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Statistical Model for Detecting Probability of Severity Level of Hemophilia A

Chris P Tsokos ¹Department of Health Science, University of South Florida, Florida, USA
^*Correspondence: AKM Raquibul Bashar, Department of Health Science, University of South Florida, Florida, USA, Tel: 8135989313, Email:

Received: 04-Dec-2019, Manuscript No. IPHSJ-19-3026; Editor assigned: 09-Dec-2019, Pre QC No. iphsj-19-3026 (PQ); Reviewed: 23-Dec-2019, QC No. iphsj-19-3026; Revised: 20-Jul-2022, Manuscript No. iphsj-19-3026 (R); Published: 17-Aug-2022

Introduction

As a rare bleeding disorder disease, it is very important to develop and identify the severity level of hemophilia [1]. Typically, this is done by doing several blood tests also termed as screening tests in medical science domain [2]. The types of screening tests include Complete Blood Count (CBC), Activated Partial Thromboplastin Time (APTT) test, Prothrombin Time (PT) test, fibrinogen test, clotting , factor Tests [3]. In case of that hemophilia study, the last test (CFT-Clotting Factor Tests), is the medical standard to detect and tag the severity level of hemophilia [4]. But, in some study, the severity level based on clotting factor-factor VIII or F8 is one of the predictors in the outcome of Immune tolerance induction [5]. Studies have been done only taking the Inhibitors and F8 into considerations just to study these attributes only on African American and Black population [6]. In other literature, the parametric study has been done only on F8 mutation type and Inhibitor development [7]. But, very little has been done on developing a statistical model that can identify and predict the severity level with the concept of probability considering the confidence limit on those probability predictions [8]. For these reasons, the main focus of this study is to develop a statistical prediction model to predict the probability of severity level based on diagnosis reports collected from different HTCs (Hemophilia Treatment Center) in the USA [9].

Hemophilia: A rare disease

Hemophilia is caused by a mutation or change in one of the genes that provide instructions for making the clottin g facto r proteins needed to form a blood clot [10]. This change or mutation can prevent the clotting protein from working properly or from being missing altogether [11]. These genes are located on the X chromosome. Males have one X, and one Y chromosome (XY) and females have two X chromosomes (XX) [12]. Males inherit the X chromosome from their mothers and the Y chromosome from their fathers [13]. Females inherit one X chromosome from each parent, as shown in the following (Figure 1) [14].

The X chromosome contains many genes that are not present on the Y chromosome [15]. This implies that males only have one copy of most of the genes on the X chromosome, whereas females have two copies [16]. Thus, males can have a disease like hemophilia if they inherit an affected X chromosome that has a mutation in either the factor VIII or factor IX gene [17].

Females can also have hemophilia, but this is much rarer. In such cases, both X chromosomes are affected, or one is affected, and the other is missing or inactive. In these females, bleeding symptoms may be similar to males with hemophilia. A female with one affected X chromosome is a carrier of hemophilia [18]. Sometimes a female who is a carrier can have symptoms of hemophilia [19]. Also, she can pass the affected X chromosome with the clotting factor gene mutation on to her children [20].

Materials and Methods

Because of the fact that the focus of this study is to identify and develop a statistical model that can predict the severity level of the patient’s/individual’s Hemophilia, we have collected data from the open and freely accessible database of CDC called CHAMP 8 database. The results produced by this study are solely based on the dataset we have at hand [21].

Data description

The CHAMP mutation list was collected and compiled by CDC and made freely accessible and available to public use by downloading at CHAMP Mutaon database for statistical reporting and analysis only [22].

In this present statistical study and analysis, we have considered the Severity level definition based on HGVSstandardized nomenclature. It states that, if the clotting factor - F8 in blood is between 50% to 100% then the person/individual is in Normal state and does not have hemophilia. On the other hand, if the F8 ranges between 5%<F8<50% then in medical science it is termed as Mild Level of Hemophilia A and if any individual’s CFT (Clotting Factor Test) shows the results as 1% ≤ F8 ≤ 5% or F8<1% then that individual is termed as having Moderate or Severe level of hemophilia respectively. This categorization/classification has been defined only by HGVSstandardized nomenclature as per medical research and study. So, we have used this as a response variable to the predictor/ attributable variables such as mutation, mechanism, subtype, domain, race, and inhibitor information as the categorical attributes and Exon, Codon as the continuous covariates. The schematic diagram of the data set is shown in the Figure 2.

From the above diagram, we see that the dataset consists of 6 categorical variables and 2 continuous variables and each of the categories for every categorical variable has some missing values. In some cases so do the continuous variables. Only the categorical variable Inhibitor does not contain any missing information and because of the nature and objectives of this study, we have considered ‘F8 Severity Level ‘the response variable and all other variables were considered as the predictor variables to be.

Statistical modeling

For the purpose of the study, we have explored the relationship and association of each of the categorical predictor variables against the response variable ‘F8 Severity Level ‘through Mosaic plot. Then we have established the statistical model through the multinomial logistic regression, generalized logistic regression and cumulative logistic regression (taking the ordered categories of the response variable-F8 Severity Level into consideration) and we have compared their results to find the better fitted model if not the best.

Results

To formulate the statistical model that predicts the probability of severity level of F8, we have started with the cross tabulation of predictor variable mutation mechanism and response variable F8 severity level presented in Table 1.

Table 1: Cross tabulation of severity level vs. mutation mechanism.

To visualize the above cross tabulation we have used the mosaic plot to have a better insight of the information presented above given in the Figure 3.

The mosaic plot shows the distribution of mutation mechanism in the DNA categories in the x-axis by dividing that axis into 5 intervals. Its shows that inversion and substitution mechanism are greatly associated to the severity level of F8 in the individuals.

Similarly, we have checked the association between Severity level and Domain of the Mutation given in the cross tabulation in Table 2.

Table 2: Cross tabulation of severity level vs. domain location of mutation.

The mosaic plot of the information given above shows there is association of some of the categories of mutation domain with the severity level of the F8 clotting factor presented in the blood (Figure 4).

It is obvious that the other signals category of domain variable has a very large proportion of population having severe level of F8 clotting factor contained in their blood than all other categories of the domain location of mutation (Figure 5).

Also we have examined the cross-table relationship between severity level and race of individuals presented in Table 3.

Table 3: Cross tabulation of severity level vs. race.

This shows very strong relationship between white nonhispanic category of race variable to the severe category of severity level of hemophilia A (Table 4). At the same time we have explored the cross-table relationship between severity level and mutation type variables in given following (Figure 6).

Table 4: Cross tabulation of severity level vs. mutation type.

Here in that of Figure above, we see the largest category of mutation type is large structural change in the protein is strongly associated to the severe level of hemophilia a bleeding disorder. It implicates the fact that large structural change catogory of mutation type covariate might have a significant effect on the outcome variable severity level (Table 5).

Table 5: Cross tabulation of severity level vs. inhibitor history of the individuals.

It represents the association of inhibitor built in protein to help the blood clotting and prohibit the mutation in the DNA causing F8 to reduce is very important variable associated with the severity level of the hemophilia A. This table shows the evidence that approximately 77% of the individuals don’t built the inhibitor in their blood that causing almost 47% to have very severe level of hemophilia A. This confirmed by the mosaic plot depicted in Figure 7.

It shows that, the biggest tile in this mosaic is at the intersection of No and severe category of inhibitor and severity level variables respectively.

After taking all of the associations presented in the cross tables and mosaic plots, we have built models from several approaches. Since, we have response variable with three categories and those categories are ordered based on the values and nature of the study dataset, the cumulative logistic regression or ordinal logistic regression is the most appropriate modeling approach suggested by some scholars in the medical research. Also, multinomial logistic regression and Generalized Estimating Equations (GEE) [10]. Are some alternative options considering the response variables are name categories only. So we have built the model considering all the modeling approaches mentioned above and we have taken the model based on the quickest convergence and highest prediction accuracy and maximum Information criterion. Considering all the factors we have started with the “Multinomial Logistic Regression”. Then we have modeled the data through the “Generalized Estimating Equation (GEE)” and after this we have modeled through “cumulative logistic regression”or in other words termed as “ordinal logistic regression”. After estimating the parameters for each of the models those were compared with each other and as a validation of the model we have comparedthe convergence, information criterion and probability prediction quality to come up with the best modeling so far by the modeling objectives and methodology.

There several types of multinomial logistic models can be used based on the type of information on response variables in data at hand. If the response variables are in Nominal scale then Generalized Logit Models (GLM) and the Conditional Logit Models (CLM) can be used. The GLM consists of estimating parameters of several binary logistic models simultaneously. On the other hand, the CLM is used in biomedical research in order to estimate relative risks in matched case-control studies. Since, we have polytomous response variable which is in ordered structure with a set of regressors (attributes), the most appropriate modeling method would be the cumulative logistic regression. But to have the best approximate model of the real world phenomena, we have estimated all the aforementioned models and compared the statistic to make our final decisions and the following sections will be discussed on our model building methods.

Generalized Logit Model (GLM)

The generalized logistic model focuses on the individual that is considered the unit of analysis and this GLM model uses individual characteristics as explanatory variables. The explanatory variables that being characteristics of an individual are constant over the alternative choices of the response variable. Considering m nominal choices of the response variable, jk denote the probability that the individual j falls in category k. Also, let X_j represent the characteristic of individual j. The probability of the individual

j falling in category k is given by

Here, γ₁ γ_m are m vectors of unknown parameters where each of the estimates are different even though X_j is constant accross other categories or choices. The model to be fit.

Modeling our hemophilia a data by GLM

Using GLM method, the list significantly effective variables ordered as per the p-value of wald chi-square from smallest to largest in the model are shown in the following Table 6.

Table 6: Ranking of covariates in the generalized logistic regression.

Taking statistically significant covariates as per the above into consideration and letting severe as the reference category of response variable, the final generalized models are:

The equation for mild category of response variable where severe as the base reference category:

Generalized Estimating Equations (GEE)

This modeling technique is widely used in analyzing longitudinal data when the average effect of population is the primary interest of the study objectives. Let, y_ij, j=1 n_i and i=1k represents jth response of the i^th subject and has the vector of attributes x_ij, then, there are ni measurements on subject i and maximum number of measurements per subject is T. Also, if μij is the marginal mean of response yij which is related to a linear predictor through a link function g(μ_ij)=xJ_ijθ, then the variance of y_ij depends on the mean through a variance function v(μ_ij). An estimate of the parameter θ can be solved by generalized estimating equations.

Here, V_i is the covariance matrix of Y_i.

Cumulative Logistic Model (CLM)

Suppose, Y takes values of y₁, y₂,…ym such that y₁<y₂,<,<y_m, it is assumed that the observed variable is categorized through a continuous latent variable U such that, Y=y_i ⇔ α_i-1<U ≤ α_i, i=1, m where −∞=α₀<α₁<α_m=∞. The assumption on U is that the value will be determined by the attributable variable vector X in the form of a linear function U=−λjx+ϵ where, λ is vector of regression coefficients and ϵ is a random variable with a distribution function F that assumed to follow P(Y ≤ y_i|x)=F(α_i+j_x) logistic distribution. Moreover, in the CLM concept, alternatively known as proportion odds model, it is assumed that the predictor variable, let X, takes different values for each of the alternative categories and effect of a unit of X is assumed constant across different alternative categories. Under these assumptions, the probability that an individual j will fall into category k is:

Modeling our hemophilia a data by CLM

In order to appy CLM method to our data set at hand, let our response variable be F8 Severity Level=Y_j={y_j1, y_j2, y_j3} where the ordering is in reverse order of category 1 indicates the most severe case (Severe) and category 3 indicates least severe (mild). The associated probabilities are {π_j1, π_j2, π_j3}, and a cumulative probability of a response less than equal to Y_j is:

Then the CLM would be:

The sequence of cumulative logit models will be:

Or alternatively the simplified versions of the above models can be expressed as by the following equations as well:

Here, γ is indicator for categories of each categorical variable and other notations:

Here, we should notice that the intercepts are changing from one model to another but the slopes are equal for all the models. Also, we need to estimate 2-intercepts and p-slopes. In our case p=7, so we will have 7 slopes for 7 covariates and 2 intercepts should make the full model. The estimated final model is given in equation.

The model given in equation, is formulated considering all the attributable variables in the given data set of hemophilia A disease. But if the covariates are ranked based on their effects on the model, then there are some v ariables which are not statistically significant. The following represents the ranking of statistically significant covariates ranked f rom that of the most statistically significant to statisti cally insignificant variables of (Table 7).

Table 7: Ranking of covariates in the cumulative logistic regression.

Validation of the proposed model

From the previous section, it is recommended that, in the modeling purpose with the data set at hand, we should include only 3 IVs (Independent Variables) as per Table , where as that thestatistically significant variables are ranked as per their corresponding p-values.

So, by taking this with the significance into considerations the final proposed model for the hemophilia dataset we have at our hand is given in the Equation below:

As a part of validation of the model presented in equation, from it clearly shows the insignificance of the proportional odds assumption of the cumulative logistics Model and from this fact it can be concluded that the relationship between Severe category and mild category or the severe and moderate category are the same with respect to the proportional odds. In other words, the co-efficients that describe the relationship between Severe level of disease vs. mild and moderate categories of the response variable (severity level) are the same as those coefficients that describe the relationship between Moderate and mild category of response variable which is in other words called parallel regression assumption as shown in the Figure 8.

From the above, it indicates that the proportional odds assumption is not violated and the diagonal parallel lines indicate the parallel regression assumptions as well. This is also confirmed by the chi-square test of the model given in the Table 8.

Table 8: Testing for proportional odds assumption.

Concerning the model fit, AIC, SC, and LogL given in Table 9 indicate that the fitted model is statistically significant also From the model given in equation, it indicates that 1 unit change of domain protein (A2), we expect that there will be approximately 42% increase in the log odds being in the lower level of the severity level from sever to mild when all other covariates are held constant. So, there are always a positive increase in the log odds for one unit increase in each of the domain protein category. On the other hand, one unit increase or change from one category of race to other category of race will have negative effect in the log odds of being from mild to moderate category of severity level so is the third categorical variable, mutation.

Table 9: Model fit statistics for hemophilia A data.

Now, we want to rank the attributable variables with respect to the effects in the model. Table th at shows the ranking of attributable variables with respect to their p-value in the model. The first variable in case of GLM is Mutation and last variable is Codon and it is also the same when the modeling is done with CLM also. But things to be noted in CLM method is that the ranking of statistically significant variables changes as the modeling technique changes. For example, the third variable in GLM from top is Race of individuals, but in CLM method, the third variable from top is Domain (Location of Mutation Domain) at 5% level of significance.

Here, for instance, it is very important to discuss the proposed model given in equation and model in equation the following table shows the comparison of some vital information about the full model and the model that is built considering the statistically significant covariates only (Table 10).

Table 10: Comparison of association of predicted probabilities and observed responses.

Discussion

From the table above, we can see that the percentage of concordant pairs of CLM while considering all the covariates in the model is approximately 4% less than that of the CLM built with statistically significant covariates. Also, % of area covered under the ROC curve is 6% higher in CLM with significant variables than that of the CLM with all covariates. After comparing the statistic given in it is statistically conclusive that the model given in equation is a better predictive model to predict the probability of the severity level of hemophilia A with F8 as per our dataset at hand.In terms of finding the best model driven by our data at hand, the Cumulative Logistic Regression considering the statistically significant covariates only as the independent variables has fast convergence rate and best results based on our data apprently. This model is predicting the probability for each of the response category with about 87% accuracy under the ROC as per C statistic.

In the context of the disease, the model given in Equation, has not violated the proportional odds assumption of the CLM. In brief, as per model given in the equ ation it indicates that, protiens A1, A2, A3, B, C1 of Domain mutation will effect the probability of severity level of any individual in an increasing manner, i. e., if the doctors and scientists can identify these protiens in the blood then they have to provide some treatments that will locate these protiens and reduce their positive effects to increase patients probability of being in the mild category (lowest category of severity level in hemophilia A) from very severe category of the disease and it has to be opposite in case of C2 protien to improve or alleviate the severity level of any patient. As per model in Equat ion race is affecting the probability of individuals being in one of the three categories of response variable and there are no real life treatment of to chang race of individuals we might counclude that being in the different categories of Severity level by race categories are totally in hands of mother nature. But, it is conclusive that majority of patients in sever cateogry comes from white non-hispanic rather than other categories of race. On the other hand, the (Nonsense) has the smallest coefficient in the model mentioned above indicates that Nonsense type mutation change in the gene of individual will have maximum effect on the response outcome and to change/alleviate the severity level of any individual, it is suggested to attack/reverse this particular type of mutation cause and consequently change other types of mutaions.

Conclusion

The objectives of this study were to indentify statistically signifi--cant variables that effects the outcome variable “Severity Level”. Also, we wanted to see statistically significant categories of variables interacting with each other effecting the categories of response variable if any. At the same time we wanted to rank the main effect variables that are contributing to the response and eventually come upe with a model that is statistically siginificant and robust with high degree of prediction porbability prediction and convergence. So, we have indentified statistically significant variables that effects the Severity Level by implementing various methodology for model building and we have compared them through some statistic. It turns out that the attributable variables Race, Domain and Mutation type are the most significant variables to predict the probability of the severity level of hemophilia A statistically shown in Tables. Also, we have investigated the interaction terms among the attributable variables and it turns out that there were no interactions among variates as per out data concerns.

In terms of practical relevancies, our model will predict the probability of severity level of any individuals with 87% accuracy. So, if any individual goes to Medical doctor and after getting the results of blood diagnosis and if the individual provides available information on Domain (Location of mutation change), Mutation (Type of Protein change) and Race of individuals then Medical doctors will be able to identify the Level of Severity for that particular individual and based on this medical physician will be able decide proper treatment program for that individual afetr having the genetic profile analyzed.

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