Fisher’s discriminant modeling for human platelet parameters

Introduction

Platelets, crucial blood components, play a fundamental role in various physiological processes, including clotting and immune response. Understanding platelet parameters and their associations with demographic factors is essential for medical research and clinical practice. Platelet parameters, such as distribution width, volume, large cell ratio, and count, reflect platelet morphology and function, providing valuable insights into an individual's health status. Discriminant analysis, a multivariate technique, has been widely employed to explore relationships between platelet parameters and demographic variables. This method allows for classifying individuals based on observed characteristics, facilitating the identification of factors influencing platelet biology. Discriminant analysis (DA) is one of the methods used in multivariate analysis, along with the dependency method. It is used in cases where the independent variable is matrix data and the dependent variable is non-metric data. DA undertakes the same task as multiple linear regression by predicting an outcome, and it is used to build a predictive model of group membership based on the observed characteristics of each case.

The main objectives of the study are  To identify linear combinations of platelet parameters that effectively discriminate between demographic groups.  To identify most significant platelet parameters contribute to the discrimination between demographic categories.  To assess the statistical significance of the discriminant functions and ascertain whether significant differences exist among demographic groups based on platelet parameters.

Review of literature

Durrant and Kaban (2010) [7] utilized a novel approach incorporating random projections with Fisher's Linear Discriminant (FLD) classifier for classification tasks. Unlike previous methods focusing solely on preserving pairwise distances under projection, the authors' approach emphasizes leveraging the inherent class structure within the data.

Ramayah et al. (2010) [19] present a step-by-step example, making it easier for readers to comprehend the intricacies of discriminant analysis. They carefully explain the necessary assumptions and procedures involved in discriminant analysis, including data preparation, model estimation, and interpretation of results. Nainggolan et al. (2018) [15] used discriminant analysis and classified Hypertension women aged 27 to 54 years living in the village in the central district of Bogor. The results of the multivariate discriminant analysis showed that the level of Vo2 max is the only distinction maker in the incidence of hypertension.

ALKubaisi et al. (2019) [1] used discriminant analysis with three criteria to test the developed model, producing excellent projecting precision. The discriminant function properly assessed and classified about 67% of the cases in the analysis. Also, the analysis produced two discriminant functions: the first explained 77% and the second explained 23% of the Variance.

Dibal and Abraham (2020) [5] [18] utilized discriminant analysis to distinguish between two types of Bowel and Esophageal cancer in Jordan, identifying significant variables such as sex, weight, and Platelets Count P.C. The correct classification rates for the first and second groups were 62.8% and 77% respectively, with misclassification rates of 37.2% and 23%. The proper classification ratio was 71.6%, with a false classification ratio of 28.4%. The method effectively identified vital independent variables for diagnosing both cancer types, with correct classification probabilities of 66.4% and 77.6% for the first and second groups, respectively. Henry et al. (2023) [10] investigated the spectral differences of tobacco leaves under macronutrient deficiencies. They employed information entropy and spectral derivatives methods to identify the most effective wavelengths for discrimination. Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA) algorithms were utilized to reduce data dimensionality and classify the symptoms. The study's findings revealed that the overall accuracy for classifying young, intermediate, and mature plants was 92%, 82%, and 75%, respectively. The results also indicated that nitrogen, sulfur, and magnesium deficiencies significantly impacted the classification accuracy. In contrast, deficiencies in phosphorus and potassium had minimal effect on the classification outcomes.

Materials and Methods

Materials

The dataset under investigation is collected through Mendeley Data (https://data.mendeley.com/datasets/5t8dr6d73f/1). The dataset comprises comprehensive information on platelet parameters and red cell distribution width (RDW) for 1883 samples. It includes measurements obtained from individuals across various demographics, encompassing platelet distribution width, mean platelet volume, platelet large cell ratio, plateletcrit, total platelet count, and RDW values. Each entry in the dataset provides specific values for these parameters, allowing for a detailed analysis of their distributions, variability, and potential interrelationships.

Methods

Box-Cox method

The Box-Cox method is used in statistics and econometrics to transform non-normal data into approximately normal data. It is named after statisticians George Box and Sir David Cox and was introduced in 1964. Let be the date on which the Box-Cox transformation is applied. Box and Cox (1964) defined their transformation as (1) Such that for unknown (2) Where is the transformed data, X is the design matrix (possible covariates of interest), is the set of parameters associated with the transformed data, and is the error term. Since the aim of Equation ( 1) is that ' 12 ( , ,..., )

The transformation in Equation ( 1) is only valid for modifications to be made when negative observations are present (Vélez et al., (2015)).

Multivariate Analysis of Variance

A MANOVA technique (Johnson & Wichern, (1998)) [12] is employed to test the significance of variation among all the five parameters considered simultaneously. The MANOVA model for comparing the population means vectors is as follows:

Where, is a vector of random error distributed as .

Here, the parameter vector is the overall mean and represents the model's status in (4); each component of the observation vector satisfies the univariate model, and the variance-covariance matrix is the same for all populations.

Variance Inflation Factor

The variance inflation factor is used to measure how much the Variance of the estimated regression coefficient is inflated if the independent variables are correlated. VIF is calculated as

Where the tolerance is simply the inverse of the VIF; the lower the tolerance, the more likely the multicollinearity among the variables. The value of VIF=1 indicates that the independent variables are not correlated. If the value of VIF is 1 < VIF < 5, it specifies that the variables are moderately correlated. If the VIF value is above 5, there will be multicollinearity among the predictors in the regression model (Goldstein, (1993) [9] and Shrestha, (2020) [21] . Another one is the scatterplot graphical method, which signifies the linear relationship between pairs of independent variables. It is essential to look for scatterplots that indicate a linear relationship between pairs of independent variables. The correlation coefficient is calculated using the formula:

Where, r is the correlation coefficient, n is the number of observations, X represents the first variable in the context, and Y is the second variable in the context. If the correlation coefficient value is higher with the pairwise variables, it indicates the possibility of collinearity.

Box's-M test

The Test for homogeneity of covariance matrices, introduced in 1949, examines the covariance matrices derived from multivariate normal data considering one or more classification factors. This test assesses the similarity between the separate covariance matrices by comparing the product of their log determinants to the log determinant of the combined covariance matrix, similar to a likelihood ratio test. The test statistic employs a chi-square approximation.

Wilk's lambda

In discriminant analysis, Wilk's lambda is utilized to assess the contribution of each level of an independent variable to the model. This scale ranges from 0 to 1, where a value of 0 indicates complete discrimination, while a value of 1 signifies no discrimination. To test the impact of each independent variable, it is successively included and excluded from the model, generating a Λ statistic. The significance of the change in Λ is evaluated using an F-test; if the computed F-value exceeds the critical value, the variable is retained in the model (Onwukwe, (2014) [17] . Thus, a non-significant Wilks' lambda value is always preferred.

B is the between-groups matrix, and W is the within-group matrix. The Eigenvalue can be explained as the ratio of the between-groups sum of squares to the within-group sum of squares (McGarigal et al., (2000) [14] .

Multiple Discriminant Analysis

Multiple Discriminant Analysis (MDA) is an extension of discriminant analysis; it shares ideas and techniques with various analyses of Variance (MANOVA). MDA aims to classify cases into three or more categories using continuous or dummy categorical variables as predictors (Cramer, (2003) [4] , Jang et al., (2015) [11] . The term DA refers to numerous types of analyses. DA is the most popular statistical technique to classify individuals or observations into non-overlapping groups based on scores derived from a suitable "statistical decision function" constructed from one or more continuous predictor variables.

While investigating the differences between the groups or categories, the necessary step is to identify the attributes with the most contributions to maximum reparability between known groups or categories to classify a given observation into one of the groups. For that purpose, DA successively identifies the linear combination of attributes known as canonical discriminant functions (equations) that contribute maximally to group separation. Predictive DA addresses the question of how to assign new cases to groups. The form of the Equation or function is:

Where D is an independent variable and is the value of discriminant score from the i th category (i=1,2,…n), is the discriminant coefficient of j th attributes (j=0,1,2,…,k), and is the k th independent variable of the i th category. This function is similar to a regression equation or function. The re are unstandardized discriminant coefficients analogous to the ones in the regression equation. These maximize the distance between the means of the dependent variable, and the standardized discriminant coefficients can also be used, like beta

Results and Discussion

The results presented in Table 1 In Figure 1 Various multivariate tests, such as Pillai's trace, Wilks' lambda, Hotelling's trace, and Roy's most significant root tests, were utilized to assess the collective variation of all five platelet parameters across the gender groups. The outcomes of these tests are presented in the Table 3. These MANOVA statistics provide insights into the multivariate effects of the analysis. Pillai's Trace, Wilks' Lambda, Hotelling's Trace, and Roy's Largest Root are all measures of the significance of the overall model. In this case, the extremely low p-value (0.00) indicates that the model has a significant overall effect. The values of these statistics (ranging from 0.09 to 0.10) suggest the proportion of Variance in the dependent variables explained by the independent variables in the model.

Conclusion

In this study, the analysis of the platelet parameter dataset reveals valuable insights into population characteristics and their associations with platelet biology. The dataset represents a diverse population and includes comprehensive information on platelet parameters and red cell distribution width. Descriptive statistics highlight variability and distribution characteristics, while normality tests indicate the effectiveness of transformation methods. Multivariate analysis confirms significant overall effects of platelet parameters, and correlation analysis identifies potential interdependencies among variables. Fisher Linear discriminant analysis underscores the influence of platelet parameters on gender classification, with acceptable levels of multicollinearity ensuring reliability. Moreover, tests for covariance matrix equality and canonical analysis provide further insights. These findings contribute to a better understanding of platelet biology's importance in medical research and clinical practice. The Platelet parameters exhibit significant associations with population characteristics and gender, which are crucial for medical research and clinical applications.

References

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