# Simulating Choices

Choice simulation converts experimental designs into realistic choice data by predicting how respondents would answer choice questions. This is essential for testing designs, conducting power analyses, and validating experimental assumptions before data collection. This article shows how to use `cbc_choices()` to simulate choice patterns.

Before starting, let’s define some basic profiles and a basic random design to work with:

    library(cbcTools)

    profiles <- cbc_profiles(
      price = c(1, 1.5, 2, 2.5, 3),
      type = c('Fuji', 'Gala', 'Honeycrisp'),
      freshness = c('Poor', 'Average', 'Excellent')
    )

    design <- cbc_design(
      profiles = profiles,
      method = "random",
      n_alts = 2,
      n_q = 6,
      n_resp = 100
    )

    design
    #> Design method: random
    #> Structure: 100 respondents × 6 questions × 2 alternatives
    #> Profile usage: 45/45 (100.0%)
    #>
    #> 💡 Use cbc_inspect() for a more detailed summary
    #>
    #> First few rows of design:
    #>   profileID respID qID altID obsID price typeGala typeHoneycrisp
    #> 1        31      1   1     1     1   1.0        0              0
    #> 2        15      1   1     2     1   3.0        0              1
    #> 3        14      1   2     1     2   2.5        0              1
    #> 4         3      1   2     2     2   2.0        0              0
    #> 5        42      1   3     1     3   1.5        0              1
    #> 6        43      1   3     2     3   2.0        0              1
    #>   freshnessAverage freshnessExcellent
    #> 1                0                  1
    #> 2                0                  0
    #> 3                0                  0
    #> 4                0                  0
    #> 5                0                  1
    #> 6                0                  1
    #> ... and 1194 more rows

# Choice Simulation Approaches

`cbc_choices()` supports two simulation approaches:

1. Random simulation: Each alternative has equal probability of being chosen
2. Utility-based simulation: Choice probabilities based on multinomial logit model with specified priors

## Random Choices

Without priors, choices are simulated randomly with equal probabilities:

    # Random choice simulation (default)
    choices_random <- cbc_choices(design)

    head(choices_random)
    #> CBC Choice Data
    #> ===============
    #> Observations: 3 choice tasks
    #> Alternatives per task: 2
    #> Total choices made: 3
    #>
    #> Simulation method: random
    #> Priors: None (random choices)
    #> Simulated at: 2025-07-14 10:45:28
    #>
    #> Choice rates by alternative:
    #>   Alt 1: 33.3% (1 choices)
    #>   Alt 2: 66.7% (2 choices)
    #>
    #> First few rows:
    #>   profileID respID qID altID obsID price typeGala typeHoneycrisp
    #> 1        31      1   1     1     1   1.0        0              0
    #> 2        15      1   1     2     1   3.0        0              1
    #> 3        14      1   2     1     2   2.5        0              1
    #> 4         3      1   2     2     2   2.0        0              0
    #> 5        42      1   3     1     3   1.5        0              1
    #> 6        43      1   3     2     3   2.0        0              1
    #>   freshnessAverage freshnessExcellent choice
    #> 1                0                  1      0
    #> 2                0                  0      1
    #> 3                0                  0      0
    #> 4                0                  0      1
    #> 5                0                  1      1
    #> 6                0                  1      0

    # Check choice distribution
    table(choices_random$choice, choices_random$altID)
    #>
    #>       1   2
    #>   0 294 306
    #>   1 306 294

Random simulation is useful for:

- Quick testing of design structure
- Conservative power analysis (worst-case scenario)
- Baseline comparisons

## Utility-Based Choices

With priors, choices follow realistic utility-based patterns:

    # Create priors for utility-based simulation
    priors <- cbc_priors(
      profiles = profiles,
      price = -0.25, # Negative preference for higher prices
      type = c(0.5, 1), # Gala and Honeycrisp preferred over Fuji
      freshness = c(0.6, 1.2) # Average and Excellent preferred over Poor
    )

    # Utility-based choice simulation
    choices_utility <- cbc_choices(design, priors = priors)

    head(choices_utility)
    #> CBC Choice Data
    #> ===============
    #> Observations: 3 choice tasks
    #> Alternatives per task: 2
    #> Total choices made: 3
    #>
    #> Simulation method: utility_based
    #> Priors: Used for utility-based simulation
    #> Simulated at: 2025-07-14 10:45:28
    #>
    #> Choice rates by alternative:
    #>   Alt 1: 66.7% (2 choices)
    #>   Alt 2: 33.3% (1 choices)
    #>
    #> First few rows:
    #>   profileID respID qID altID obsID price typeGala typeHoneycrisp
    #> 1        31      1   1     1     1   1.0        0              0
    #> 2        15      1   1     2     1   3.0        0              1
    #> 3        14      1   2     1     2   2.5        0              1
    #> 4         3      1   2     2     2   2.0        0              0
    #> 5        42      1   3     1     3   1.5        0              1
    #> 6        43      1   3     2     3   2.0        0              1
    #>   freshnessAverage freshnessExcellent choice
    #> 1                0                  1      1
    #> 2                0                  0      0
    #> 3                0                  0      0
    #> 4                0                  0      1
    #> 5                0                  1      1
    #> 6                0                  1      0

## Choice Data Format

The simulated choice data includes all design columns plus a `choice` column:

    head(choices_utility)
    #> CBC Choice Data
    #> ===============
    #> Observations: 3 choice tasks
    #> Alternatives per task: 2
    #> Total choices made: 3
    #>
    #> Simulation method: utility_based
    #> Priors: Used for utility-based simulation
    #> Simulated at: 2025-07-14 10:45:28
    #>
    #> Choice rates by alternative:
    #>   Alt 1: 66.7% (2 choices)
    #>   Alt 2: 33.3% (1 choices)
    #>
    #> First few rows:
    #>   profileID respID qID altID obsID price typeGala typeHoneycrisp
    #> 1        31      1   1     1     1   1.0        0              0
    #> 2        15      1   1     2     1   3.0        0              1
    #> 3        14      1   2     1     2   2.5        0              1
    #> 4         3      1   2     2     2   2.0        0              0
    #> 5        42      1   3     1     3   1.5        0              1
    #> 6        43      1   3     2     3   2.0        0              1
    #>   freshnessAverage freshnessExcellent choice
    #> 1                0                  1      1
    #> 2                0                  0      0
    #> 3                0                  0      0
    #> 4                0                  0      1
    #> 5                0                  1      1
    #> 6                0                  1      0

# Advanced Simulation Options

## Designs with No-Choice

For designs with no-choice options, specify no-choice priors:

    # Create design with no-choice option
    design_nochoice <- cbc_design(
      profiles = profiles,
      n_alts = 2,
      n_q = 6,
      n_resp = 100,
      no_choice = TRUE,
      method = "random"
    )

    # Create priors including no-choice utility
    priors_nochoice <- cbc_priors(
      profiles = profiles,
      price = -0.25,
      type = c(0.5, 1.0),
      freshness = c(0.6, 1.2),
      no_choice = -0.5 # Negative = no-choice less attractive
    )

    # Simulate choices
    choices_nochoice <- cbc_choices(
      design_nochoice,
      priors = priors_nochoice
    )

    # Examine no-choice rates
    nochoice_rate <- mean(choices_nochoice$choice[choices_nochoice$no_choice == 1])
    cat("No-choice selection rate:", round(nochoice_rate * 100, 1), "%\n")
    #> No-choice selection rate: 13.3 %

## Random Parameters (Mixed Logit)

Simulate heterogeneous preferences using random parameters:

    # Create priors with random parameters
    priors_random <- cbc_priors(
      profiles = profiles,
      price = rand_spec(dist = "n", mean = -0.1, sd = 0.05),
      type = rand_spec(dist = "n", mean = c(0.1, 0.2), sd = c(0.05, 0.1)),
      freshness = c(0.1, 0.2), # Keep some parameters fixed
      n_draws = 100
    )

    # Simulate choices with preference heterogeneity
    choices_mixed <- cbc_choices(design, priors = priors_random)

## Interaction Effects

Include interaction effects in choice simulation:

    # Create priors with interactions
    priors_interactions <- cbc_priors(
      profiles = profiles,
      price = -0.1,
      type = c("Fuji" = 0.5, "Gala" = 1),
      freshness = c("Average" = 0.6, "Excellent" = 1.2),
      interactions = list(
        # Price sensitivity varies by apple type
        int_spec(
          between = c("price", "type"),
          with_level = "Fuji",
          value = 0.5
        ),
        int_spec(
          between = c("price", "type"),
          with_level = "Gala",
          value = 0.2
        )
      )
    )

    # Simulate choices with interaction effects
    choices_interactions <- cbc_choices(
      design,
      priors = priors_interactions
    )

# Validating Choice Patterns

## Overall Choice Frequencies

Based on the priors used, we expect:

- Lower prices preferred (negative price coefficient)
- Honeycrisp > Gala > Fuji (type coefficients: 0.2 > 0.1 > 0)
- Excellent > Average > Poor (freshness coefficients: 0.2 > 0.1 > 0)

Examine aggregate choice patterns to validate simulation:

    # Decode the choice data first to get categorical variables
    choices_decoded <- cbc_decode(choices_utility)

    # Aggregate attribute choices across all respondents
    choices <- choices_decoded

    # Price choices
    price_choices <- aggregate(choice ~ price, data = choices, sum)
    price_choices$prop <- price_choices$choice / sum(price_choices$choice)
    print(price_choices)
    #>   price choice      prop
    #> 1   1.0    123 0.2050000
    #> 2   1.5    119 0.1983333
    #> 3   2.0    121 0.2016667
    #> 4   2.5    107 0.1783333
    #> 5   3.0    130 0.2166667

    # Type choices
    type_choices <- aggregate(choice ~ type, data = choices, sum)
    type_choices$prop <- type_choices$choice / sum(type_choices$choice)
    print(type_choices)
    #>         type choice  prop
    #> 1       Fuji    171 0.285
    #> 2       Gala    213 0.355
    #> 3 Honeycrisp    216 0.360

    # Freshness choices
    freshness_choices <- aggregate(choice ~ freshness, data = choices, sum)
    freshness_choices$prop <- freshness_choices$choice /
      sum(freshness_choices$choice)
    print(freshness_choices)
    #>   freshness choice      prop
    #> 1      Poor    145 0.2416667
    #> 2   Average    210 0.3500000
    #> 3 Excellent    245 0.4083333

## Respondent Heterogeneity

For random parameter models, examine variation across respondents:

    # Create dataset with only chosen alternatives
    chosen_alts <- choices_mixed[choices_mixed$choice == 1, ]

    # Mean attribute levels chosen by each respondent
    resp_means <- aggregate(
      cbind(
        price,
        typeGala,
        typeHoneycrisp,
        freshnessAverage,
        freshnessExcellent
      ) ~
        respID,
      data = chosen_alts,
      mean
    )

    # Look at variation across respondents
    cat("Price variation across respondents:\n")
    #> Price variation across respondents:
    cat("Mean:", round(mean(resp_means$price), 2), "\n")
    #> Mean: 2.01
    cat("SD:", round(sd(resp_means$price), 2), "\n")
    #> SD: 0.28

    cat("\nHoneycrisp choice rate variation:\n")
    #>
    #> Honeycrisp choice rate variation:
    cat("Mean:", round(mean(resp_means$typeHoneycrisp), 2), "\n")
    #> Mean: 0.32
    cat("SD:", round(sd(resp_means$typeHoneycrisp), 2), "\n")
    #> SD: 0.2

# Design Consistency

## Using Consistent Priors

For D-optimal designs created with priors, use the same priors for choice simulation:

    # Create D-optimal design with priors
    design_optimal <- cbc_design(
      profiles = profiles,
      n_alts = 2,
      n_q = 6,
      n_resp = 100,
      priors = priors,
      method = "stochastic"
    )

    # Use SAME priors for choice simulation
    choices_consistent <- cbc_choices(
      design_optimal,
      priors = priors
    )

## Prior Consistency Warnings

cbcTools warns when different priors are used:

    # Create different priors
    different_priors <- cbc_priors(
      profiles = profiles,
      price = -0.2, # Different from design optimization
      type = c(0.2, 0.4),
      freshness = c(0.2, 0.4)
    )

    # This will generate a warning about inconsistent priors
    choices_inconsistent <- cbc_choices(
      design_optimal,
      priors = different_priors
    )

# Best Practices

## Prior Specification

- Use realistic priors: Base on literature, pilot studies, or expert judgment
- Match design priors: Use same priors for design optimization and choice simulation
- Test multiple scenarios: Simulate under optimistic and conservative assumptions
- Include heterogeneity: Use random parameters when appropriate

## Validation Steps

1. Check choice counts: Verify one choice per question
2. Examine patterns: Ensure choices align with prior expectations
3. Test extremes: Simulate with very strong/weak preferences
4. Compare methods: Test different simulation approaches

# Next Steps

After simulating choices:

1. Conduct power analysis using `cbc_power()` to determine sample size requirements
2. Compare designs by simulating choices for different design methods
3. Validate assumptions by checking if simulated patterns match expectations
4. Refine priors based on simulation results before data collection

For details on power analysis, see the Power Analysis vignette.