Polycystic ovary syndrome (PCOS) is a usual endocrine medical situation that assails a large number of women in adolescence and reproductive age. The PCOS distribution over different body mass index (BMI) grades can vary, and the current research has shown the effects of BMI on PCOS women, using 1000 real observations, and the data set is in the site: https://www.kaggle.com/datasets/samikshadalvi/pcos-diagnosis-dataset The BMI values analysis findings are developed herein applying statistical joint generalized linear models (JGLMs). It is developed herein that mean BMI is negatively associated with the joint interaction effect (JIE) of the subject’s testosterone (TET) levels and menstrual irregularity (MIT) i.e., TET*MIT (P<0.0001), while it is positively associated with both TET (P=0.0025) and MIT (P<0.0001). Mean BMI is negatively associated with the JIE of the subject’s antral follicle count (AFC) values and MIT i.e., AFC*MIT (P<0.0001), while it is positively associated with both AFC (P=0.0387) and MIT (P<0.0001).  Mean BMI is negatively associated with the JIE of the subject’s TET levels and AFC values i.e., TET*AFC (P=0.0253), while it is positively associated with both TET (P=0.0025) and AFC (P=0.0387). Mean BIM value is positively associated with the JIE of TET levels and the subject’s PCOS diagnostic status i.e., TET*PCOS (P<0.0001), while it is positively associated with TET (P=0.0025) and negatively with PCOS (P=0.0661). Mean BIM value is positively associated with the JIE of AFC values and the subject’s PCOS diagnostic status i.e., AFC*PCOS (P<0.0001), while it is positively associated with AFC (P=0.0387) and negatively with PCOS (P=0.0661). BMI values’ variance is negatively associated with age (P=0.0839), PCOS (P<0.0001) and AFC (P=0.1170). The article has shown that BMI value has different significant JIEs on PCOS women. The current outcomes regarding the BMI values may be instrumental for the PCOS women, practitioners and researchers. It concludes that BMI along with MIT, TET and AFC has multiple effects on PCOS women.

Polycystic ovary syndrome (PCOS) is a usual endocrine medical situation that assails a large number of women in adolescence and reproductive age. Many articles [1,2] have pointed out that approximately 8-16% of women are affected by PCOS globally in their reproductive age group. The PCOS distribution over different body mass index (BMI) grades can vary, many research articles have tried to focus on the association between BMI and PCOS women [3-5]. 

PCOS can have very complex interaction effects with various factors such as BMI, testosterone (TET) levels, menstrual irregularity (MIT), antral follicle count (AFC), age including lifestyle choices [6,7]. Obesity is highly prevalent in PCOS women. There is a preferential androgenic distribution pattern of body fat in PCOS women [8].  Reproductive functional problems such as MIT and infertility are more prevalent in obese women [8-10]. Obesity is also directly associated with PCOS, which assails 6–12% of women of their reproductive age [11].

The association between elevated BMI and individual phenotypic characteristics of the Rotterdam criteria that discriminate against PCOS remains hazy [12]. For instance, a meta-analysis among PCOS women shows that hirsutism, as quantified by modified Ferriman-Gallwey score, was elevated only when comparing obese women versus overweight, but not when comparing obese women versus normal weight [13]. Roles of BMI on features such as MIT, and AFC remain unclear, especially in healthy women. From many articles [7,9,10,11, 14], it is suspected that BMI affects these above-mentioned individual phenotypic features (MIT, AFC, TET, PCOS diagnosis status) of women with PCOS or without PCOS.

In previous research articles [8,12,13], the effects of BMI on PCOS women are unclear. Presently, some advanced research tools such as machine learning, statistical modelling, data mining etc. are employed in PCOS data analysis [10,12,15]. Several machine learning algorithms such as Random Forest, locally weighted learning, Decision table, Multilayer perceptron, Random tree, etc. are employed in the PCOS data analysis [2,15,16].  Several common statistical techniques such as testing of hypotheses, simple correlation & regression, analysis of variance etc. are applied in PCOS data analysis that are not suitable for positive, non-normal and non-constant variance PCOS data sets [2,4,8,11].

The present BMI response in the considered PCOS data set is a non-constant variance dependent variable, which is positive and non-normal. Best of our knowledge, most of the previous manuscripts did not consider the response BMI in PCOS data sets as a heteroscedastic and non-normal random variable. Therefore, most of the BMI analysis reports of PCOS data sets invite several doubts and debates. The effects of BMI on PCOS women are little studied adopting advanced probabilistic modeling. The present BMI study manuscript for PCOS data set searches the following research statistical hypotheses that are connected with and without PCOS women.

•               Does BMI associate with irregular menstrual cycles, age, TET levels, AFC values and polycystic ovarian morphology of PCOS women? 

• If it is affirmative, how can one derive the most probable BMI grades association model? 
• What are the most probable BMI grades statistical model?
• What are the effects of BMI grades on PCOS women? 

The article studies the above BMI grades examination research hypotheses adopting the following paragraphs such as materials & methods, statistical analysis & results, discussions, and conclusions. Statistical mean & variance models of the response BMI are revealed in Table 1, based on the PCOS data set, which is marked in the materials section. Mean and variance joint statistical model of the response BMI is obtained using joint generalized linear models (JGLMs), which is shortly revealed in the methods section. Response BMI modelling outcomes are presented in the result section, while the BMI modelling outcomes are presented in the discussion section. The BMI analysis’s main information is noted in the conclusions section. 

2.1. Materials 

The current BMI study dataset is a sample of 1000 women subjects with PCOS and without PCOS, while PCOS is a usual hormonal endocrine disorder assailing woman of their reproductive age. The considered PCOS sample data set contains six correlated characters which are primarily connected with PCOS diagnosis. These six characters are considered as the valued insights into the subjects’ medical health situations, and they can be employed for exploratory data analysis such as statistical modelling, feature engineering and machine learning for identifying PCOS diagnosis status. The considered PCOS data is available in the site : https://www.kaggle.com/datasets/samikshadalvi/pcos-diagnosis-dataset 

The under study PCOS data set comprises six features such as the sample unit woman’s body mass index (BMI), age, testosterone level (TET), antral follicle count (AFC), menstrual irregularity (MIT) (0=No, 1= Yes) and polycystic ovary syndrome (PCOS) (0=No, 1=Yes) diagnosis status. The sample study women are taken in their reproductive age. In the present study, BMI is the response variable, which is a body fat measure that is computed based on height and weight, and it is commonly ranging from 18 to 35. It is computed using the weight (in Kg) and height (in meter), and it is defined as BMI= Weight (kg) / Height (m2). BMI is a widely used screening tool that provides a simple numeric measure of an individual’s weight in relation to their height as stated above. It is generally used to categorize individuals as underweight, normal weight, overweight, or obese. BMI is an important risk factor for many diseases such as diabetes, cardiovascular disease, and hormonal disorders such as PCOS etc. The different categories of individuals based on BMI are as follows. Individuals are categorized as underweight class when BMI values < 18> 39.8. The five BMI’s explanatory variables are age, MIT (0 = No, 1 = Yes), TET levels, AFC values and the subject’s PCOS diagnosis status (0=No, 1=Yes).

2.2 Statistical Methods 

The current study takes into account the BMI grades as the response random variable, and it is to be modeled with the remaining five variables such as age, TET levels, MIT, AFC values and PCOS diagnosis status.  The response BMI is identified as a non-normally and non-constant variance distributed random variable. The BIM’s variation can’t be stabilized by any proper transformation, so BIM value is modeled herein using joint generalized linear models (JGLMs) considering both the Gamma and Log-normal distributions, which is well described in [17-20]. Joint mean & variance models i.e., JGLMs are well described in the book by Lee et al. [17] and in the book by Das [18]. A short note of JGLMs for BMI values under both the Log-normal and Gamma distribution is displayed as follows.

JGLMs for Log-normal distribution: For the positive response Yi (=BMI) with E(Yi=BMI) = µi (mean) and Var(Yi=BMI) = µi2 =  say, where  ’s are dispersion parameters and V () reveals the variance function. Generally, log transformation Zi = log (Yi=BMI) is adopted to stabilize the variance Var (Zi) ≈ , but the variance may not always be stabilized [21]. For developing a BMI improved model, JGLMs for the mean and dispersion are considered. For the response BMI, assuming log-normal distribution, JGL mean and dispersion models (with Zi = log (Yi=BMI)) are as follows: 

 E(Zi)= µzi and Var (Zi) = σzi2,

µzi=xit β    and  log (σzi2)= git γ,                                       

where xit and git are the explanatory factors/variables vectors of BMI values associated with the mean regression coefficients β and dispersion regression coefficients γ, respectively. 

3.1 Statistical Analysis 

The manuscript has developed the associations of BMI values with five explanatory variables such as age, AFC, TET, MIT and PCOS diagnosis status of the study subject.  Joint generalized linear BMI values model has been obtained on the five explanatory variables such as age, AFC, TET, MIT and PCOS diagnosis status. Final BMI values model has been taken on the basis of lowest Akaike information criterion (AIC) value (within each class) that minimizes both the squared error loss and predicted additive errors [22, p. 203--204]. According to the AIC rules, JGLMs Gamma fit (AIC= 5662.777) and Log-normal fit (AIC=5663) are almost the same as the AIC difference is smaller than one. In the BMI mean model, all the included marginal and joint interaction effects are significant. Note that if any interaction effect is significant, then all its lower order interaction effects and marginal effects should be allowed in the model even if they are insignificant by marginality rule by Nelder [23].  Here all the included effects of BMI mean model are significant. Two partial marginal effects such as AGE (P=0.0839) and AFC (P=0.1170) are included in the BMI’s dispersion model for better model improvement [22].  It is pointed out in Epidemiology that partial significant effects are referred as confounders that may have some influence on the risk factor or marker.
 

                               Figure 1(a)                                                                                           Figure 1(b)

Figure 1: For the joint Gamma fitted models of Body Mass Index (Table 1), the (a) absolute residuals plot with the fitted values, and (b) the normal probability plot for mean model

The developed BMI values Gamma fitted JGLM (Table 1) is a data extracted model that is to be tested by model checking plots. The interpretations about BMI values are taken from the data exhibited in the Gamma fitted BMI values probabilistic model (Table 1), which should be accepted based on graphical diagnostic plots in Figure 1. Figure 1(a) presents the absolute residuals plot for the Gamma fitted BMI values model (Table 1) with respect to the fitted values, which is almost flat linear, indicating that variance is constant with the running means.  Figure 1(b) shows the normal probability plot for the Gamma fitted BMI values mean model (Table 1) that does not reveal any lack of fit. So, both the figures 1(a) and (1b) do not present any discrepancy in the Gamma fitted BMI values models (Table 1). The above two figures confirm that the Gamma fitted BMI values model is an approximate form of the unknown true BMI values model.