Temperature and day-length are considered to be the 2 important factors that can significantly affect insect diapause, which is a typical proportional dataset. In the previous studies, the method of arcsine square root transformation is widely used to analyze the effect of temperature or day-length or their joint effects on diapause in insects. However, this method has many limitations, for example, the proportional data should be normally distributed. The logistic regression in generalized additive models is a promising method for analyzing the effects of temperature and day-length on diapause. Compared to the arcsine square root transformation method, this method does not require normal distribution of proportional diapause data. The logistic regression also provides better goodness-of-fit by using the non-parametric fitting technique. In this report, we used the diapause data of Pieris melete (Xiao et al. 2012) to compare the fitted results of the logistic regression in generalized additive models with arcsine square root transformation. We found that the logistic regression in generalized additive models is better than linear regression of arcsine square root transformed data in following ways: (1) reasonable predictions about diapause ranging from 0 to 1 can be made without transforming the proportional data; (2) non-linear effects of temperature and day-length on diapause can be determined; (3) the goodness-of-fit can be substantially improved.
Recently, Xiao et al. (2012) published a study on the effects of daily average temperature and natural day-length on the incidence of summer and winter diapause in the Cabbage Butterfly, Pieris melete Ménétriés (Lepidoptera: Pieridae). Under field conditions, the cabbage butterfly, Pieris melete, displays a pupal summer diapause in response to relatively low daily temperatures and gradually increasing day-length during spring, and a pupal winter diapause in response to the progressively shorter day-length. To determine whether photoperiod has a stronger role than temperature in the determination of the summer and winter diapause, or vice versa, the effects of the naturally changing day-length and temperature on the initiation of summer and winter diapause were systematically investigated under field conditions for 5 successive years. Field results showed that the incidence of summer diapause significantly declined with the naturally increasing temperature in spring and summer generations. Path coefficient analysis showed that the effect of temperature was much greater than that of photoperiod in the determination of summer diapause. In autumn, the incidence of diapause was extremely low when larvae developed under gradually shortening day-length and high temperatures. However the incidence of winter diapause increased to 60–90% or higher with gradually shortening day-length combined with lower temperatures, i.e., between 20.0 °C and 22.0 °C. Decreasing day-length played a more important role in the determination of winter diapause induction than temperature.
Thus Xiao et al. (2012) provided 5-year proportional data of cabbage butterfly diapause and corresponding temperature and day-length data (Tables 1 and 2 in Xiao et al. 2012). However, only linear regression was used to describe the effects of these 2 predictors on diapause (Table 3 in Xiao et al., 2012). Linear regression simply hypothesizes that the effects of these predictors on the response variable are linear. However, nonlinear effects are ubiquitous in nature. Therefore, linear regression actually neglected the nonlinear effects, which led to low goodness-of-fit to the objective data. Additionally, the proportional data were arcsine square root transformed before performing the linear regression. Although this transformation has long been standard procedure in analyzing proportional data in ecology, logistic regression has greater interpretability and higher power than transformation in data containing binomial and non-binomial response variables (Warton & Hui 2011). The detailed problems with arcsine transformation can be found in Wilson et al. (online), and logistic regression was strongly recommended as an alternative to the arcsine transformation in biological analysis. If the distribution of proportional data is not normal, then the use of arcsine transformation is problematic. In fact, the distribution of the transformed diapause data of Xiao et al. (2012) was still not normalized (W = 0.8491, P-value < 0.05) as revealed by the Shapiro-Wilk normality test (Faraway 2005). Thus, we suggest using logistic regression to fit the proportional data of diapause. The following analysis can be considered as an alternative to the analysis performed by Xiao et al. (2012), and also for similar data in future studies.
Diapause and non-diapause response to a combination of temperature and day-length can be exactly described either by the generalized linear model or the generalized additive model (Hastie & Tibshirani 1990). The latter is more flexible in fitting the data. We used the following generalized additive model to describe the effects of temperature and day-length on diapause: logit(Diapause) = α+ f1(Temperature, + f2 (Day-length),(1) where f1 (i = 1, 2) are smooth functions.
We pooled and fitted the data of summer and winter diapause, and found that temperature and day-length both could significantly affect diapause (P < 2e-16) (Fig. 1). Fig. 2 exhibits the fitted surface of diapause. We found that the temperature > 23 °C led to a very low diapause. Relative to day-length, temperature appeared to be more important in determining diapause of the cabbage butterfly. The goodness-of-fit obtained by using the generalized additive model is satisfactory with r2 = 0.95. It is higher than the r2 value calculated in Xiao et al. (2012) by using 2 predictors. Considering the important effect of temperature on diapause, we also explored the effect of a single predictor, i.e., temperature on diapause. The prediction by using temperature only is also satisfactory with r2 = 0.90, which is greater than the r2 calculated in Xiao et al. (2012) by using temperature only.