On the Efficiency of Score Tests for Homogeneity in Two‐Component Parametric Models for Discrete Data

Abstract

Summary In many applications of two-component mixture models for discrete data such as zero-inflated models, it is often of interest to conduct inferences for the mixing weights. Score tests derived from the marginal model that allows for negative mixing weights have been particularly useful for this purpose. But the existing testing procedures often rely on restrictive assumptions such as the constancy of the mixing weights and typically ignore the structural constraints of the marginal model. In this article, we develop a score test of homogeneity that overcomes the limitations of existing procedures. The technique is based on a decomposition of the mixing weights into terms that have an obvious statistical interpretation. We exploit this decomposition to lay the foundation of the test. Simulation results show that the proposed covariate-adjusted test statistic can greatly improve the efficiency over test statistics based on constant mixing weights. A real-life example in dental caries research is used to illustrate the methodology.

Generalized endpoint-inflated binomial model

“…(2.4) 19 Now let z * = (Z * 0 , Z * 1 , Z * 2 ) ⊤ ∼ Multinomial (1; φ * 0 , φ * 1 , φ * 2 ), V 0 ∼ ZTB(m, p) and z * V 0 . Then…”

“…In such cases, we could employ the EM algorithm. The zero observations 19 from an EIB distribution can be classified into the extra zeros resulted from the degenerate distribution with all mass at .2) is the first and second terms on the right-hand-side (RHS) of (4.2). To overcome this difficulty, we 24 augment Y obs with 2n latent binary variables…”

“…Application to the whitefly data zero-inflated over-dispersed binomial models to this response because the presence of an upper bound. But as pointed by 19 Deng and Zhang (in press), the responses not only contain many extra zeros but also contain many excessive right endpoints. 20 They also proposed score tests to test either zero-inflation or right-endpoint inflation, and the results showed that there is 21 inflation at both endpoints.…”

“…(5.8) 18 Therefore, the Newton-Raphson algorithm or the quadratic lower bound (QLB) algorithm (Böhning and Lindsay, 1988; 19 Böhning, 1992) can be applied to (5.7) and (5.8). …”

Score Tests for Both Extra Zeros and Extra Ones in Binomial Mixed Regression Models

“…with two degenerate probability functions f 1 (y) = 1 for y = 0 and f 2 (y) = 1 for y = m, a binomial probability function f 3 (y) = m y π y (1 − π ) m−y for y = 0, 1, ..., m. As Todem et al (2012) pointed out, the mixture model (2.1) allows for two types of representations, in terms of the supports of ω and φ. Under the hierarchical representation, the mixing weights are the probability masses, requiring the distribution constraints 0 ≤ ω ≤ 1, 0 ≤ φ ≤ 1.…”