bizicount (version 1.3.3)

zi_test: He's (2019) test for zero-modification

Description

This is an implementation of He et al. (2019)'s test for zero-modification (discussed further in Tang & Tang (2019)). This is a test of zero-modification instead of inflation, because the test is capable of detecting both excessive or lack of zeros, but cannot determine the cause. For example, a mixed data generating process could be generating structural zeros, implying a zero-inflated distribution. However, overdispersion via a negative binomial may also result in excessive zeros. Thus, the test merely determines whether there are excessive (or lacking) zeros, but does not determine the process generating this pattern. That in mind, typical tests in the literature are inappropriate for zero-modified regression models, namely the Vuong, Wald, score, and likelihood ratio tests. See the references below for more information on this claim.

Usage

zi_test(model, alternative = "inflated")

Arguments

model

A model object of class bizicount or glm. If a bizicount model, then at least one margin must be specified as "pois". If a glm model, then the family must be poisson.

alternative

The alternative hypothesis. One of c("inflated", "deflated", "both"). These correspond to an upper tail, lower tail, or two-tailed test, respectively. Default is "inflated". Partial matching supported.

Author

John Niehaus

Details

The test compares the observed proportion of zeros in the data to the expected proportion of zeros under the null hypothesis of a Poisson distribution. This is done using estimating equations to account for the fact that the expected proportion is based on an estimated parameter vector, rather than the true parameter vector. The test statistic is

\(\hat s = 1/n\sum_i (r_i - \hat p_i)\)

where \(r_i = 1\) if \(y_i = 0\), otherwise \(r_i = 0\), and \(\hat p = dpois(0, exp(X\hat\beta)) = \hat E(r_i)\) is the estimated proportion of zeros under the assumption of a Poisson distribution generated with covariates \(X\) and parameter vector \(\hat\beta\).

By the central limit theorem, \(\hat s \sim AN(0, \sigma^2_s)\). However, estimating \(\hat \sigma_s\) by a plug-in estimate using \(\hat\beta\) is inefficient due to \(\hat \beta\) being an random variable with its own variance. Thus, \(\hat\sigma\) is estimated via estimating equations in order to account for the variance in \(\hat \beta\).

See the references below for more discussion and proofs.

References

He, H., Zhang, H., Ye, P., & Tang, W. (2019). A test of inflated zeros for Poisson regression models. Statistical methods in medical research, 28(4), 1157-1169.

Tang, Y., & Tang, W. (2019). Testing modified zeros for Poisson regression models. Statistical Methods in Medical Research, 28(10-11), 3123-3141.

Examples

set.seed(123)
n = 500
u = rpois(n, 3)
y1 = rzip(n, 12, .2) + u
y2 = rpois(n, 8) + u

# Single parameter test, covariates can be added though.
uni1 = glm(y1 ~ 1, family = poisson())
uni2 = glm(y2 ~ 1, family = poisson())

biv = bizicount(y1~1, y2~1, margins = c("pois", "pois"), keep = TRUE)

zi_test(uni1)
zi_test(uni2)

zi_test(biv)