On testing for homogeneity with zero-inflated models through the lens of model misspecification.

In many applications of two-component mixture models such as the popular zero-inflated model for discrete-valued data, it is customary for the data analyst to evaluate the inherent heterogeneity in view of observed data. To this end, the score test, acclaimed for its simplicity, is routinely performed. It has long been recognized that this test may behave erratically under model misspecification, but the implications of this behavior remain poorly understood for popular two-component mixture models. For the special case of zero-inflated count models, we use data simulations and theoretical arguments to evaluate this behavior and discuss its implications in settings where the working model is restrictive with regard to the true data generating mechanism. We enrich this discussion with an analysis of count data in HIV research, where a one-component model is shown to fit the data reasonably well despite apparent extra zeros. These results suggest that a rejection of homogeneity does not imply that the underlying mixture model is appropriate. Rather, such a rejection simply implies that the mixture model should be carefully interpreted in the light of potential model misspecifications, and further evaluated against other competing models.

A robust score test of homogeneity for zero-inflated count data.

In many applications of zero-inflated models, score tests are often used to evaluate whether the population heterogeneity as implied by these models is consistent with the data. The most frequently cited justification for using score tests is that they only require estimation under the null hypothesis. Because this estimation involves specifying a plausible model consistent with the null hypothesis, the testing procedure could lead to unreliable inferences under model misspecification. In this paper, we propose a score test of homogeneity for zero-inflated models that is robust against certain model misspecifications. Due to the true model being unknown in practical settings, our proposal is developed under a general framework of mixture models for which a layer of randomness is imposed on the model to account for uncertainty in the model specification. We exemplify this approach on the class of zero-inflated Poisson models, where a random term is imposed on the Poisson mean to adjust for relevant covariates missing from the mean model or a misspecified functional form. For this example, we show through simulations that the resulting score test of zero inflation maintains its empirical size at all levels, albeit a loss of power for the well-specified non-random mean model under the null. Frequencies of health promotion activities among young Girl Scouts and dental caries indices among inner-city children are used to illustrate the robustness of the proposed testing procedure.

Associations between quality of life and duration and frequency of physical activity and sedentary behaviour: Baseline findings from the WALK 2.0 randomised controlled trial.

While physical and mental health benefits of regular physical activity are well known, increasing evidence suggests that limiting sedentary behaviour is also important for health. Evidence shows associations of physical activity and sedentary behaviour with health-related quality of life (HRQoL), however, these findings are based predominantly on duration measures of physical activity and sedentary behaviour (e.g., minutes/week), with less attention on frequency measures (e.g., number of bouts). We examined the association of HRQoL with physical activity and sedentary behaviour, using both continuous duration (average daily minutes) and frequency (average daily bouts≥10 min) measures. Baseline data from the WALK 2.0 trial were analysed. WALK 2.0 is a randomised controlled trial investigating the effects of Web 2.0 applications on engagement, retention, and subsequent physical activity change. Daily physical activity and sedentary behaviour (duration = average minutes, frequency = average number of bouts ≥10 minutes) were measured (ActiGraph GT3X) across one week, and HRQoL was assessed with the 'general health' subscale of the RAND 36-Item Health Survey. Structural equation modelling was used to evaluate associations. Participants (N = 504) were 50.8±13.1 (mean±SD) years old with a BMI of 29.3±6.0. The 465 participants with valid accelerometer data engaged in an average of 24.0±18.3 minutes and 0.64±0.74 bouts of moderate-vigorous physical activity per day, 535.2±83.8 minutes and 17.0±3.4 bouts of sedentary behaviour per day, and reported moderate-high general HRQoL (64.5±20.0). After adjusting for covariates, the duration measures of physical activity (path correlation = 0.294, p<0.05) and sedentary behaviour were related to general HRQoL (path coefficient = -0.217, p<0.05). The frequency measure of physical activity was also significant (path coefficient = -0.226, p<0.05) but the frequency of sedentary behaviour was not significantly associated with general HRQoL. Higher duration levels of physical activity in fewer bouts, and lower duration of sedentary behaviour are associated with better general HRQoL. Further prospective studies are required to investigate these associations in different population groups over time.