Expected Value for One-Inflated Zero-Truncated Negative Binomial Distribution

Description

Computes the expected value from a one-inflated, zero-truncated negative binomial (OIZTNB) distribution. This function calculates the expected value based on the model parameters and covariates.

Usage

E_negbin(b, g, a, X, Z)

Arguments

Value

A numeric vector of expected values for each observation in the dataset.

See Also

dEdq_nb for computing partial derivatives of expected values in the OIZTNB model.

Expected Value for Zero-Truncated Negative Binomial Distribution

Description

Computes the expected value from a zero-truncated negative binomial (ZTNB) distribution based on the model parameters and covariates.

Usage

E_negbin_noinfl(b, a, X)

Arguments

Value

A numeric vector of expected values for each observation in the dataset.

See Also

dEdq_nb_noinfl for computing partial derivatives of expected values in the ZTNB model.

Expected Value for One-Inflated Poisson Distribution

Description

Computes the expected value from a one-inflated Poisson (OIPP) distribution based on the model parameters and covariates.

Usage

E_pois(b, g, X, Z)

Arguments

Value

A numeric vector of expected values for each observation in the dataset.

See Also

dEdq_pois for computing partial derivatives of expected values in the OIPP model.

Expected Value for Positive Poisson Distribution

Description

Computes the expected value from a Poisson (PP) distribution based on the model parameters and covariates.

Usage

E_pois_noinfl(b, X)

Arguments

Value

A numeric vector of expected values for each observation in the dataset.

See Also

dEdq_pois_noinfl for computing partial derivatives of expected values in the Poisson model.

Compute Partial Derivatives of Expected Values for a One-inflated Zero-truncated Negative Binomial Model

Description

This internal function computes the partial derivatives of expected values for a one-inflated zero-truncated negative binomial regression model with respect to covariates. It also accounts for marginal effects of dummy variables.

Usage

dEdq_nb(b, g, a, X, Z, dummies, formula)

Arguments

Details

This function performs the following tasks: - Computes partial derivatives of expected values with respect to covariates in 'X' and 'Z'. - Handles marginal effects for dummy variables by comparing expected values when the dummy variable is set to 0 versus 1.

The function is designed for internal use in the package and assumes that all input matrices and vectors are correctly specified. Any unexpected input structure may result in errors.

Value

A matrix of partial derivatives (or marginal effects) with rows corresponding to observations and columns to covariates. For dummy variables, marginal effects are calculated directly.

See Also

E_negbin for expected values in the negative binomial model.

Compute Partial Derivatives of Expected Values for Zero-truncated Negative Binomial Model

Description

This internal function calculates the partial derivatives of expected values for a regular truncated negative binomial regression model with respect to covariates. It also adjusts for marginal effects of dummy variables when specified.

Usage

dEdq_nb_noinfl(b, a, X, dummies, formula)

Arguments

Details

This function performs the following tasks: - Computes partial derivatives of expected values with respect to covariates in 'X'. - Adjusts for marginal effects of dummy variables by altering their values in the design matrix and computing the difference in expected values.

It is designed for internal use and assumes correct input structure. Improper inputs may result in errors or unexpected behavior.

Value

A matrix of partial derivatives (or marginal effects) with rows corresponding to observations and columns to covariates. Marginal effects for dummy variables are calculated by contrasting expected values when the dummy variable is set to 0 versus 1.

See Also

E_negbin_noinfl for computing expected values in the regular truncated negative binomial model.

Compute Partial Derivatives of Expected Values for a One-Inflated Positive Poisson Model

Description

This internal function calculates the partial derivatives of expected values for a one-inflated Poisson regression model with respect to covariates. It also computes marginal effects for specified dummy variables.

Usage

dEdq_pois(b, g, X, Z, dummies, formula)

Arguments

Details

This function: - Computes partial derivatives of expected values with respect to covariates in 'X' and 'Z'. - Handles marginal effects for dummy variables by modifying their values in the design matrices and computing the difference in expected values.

It is designed for internal use and assumes correct input structure. Improper inputs may result in errors or unexpected behavior.

Value

A matrix of partial derivatives (or marginal effects) with rows corresponding to observations and columns to covariates. For dummy variables, marginal effects are calculated by contrasting expected values when the dummy is set to 0 versus 1.

See Also

E_pois for computing expected values in the one-inflated Poisson model.

Compute Partial Derivatives of Expected Values for a Positive Poisson Model

Description

This internal function calculates the partial derivatives of expected values for a positive Poisson regression model with respect to covariates. It also computes marginal effects for specified dummy variables.

Usage

dEdq_pois_noinfl(b, X, dummies, formula)

Arguments

Details

This function: - Computes partial derivatives of expected values with respect to covariates in 'X'. - Handles marginal effects for dummy variables by modifying their values in the design matrix and computing the difference in expected values.

It is designed for internal use and assumes correct input structure. Improper inputs may result in errors or unexpected behavior.

Value

A matrix of partial derivatives (or marginal effects) with rows corresponding to observations and columns to covariates. Marginal effects for dummy variables are calculated by contrasting expected values when the dummy is set to 0 versus 1.

See Also

E_pois_noinfl for computing expected values in the positive Poisson model.

Prepare Design Matrices and Response Vector

Description

Processes a model formula and a data frame to generate design matrices ('X' and 'Z') and a response vector ('y') for regression models, including support for complex formulas with '|' operators.

Usage

makeXZy(formula, df)

Arguments

Details

This function processes the formula to extract and construct: - 'X': The main design matrix. - 'Z': A secondary design matrix (if a '|' operator is used in the formula, separating components). - 'y': The response variable.

It handles cases where the formula specifies: - Only the main component (e.g., y ~ x1 + x2). - A secondary component using the '|' operator (e.g., y ~ x1 + x2 | z1 + z2).

Value

A list containing the following components:

X

A design matrix for the main predictors.

Z

A design matrix for additional predictors (e.g., for a secondary process in a two-component model).

y

The response vector extracted from the formula.

See Also

model.matrix, model.frame, model.response

Compute Marginal Effects for One-inflated models

Description

This wrapper function calls a different function to calculate marginal effects depending on the model type. The marginal effects of the variables are evaluated at specified points, such as the sample means or averages, or at custom-defined cases.

Usage

margins(model, df, at = "AE", verbose = TRUE)

Arguments

Details

The function computes marginal effects for zero-truncated Poisson or negative binomial regression models. It handles different model types; 'oneinflmodel' for one-inflated models, and 'truncmodel' for standard count models. The marginal effects are evaluated at either all data points and averaged ('AE', the default), at the sample means of the variables ('EM'), or at a custom case. The marginal effects for dummy variables are actually the differences in expected outcomes for values of the dummy of 1 and 0. The marginal effects are displayed along with their statistical significance, evaluated based on the chosen 'at' parameter.

Value

If verbose=TRUE (default), prints the marginal effects, their standard errors, z-values, p-values, and significance levels.

If verbose=FALSE, returns a list containing the following components:

where

A description of how the marginal effects have been evaluated.

dEdq

The marginal effect. The partial derivative of the expected count with respect to a variable q in the X and or Z matrix, or the difference in expectation if q is binary.

se

The standard errors of the marginal effects evaluated numerically and using a Jacobian via the delta method.

See Also

dEdq_nb, dEdq_nb_noinfl, dEdq_pois, dEdq_pois_noinfl, model.frame, model.matrix, numericDeriv

Examples

df <- data.frame(x = rnorm(100), z = rnorm(100), y = rpois(100, lambda = 5))
model <- oneinfl(y ~ x | z, df = df, dist = "Poisson")
margins(model, df, at = "AE") # Average Effect
margins(model, df, at = "EM", verbose=FALSE) # Effect at Means, suppress printing
margins(model, df, at = list(x = 1, z = 0)) # Custom case

Likelihood Ratio Test for Nested Models

Description

Performs a likelihood ratio test (LRT) to compare two nested models estimated by oneinfl or truncreg. It calculates the LRT statistic and its associated p-value, testing whether the more complex model provides a significantly better fit to the data than the simpler model.

Usage

oneLRT(mod0, mod1)

Arguments

Details

The function extracts the log-likelihoods and number of parameters from the two models. It then calculates the LRT statistic:

-2 \cdot (\ell_0 - \ell_1)

where \ell_0 and \ell_1 are the log-likelihoods of the simpler and more complex models, respectively. The degrees of freedom for the test are equal to the difference in the number of parameters between the models.

The likelihood ratio test is commonly used to test for: - Overdispersion: Comparing a Poisson model to a negative binomial model. - One-inflation: Comparing a one-inflated model to a non-one-inflated model.

Value

A list with the following components:

LRTstat

The likelihood ratio test statistic.

pval

The p-value associated with the test statistic, based on a chi-squared distribution.

See Also

oneinfl for fitting one-inflated models. truncreg for fitting zero-truncated models. pchisq for the chi-squared distribution.

Examples

# Example: One-inflation test
df <- data.frame(y = rpois(100, lambda = 5), x = rnorm(100), z = rnorm(100))
OIZTNB <- oneinfl(y ~ x | z, df = df, dist = "negbin")
ZTNB <- truncreg(y ~ x, df = df, dist = "negbin")
oneLRT(OIZTNB, ZTNB)

# Example: Overdispersion test
OIPP <- oneinfl(y ~ x | z, df = df, dist = "Poisson")
oneLRT(OIZTNB, OIPP)

Wald Test for One-Inflation

Description

Performs a Wald test to evaluate the significance of the one-inflation parameters in a model estimated using oneinfl.

Usage

oneWald(model)

Arguments

Details

The Wald test evaluates the null hypothesis that all one-inflation parameters (gamma) are equal to zero, indicating no one-inflation. The test statistic is calculated as:

W = \gamma^\top V^{-1} \gamma

where \gamma is the vector of one-inflation parameters and V is their variance-covariance matrix. The p-value is computed using a chi-squared distribution with degrees of freedom equal to the length of \gamma.

This test is commonly used to determine whether a one-inflated model provides a significantly better fit than a non-one-inflated counterpart.

Value

A list with the following components:

W

The Wald test statistic.

pval

The p-value associated with the test statistic, based on a chi-squared distribution.

See Also

oneinfl for fitting one-inflated models. oneLRT for a likelihood ratio test of nested models. pchisq for the chi-squared distribution.

Examples

# Example usage
df <- data.frame(y = rpois(100, lambda = 5), x = rnorm(100), z = rnorm(100))
OIZTNB <- oneinfl(y ~ x | z, df = df, dist = "negbin")
oneWald(OIZTNB)

One-Inflated Regression Model

Description

Fits a one-inflated positive Poisson (OIPP) or one-inflated zero-truncated negative binomial (OIZTNB) regression model.

Usage

oneinfl(formula, df, dist = "negbin", start = NULL, method = "BFGS")

Arguments

Details

This function fits a regression model for one-inflated counts. One-inflated models are used when there are an excess number of ones, relative to a Poisson or negative binomial process.

The function supports two distributions: - '"Poisson"': One-inflated Poisson regression. - '"negbin"': One-inflated negative binomial regression.

The function uses numerical optimization via optim to estimate the parameters.

Value

An object of class '"oneinflmodel"' containing the following components:

beta

Estimated coefficients for the rate component of the model.

gamma

Estimated coefficients for the one-inflation component of the model.

alpha

Dispersion parameter (only for negative binomial distribution).

vc

Variance-covariance matrix of the estimated parameters.

logl

Log-likelihood of the fitted model.

avgw

Average one-inflation probability.

absw

Mean absolute one-inflation probability.

dist

The distribution used for the model ("Poisson" or "negbin").

formula

The formula used for the model.

See Also

summary for summarizing the fitted model. margins for calculating the marginal effects of regressors. oneWald to test for no one-inflation. signifWald for testing the joint significance of a single regressor that appears before and after the pipe '|'. oneplot for plotting actual and predicted counts. predict for expected response/dependent variable at each observation. truncreg for fitting positive Poisson (PP) and zero-truncated negative binomial (ZTNB) models. oneLRT to test for no one-inflation or no overdispersion using a nested PP, OIPP, or ZTNB model.

Examples

# Example usage
df <- data.frame(x = rnorm(100), z = rnorm(100), y = rpois(100, lambda = 1) + 1)
model <- oneinfl(y ~ x | z, df = df, dist = "Poisson")
summary(model)
margins(model, df)
oneWald(model)
predict(model, df=df)

Plot Observed Data and Model Predictions

Description

Generates a bar plot of observed count data and overlays predicted values from one or more models fitted using oneinfl or truncreg.

Usage

oneplot(
  model1,
  model2 = NULL,
  model3 = NULL,
  model4 = NULL,
  df,
  maxpred = NULL,
  ylimit = NULL,
  ccex = 1.5
)

Arguments

Details

This function visualizes observed count data as a bar plot and overlays predicted values from up to four models. The function automatically detects the type of model (Poisson or negative binomial; one-inflated or truncated) and adjusts the plot accordingly. Predictions are generated using the pred function.

Model types are distinguished by different point and line styles:

• Poisson (PP): Dark magenta, triangle-down

• Zero-truncated negative binomial (ZTNB): Red, diamond

• One-inflated Poisson (OIPP): Green, triangle-up

• One-inflated zero-truncated negative binomial (OIZTNB): Blue, circle

The legend in the top-right corner of the plot indicates the models displayed.

Value

A plot is generated but no values are returned.

See Also

oneinfl for fitting one-inflated models. truncreg for fitting truncated models. pred for generating predictions used in the plot.

Examples

# Example usage
df <- data.frame(x = rnorm(100), z = rnorm(100), y = rpois(100, lambda = 5) + 1)
model1 <- oneinfl(y ~ x | z, df = df, dist = "Poisson")
model2 <- truncreg(y ~ x, df = df, dist = "Poisson")
oneplot(model1, model2, df = df, maxpred = 10)

Generate Predicted Frequencies for Models

Description

Computes the predicted frequencies for a specified model, given the observed data and maximum count value. This function supports models estimated using oneinfl or truncreg.

Usage

pred(model, df, maxpred)

Arguments

Details

The function computes predicted frequencies based on the type of model and its distribution:

• PP (Poisson, zero-truncated): Computes predictions for a zero-truncated Poisson model.

• ZTNB (Negative Binomial, zero-truncated): Computes predictions for a zero-truncated negative binomial model.

• OIPP (Poisson, one-inflated): Computes predictions for a one-inflated Poisson model.

• OIZTNB (Negative Binomial, one-inflated, zero-truncated): Computes predictions for a one-inflated, zero-truncated negative binomial model.

The predictions are generated for count values from 1 to maxpred. For one-inflated models, the predictions account for the one-inflation probabilities.

This is an internal function primarily used by oneplot for visualization purposes.

Value

A numeric vector of predicted frequencies for count values from 1 to maxpred.

See Also

oneplot for visualizing observed and predicted frequencies. oneinfl for fitting one-inflated models. truncreg for fitting zero-truncated models.

Predicted Expected Response for One-Inflated or Truncated Models

Description

Calculates the predicted expected response for a model fitted using oneinfl or truncreg.

Usage

## S3 method for class 'oneinflmodel'
predict(object, ...)

Arguments

Details

This function computes the expected response based on the fitted model. The computation differs depending on the distribution. For Poisson (OIPP), predicted values are computed using E_pois. For Negative Binomial (OIZTNB), predicted values are computed using E_negbin.

Value

A numeric vector of predicted expected responses for the observations in df.

See Also

oneinfl for fitting one-inflated models. E_pois, E_negbin, for the expected value calculations.

Examples

# Example usage
df <- data.frame(x = rnorm(100), z = rnorm(100), y = rpois(100, lambda = 5))
model <- oneinfl(y ~ x | z, df = df, dist = "Poisson")
predict(model, df = df)

Predicted Expected Response for One-Inflated or Truncated Models

Description

Calculates the predicted expected response for a model fitted using oneinfl or truncreg.

Usage

## S3 method for class 'truncmodel'
predict(object, ...)

Arguments

Details

This function computes the expected response based on the fitted model. The computation differs depending on the distribution. For Poisson (PP), predicted values are computed using E_pois_noinfl. For Negative Binomial (ZTNB), predicted values are computed using E_negbin_noinfl.

Value

A numeric vector of predicted expected responses for the observations in df.

See Also

oneinfl for fitting one-inflated models. truncreg for fitting truncated models. E_pois_noinfl, E_negbin_noinfl for the expected value calculations.

Examples

# Example usage
df <- data.frame(x = rnorm(100), y = rpois(100, lambda = 5))
model <- truncreg(y ~ x, df = df, dist = "Poisson")
predict(model, df = df)

Generate Random Counts from a One-Inflated Poisson Process

Description

Simulates count data from a one-inflated Poisson process using specified parameters for the rate and one-inflation components.

Usage

roipp(b, g, X, Z)

Arguments

Details

This function generates count data from a one-inflated Poisson process. The process combines:

• A Poisson distribution for counts greater than one.

• A one-inflation component that adjusts the probability of observing a count of one.

The algorithm:

1. Calculates the rate parameter (\lambda) as \exp(X \cdot b).

2. Computes the one-inflation probabilities (\omega) based on Z \cdot g.

3. Simulates counts for each observation:

   • Draws a random number to determine whether the count is one.

   • Iteratively calculates probabilities for higher counts until the random number is matched.

This function is useful for generating synthetic data for testing or simulation studies involving one-inflated Poisson models.

Value

A numeric vector of simulated count data.

See Also

oneinfl for fitting one-inflated models.

Examples

# Example usage
set.seed(123)
X <- matrix(rnorm(100), ncol = 2)
Z <- matrix(rnorm(100), ncol = 2)
b <- c(0.5, -0.2)
g <- c(1.0, 0.3)
simulated_data <- roipp(b, g, X, Z)
print(simulated_data)

Generate Random Counts from a One-Inflated Zero-Truncated Negative Binomial Process

Description

Simulates count data from a one-inflated, zero-truncated negative binomial (OIZTNB) process using specified parameters for the rate, one-inflation, and dispersion components.

Usage

roiztnb(b, g, alpha, X, Z)

Arguments

Details

This function generates count data from a one-inflated, zero-truncated negative binomial process. The process combines:

• A negative binomial distribution for counts greater than one.

• A one-inflation component that adjusts the probability of observing a count of one.

The algorithm:

1. Calculates the rate parameter (\lambda) as \exp(X \cdot b).

2. Computes the one-inflation probabilities (\omega) based on Z \cdot g.

3. Computes the negative binomial dispersion parameter (\theta = \lambda / \alpha).

4. Simulates counts for each observation:

   • Draws a random number to determine whether the count is one.

   • Iteratively calculates probabilities for higher counts until the random number is matched.

This function is useful for generating synthetic data for testing or simulation studies involving one-inflated, zero-truncated negative binomial models.

Value

A numeric vector of simulated count data.

See Also

oneinfl for fitting one-inflated models.

Examples

# Example usage
set.seed(123)
X <- matrix(rnorm(100), ncol = 2)
Z <- matrix(rnorm(100), ncol = 2)
b <- c(0.5, -0.2)
g <- c(1.0, 0.3)
alpha <- 1.5
simulated_data <- roiztnb(b, g, alpha, X, Z)
print(simulated_data)

Generate Random Counts from a Zero-Truncated Poisson Process

Description

Simulates count data from a zero-truncated Poisson process using specified parameters for the rate component.

Usage

rpp(b, X)

Arguments

Details

This function generates count data from a zero-truncated Poisson process, which models count data without zeros. The process involves:

• Calculating the rate parameter (\lambda) as \exp(X \cdot b).

• Iteratively computing probabilities for counts starting from 1 and adding to the cumulative probability until a randomly drawn value is matched.

This function is useful for generating synthetic data for testing or simulation studies involving zero-truncated Poisson models.

Value

A numeric vector of simulated count data.

See Also

truncreg for fitting zero-truncated models.

Examples

# Example usage
set.seed(123)
X <- matrix(rnorm(100), ncol = 2)
b <- c(0.5, -0.2)
simulated_data <- rpp(b, X)
print(simulated_data)

Wald Test for Significance of a Predictor Variable

Description

Performs a Wald test to evaluate the joint significance of a predictor variable in both the rate and one-inflation components of a model.

Usage

signifWald(model, varname)

Arguments

Details

This function tests the null hypothesis that the coefficients for the specified predictor variable are jointly equal to zero in both the rate (beta) and one-inflation (gamma) components of the model. The test statistic is calculated as:

W = \mathbf{c}^\top V^{-1} \mathbf{c}

where \mathbf{c} is the vector of coefficients for the predictor in the rate and one-inflation components, and V is their variance-covariance matrix. The p-value is computed using a chi-squared distribution with 2 degrees of freedom.

Value

A list with the following components:

W

The Wald test statistic.

pval

The p-value associated with the test statistic, based on a chi-squared distribution with 2 degrees of freedom.

See Also

oneinfl for fitting one-inflated models. oneWald for a general Wald test of one-inflation parameters.

Examples

# Example usage
set.seed(123)
df <- data.frame(x = rnorm(100), z = rnorm(100), y = rpois(100, lambda = 5))
model <- oneinfl(y ~ x | z, df = df, dist = "Poisson")
result <- signifWald(model, varname = "x")
print(result$W)    # Wald test statistic
print(result$pval) # p-value

Summarize One-Inflated Regression Models

Description

Provides a summary of the fitted model, including estimated coefficients, standard errors, significance levels, and other relevant statistics.

Usage

## S3 method for class 'oneinflmodel'
summary(object, ...)

Arguments

Details

This function generates a detailed summary of the fitted model, including: - Estimated coefficients for the rate component (beta). - Estimated coefficients for the one-inflation component (gamma). - Standard errors. - z-statistics, associated p-values, and corresponding significance codes. - Average, and average absolute, one-inflation. - Log-likelihood of the fitted model.

Value

Prints a summary table of coefficients, standard errors, z-values, p-values, significance codes, one-inflation probabilities, and log-likelihood.

See Also

oneinfl for fitting one-inflated models.

Examples

# Example usage
df <- data.frame(x = rnorm(100), z = rnorm(100), y = rpois(100, lambda = 5))
model <- oneinfl(y ~ x | z, df = df, dist = "Poisson")
summary(model)

Summarize Truncated Regression Models

Description

Provides a summary of the fitted model, including estimated coefficients, standard errors, significance levels, and other relevant statistics.

Usage

## S3 method for class 'truncmodel'
summary(object, ...)

Arguments

Details

This function generates a detailed summary of the fitted model, including: - Estimated coefficients for the rate component (beta). - Standard errors. - z-statistics, associated p-values, and corresponding significance codes. - Log-likelihood of the fitted model.

Value

Prints a summary table of coefficients, standard errors, z-values, p-values, significance codes, and log-likelihood.

See Also

truncreg for fitting truncated regression models.

Examples

# Example usage
df <- data.frame(x = rnorm(100), y = rpois(100, lambda = 5))
model <- truncreg(y ~ x, df = df, dist = "Poisson")
summary(model)

Truncated Regression Model

Description

Fits a positive Poisson (PP) or zero-truncated negative binomial (ZTNB) regression model.

Usage

truncreg(formula, df, dist = "negbin", start = NULL, method = "BFGS")

Arguments

Details

This function fits a regression model for zero-truncated counts. Zero-truncated models are used when the count data does not include zeros, such as in cases where only positive counts are observed.

The function supports two distributions: - '"Poisson"': Zero-truncated Poisson regression. - '"negbin"': Zero-truncated negative binomial regression.

The function uses numerical optimization via optim to estimate the parameters.

Value

An object of class '"truncmodel"' containing the following components:

beta

Estimated coefficients for the regression model.

alpha

Dispersion parameter (only for negative binomial distribution).

vc

Variance-covariance matrix of the estimated parameters.

logl

Log-likelihood of the fitted model.

dist

The distribution used for the model ("Poisson" or "negbin").

formula

The formula used for the model.

See Also

summary for summarizing the fitted model.

Examples

# Example usage
df <- data.frame(x = rnorm(100), y = rpois(100, lambda = 1) + 1)
model <- truncreg(y ~ x, df = df, dist = "Poisson")
summary(model)