Statistical analysis (GLMM) for: Implications of the existence of separated sexual forms of Opuntia robusta for plant defense hypotheses

We analyze the effect of different variables on the concentrations of 4-Hydroxybenzoic acid (4-HBA), chlorogenic acid (CGA) and quercetin (QUE) in female and hermaphrodite individuals of Opuntia robusta from San Nicolas Tecoaso, Municipality of Singuilucan, Hidalgo State, Central Mexico, using general linear mixed models (GLMM). To analyze the effect of months and age on 4-HBA concentrations, we used the Gamma error distribution (GED) and the inverse link function. For the effect of months on CGA concentrations, we used GED and log link functions. For the effect of cladode age on CGA concentrations, GED and inverse link functions. For the effect of months on QUE concentration, GED and log link functions. For the effect of cladode age on QUE concentration, GED and inverse link function. For the models that related substances with cladode length, width and thickness, we used different models: for 4-HBA concentrations we used a GED with square root link function. For CGA concentrations, GED and identity link function, and for QUE concentrations, GED and inverse link function. To test the effect of cladodes order from soil on the concentrations of the three substances, we used GED and square root link function. To test the effect of the number of cladodes above a given cladode on the concentrations of the three substances, we used GED and inverse link function. We checked the goodness of fit of the linear models (GLMM) by plotting the standardized residuals against fitted values and by normal QQ‐plots, and revised the assumptions of the homoscedasticity, proper distribution used, and independence. We chose the best GLM model used in this study according to the lowest AIC (Akaike, 1974; Akaike, 1987).本研究采用广义线性混合模型（general linear mixed models，GLMM），针对墨西哥中部伊达尔戈州辛吉卢坎市圣尼古拉斯特科阿索地区健壮仙人掌（Opuntia robusta）的雌性与雌雄同株个体，分析不同变量对其体内对羟基苯甲酸（4-Hydroxybenzoic acid, 4-HBA）、绿原酸（chlorogenic acid, CGA）及槲皮素（quercetin, QUE）浓度的影响效应。为分析月份与茎节龄期对4-HBA浓度的影响，本研究采用Gamma误差分布（Gamma Error Distribution, GED）与逆连接函数。针对月份对CGA浓度的影响，采用GED与对数连接函数；针对茎节龄期对CGA浓度的影响，采用GED与逆连接函数。针对月份对QUE浓度的影响，采用GED与对数连接函数；针对茎节龄期对QUE浓度的影响，采用GED与逆连接函数。针对物质浓度与茎节长、宽、厚度的关联模型，本研究采用不同建模方案：对于4-HBA浓度，采用带有平方根连接函数的GED模型；对于CGA浓度，采用GED与恒等连接函数；对于QUE浓度，采用GED与逆连接函数。为检验茎节离地层级对三种物质浓度的影响，本研究采用GED与平方根连接函数。为检验指定茎节上方的茎节数量对三种物质浓度的影响，本研究采用GED与逆连接函数。本研究通过绘制标准化残差与拟合值的散点图以及正态QQ图，检验广义线性混合模型的拟合优度，并验证了同方差性、分布适用性以及独立性等模型假设。本研究依据最低赤池信息准则（Akaike Information Criterion, AIC）值（Akaike, 1974; Akaike, 1987）选取本研究的最优广义线性模型。