   0.02      2.0   0.08    0.08     1.00         95.8 95
    #        logQ0  2.3     2.29  -0.01     -0.4   0.05    0.05     1.00         94.7 95
    #            k  2.0     2.03   0.03      1.5   0.15    0.14     0.93         93.7 95
    #        alpha  0.5     0.51   0.01      2.0   0.04    0.04     1.00         95.8 95
    # ...

### Interpreting Monte Carlo Results

Metric | Target | Interpretation
--- | --- | ---
Bias | ~0 | Estimates should be unbiased
Relative Bias | < 5% | Acceptable bias relative to true value
SE Ratio | ~1.0 | Model SEs match empirical variability
Coverage 95% | ~95% | CIs should contain true value 95% of time

## Integration with beezdemand Workflow

### Combining with Other Analyses

    # Fit hurdle model
    hurdle_fit <- fit_demand_hurdle(data,
        y_var = "y", x_var = "x", id_var = "id",
        random_effects = c("zeros", "q0"), verbose = 0
    )

    # Extract subject parameters
    hurdle_pars <- get_subject_pars(hurdle_fit)

    # These can be merged with other analyses
    # e.g., correlate with individual characteristics
    cor(hurdle_pars$Q0, subject_characteristics$age)
    cor(hurdle_pars$breakpoint, subject_characteristics$dependence_score)

### Exporting Results

    # Subject parameters
    write.csv(get_subject_pars(hurdle_fit), "hurdle_subject_parameters.csv")

    # Summary statistics
    summ <- get_hurdle_param_summary(hurdle_fit)
    print(summ)

## Technical Details

### TMB Backend

The hurdle model is implemented using Template Model Builder (TMB), which provides:

- Efficient Laplace approximation for marginal likelihood
- Automatic differentiation for fast optimization
- Standard errors via the delta method

### Control Parameters

    fit <- fit_demand_hurdle(
        data,
        y_var = "y",
        x_var = "x",
        id_var = "id",
        epsilon = 0.001, # Constant for log(price + epsilon)
        tmb_control = list(
            max_iter = 200, # Maximum iterations
            eval_max = 1000, # Maximum function evaluations
            trace = 0 # Optimization trace level
        ),
        verbose = 1 # 0 = silent, 1 = progress, 2 = detailed
    )

### Custom Starting Values

For difficult optimization problems:

    custom_starts <- list(
        beta0 = -3.0,
        beta1 = 1.5,
        logQ0 = log(8),
        k = 2.5,
        alpha = 0.1,
        logsigma_a = 0.5,
        logsigma_b = -0.5,
        logsigma_e = -1.0,
        rho_ab_raw = 0
    )

    fit <- fit_demand_hurdle(data,
        y_var = "y", x_var = "x", id_var = "id",
        start_values = custom_starts, verbose = 1
    )

## References

Zhao, T., Luo, X., Chu, H., Le, C.T., Epstein, L.H. and Thomas, J.L. (2016), A two-part mixed effects model for cigarette purchase task data. Jrnl Exper Analysis Behavior, 106: 242-253. https://doi.org/10.1002/jeab.228

Hursh, S. R., & Silberberg, A. (2008). Economic demand and essential value. Psychological Review, 115(1), 186-198.

## See Also

- `vignette("beezdemand")` – Getting started with beezdemand
- `vignette("model-selection")` – Choosing the right model class
- `vignette("fixed-demand")` – Fixed-effect demand modeling
- `vignette("mixed-demand")` – Mixed-effects nonlinear demand models
- `vignette("mixed-demand-advanced")` – Advanced mixed-effects topics
- `vignette("cross-price-models")` – Cross-price demand analysis
- `vignette("group-comparisons")` – Group comparisons
- `vignette("migration-guide")` – Migrating from `FitCurves()`