1. Introduction

The classification technique is crucial part of classifying different groups based upon defined characteristics, especially in the medical field. Multivariate data analysis is widely employed to classify this type of data. There are two well-known models (Multivariate Logistic regression (MLR) and linear discriminant analysis (LDA)) to predict relations between two or more groups, using a set of predictors. Alkarkhi et al. [1] and Krieng [6]. Both techniques were used and analyzed in many previous articles, books and papers. For example, compared DA and the LRA model in predicting method of surrender of an expectant mother, natural birth and caesarian section Montgomery. Balogun et al. [5]. Various real data sets that are different in terms of normality, the performance of both methods were studied by a number of independent variables and sample size.

The two used methods are compare between the percentage of correct classification and B index. conclusively, LR is distinguished with better results, a way from the data distribution type. The aim of the study is investigating predictive group discriminant using (LRA) and (DA) to predict probability of patients with cancer who already had a medical check to determine the probability of having a breast cancer or not breast cancer. Krieng [6].

Undoubtedly, incomplete data can cause a real problem for most applied researchers. Many mechanisms were and still under developing up till now will continue to be developed to reach conclusions from data sets with missing values Balogun et al. [5]. Handling data requires a real and precise data resources to analyze it and to make deductions. Using regular pattern to collect with no outliers or missing values, which is not feasible all the time. Accordingly, it is important to assess the used information to achieve an dependent data analysis.

The present investigation compares between MLR and LDA for classification of objects to groups after handling missing data. The accuracy of the prediction is mainly decided by apparent error rate (AER) and apparent correct classification rate (ACCR), and will be applied to real data of children patients who suffer from Chronic kidney disease.

2. Objectives of Study

The objectives of the present study may be summarized in the following four points:

1- Handling missing data using (Fully conditional specification).

2- Investigating differences between groups of the anemia, patient and to identify important discriminating variables of the anemia, on perform hypothesis testing once both those significance of the model and the significance of the independent variables Also to classify new observations under pre-existing groups.

3- Estimating the relation between dependent and independent (predictors) variables and for the logistic function binary logistic function was used to detect the relation and determine the best predictors for anemia.

4- Judging the accuracy of classification.

3. Data and Variables

The 344 subjects selected for the conduction of this study were patients in the chronic kidney disease department of Mansoura University Hospital. All subjects who participated in this research were selected as they started to be admitted to hospital at the same time. The patients had incomplete data that will be treated using the method of (FSC).

The independent variables were (sex, age group, Body Mass Index(BMI), CKD stage, time, fate, ultrasound) and the presence of anemia (dependent variable data was coded as 1 for anemic and 2 for non anemic). Data analysis was done using two approaches LDA and LR from SPSS software (Statistical Package for the Social Sciences), version 16, and for handled missing data was done using (FCS) from SPSS, version 20.Variables are explained in table (1). Glomerular filtrations rate (GFR) ml/min per 1.73m² is used to determine (CKD stage).

Table (1). Explanations of variables

4. Methodology

4.1. Fully Conditional Specification

The most popular method to multiply impute multivariate data having an arbitrary pattern of incomplete data is the chained equation method also known as fully conditional specification (FCS) that evenly imputes one variable at a time. The FCS approach is to impute the data on a variable-by-variable based on specifying an imputation model per variable. Elhabil et al [9]. In fact, there are three main episodes to treat the missing data:

1) setting initial values for missing values in all variables

2) At iteration t, for j=1 to k: Given that is, most recently impute values of all other variables, and for , and then using a univariate method to impute all missing values in j^th variable,

3) To continue iteration until the maximum number of iteration is reached.

4.2. Discriminant Analysis

Discriminant analysis (DA) is a multivariate parametric statistical approach employed to establish a predictive model of group discrimination basis by observed predictor variables (factors), and classifying each observation to one of the groups to be discriminated. The analysis makes a discriminant function, which is a linear combination of the weightings and scores of variables that are considered. Van. [7]. This combination for a discriminant analysis, also called the discriminant function is derived from an equation which defined as the following:

where:

discriminant score of discriminant function i for object k,

independent variable j for object

discriminant weight for independent variable j and discriminant function i

constant of discriminant functionand computational method is happening at about the same time estimation which includes computing the discriminant function so that all of the independent variables are considered concurrently.

4.3. Multivariate Logistic Regression

In fact, Logistic regression is the most popular modeling method when the dependent variable is dichotomous. The binary LR model is application when the response variable is divided into two categories. This model is mainly employed to figure out the relationship between one or more explanatory variables and the dependent variable This model is determined by the following equation:

where, (explains the ratio of the probability of a success to the probability of a failure, is known as odds, are parameters to be estimated, and is the response probability for ith group, (j) expresses number of variables [1].

4.4. Criteria of Efficiency

The most leading thing when building a classification rule is to correctly. Little et al. [3]. An estimate of the error rate can be acquired by testing the classification procedure on alike data set which has been used to calculate the classification functions. This technique is usually make reference to as re-substitution. Ramayah et al. [8]. For two groups, around then observations in G1, n₁₁ would effectively ordered under G1, and n₁₂ are misclassified into G2, where n₁ = n₁₁ + n₁₂. Similarly, of the n₂ observations in G2, n₂₁ need aid misclassified under G1 furthermore n₂₂ need aid effectively classified into G2, where n₂ = n₂₁ + n₂₂. Thus, the apparent error rate AER is presented as:

5. Results

The following chart shows the type of the missing data. Based on this figure, it can be noticed that the missing is (non monotone or Arbitrary). So, the suitable method to treat is (FCS) which was previously identified.

Figure (1). Represents the type of missing data

As is demonstrated in table 2, the sample size of the data is 344 observations and the data set were divided in to two groups where, the first group anemic with n₁ = 123 which represents the 35.8% of observations, and the second group is not anemic with n₂ = 221 which represents the 64.2% 0f observations.

Table (2). Descriptive analysis for the variables

The main assumptions of DA were tested. Kolmogorov-Smirnov test statistic was used for testing normality of data and had a value of 0.414 with p-value <0.000. Depending on the significance level = 0.05, so, the data are not normally distributed and Box's M test used here to test the assumption of equality of covariance matrices and had a value of 9.12 with p value 0.000 indicates that the data do not differ significantly from multivariate normal.

Table (3). Means and standard deviation for all variables

Table (4). VIF Values of Predictor Variables

Concerning of multicollinearity, high correlations should not be present among variables of interest. To perform this, the Variance Inflation Factor (VIF) index is employed, and a value of VIF >10 illustrate that multicollinearity is present all the VIF for the predictor is less than 10 which shows that there is no evidence of multicollinearity among the set of predictor variables or predictors. This means one can proceed with the analysis.

5.1. For Discriminant Analysis Model

Wilks' Lambda can be used to measure of how well each function separates cases into births groups. From table 5, we can conclude that the discriminant function is significant in case of X₂, X₄, X₅ and their function explains the group membership well.

Table (5). Selection of Discriminating Variables Depending on Simultaneous Method

Table (6). Wilks' Lambda Table

Table (7). Unstandardized Canonical Discriminant Model Coefficients

Unstandardized canonical discriminant function coefficients are used in the formula for making the classifications in DA, and function will be as follows:

Z = 0.892 - 0.036(X₁) + 0.775(X₂) - 0.038(X₃) - 0.366(X₄) + 0.167(X₅) + 0.024(X₆) - 1.260(X₇)

The two performance criteria AER and ACCR were used to evaluate the efficiency of the discriminatory model of the estimated function. From table 8, it could be showed that 47 of 123 children from the anemic group have correctly been classified and 190 of 221children from the non anemic group have correctly been classified. It can be concluded that the DA was able to correctly classify 237 cases of patients out of 344 cases while, The AER was 29.8% and the ACCR was 69.2% indicating that the model has ability on classification.

Table (8). The Final Classification Results

5.2. For Logistic Model

Table 9 clearly shows that -2log likelihood value of basic model was 384.13. The value of the Chi-square was 64.60 against the probability 0.000 showing that the model is significant and for this reason, the null hypothesis is rejected and the alternative hypothesis accepted. There is an obvious relationship between the predictors and the dependent variables.

Table (9). Model Fitting Information

Estimation of the model parameters obtained by using Wald statistic for the final model are presented in the table 10 which gives the results of fitting the LRA model to anemia presence and presenting coefficients which are used in the formula for making the classifications in LRA. Also, the estimated logistic regression model is:

Ln (π /1 – π) = 0.981 + 0.045(X₁) − 0.672(X₂) - 0.037(X₃) − 0.423(X₄) − 0.031(X₅) + 1.137(X₆) + 0.363(X₇).

Table (10). Result of LRT

Table (11). Results of Fitting the LRA Model to Births Data

Table (12). The Final Classification Results Using LRA model

The two criteria of AER and ACCR have been used to evaluate the LRA model efficiency of the estimated function and we can see that 55 of 123 children from the anemic group have been correctly classified, and 182 of 221 children from the non anemic group have been correctly classified, it can be concluded that the LRA was able to classify cases of patient out of 237cases correctly. The AER was 31.1% and the ACCR was 68.9% indicating that the model has ability on classification.

6. Conclusions

The result of this study illustrated that LDA is significantly more efficient than MLR in the accuracy of the prediction. The missing data were treated by using fully conditional specification (FCS). The comparison between groups depended mainly on statistical criteria; apparent error rate AER and apparent correct classification rate ACCR. Also, a simultaneous method was used in case of discriminant analysis model to estimate the relation between dependent and independent variables and for the binary logistic function. The sample size of the data was 344 observations and the data set were divided in two groups where, the first group anemic with n₁ = 123 which represents the 35.8% of observations, and the second group is not anemic with n₂ = 221 which represents the 64.2% 0f observations. Kolmogorov-Smirnov test statistic was used for testing normality of data and had a value of 0.414 with p-value <0.000. The DA was able to correctly classify 237 cases of patients out of 344 cases. while, the AER was 29.8% and the ACCR was 69.2% indicating that the model has ability on classification. The LRA was able to classify cases of patient out of 237cases correctly. The AER was 31.1% and the ACCR was 68.9% indicating that the model has ability on classification.