### Build Transform Example

Source: https://sciminds.ucsd.edu/bossanova/api/maths/transforms

Example of creating a stateful transform instance by name, such as 'center'.

```python
>>> transform = build_transform("center")
>>> transform.fit_transform(np.array([1.0, 2.0, 3.0]))
array([-1.,  0.,  1.])
```

--------------------------------

### MathDisplay to_markdown Examples

Source: https://sciminds.ucsd.edu/bossanova/api/containers/display

Shows how to export equations as Quarto-compatible display math markdown, with and without explanations. The first example writes to a file, while the second returns the markdown string.

```python
>>> m.show_math().to_markdown("equations/model_eq.md")
```

```python
>>> md = m.show_math().to_markdown(include_explanations=False)
```

--------------------------------

### Install and Verify Project Dependencies

Source: https://sciminds.ucsd.edu/bossanova/engineering/developer-guide

Use pixi to install project dependencies, run linters, execute unit tests, and perform R parity tests. Ensure R and the lme4 package are installed for parity tests.

```bash
# Clone and install
git clone https://github.com/sciminds/bossanova.git
cd bossanova
pixi install

# Verify everything works
pixi run lint          # ruff + ty check
pixi run test          # unit tests
pixi run parity        # R parity tests (requires R + lme4)

```

--------------------------------

### MathDisplay Examples

Source: https://sciminds.ucsd.edu/bossanova/api/containers/display

Demonstrates how to obtain the raw LaTeX string and how the display object renders in a Jupyter environment.

```python
>>> display = m.show_math()
>>> display.to_latex()   # raw LaTeX string
>>> display              # renders in Jupyter
```

--------------------------------

### Example: Parsing with Conditioning

Source: https://sciminds.ucsd.edu/bossanova/api/domain/marginal

Illustrates parsing an explore formula that includes conditioning variables.

```python
>>> parse_explore_formula("treatment ~ age@50")
ExploreFormulaSpec(
    focal_var='treatment',
    contrast_type=None,
    conditions=(Condition(var='age', at_values=(50.0,)))
)
```

--------------------------------

### Setup for Statistical Calculations

Source: https://sciminds.ucsd.edu/bossanova/maths/inference

Initializes numpy and scipy for numerical operations and sets print options for clarity.

```python
import numpy as np
from scipy import stats
np.random.seed(42)
np.set_printoptions(precision=4, suppress=True)
```

--------------------------------

### Example: Contrast Function Parsing

Source: https://sciminds.ucsd.edu/bossanova/api/domain/marginal

Shows how to parse an explore formula that includes a contrast function.

```python
>>> parse_explore_formula("pairwise(treatment)")
ExploreFormulaSpec(focal_var='treatment', contrast_type='pairwise', conditions=())
```

--------------------------------

### Example: Build Gaussian Family

Source: https://sciminds.ucsd.edu/bossanova/api/maths/family

Demonstrates creating a Gaussian family object using the build_family function with the default link.

```python
>>> fam = build_family("gaussian")
>>> fam.name
'gaussian'
```

--------------------------------

### Install Bossanova with Sparse Support

Source: https://sciminds.ucsd.edu/bossanova

Install bossanova with optional sparse support for faster mixed-effects models. This requires additional system libraries on macOS and Ubuntu/Debian.

```bash
# uv
uv add "bossanova[sparse]"

# pip
pip install "bossanova[sparse]"
```

--------------------------------

### Binomial Distribution Examples

Source: https://sciminds.ucsd.edu/bossanova/api/distributions

Create default Bernoulli and custom Binomial distributions, noting their mean.

```python
dist = binomial()              # Bernoulli(0.5)
dist = binomial(n=10, p=0.3)   # Binomial(10, 0.3), mean = 3
```

--------------------------------

### Example Distribution Class Implementing Protocol

Source: https://sciminds.ucsd.edu/bossanova/api/containers/specs

Demonstrates how to implement the Distribution protocol for generating random samples. Requires a sample method.

```python
>>> class Normal:
...     def __init__(self, mean: float = 0.0, sd: float = 1.0):
...         self.mean = mean
...         self.sd = sd
...     def sample(self, n: int, rng: np.random.Generator) -> np.ndarray:
...         return rng.normal(self.mean, self.sd, size=n)
...
>>> dist = Normal(0, 1)
>>> isinstance(dist, Distribution)
True
```

--------------------------------

### Setup for Mixed Model Analysis

Source: https://sciminds.ucsd.edu/bossanova/maths/marginal-conditional

Imports necessary libraries and sets random seed and print options for reproducible numerical analysis.

```python
import numpy as np
from scipy.special import expit
np.random.seed(42)
np.set_printoptions(precision=4, suppress=True)
```

--------------------------------

### Example: Simple Term Parsing

Source: https://sciminds.ucsd.edu/bossanova/api/domain/marginal

Demonstrates parsing a basic explore formula with a single term.

```python
>>> parse_explore_formula("treatment")
ExploreFormulaSpec(focal_var='treatment', contrast_type=None, conditions=())
```

--------------------------------

### Build Ablated Formula Examples

Source: https://sciminds.ucsd.edu/bossanova/api/domain/infer

Examples demonstrating how to build a formula with one source term ablated. This utility function handles main effects and interactions, and can optionally include random effects.

```python
>>> build_ablated_formula("y", ["x", "z", "x:z"], "x")
'y ~ z'
```

```python
>>> build_ablated_formula("y", ["x", "z", "x:z"], "x:z")
'y ~ x + z'
```

```python
>>> build_ablated_formula("y", ["x"], "x")
'y ~ 1'
```

```python
>>> build_ablated_formula("y", ["x", "z"], "x", ("(1|group)",))
'y ~ z + (1|group)'
```

--------------------------------

### Google-Style Docstring Example

Source: https://sciminds.ucsd.edu/bossanova/engineering/developer-guide

Implement Google-style docstrings for functions, including sections for arguments, returns, and other relevant information.

```python
def compute_emm(
    spec: ModelSpec,
    bundle: DataBundle,
    fit: FitState,
    parsed: ExploreFormula,
) -> MeeState:
    """Compute estimated marginal means.

    Args:
        spec: Model configuration (family, link, formula).
        bundle: Validated model data (X, y, factor levels).
        fit: Fitted model state (coefficients, vcov).
        parsed: Parsed explore formula (focal variable, conditions).

    Returns:
        MeeState with grid, estimates, and metadata.
    """

```

--------------------------------

### Example: Build Gamma Family with Log Link

Source: https://sciminds.ucsd.edu/bossanova/api/maths/family

Illustrates creating a Gamma family object and explicitly setting the 'log' link function.

```python
>>> fam = build_family("gamma", "log")
>>> fam.name
'gamma'
```

--------------------------------

### Install R Packages

Source: https://sciminds.ucsd.edu/bossanova/cheatsheet

Installs necessary R packages for statistical modeling, including mixed models, marginal means, robust standard errors, and power analysis.

```r
install.packages(c("lme4", "lmerTest", "emmeans", "sandwich",
                    "lmtest", "car", "boot", "broom", "simr"))
```

--------------------------------

### Example: Build Binomial Family with Probit Link

Source: https://sciminds.ucsd.edu/bossanova/api/maths/family

Shows how to create a Binomial family object specifying a 'probit' link function.

```python
>>> fam = build_family("binomial", "probit")
>>> fam.link_name
'probit'
```

--------------------------------

### Module Section Marker Example

Source: https://sciminds.ucsd.edu/bossanova/engineering/developer-guide

Use section markers for navigation in files over 300 lines to improve readability.

```python
# =============================================================================
# Core Functions
# =============================================================================

```

--------------------------------

### Distribution Algebra Examples

Source: https://sciminds.ucsd.edu/bossanova/api/maths/distributions

Demonstrates algebraic operations on distributions, including affine transforms, sums of random variables, and probability queries.

```python
d = 2 * normal(5, 1) + 3  # Affine transform → Normal(13, 2)
z = normal() + normal()   # Sum of RVs → Normal(0, √2)
p = normal() > 1.96       # Probability query → P(X > 1.96) = 0.025
```

--------------------------------

### Example: Gamma Family with Log Link

Source: https://sciminds.ucsd.edu/bossanova/api/maths/family

Shows how to create a Gamma family object explicitly using the 'log' link function.

```python
>>> # Gamma with log link
>>> fam = gamma("log")
>>> fam.link_name
'log'
```

--------------------------------

### ModelSpec Initialization Example

Source: https://sciminds.ucsd.edu/bossanova/api/containers/specs

Shows how to create a ModelSpec instance with a formula and response variable. The family defaults to 'gaussian' if not specified.

```python
>>> spec = ModelSpec(
...     formula="y ~ x",
...     response_var="y",
...     fixed_terms=("Intercept", "x"),
... )
>>> spec.family
'gaussian'
```

--------------------------------

### Example: Single Contrast Conversion

Source: https://sciminds.ucsd.edu/bossanova/api/domain/design

Demonstrates converting a single contrast vector (A vs average(B, C)) into a coding matrix.

```python
>>> # Single contrast: A vs average(B, C)
>>> array_to_coding_matrix([-1, 0.5, 0.5], n_levels=3)
array([[-0.81649658,  0.        ],
       [ 0.40824829, -0.70710678],
       [ 0.40824829,  0.70710678]])
```

--------------------------------

### Example Formula with Sum Coding and Custom Contrast

Source: https://sciminds.ucsd.edu/bossanova/quickstart

Demonstrates fitting models with different contrast coding schemes directly in the formula or via a contrasts dictionary.

```python
model("y ~ sum(group) + x", data)                    # sum coding
model("y ~ treatment(group, ref=B) + x", data)       # treatment with ref="B"
model("y ~ group + x", data, contrasts={\"group\": M}) # custom ndarray matrix
```

--------------------------------

### Explore with Automatic Transforms

Source: https://sciminds.ucsd.edu/bossanova/api/model

This example demonstrates how `explore` automatically handles transformations like centering. Raw variable names in the formula are resolved.

```python
m = model("y ~ center(x) + group", data).fit()
m.explore("group ~ x@[50]")  # auto-centers 50
```

--------------------------------

### Example: Gamma Family with Inverse Link

Source: https://sciminds.ucsd.edu/bossanova/api/maths/family

Demonstrates creating a Gamma family object with its canonical 'inverse' link function.

```python
>>> # Gamma with inverse link (canonical)
>>> fam = gamma()
>>> fam.name
'gamma'
>>> fam.link_name
'inverse'
```

--------------------------------

### Example Usage of solve_rtol

Source: https://sciminds.ucsd.edu/bossanova/api/maths/tolerances

Demonstrates how to use `solve_rtol` to obtain a relative tolerance and assert its correctness using `np.testing.assert_allclose`.

```python
cond = 1e6
rtol = solve_rtol(cond)
np.testing.assert_allclose(computed, expected, rtol=rtol)
```

--------------------------------

### Example: Multiple Contrasts Conversion

Source: https://sciminds.ucsd.edu/bossanova/api/domain/design

Demonstrates converting multiple contrast vectors (A vs B, and (A,B) vs C) into a coding matrix.

```python
>>> # Multiple contrasts: A vs B, and (A,B) vs C
>>> array_to_coding_matrix([[-1, 1, 0], [-0.5, -0.5, 1]], n_levels=3)
array([[-0.5       , -0.28867513],
       [ 0.5       , -0.28867513],
       [ 0.        ,  0.57735027]])
```

--------------------------------

### Example: Create Student-t Family

Source: https://sciminds.ucsd.edu/bossanova/api/maths/family

Demonstrates how to instantiate a Student-t family with a specified degrees of freedom (df). The 'name' attribute confirms the family type.

```python
>>> # Robust regression with df=10
>>> fam = tdist(df=10)
>>> fam.name
'tdist'
```

--------------------------------

### Load Data for One-Sample Tests

Source: https://sciminds.ucsd.edu/bossanova/linear-models/one-sample

Imports necessary libraries and loads the penguins dataset, dropping any rows with null values. This setup is common for both t-test and Wilcoxon signed-rank test examples.

```python
import numpy as np
import polars as pl
from bossanova.model import model
from bossanova import load_dataset

np.random.seed(42)

# Load penguins dataset
penguins = load_dataset("penguins").drop_nulls()
```

--------------------------------

### Unit Test Example

Source: https://sciminds.ucsd.edu/bossanova/engineering

An example of a unit test function name, indicating a test for the 'lmer' fit function returning parameters. Unit tests verify internal correctness.

```python
test_lmer_fit_returns_params()
```

--------------------------------

### get_varying_random_terms

Source: https://sciminds.ucsd.edu/bossanova/api/containers/builders

Get all random terms (Intercept + slope terms) for a VaryingSpec.

```APIDOC
## get_varying_random_terms

### Description
Get all random terms (Intercept + slope terms) for a VaryingSpec.

### Method
get_varying_random_terms

### Parameters
#### Required Parameters
- **spec** (VaryingSpec) - A VaryingSpec instance.

### Returns
- **tuple[str, ...]** - Tuple of random term names, starting with “Intercept”.
```

--------------------------------

### Instantiate Model with Different Contrast Codings

Source: https://sciminds.ucsd.edu/bossanova/api/model

Demonstrates how to instantiate the model using different contrast coding schemes directly within the formula or by providing custom contrast matrices.

```python
model("y ~ sum(group) + x", data)                    # sum coding
```

```python
model("y ~ treatment(group, ref=B) + x", data)       # treatment with ref="B"
```

```python
model("y ~ group + x", data, contrasts={"group": M}) # custom ndarray matrix
```

--------------------------------

### Get Samples

Source: https://sciminds.ucsd.edu/bossanova/api/maths/distributions

Retrieves or generates cached Monte Carlo samples from the distribution.

```APIDOC
## get_samples

### Description
Get or generate cached samples.

### Returns
- Type: `ndarray`
- Description: Sorted array of Monte Carlo samples from the distribution.
```

--------------------------------

### Instantiate FitState

Source: https://sciminds.ucsd.edu/bossanova/api/containers/fit

Demonstrates how to create a FitState object with various fitting results. Ensure all required attributes are provided.

```python
>>> import numpy as np
>>> from fit import FitState
>>> state = FitState(
...     coef=np.array([1.0, 2.0]),
...     vcov=np.eye(2),
...     fitted=np.array([1.0, 2.0, 3.0]),
...     residuals=np.array([0.1, -0.1, 0.0]),
...     leverage=np.array([0.3, 0.3, 0.4]),
...     df_resid=1.0,
...     loglik=-10.0,
... )
```

--------------------------------

### Get True Parameter Value

Source: https://sciminds.ucsd.edu/bossanova/api/domain/simulation

Retrieves the true value for a specified parameter.

```python
get_true_value(param: str | int) -> float
```

--------------------------------

### Bossanova Full Workflow Example

Source: https://sciminds.ucsd.edu/bossanova/cheatsheet

Demonstrates a typical Bossanova workflow: fitting a model, inferring parameters with confidence intervals and p-values, exploring effects, and handling robust standard errors or bootstrapping.

```python
# Full workflow — one chain
m = model("y ~ x * group", data).fit().infer()
m.params       # estimates with CIs and p-values
m.summary()    # R-style summary table

# Explore + infer
m.explore("pairwise(group)").infer()
m.effects      # pairwise contrasts with CIs

# Robust standard errors
m.fit().infer(errors="HC3")

# Bootstrap with parallel workers
m.fit().infer(how="boot", n_boot=5000, n_jobs=4)

# Mixed model with profile CIs on variance components
m = model("y ~ x + (1 | subj)", data).fit().infer(how="profile")
m.varying_spread   # σ², τ² with profile CIs
```

--------------------------------

### sum_coding_labels

Source: https://sciminds.ucsd.edu/bossanova/api/domain/design

Gets column labels for sum contrast. Labels correspond to the non-omitted levels.

```APIDOC
## sum_coding_labels

### Description
Get column labels for sum contrast. This function returns labels for the non-omitted levels, which serve as column labels for the contrast matrix.

### Parameters
- **levels** (list[str]) - Required - Ordered list of categorical level names.
- **omit** (str | None) - Optional - Level to omit. Defaults to last level.

### Returns
- **list[str]** - List of non-omitted level names (column labels).

```

--------------------------------

### Set up random seed and print options

Source: https://sciminds.ucsd.edu/bossanova/maths/diagnostics

Initializes numpy for random number generation and sets print options for numerical precision.

```python
import numpy as np
np.random.seed(42)
np.set_printoptions(precision=4, suppress=True)
```

--------------------------------

### Match a string in formula

Source: https://sciminds.ucsd.edu/bossanova/api/domain/formula

Checks if the formula matches the expected string. No setup required.

```python
match(expected: str) -> bool
```

--------------------------------

### Get Samples

Source: https://sciminds.ucsd.edu/bossanova/api/maths/distributions

Retrieves cached random samples from the distribution or generates new ones if not available.

```APIDOC
## get_samples()

### Description
Gets or generates cached samples from the distribution.

### Returns
- **ndarray**: A sorted array of Monte Carlo samples from the distribution.

```

--------------------------------

### Get Parameter Index

Source: https://sciminds.ucsd.edu/bossanova/api/domain/simulation

Retrieves the array index corresponding to a given parameter name or index.

```python
get_idx(param: str | int) -> int
```

--------------------------------

### tdist_initialize

Source: https://sciminds.ucsd.edu/bossanova/api/maths/family

Initialize μ for Student-t family. Uses y directly as the starting value (like Gaussian).

```APIDOC
## tdist_initialize

### Description
Initialize μ for Student-t family. Uses y directly as the starting value (like Gaussian).

### Parameters
#### Path Parameters
- **y** (ndarray) - Required - Response values (n,)
- **weights** (ndarray | None) - Optional - Prior weights (n,). Unused for t-distribution. Defaults to None.

### Returns
- **ndarray** - Initial mean values (n,)
```

--------------------------------

### setup_ax

Source: https://sciminds.ucsd.edu/bossanova/api/viz

Sets up matplotlib axes, creating a new figure if necessary.

```APIDOC
## setup_ax

### Description
Set up matplotlib axes, creating figure if needed.

### Parameters
#### Path Parameters
- **ax** (‘Axes | None’) - Required - Existing axes or None to create new figure.
- **figsize** (tuple[float, float] | None) - Optional - Figure size if creating new figure. Default is None.

### Returns
- **tuple[Figure, Axes]** - Tuple of (figure, axes).
```

--------------------------------

### Get Previous Token

Source: https://sciminds.ucsd.edu/bossanova/api/domain/formula

Returns the last token that was consumed by the parser. Useful for context-dependent parsing rules.

```python
previous() -> Token
```

--------------------------------

### sequential_coding_labels

Source: https://sciminds.ucsd.edu/bossanova/api/domain/design

Gets column labels for sequential contrast. Labels represent successive differences between adjacent levels.

```APIDOC
## sequential_coding_labels

### Description
Get column labels for sequential contrast. This function returns labels representing the successive differences between adjacent levels.

### Parameters
- **levels** (list[str]) - Required - Ordered list of categorical level names.

### Returns
- **list[str]** - List of successive difference labels like [‘B-A’, ‘C-B’, ...].

### Examples
```python
>>> sequential_coding_labels(['A', 'B', 'C'])
['B-A', 'C-B']
```

```python
>>> sequential_coding_labels(['low', 'medium', 'high'])
['medium-low', 'high-medium']
```
```

--------------------------------

### Get Display Digits

Source: https://sciminds.ucsd.edu/bossanova/api/maths/config

Retrieves the current number of significant figures for DataFrame display. Defaults to 4.

```python
from bossanova import get_display_digits
get_display_digits()
# 4
```

--------------------------------

### Example: Fitting a Simple Linear Model

Source: https://sciminds.ucsd.edu/bossanova/api/domain/fit

Demonstrates fitting a simple linear model using `fit_model`. Requires `ModelSpec` and `DataBundle` to be constructed first. The output includes convergence status and coefficients.

```python
>>> import numpy as np
>>> from containers import build_model_spec, DataBundle
>>> spec = build_model_spec(
...     formula="y ~ x",
...     response_var="y",
...     fixed_terms=["Intercept", "x"],
... )
>>> bundle = DataBundle(
...     X=np.array([[1.0, 1.0], [1.0, 2.0], [1.0, 3.0]]),
...     y=np.array([2.0, 4.0, 6.0]),
...     X_names=["Intercept", "x"],
...     y_name="y",
...     valid_mask=np.array([True, True, True]),
...     n_total=3,
... )
>>> state = fit_model(spec, bundle)
>>> state.converged
True
>>> state.coef  # [Intercept, x] = [0, 2]
array([0., 2.])
```

--------------------------------

### setup_ax

Source: https://sciminds.ucsd.edu/bossanova/api/viz

Sets up matplotlib axes, creating a figure if needed. This is useful for preparing a plotting area before rendering visualizations.

```APIDOC
## setup_ax

### Description
Set up matplotlib axes, creating figure if needed.

### Parameters
#### Path Parameters
- **ax** (Axes | None) - Required - Existing axes or None to create new figure.
- **figsize** (tuple[float, float] | None) - Optional - Figure size if creating new figure. Default: None

### Returns
- **tuple[Figure, Axes]** - Tuple of (figure, axes).
```

--------------------------------

### tdist_initialize

Source: https://sciminds.ucsd.edu/bossanova/api/maths/family

Initializes the mean \(\mu\) for the Student-t family, using the response values directly as the starting point.

```APIDOC
## tdist_initialize

### Description
Initialize \(\mu\) for Student-t family. Uses y directly as the starting value (like Gaussian).

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Parameters
- **y** (ndarray) - Required - Response values (n,)
- **weights** (ndarray | None) - Optional - Optional prior weights (n,). Unused for t-distribution. Default: None

### Request Example
None

### Response
#### Success Response (200)
- **ndarray** - Initial mean values (n,)

#### Response Example
None
```

--------------------------------

### Model Initialization and Usage

Source: https://sciminds.ucsd.edu/bossanova/api/model

Demonstrates how to initialize the BossaNova model with different formula structures and data inputs, including examples of sum coding and treatment coding with custom references.

```APIDOC
## model(formula, data, family, link, method, missing, contrasts)

### Description
Initializes and fits a unified statistical model. The model type is inferred from the formula structure and family parameter.

### Parameters
#### Path Parameters
- **formula** (str) - R/Patsy-style formula (e.g., `"y ~ x"`, `"y ~ x + (1| group)"`).
- **data** (DataFrame | str | None) - Input data. Accepts a Polars DataFrame, a file path (str to CSV/TSV/Parquet/JSON/NDJSON), or None for simulation-first workflows.
- **family** (str) - Response distribution: `"gaussian"` (default), `"binomial"`, `"poisson"`, `"gamma"`, or `"tdist"`.
- **link** (str | None) - Link function (`"identity"`, `"logit"`, `"log"`, etc.). None uses canonical link for family.
- **method** (str | None) - Estimation method: `"ols"`, `"ml"`, or `"reml"`. None auto-selects.
- **missing** (str) - Missing value handling: `"drop"` (default) or `"fail"`.
- **contrasts** (dict | None) - Custom contrast matrices for categorical predictors. Dict mapping column names to ndarray matrices of shape `(n_levels, n_levels - 1)`. For named contrasts, use formula syntax instead: `sum(x)`, `treatment(x, ref=B)`, etc.

### Request Example
```python
model("y ~ sum(group) + x", data)                    # sum coding
model("y ~ treatment(group, ref=B) + x", data)       # treatment with ref="B"
model("y ~ group + x", data, contrasts={"group": M}) # custom ndarray matrix
```

### Attributes
- **contrasts** (dict | None) - Custom contrast matrices mapping column names to ndarray matrices.
- **data** (DataFrame | str | None) - Input data as a Polars DataFrame, file path, or None for simulation.
- **designmat** (DataFrame) - Fixed-effects design matrix as a named DataFrame.
- **diagnostics** (DataFrame) - Model-level goodness-of-fit diagnostics.
- **effects** (DataFrame) - Marginal effects or estimated marginal means (EMMs) table.
- **family** (str) - Response distribution (`"gaussian"`, `"binomial"`, `"poisson"`, `"gamma"`, `"tdist"`).
- **formula** (str) - R-style model formula string (e.g., `"y ~ x + (1 | group)"`).
- **link** (str | None) - Link function (e.g., `"identity"`, `"logit"`, `"log"`). None uses canonical link.
- **metadata** (DataFrame) - Model metadata: observation counts, parameter count, group counts.
- **method** (str | None) - Estimation method (`"ols"`, `"ml"`, `"reml"`). None auto-selects.
- **missing** (str) - Missing value handling: `"drop"` (default) or `"fail"`.
- **params** (DataFrame) - Coefficient estimates table from the fitted model.
- **power_results** (DataFrame) - Power analysis results from `simulate(power=...)`.
- **predictions** (DataFrame) - Predictions table from the fitted model.
- **resamples** (DataFrame | None) - Raw resampled values from bootstrap or permutation inference.
- **simulations** (DataFrame) - Simulations table from data generation or response simulation.
- **varying_corr** (DataFrame) - Random effect correlations for mixed models.
```

--------------------------------

### Example: Sparse Cholesky factorization and solving

Source: https://sciminds.ucsd.edu/bossanova/api/maths/linalg

Demonstrates the usage of compute_sparse_cholesky to factor a sparse SPD matrix and then solve a linear system and compute the log-determinant using the resulting factorization object.

```python
>>> import scipy.sparse as sp
>>> A = sp.eye(100, format='csc') + sp.random(100, 100, density=0.1)
>>> A = A @ A.T  # Make SPD
>>> factor = compute_sparse_cholesky(A)
>>> x = factor.solve(b)
>>> logdet = factor.logdet()
```

--------------------------------

### tdist_loglik

Source: https://sciminds.ucsd.edu/bossanova/api/maths/family

Placeholder function for tdist log-likelihood. Use tdist(df=...) factory to get the proper function.

```APIDOC
## tdist_loglik

### Description
Placeholder - use tdist(df=...) factory to get proper function.

### Parameters
#### Path Parameters
- **y** (jnp.ndarray) - Required
- **mu** (jnp.ndarray) - Required

### Returns
- **jnp.ndarray**
```

--------------------------------

### Fit Model with Dispatcher

Source: https://sciminds.ucsd.edu/bossanova/api/domain/fit

Use `fit_model` as the main entry point for fitting models. It automatically selects the appropriate solver (e.g., QR, IRLS, PLS, PIRLS) based on the `ModelSpec`. Handles rank-deficient design matrices by reducing and expanding coefficients.

```python
fit_model(spec: ModelSpec, bundle: DataBundle, *, solver: str | None = None, max_iter: int | None = None, max_outer_iter: int = 10000, tol: float | None = None, verbose: bool = False, nAGQ: int = 1, use_hessian: bool = False) -> FitState
```

```python
>>> import numpy as np
>>> from containers import build_model_spec, DataBundle
>>> spec = build_model_spec(
...     formula="y ~ x",
...     response_var="y",
...     fixed_terms=["Intercept", "x"],
... )
>>> bundle = DataBundle(
...     X=np.array([[1.0, 1.0], [1.0, 2.0], [1.0, 3.0]]),
...     y=np.array([2.0, 4.0, 6.0]),
...     X_names=["Intercept", "x"],
...     y_name="y",
...     valid_mask=np.array([True, True, True]),
...     n_total=3,
... )
>>> state = fit_model(spec, bundle)
>>> state.converged
True
>>> state.coef  # [Intercept, x] = [0, 2]
array([0., 2.])
```

--------------------------------

### tdist_deviance

Source: https://sciminds.ucsd.edu/bossanova/api/maths/family

Placeholder function for tdist deviance. Use tdist(df=...) factory to get the proper function.

```APIDOC
## tdist_deviance

### Description
Placeholder - use tdist(df=...) factory to get proper function.

### Parameters
#### Path Parameters
- **y** (jnp.ndarray) - Required
- **mu** (jnp.ndarray) - Required

### Returns
- **jnp.ndarray**
```

--------------------------------

### setup_ax

Source: https://sciminds.ucsd.edu/bossanova/api/viz

Set up matplotlib axes, creating the figure if needed. This function initializes axes for plotting.

```APIDOC
## setup_ax

### Description
Set up matplotlib axes, creating figure if needed.

### Method
setup_ax

### Parameters
No parameters are explicitly documented for this function.
```

--------------------------------

### Get Sum Coding Labels

Source: https://sciminds.ucsd.edu/bossanova/api/domain/design

Retrieves column labels for the sum contrast matrix, which correspond to the non-omitted levels.

```python
sum_coding_labels(levels: list[str], omit: str | None = None) -> list[str]
```

--------------------------------

### Get Singular Tolerance

Source: https://sciminds.ucsd.edu/bossanova/api/maths/config

Retrieves the current threshold for determining if a mixed model fit is singular. Defaults to 0.0001.

```python
from bossanova import get_singular_tolerance
get_singular_tolerance()
# 0.0001
```

--------------------------------

### Create and Inspect DataBundle

Source: https://sciminds.ucsd.edu/bossanova/api/containers/data

Instantiate a DataBundle with observation data and inspect its properties like the number of observations (n) and parameters (p).

```python
>>> import numpy as np
>>> from data import DataBundle
>>> bundle = DataBundle(
...     X=np.array([[1, 0], [1, 1], [1, 2]]),
...     y=np.array([1.0, 2.0, 3.0]),
...     X_names=["Intercept", "x"],
...     y_name="y",
...     valid_mask=np.array([True, True, True]),
...     n_total=3,
... )
>>> bundle.n
3
>>> bundle.p
2
```

--------------------------------

### arange Function Signature

Source: https://sciminds.ucsd.edu/bossanova/api/maths/backend

Creates an array with evenly spaced values. Accepts start, stop, and step parameters.

```python
arange(start: int, stop: int | None = None, step: int = 1) -> Array
```

--------------------------------

### Model Inference with CV

Source: https://sciminds.ucsd.edu/bossanova/api/domain/infer

Example of fitting a model and then inferring cross-validation diagnostics, including parameter importance and CV metrics.

```python
m = model("y ~ x + z", data).fit().infer(how="cv")
>>> m.params  # Includes pre, pre_sd columns
>>> m.diagnostics  # Includes cv_rmse, cv_mae, cv_rsquared
```

--------------------------------

### Initialize Linear Model (LM) in Python

Source: https://sciminds.ucsd.edu/bossanova/cheatsheet

Initializes a linear model using the bossanova library. Requires a formula string and a data source.

```python
from bossanova import model
m = model("y ~ x1 + x2", data)
```

--------------------------------

### Create and Use Normal Distribution

Source: https://sciminds.ucsd.edu/bossanova/api/maths/distributions

Demonstrates creating a normal distribution using a factory function and calculating its CDF and PPF values. Ensure the 'distributions' module is imported.

```python
from distributions import normal, t_dist
d = normal(mean=0, sd=1)
d.cdf(1.96)  # 0.975
d.ppf(0.975)  # 1.96
```

--------------------------------

### Get Distribution Samples

Source: https://sciminds.ucsd.edu/bossanova/api/maths/distributions

Retrieves or generates cached samples from the distribution. These samples can be used for various statistical analyses and simulations.

```python
get_samples() -> np.ndarray
```

--------------------------------

### Example: Asserting Algorithm Comparison with Absolute Tolerance

Source: https://sciminds.ucsd.edu/bossanova/api/maths/tolerances

Compares coefficients from QR and SVD solves using an absolute tolerance derived from `algorithm_comparison_atol`. This ensures that differences due to distinct rounding patterns are accounted for.

```python
result_qr = qr_solve(X, y)
result_svd = svd_solve(X, y)
cond = np.linalg.cond(X)
scale = max(np.abs(result_qr.coef).max(), 1.0)
atol = algorithm_comparison_atol(scale, cond)
np.testing.assert_allclose(result_qr.coef, result_svd.coef, atol=atol)
```

--------------------------------

### tdist_robust_weights

Source: https://sciminds.ucsd.edu/bossanova/api/maths/family

Placeholder function for tdist robust weights. Use tdist(df=...) factory to get the proper function.

```APIDOC
## tdist_robust_weights

### Description
Placeholder - use tdist(df=...) factory to get proper function.

### Parameters
#### Path Parameters
- **y** (jnp.ndarray) - Required
- **mu** (jnp.ndarray) - Required
- **scale** (float) - Required

### Returns
- **jnp.ndarray**
```

--------------------------------

### Get Sequential Coding Labels

Source: https://sciminds.ucsd.edu/bossanova/api/domain/design

Retrieves labels for sequential contrast matrix columns, representing successive differences between levels.

```python
sequential_coding_labels(levels: list[str]) -> list[str]
```

```python
>>> sequential_coding_labels(['A', 'B', 'C'])
['B-A', 'C-B']
```

```python
>>> sequential_coding_labels(['low', 'medium', 'high'])
['medium-low', 'high-medium']
```

--------------------------------

### Initialize ConvergenceMessage

Source: https://sciminds.ucsd.edu/bossanova/api/maths/convergence

Instantiate a ConvergenceMessage object with category, technical details, explanation, and an optional tip.

```python
ConvergenceMessage(category: str, technical: str, explanation: str, tip: str | None = None) -> None
```

--------------------------------

### Get Treatment Contrast Labels

Source: https://sciminds.ucsd.edu/bossanova/api/domain/design

Obtains the column labels for a treatment contrast matrix. These labels represent the non-reference levels.

```python
>>> treatment_coding_labels(['A', 'B', 'C'])
['A', 'B']
```

--------------------------------

### ExploreParser Initialization

Source: https://sciminds.ucsd.edu/bossanova/api/domain/marginal

Initialize `ExploreParser` with a list of tokens and the original formula string for error reporting. This parser handles the recursive descent parsing of explore formula syntax.

```python
ExploreParser(tokens: list[Token], formula: str) -> None
```

--------------------------------

### Get Sum Contrast Labels

Source: https://sciminds.ucsd.edu/bossanova/api/domain/design

Retrieves the column labels for a sum contrast matrix. These labels correspond to the non-omitted levels.

```python
>>> sum_coding_labels(['A', 'B', 'C'])
['A', 'B']
```

--------------------------------

### jit

Source: https://sciminds.ucsd.edu/bossanova/api/maths/backend

JIT-compile a function using JAX. For the NumPy backend, this is a no-op.

```APIDOC
## jit

### Description
JIT-compile a function using JAX. For the NumPy backend, this is a no-op.

### Signature
```python
jit(fn: Callable, *, donate_argnums: tuple[int, ...] | None = None) -> Callable
```
```

--------------------------------

### Get Model Type

Source: https://sciminds.ucsd.edu/bossanova/api/domain/compare

Retrieves the type string (e.g., 'lm', 'glm', 'lmer', 'glmer') for a fitted model instance.

```python
get_model_type(model: Any) -> str
```

--------------------------------

### Load Dataset and Initialize for Correlation Tests

Source: https://sciminds.ucsd.edu/bossanova/linear-models/correlation

This code block sets up the environment by importing necessary libraries and loading the 'penguins' dataset, dropping any rows with missing values, which is crucial for correlation analysis.

```python
import numpy as np
import polars as pl
from bossanova.model import model
from bossanova import load_dataset
np.random.seed(42)

# Load penguins dataset
penguins = load_dataset("penguins").drop_nulls()
```

--------------------------------

### Scanner Attributes

Source: https://sciminds.ucsd.edu/bossanova/api/domain/formula

Holds the input code, current scanning position, start of the current token, and the list of generated tokens.

```python
code = code
```

```python
current = 0
```

```python
start = 0
```

```python
tokens: list[Token] = []
```