# Simulate Data for Latent Class Analysis Generates synthetic multivariate categorical data from a latent class model with `L` latent classes. Each observed variable follows a multinomial distribution within classes, with flexible control over class separation via the `IQ` parameter and class size distributions. ```r sim.LCA( N = 1000, I = 10, L = 3, poly.value = 5, IQ = "random", distribution = "random", params = NULL, is.sort = TRUE ) ``` ## Arguments - `N`: Integer; total number of observations to simulate. Must be $> L$. Default: `1000`. - `I`: Integer; number of categorical observed variables. Must be $\geq 1$. Default: `10`. - `L`: Integer; number of latent classes. Must be $\geq 2$ when `IQ` is numeric. Default: `3`. - `poly.value`: Integer or integer vector; number of categories (levels) for each observed variable. If scalar, all variables share the same number of categories. If vector, must have length `I`. Minimum valid value is `2` when `IQ` is numeric. Default: `5`. - `IQ`: Character or numeric; controls category probability distributions: - **`"random"`**: (default) Dirichlet-distributed probabilities ($\alpha=3$). - **`Numeric`**: in $(0.5, 1)$. Forces high discriminative power (see details in section below). - `distribution`: Character; distribution of class sizes. Options: `"random"` (default) or `"uniform"`. - `params`: List with fixed parameters for simulation: - **`par`**: $L \times I \times K_{\max}$ array of conditional response probabilities per latent class. - **`P.Z`**: Vector of length $L$ with latent class prior probabilities. - **`Z`**: Vector of length $N$ containing the latent classes of observations. A fixed observation classes `Z` is applied directly to simulate data only when `P.Z` is `NULL` and `Z` is a `N` length vector. - `is.sort`: A logical value. If `TRUE` (Default), the latent classes will be ordered in descending order according to `P.Z`. All other parameters will be adjusted accordingly based on the reordered latent classes. ## Returns A list containing: - **response**: Integer matrix ($N \times I$) of simulated observations. Rows are observations (named `"O1"`, `"O2"`, ...), columns are variables named `"I1"`, `"I2"`, ... Values range from `0` to `poly.value[i]-1`. - **par**: Array ($L \times I \times K$) of true class-specific category probabilities, where $K = \text{max}(poly.value)$ (i.e., the maximum number of categories across variables). Dimensions: classes x variables x categories. Note: For variables with `poly.value[i] < K`, unused category dimensions contain `NA`. Dimension names: `"L1"`, `"L2"`, ... (classes); `"I1"`, `"I2"`, ... (variables); `"poly0"`, `"poly1"`, ... (categories). - **Z**: Integer vector (length $N$) of true class assignments (1 to L). Named with observation IDs (e.g., `"O1"`). - **P.Z**: Numeric vector (length $L$) of true class proportions. Named with class labels (e.g., `"L1"`). - **poly.value**: Integer vector (length $I$) specifying number of categories per variable. - **P.Z.Xn**: Binary matrix ($N \times L$) of true class membership indicators (one-hot encoded). Row `i`, column `l` = 1 if observation `i` belongs to class `l`, else 0. Row/column names match `Z` and class labels. - **arguments**: A list containing all input arguments. ## Details Probability Generation: * Dirichlet Sampling (`IQ="random"`): For each variable-class combination, probabilities are drawn from $\text{Dirichlet}(\alpha_1=3, \dots, \alpha_k=3)$ where $k = \text{poly.value}[i]$. * High-Discrimination Mode (`IQ=numeric`): For each variable `i`: 1. Generate special probabilities `par.special` of length `L` containing: $IQ$, $1-IQ$, and $L-2$ values uniformly sampled from $[1-IQ, IQ]$. 2. For each class `l`, assign `par.special[l]` to one category, distribute remaining probability $1 - \text{par.special}[l]$ uniformly (via Dirichlet) across other categories. 3. Shuffle category assignments to avoid position bias. Data Generation: * Class assignments `Z` are generated first according to `distribution`. * For each observation `p` and variable `i`: 1. Retrieve cumulative probabilities for class `Z[p]` 2. Draw uniform random number $u \sim \text{Uniform}(0,1)$ 3. Assign category `k` where $P(\text{category} \leq k-1) \\< u \leq P(\text{category} \leq k)$ * Entire dataset is regenerated if any category of any variable has zero observations. Critical Constraints: * When `IQ` is numeric: $0.5 \< IQ \< 1$ and `min(poly.value) \>= 2` * `N` must be sufficiently large to observe all categories, especially when `IQ` is high or `poly.value` is large. Simulation may fail for small `N`. * For `distribution="uniform"`, empty classes possible when $N \< L$. ## Item Quality (IQ) Parameter Controls the discriminative power of observed variables: - **`IQ = "random"`**: (Default) Category probabilities for each variable-class combination are drawn from a symmetric Dirichlet distribution ($\alpha = 3$), resulting in moderate class separation. - **`IQ = numeric`**: (0.5 < IQ < 1) Forces high discriminative power for each variable: 1. For each variable, two categories per class are assigned extreme probabilities: one category gets probability $IQ$, another gets $1-IQ$. 2. Remaining categories share the residual probability $1 - IQ - (1-IQ) = 0$. **Note: This requires `poly.value` \>= 2 for all variables.** 3. Category assignments are randomized within classes to avoid structural patterns. Higher `IQ` values (closer to 1) yield stronger class separation but increase simulation failure risk. ## Class Size Distribution - **`"random"`**: (Default) Class proportions drawn from Dirichlet distribution ($\alpha = 3$ for all classes), ensuring no empty classes. Sizes are rounded to integers with adjustment for exact `N`. - **`"uniform"`**: Equal probability of class membership ($1/L$ per class), sampled with replacement. May produce empty classes if `N` is small relative to `L`. ## Response Validation The simulation enforces a critical constraint: **every category of every observed variable must appear at least once in the dataset**. If initial generation violates this (e.g., a rare category is missing), parameters and responses are regenerated until satisfied. This ensures compatibility with standard LCA estimation. ## Examples ```r # Example 1: Default settings (moderate separation, random class sizes) sim_data <- sim.LCA(N = 30, I = 5, L = 3) # Example 2: High-discrimination items (IQ=0.85), uniform class sizes sim_high_disc <- sim.LCA( N = 30, I = 4, L = 2, poly.value = c(3,4,3,5), # Variable category counts IQ = 0.85, distribution = "uniform" ) # Example 3: Binary items (poly.value=2) with high separation sim_binary <- sim.LCA(N = 300, I = 10, L = 2, poly.value = 2, IQ = 0.9) ```

sim.LCA function