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December 28, 2023

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MLA

Das, Mahashweta, et al. Role of Serum Sodium on Heart, Anemia, and Diabetes Patients. Vol. 1, no. 1, Academia Medicine, 2023. https://doi.org/10.20935/AcadMed6167

APA

Das, M., Medda, S. K., Banik, S., & Das, R. N. (2023). Role of serum sodium on heart, anemia, and diabetes patients, 1(1). https://doi.org/10.20935/AcadMed6167

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Das, Mahashweta, Sunit Kumar Medda, Shipra Banik, and Rabindra Nath Das. “Role of Serum Sodium on Heart, Anemia, and Diabetes Patients” 1, no. 1 (2023). doi:10.20935/AcadMed6167.

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Das M, Medda SK, Banik S, Das RN. Role of serum sodium on heart, anemia, and diabetes patients. 2023;1(1). https://doi.org/10.20935/AcadMed6167

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Das, M. et al. (2023) “Role of serum sodium on heart, anemia, and diabetes patients.” Academia Medicine, 1(1). doi: 10.20935/AcadMed6167.

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November 22, 2023

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December 16, 2023

Published

December 28, 2023

DOI

doi.org/10.20935/AcadMed6167

Das, Medda, Banik, Das, Amugsi, and Ansaloni: Role of serum sodium on heart, anemia, and diabetes patients

Role of serum sodium on heart, anemia, and diabetes patients

Mahashweta Das [1]

Sunit Kumar Medda [2]

Shipra Banik [3]

Rabindra Nath Das* [4]

Author Affiliations

¹Department of History, The University of Burdwan, Burdwan, 713104, West Bengal, India.

²Department of Surgery, Kalyani J.N.M. Hospital, Kalyani, 741235, West Bengal, India.

³Department of Physical Sciences, Independent University, Dhaka, 1510, Bangladesh.

⁴Department of Statistics, The University of Burdwan, Burdwan, 713104, West Bengal, India.

*Correspondence to rndas@stat.buruniv.ac.in

Open Access

Abstract

This article examines the role of serum sodium (SNa) on heart, anemia, and diabetes patients based on a real data set of 299 patients and probabilistic modeling. It is derived herein that mean SNa is positively associated with the marginal effects of platelets count (PLC) (P = 0.0019), anemia status (ANS) (P = 0.1053) (partially), creatinine phosphokinase (CRP) (P = 0.0462), smoking habit (SMH) (P= 0.1017), SEX (P = 0.0029), and the joint interaction effects (JIEs) of ejection fraction (EJF) and the status of high blood pressure (HBP), i.e., EJF*HBP (P = 0.1398); ANS and death event (DEE), i.e., ANS*DEE (P = 0.0009); age and time to follow-up (TTF), i.e., AGE*TTF (P = 0.0700); age and serum creatinine (SEC), i.e., AGE*SEC (P = 0.0901); EJF and SEC, i.e., EJF*SEC (P = 0.0044); and CRP and TTF, i.e., CRP*TTF (P = 0.0021). Mean SNa is negatively associated with the marginal effects of DEE (P = 0.0001), TTF (P= 0.0521), SEC (P = 0.0028), diabetes status (DIS) (P = 0.0121), and the JIEs of PLC*ANS (P = 0.1064), CRP*SMH (P = 0.0274), CRP*HBP (P = 0.0044), AGE*PLC (P = 0.0811), AGE*SEX (P = 0.0008), AGE*CRP (P = 0.0463), and EJF*PLC (P = 0.0420). The variance of SNa is marginally positively associated with AGE (P = 0.0933), DEE (P = 0.1408), and the JIEs of AGE*ANS (P = 0.0575), EJF*SEX (P = 0.1001), SEC*SMH (P = 0.0091), EJF*TTF (P = 0.0008), CRP*DIS (P = 0.0039), EJF*DEE (P = 0.0022), and SEC*DIS (P = 0.0001). Also the variance of SNa is marginally negatively associated with ANS (P = 0.0631), EJF (P = 0.0008), DIS (P = 0.0548), PLC (P = 0.0023), SMH (P = 0.0242), TTF (P = 0.0018), CRP (P = 0.0027), and the JIEs of AGE*DEE (P = 0.0003) and SEC*SEX (P = 0.1327). SNa maintains a complex relationship with different patients and factors, and they have both marginal and JIEs on SNa. Medical practitioners and researchers should care about the complicated functional roles of SNa.

1. Introduction

Serum sodium (SNa) disorders are common situations in hospitalized patients. It is an important electrolyte for human beings. It has many roles in the human body. However, the associations of SNa with heart, diabetes, and anemia patients are not clear. SNa concentration is a general index in intensive care units, and disturbances in Na concentration are connected with increased mortality in critically sick subjects [1, 2]. Dysnatremias [hyponatremia (<136 mmol/L) and hypernatremia (>145 mmol/L)] can severely affect several physiologic organ systems and functions [3–5]. Many previous articles have pointed out that hyponatremia and hypernatremia are independent mortality risk factors in patients with acute subarachnoid hemorrhage [6], kidney disease [7], pneumonia [4, 8], etc.

Diabetes is connected with several significant electrolyte disorders, predominantly affecting SNa, magnesium, and potassium [9]. SNa is a component of normal physiological systems, and type-2 diabetes (T2D) patients may feel osmotic diuresis as a consequence of disease-connected hyperglycemia, contributing to the excess Na excretion in the urine and resulting in hyponatremia [10]. This type of hyponatremia can bring a wide range of severe clinical findings and pathophysiological changes in T2D patients [11]. Electrolyte abnormality hyponatremia is most often encountered in patients admitted to hospitals for heart failure [12–14]. Most research on the prognostic value of blood sodium levels in heart failure patients has shown that hyponatremia is related to significantly lower blood pressure (BP) value, which is also independently related to higher risk [15]. One-third of the global population is suffering from anemia disease, which affects more than two billion people worldwide with mortality around 800,000 per year [16]. Some studies have shown that anemia is an imbalance in the serum electrolyte levels, which is the alteration in red cell membrane-bound sodium-potassium adenosine triphosphatase pump activity that governs intra- and extracellular cation homeostasis [17]. Most of the earlier studies are based on meta-analysis or multivariate analysis that may not be able to identify the effects of SNa on the heart, diabetes, and anemia patients [18, 19].

Some previous articles have examined the effects of SNa on anemia, heart, and diabetes patients applying simple bivariate correlation, simple/multiple regression analysis, and machine learning approaches [10, 12, 15, 19]. Most of the previous articles invite some doubts and debates, as the analysis approaches that are used in the previous articles are not appropriate. In addition, appropriate model fitting diagnostics of the used methods are not discussed in the earlier articles. The effects of SNa are little investigated based on probabilistic modeling. The present article searches for the following research problems.

Is there any association of SNa with anemia, heart, and diabetes patients along with other biochemical/physical factors? If it is affirmative, what is the most probable SNa association model?
How can we derive the most probable SNa association model?
What are the effects of SNa on the different patients and the explanatory factors?

The article examines the above research queries considering the following sections such as materials and methods, statistical analysis and results, discussions, and conclusions. The derived SNa probabilistic model is presented in Table 1 based on the data set mentioned in the materials section. The probabilistic SNa mean and dispersion models are obtained by joint generalized linear models (JGLMs), which are described in the methods section. The derived findings are illustrated in the results section, while the results are described in the discussion section. Based on the derived SNa mean and variance probabilistic models, the current article has collected some information that are discussed in the conclusions section.

Table 1

Results for mean and dispersion models for serum sodium from gamma and log-normal fit

Model	Covariates	Gamma fitted model				Log-normal fitted model
Model	Covariates	Estimate	S.E.	t(273)	P-value	Estimate	S.E.	t(273)	P-value
Mean	Constant	4.9016	0.0314	155.8	<0.0001	4.9015	0.0316	155.7	<0.0001
	EJF (v5)	0.0002	0.0003	0.496	0.6203	0.0002	0.0004	0.521	0.6027
	HBP (Fv6)	−0.0020	0.0088	−0.228	0.8198	−0.0019	0.0088	−0.211	0.8330
	EJFHBP (v5Fv6)	0.0003	0.0002	1.481	0.1398	0.0003	0.0002	1.465	0.1441
	PLC (v7)	0.0001	0.0001	3.127	0.0019	0.0001	0.0001	3.116	0.0020
	ANS (Fv2)	0.0143	0.0088	1.625	0.1053	0.0146	0.0088	1.657	0.0986
	PLCANS (v7.Fv2)	−0.0001	0.0001	−1.620	0.1064	−0.0001	0.0001	−1.654	0.0993
	CRP (v3)	0.0001	0.0001	2.002	0.0462	0.0001	0.0001	1.985	0.0481
	SMH (Fv11)	0.0057	0.0034	1.642	0.1017	0.0057	0.0034	1.640	0.1021
	CRPSMH (v3Fv11)	−0.0001	0.0001	−2.217	0.0274	−0.0001	0.0001	−2.206	0.0282
	CRPHBP (v3Fv6)	−0.0001	0.0001	−2.868	0.0044	−0.0001	0.0003	−2.882	0.0042
	DEE (Fv13)	−0.0145	0.0037	−3.870	0.0001	−0.0145	0.0037	−3.860	0.0001
	ANSDEE (Fv2Fv13)	0.0179	0.0053	3.348	0.0009	0.0179	0.0053	3.356	0.0009
	AGE (v1)	0.0002	0.0004	0.505	0.6139	0.0002	0.0004	0.475	0.6351
	TTF (v12)	−0.0002	0.0001	−1.948	0.0521	−0.0002	0.0001	−1.953	0.0518
	AGETTF (v1v12)	0.0001	0.0001	1.819	0.0700	0.0001	0.0001	1.826	0.0689
	AGEPLC (v1v7)	−0.0001	0.0001	−1.751	0.0811	−0.0001	0.0001	−1.726	0.0855
	SEC (v8)	−0.0324	0.0107	−3.016	0.0028	−0.0322	0.0107	−3.006	0.0029
	AGESEC (v1v8)	0.0002	0.0001	1.701	0.0901	0.0002	0.0001	1.711	0.0882
	SEX (Fv10)	0.0400	0.0133	2.999	0.0029	0.0400	0.0133	2.998	0.0030
	AGESEX (v1Fv10)	−0.0007	0.0002	−3.371	0.0008	−0.0007	0.0002	−3.368	0.0008
	AGECRP (v1v3)	−0.0001	0.0001	−2.001	0.0463	−0.0001	0.0001	−1.977	0.0490
	EJFPLC (v5v7)	−0.0001	0.0001	−2.043	0.0420	- 0.0001	0.0001	−2.038	0.0425
	EJFSEC (v5v8)	0.0003	0.0001	2.872	0.0044	0.0003	0.0001	2.831	0.0049
	DIS (Fv4)	−0.0079	0.0031	−2.524	0.0121	−0.0082	0.0031	−2.602	0.0097
	CRPTTF (v3v12)	0.0001	0.0001	3.105	0.0021	0.0001	0.0001	3.112	0.0020
Dispersion	Constant	−4.2727	1.4067	−3.037	0.0026	−4.2843	1.4065	−3.046	0.0025
	AGE (v1)	0.0206	0.0123	1.684	0.0933	0.0207	0.0123	1.682	0.0937
	DEE (Fv13)	1.8254	1.2360	1.477	0.1408	1.8360	1.2380	1.483	0.1392
	AGEDEE (v1Fv13)	−0.0639	0.0174	−3.680	0.0003	−0.0642	0.0174	−3.694	0.0003
	ANS (Fv2)	−1.9406	1.0397	−1.866	0.0631	−1.9536	1.0406	−1.877	0.0615
	AGEANS (v1Fv2)	0.0322	0.0169	1.907	0.0575	0.0324	0.0169	1.918	0.0561
	EJF (v5)	−0.0872	0.0257	−3.389	0.0008	−0.0870	0.0257	−3.381	0.0008
	DIS (Fv4)	−0.8928	0.4630	−1.928	0.0548	−0.9170	0.4645	−1.974	0.0493
	SEX (Fv10)	−0.4122	0.8363	−0.493	0.6224	−0.3816	0.8396	−0.455	0.6494
	EJF (v5*Fv10)	0.0270	0.0164	1.650	0.1001	0.0265	0.0164	1.619	0.1065
	PLC (v7)	−0.0001	0.0001	−3.077	0.0023	−0.0001	0.0001	−3.071	0.0023
	SEC (v8)	−0.1810	0.1705	−1.061	0.2896	−0.1801	0.1715	−1.050	0.2946
	SECSEX (v8Fv10)	−0.5337	0.3540	−1.508	0.1327	−0.5457	0.3593	−1.519	0.1299
	SMH (Fv11)	−1.3067	0.5768	−2.265	0.0242	−1.3169	0.5815	−2.265	0.0243
	SECSMH (v8Fv11)	1.0433	0.3972	2.626	0.0091	1.0575	0.4016	2.633	0.0089
	TTF (v12)	−0.0158	0.0050	−3.146	0.0018	−0.0159	0.0050	−3.146	0.0018
	EJF (v5*v12)	0.0005	0.0001	3.369	0.0008	0.0005	0.0001	3.366	0.0008
	CRP (v3)	−0.0007	0.0002	−3.024	0.0027	−0.0007	0.0002	−3.024	0.0027
	CRPDIS (v3Fv4)	0.0008	0.0003	2.906	0.0039	0.0008	0.0003	2.920	0.0038
	EJFDEE (v5Fv13)	0.0657	0.0212	3.095	0.0022	0.0658	0.0212	3.098	0.0022
	SECDIS (v8Fv4)	1.1370	0.2918	3.896	0.0001	1.1627	0.2938	3.958	<0.0001
AIC		1,547.205				1,548

2. Materials and statistical methods

2.1. Materials

The SNa probabilistic model is obtained herein from a data set of 299 heart failure patients obtained from the Faisalabad Cardiology Institute under the Allied Hospital at Faisalabad in Punjab province, Pakistan, for nine months, i.e., April to December 2015 [20, 21]. The data set is available on the site https://archive.ics.uci.edu/ml/datasets/Heart+failure+clinical+records (accessed on 22 November 2023). The data set is clearly described by the authors in the article [21]. The considered study [20] contained 105 female and 194 male study subjects, who had left ventricular systolic dysfunction. The considered study subjects had previous heart failures and were grouped in III or IV heart failure stages of the New York Heart Association (NYHA) classification [22]. The study ethics approval and the subject’s consent are clearly described in the original article [20], which are not reproduced herein. An article by Chicco and Jurman [23] describes clearly the considered heart failure data set using two tables that illustrate attribute and continuous characters separately.

The present study includes 13 characters out of which 6 are attribute characters and 7 are continuous variables. The attribute characters are anemia status (ANS) patients (0 = no anemia, 1 = anemia) (Fv2), diabetes status (DIS) patients (0 = no diabetes, 1 = diabetes) (Fv4), high blood pressure (HBP) patients (0 = normal BP, 1 = high BP) (Fv6), gender (0 = female, 1 = male) (Fv10), smoking habit (SMH) (0 = no smoking, 1 = smoking) (Fv11), death event (DEE) (0 = alive, 1 = death) (Fv13), while the continuous study characters are age (v1), creatinine phosphokinase (CRP) (v3), ejection fraction (EJF) (v5), platelets count (PLC) (v7), serum creatinine (SEC) (v8), SNa (v9 = x9), time up to the end of the follow-up period (TTF) (v12). A study subject was grouped as an anemia patient (Fv2) if his/her hematocrit levels were lower than 36% [20]. The original article [20] does not provide any information regarding the classification of high BP (Fv6) patients. The DEE (Fv13) indicates that the patient died (=1) or survived (=0) up to the end of the follow-up period, which was 130 days on average [20].

The CRP shows the CRP enzyme levels in the blood. When a muscle tissue gets damaged, CRP flows into the blood. Therefore, high CRP enzyme levels in the blood of a subject might cause heart failure or injury [24]. The SEC is a waste product produced by creatine, when a muscle breaks down. High SEC levels in the blood may indicate renal dysfunction [25]. Medical doctors/practitioners focus on SEC in blood to examine kidney function. The main article [20] containing the data source unfortunately does not provide any information whether any patient had primary kidney disease or not. In addition, it does not give any further information about the type of follow-up carried out for locating kidney disease subjects. The EJF shows the percentage of how much blood the left ventricle pumps out with each contraction. SNa is a mineral substance, which is responsible for the correct functioning of nerves and muscles. The SNa blood routine test of a subject shows the normal or abnormal levels of sodium in the blood. An abnormally low level of SNa in the blood might be caused by heart failure [26]. Additional information about the whole data set can be found easily in the main article [20], and interested readers are suggested to go through it.

2.2. Statistical methods

The present study considers SNa as the targeted response random variable, which is to be modeled with the remaining cardiac/diabetes/anemia disease status and biochemical and physiological characteristics. It is observed that the response SNa is a non-normally and heteroscedastic distributed random variable. The variance of SNa can’t be stabilized with the help of any suitable transformation; therefore, it is modeled in the report (herein) using JGLMs under both the gamma and log-normal distributions, which are clearly described in references [27–30]. For elaborate discussion regarding JGLMs, interested readers may go through the book by Lee, Nelder, and Pawitan [27]. JGLMs for both the log-normal and gamma distributions are shortly reported herein.

JGLMs for log-normal distribution

For the positive response Y_i (=SNa) with E(Y_i = SNa) = µ_i (mean) and Var(Y_i = SNa) = $σ_{i}^{2} μ_{i}^{2} = σ_{i}^{2} V (μ_{i})$ say, where $σ_{i}^{2}$ ’s are dispersion parameters and V (∙) reveals the variance function. Generally, log transformation Z_i = log(Y_i = SNa) is adopted to stabilize the variance Var(Z_i) ≈ $σ_{i}^{2}$ , but the variance may not always be stabilized [31]. For deriving an SNa improved model, JGLMs for the mean and dispersion are considered. For the response SNa, assuming log-normal distribution, JGL mean and dispersion models (with Z_i = log(Y_i = SNa)) are as follows:

E (Z_{i}) = μ_{Z_{i}} and var (Z_{i}) = σ_{Z_{i}}^{2},

μ_{Z_{i}} = x_{i}^{t} β and \log (σ_{Z_{i}}^{2}) = g_{i}^{t} γ,

where x_i^t and g_i^t are the explanatory factors/variables vectors of SNa associated with the mean regression coefficients β and dispersion regression coefficients γ, respectively.

JGLMs for gamma distribution

In the above-stated Y_i’s (=SNa), the variance has two portions such as $V (μ_{i})$ (based on the mean parameters µ_i’s) and $σ_{i}^{2}$ (free of µ_i’s). The variance function V (∙) displays the GLM family distributions. For instance, if V( $μ$ ) = 1, it is normal, Poisson if $V (μ) = μ$ , and gamma if $V (μ) = μ^{2}$ . Gamma JGLMs mean and dispersion models of SNa are as follows:

η_{i} = g (μ_{i}) = x_{i}^{t} β and ε_{i} = h (σ_{i}^{2}) = w_{i}^{t} γ,

where $g (\cdot)$ and $h (\cdot)$ are the GLM link functions attached with the mean and dispersion linear predictors, respectively, and $x_{i}^{t}$ and $w_{i}^{t}$ are the explanatory factors/variables vectors of SNa attached with the mean and dispersion parameters, respectively. The maximum likelihood (ML) method is used for estimating the mean parameters, while the restricted ML (REML) method is applied for estimating the dispersion parameters, which are explicitly stated in the book by Lee, Nelder, and Pawitan [27].

3. Statistical analysis and results

3.1. Statistical analysis

The response SNa is modeled by JGLMs with both the gamma and log-normal distributions. In the statistical analysis, SNa is considered as the dependent (response) variable, and the remaining 12 variables are considered as the explanatory variables of SNa. The final model of SNa has been chosen based on the lowest Akaike information criterion (AIC) value (within each class), which minimizes both the squared error loss and predicted additive errors [31]. According to the AIC criterion, JGLMs gamma fit (AIC = 1,547.205) and log-normal fit (AIC = 1,548) are identical, as the AIC difference is less than one.

In the mean model, some partially significant effects (<14%) such as EJF*HBP, PLC*ANS, AGE*TTF, AGE*PLC, and AGE*SEC are included. Note that the partially significant effects are termed as confounders in Epidemiology, which are important in the model for better fitting [31]. Based on the marginality rule by Nelder [32], namely that if an interaction effect (e.g., EJF*HBP) is significant or partially significant, all its related lower-order effects (e.g., EJF and HBP) should be included in the model even insignificant. Similarly, in the dispersion model, two partially significant effects such as EJF*SEX and SEC*SEX are included as the confounders. Both the fitted JGLMs analysis results are presented in Table 1.

The developed SNa gamma fitted probabilistic JGLM (Table 1) is a data-generated model that is to be verified by model-checking tools. All the valid interpretations regarding SNa are obtained from the data-generated gamma fitted SNa probabilistic model (Table 1), which should be accepted based on suitable diagnostic checking, that is displayed in Figure 1. In Figure 1a, absolute residuals for the gamma fitted SNa model (Table 1) are plotted with respect to the fitted values, which are almost flat linear, indicating that variance is constant with the running means. Figure 1b presents the normal probability plot for the gamma fitted SNa mean model (Table 1), which does not indicate any lack of fit. Therefore, Figure 1a and b does not indicate any discrepancy in the gamma fitted SNa models (Table 1). The figure indicates that the gamma fitted SNa model is an approximate form of the unknown true SNa model.

3.2. Results

According to the AIC rule, both the gamma fitted and the log-normal models give identical results. There is no discrepancy between these two fitted SNa models [33]. Herein the gamma fitted model outcomes are presented, as its AIC value is lower than the log-normal fit. For the gamma fitted SNa mean model, mean SNa is negatively associated with DIS (P = 0.0121), and the joint interaction effects (JIEs) of EJF and HBP, i.e., EJF*HBP (P = 0.1398) is partially positively associated with the mean SNa, while their marginal effects of EJF and HBP are insignificant. Mean SNa is partially negatively associated with the JIEs of PLC and ANS, i.e., PLC*ANS (P = 0.1064), while it is positively associated with both the marginal effects of PLC (P = 0.0019) and ANS (P = 0.1053). Mean SNa is negatively associated with the JIEs of CRP and SMH, i.e., CRP*SMH (P = 0.0274), while it is positively associated with both the marginal effects of CRP (P = 0.0462) and SMH (P = 0.1017). Mean SNa is negatively associated with the JIEs of CRP and HBP, i.e., CRP*HBP (P = 0.0044), while it is positively associated with the marginal effects of CRP (P = 0.0462) and indifferent to HBP. Mean SNa is positively associated with the JIEs of ANS and DEE, i.e., ANS*DEE (P = 0.0009), while it is negatively associated with the marginal effect of DEE (P = 0.0001) and partially positively associated with ANS. Mean SNa is partially positively associated with the JIEs of AGE and TTF, i.e., AGE*TTF (P = 0.0700), while it is negatively associated with the marginal effect of TTF (P = 0.0521) and indifferent to AGE. Mean SNa is partially negatively associated with the JIEs of AGE and PLC, i.e., AGE*PLC (P = 0.0811), while it is positively associated with the marginal effects of PLC (P = 0.0019) and indifferent to AGE. Mean SNa is partially positively associated with the JIEs of AGE and SEC, i.e., AGE*SEC (P = 0.0901), while it is negatively associated with the marginal effect of SEC (P = 0.0028) and indifferent to AGE. Mean SNa is negatively associated with the JIEs of AGE and SEX, i.e., AGE*SEX (P = 0.0008), while it is positively associated with the marginal effect of SEX (P = 0.0029) and indifferent to AGE. Mean SNa is negatively associated with the JIEs of AGE and CRP, i.e., AGE*CRP (P = 0.0463), while it is positively associated with the marginal effect of CRP (P = 0.0462) and indifferent to AGE. Mean SNa is negatively associated with the JIEs of EJF and PLC, i.e., EJF*PLC (P = 0.0420), while it is positively associated with the marginal effect of PLC (P = 0.0019) and indifferent to EJF. Mean SNa is positively associated with the JIEs of EJF and SEC, i.e., EJF*SEC (P = 0.0044), while it is negatively associated with the marginal effect of SEC (P = 0.0028) and indifferent to EJF. Mean SNa is positively associated with the JIEs of CRP and TTF, i.e., CRP*TTF (P = 0.0021), while it is positively associated with the marginal effect of CRP (P = 0.0462) and negatively with TTF (P = 0.0521).

Figure 1

For the joint gamma fitted models of serum sodium (Table 1), (a) the absolute residuals plot with the fitted values and (b) the normal probability plot for the mean model.

The variance of SNa is negatively associated with PLC (P = 0.0001). SNa variance is negatively associated with the JIEs of AGE and DEE, i.e., AGE*DEE (P = 0.0003), while it is partially positively associated with both the marginal effects of AGE (P = 0.0933) and DEE (P = 0.1408). SNa variance is positively associated with the JIEs of AGE and ANS, i.e., AGE*ANS (P = 0.0575), while it is partially positively associated with the marginal effect of AGE (P = 0.0933) and negatively with ANS (P = 0.0631). SNa variance is partially positively associated with the JIEs of EJF and SEX, i.e., EJF*SEX (P = 0.1001), while it is negatively associated with the marginal effect of EJF (P = 0.0008) and indifferent to SEX. SNa variance is partially negatively associated with the JIEs of SEC and SEX, i.e., SEC*SEX (P = 0.1327), while it is indifferent to both SEC and SEX. SNa variance is positively associated with the JIEs of SEC and SMH, i.e., SEC*SMH (P = 0.0091), while it is negatively associated with the marginal effect of SMH (P = 0.0242) and indifferent to SEC. SNa variance is positively associated with the JIEs of EJF and TTF, i.e., EJF*TTF (P = 0.0008), while it is negatively associated with both the marginal effects of EJF (P = 0.0008) and TTF (P = 0.0018). SNa variance is positively associated with the JIEs of CRP and DIS, i.e., CRP*DIS (P = 0.0039), while it is negatively associated with both the marginal effects of CRP (P = 0.0027) and DIS (P = 0.0548). SNa variance is positively associated with the JIEs of EJF and DEE, i.e., EJF*DEE (P = 0.0022), while it is negatively associated with the marginal effect of EJF (P = 0.0008) and partially positively associated with DEE (P = 0.1408). SNa variance is positively associated with the JIEs of SNa and DIS, i.e., SEC*DIS (P = 0.0001), while it is negatively associated with the marginal effect of DIS (P = 0.0548) and indifferent to SEC.

Gamma fitted SNa mean ( $\hat{μ}$ ) model (Table 1) is

$\hat{μ}$ = exp(4.9016 + 0.0002EJF − 0.0020 HBP + 0.0003 EJF*HBP + 0.0001 PLC + 0.0143 ANS − 0.0001 PLC*ANS + 0.0001 CRP + 0.0057 SMH − 0.0001 CRP*SMH − 0.0001 CRP*HBP − 0.0145 DEE + 0.0179 ANS*DEE + 0.0002 AGE − 0.0002 TTF + 0.0001 AGE*TTF − 0.0001 AGE*PLC − 0.0324 SEC + 0.0002 AGE*SEC + 0.0400 SEX − 0.0007 AGE*SEX − 0.0001 AGE*CRP − 0.0001 EJF*PLC + 0.0003 EJF*SEC − 0.0079 DIS + 0.0001 CRP*TTF),

and the fitted SNa variance ( ${\hat{σ}}^{2}$ ) model is

${\hat{σ}}^{2}$ = exp(−4.2727 + 0.0206 AGE + 1.8254 DEE − 0.0639 AGE*DEE − 1.9406 ANS + 0.0322 AGE*ANS − 0.0872 EJF − 0.8928 DIS − 0.4122 SEX + 0.0270 EJF*SEX − 0.0001 PLC − 0.1810 SEC − 0.5337 SEC*SEX − 1.3067 SMH + 1.0433 SEC*SMH − 0.0158 TTF + 0.0005 EJF*TTF − 0.0007 CRP + 0.0008 CRP*DIS + 0.0657 EJF*DEE + 1.1370 SEC*DIS).

From the above mean and variance equations of SNa, it is observed that mean SNa is explained by EJF, HBP, PLC, ANS, CRP, SMH, DEE, AGE, TTF, SEC, SEX, DIS, EJF*HBP, PLC*ANS, CRP*SMH, CRP*HBP, ANS*DEE, AGE*TTF, AGE*PLC, AGE*SEC, AGE*SEX, AGE*CRP, EJF*PLC, EJF*SEC, CRP*TTF, while the variance of SNa is explained by AGE, DEE, ANS, EJF, DIS, SEX, PLC, SEC, SMH, TTF, CRP, AGE*DEE, AGE*ANS, EJF*SEX, SEC*SEX, SEC*SMH, EJF*TTF, CRP*DIS, EJF*DEE, SEC*DIS,

4. Discussion

The summarized SNa analysis outcomes are presented in Table 1. The above two gamma fitted SNa equations show mean and variance models (Table 1), which are very complicated. The mean SNa model shows that mean SNa is negatively associated with DIS (0 = no diabetes, 1 = diabetes) (P = 0.0121), implying that SNa level is higher for non-diabetic subjects than diabetic. This is observed in practice, and it confirms the earlier published outcomes [10, 11]. Therefore, diabetes patients should always care on the SNa level.

Mean SNa is partially positively associated with the JIEs of EJF and HBP (0 = normal BP, 1 = high BP), i.e., EJF*HBP (P = 0.1398), while the marginal effects of EJF and HBP are insignificant. This implies that high SNa level increases the subject’s BP level along with the EJF. However, EJF and HBP are not associated with SNa. It is well known that high SNa levels increase the BP [12–14]. Here it is shown that high SNa level increases the BP along with the joint effect of EJF.

Mean SNa level is partially negatively associated with the JIEs of PLC and ANS, i.e., PLC*ANS (P = 0.1064), while it is positively associated with both the marginal effects of PLC (P = 0.0019) (significantly) and ANS (0 = no anemia, 1 = anemia) (P = 0.1053) (partially). This implies that the mean SNa level is higher for non-anemia subjects with higher PLC level than anemia subjects with lower PLC level. In addition, mean SNa level is significantly positively associated with PLC (P = 0.0019) level, which indicates that SNa level increases as PLC increases. This is observed in practice. Also, mean SNa level is partially positively associated with ANS (0 = no anemia, 1 = anemia) (P = 0.1053) that acts as a confounder in the model. Mean SNa level is partially negatively associated with the JIEs of AGE and PLC, i.e., AGE*PLC (P = 0.0811), while it is positively associated with the marginal effect of PLC (P = 0.0019) and it is insignificant of AGE (P = 0.6139). This implies that SNa level is higher at younger ages with higher PLC level than at older ages with lower PLC level. Mean SNa level is significantly negatively associated with the JIEs of EJF and PLC, i.e., EJF*PLC (P = 0.0420), while it is positively associated with the marginal effect of PLC (P = 0.0019) and indifferent to EJF (P = 0.6203). This implies that the SNa level is higher at the lower joint effect of EJF and PLC.

Mean SNa level is significantly negatively associated with the JIEs of CRF and SMH, i.e., CRP*SMH (P = 0.0274), while it is positively associated with both the marginal effects of CRP (P = 0.0462) (significantly) and SMH (0 = no smoking, 1 = smoking) (P = 0.1017) (partially). It indicates that mean SNa level is higher for non-smoker subjects along with higher CRP level than smokers with lower CRP level. In addition, mean SNa level is significantly positively associated with the CRP (P = 0.0462) level, implying that SNa level increases as CRP level rises. Note that mean SNa level is partially positively associated with SMH (P = 0.1017), which acts as a confounder in the model. Mean SNa is negatively associated with the JIEs of CRP and HBP (0 = normal BP, 1 = high BP), i.e., CRP*HBP (P = 0.0044), while it is positively associated with the marginal effect of CRP (P = 0.0462) and indifferent to HBP. It indicates that mean SNa level is higher for the subjects with normal BP along with high CRP level. Also, mean SNa is negatively associated with the JIEs of AGE and CRP, i.e., AGE*CRP (P = 0.0463), while it is positively associated with the marginal effect of CRP (P = 0.0462) and indifferent to AGE. It shows that mean SNa level is higher for the subjects at younger ages along with high CRP level than older ages with low CRP level. Further, mean SNa is positively associated with the JIEs of CRP and TTF, i.e., CRP*TTF (P = 0.0021), while it is positively associated with the marginal effect of CRP (P = 0.0462) and negatively with TTF (P = 0.0521). It implies that mean SNa level is higher for the subjects with higher TTF and high CRP level than lower TTF with low CRP level.

Mean SNa is positively associated with the JIEs of AGE and TTF, i.e., AGE*TTF (P = 0.0700), while it is negatively associated with the marginal effect of TTF (P = 0.0521) and indifferent to AGE. This indicates that mean SNa level increases as the joint effect of AGE*TTF increases. Note that mean SNa level is negatively associated with the marginal effect of TTF (P = 0.0521), which implies that the subjects with lower TTF may have higher SNa level. Mean SNa is negatively associated with the JIEs of AGE and SEX, i.e., AGE*SEX (P = 0.0008), while it is positively associated with the marginal effect of SEX (0 = female, 1 = male) (P = 0.0029) and indifferent to AGE. This indicates that mean SNa level is higher for women subjects considering age than male subjects. It is indifferent to age, but it is positively associated with the marginal effect of SEX (0 = female, 1 = male) (P = 0.0029), which shows that mean SNa level is higher for male subjects than females without considering age.

Mean SNa is partially positively associated with the JIEs of AGE and SEC, i.e., AGE*SEC (P = 0.0901), while it is significantly negatively associated with the marginal effect of SEC (P = 0.0028) and indifferent to AGE. Note that AGE*SEC is a confounder in the model. This indicates that mean SNa level increases as the joint effect of AGE*SEC increases. In addition, mean SNa level is negatively associated with the marginal effect of SEC (P = 0.0028), indicating that SNa level increases as SEC decreases and vice versa. This is observed for kidney disease patients. Their SEC level is very high, so the mean SNa level is very low, which is observed in practice. Mean SNa is positively associated with the JIEs of EJF and SEC, i.e., EJF*SEC (P = 0.0044), while it is negatively associated with the marginal effect of SEC (P = 0.0028) and indifferent to EJF. This implies that mean SNa level increases as the joint effect EJF*SEC increases. Mean SNa is positively associated with the JIEs of ANS and DEE, i.e., ANS*DEE (P = 0.0009), while it is negatively associated with the marginal effect of DEE (P = 0.0001) and partially positively associated with ANS. This implies that mean SNa level increases as the joint effect ANS*DEE increases. In addition, mean SNa is negatively associated with the marginal effect of DEE (0 = survive or 1 = died) (P = 0.0001), which indicates that mean SNa level is higher for surviving subjects than for died.

The variance of SNa is negatively associated with PLC (P = 0.0001), implying that SNa levels are highly scattered when PLC level decreases and vice versa. The variance of SNa is negatively associated with the JIEs of AGE and DEE, i.e., AGE*DEE (P = 0.0003), while it is partially positively associated with both the marginal effects of AGE (P = 0.0933) and DEE (0 = survive or 1 = died) (P = 0.1408). This implies that SNa levels are highly scattered for the surviving subjects with younger ages than dead subjects. In addition, SNa level’s variance is partially positively associated with the marginal effect of AGE (P = 0.0933), indicating that SNa levels are highly scattered at older ages than younger. Similarly, SNa level’s variance is partially positively associated with the marginal effect of DEE (P = 0.1408), implying that SNa levels are more scattered for dead subjects than surviving subjects. SNa level’s variance is positively associated with the JIEs of AGE and ANS, i.e., AGE*ANS (P = 0.0575), while it is partially positively associated with the marginal effect of AGE (P = 0.0933) and negatively with ANS (0 = no anemia, 1 = anemia) (P = 0.0631). This implies that SNa levels are highly scattered for anemia subjects at older ages than non-anemia subjects at younger ages. SNa level’s variance is partially negatively associated with the marginal effect of ANS (P = 0.0631), implying that SNa levels are highly scattered for non-anemia subjects than anemia without considering ages.

SNa variance is partially positively associated with the JIEs of EJF and SEX, i.e., EJF*SEX (P = 0.1001), while it is negatively associated with the marginal effect of EJF (P = 0.0008) and indifferent to SEX (0 = female, 1 = male) (P = 0.6224). This indicates that SNa levels are highly scattered for male subjects with higher EJF than females with lower EJF. In addition, SNa level’s variance is negatively associated with the marginal effect of EJF (P = 0.0008), indicating that SNa levels are highly scattered if EJF decreases, indifferent to sex. SNa variance is positively associated with the JIEs of EJF and TTF, i.e., EJF*TTF (P = 0.0008), while it is negatively associated with both the marginal effects of EJF (P = 0.0008) and TTF (P = 0.0018). This shows that SNa levels are highly scattered if the joint effect EJF*TTF increases. SNa level’s variance is negatively associated with the marginal effect of TTF (P = 0.0018), implying that SNa levels are highly scattered if TTF decreases, indifferent to EJF. SNa variance is positively associated with the JIEs of EJF and DEE, i.e., EJF*DEE (P = 0.0022), while it is negatively associated with the marginal effect of EJF (P = 0.0008) and partially positively associated with DEE (0 = survive or 1 = died) (P = 0.1408). This implies that SNa levels are highly scattered for the dead subjects with higher EJF than the surviving subjects. Note that DDE acts as a confounder in the model.

SNa variance is partially negatively associated with the JIEs of SEC and SEX, i.e., SEC*SEX (P = 0.1327), while it is indifferent to both SEC and SEX (0 = female, 1 = male). This implies that SNa levels are highly scattered for the female subject with a lower SEC level than male. SNa variance is positively associated with the JIEs of SEC and SMH, i.e., SEC*SMH (P = 0.0091), while it is negatively associated with the marginal effect of SMH (0 = no smoking, 1 = smoking) (P = 0.0242) and indifferent to SEC. This indicates that SNa levels are highly scattered for the smoker subjects with high SEC level than non-smokers with low SEC level. Again, SNa level’s variance is negatively associated with the marginal effect of SMH (P = 0.0242), implying that SNa levels are highly scattered for the non-smoker subjects than smokers without considering the effect of SEC. Also, SNa variance is positively associated with the JIEs of SEC and DIS, i.e., SEC*DIS (P = 0.0001), while it is negatively associated with the marginal effect of DIS (0 = no diabetes, 1 = diabetes) (P = 0.0548) and indifferent to SEC. This implies that SNa levels are highly scattered for the diabetic subjects with higher SEC level than non-diabetic subjects with lower SEC level. In addition, it is negatively associated with the marginal effect of DIS (P = 0.0548), implying that SNa levels are highly scattered for the non-diabetic subjects than diabetic, without considering SEC effect. Also, SNa variance is positively associated with the JIEs of CRP and DIS, i.e., CRP*DIS (P = 0.0039), while it is negatively associated with both the marginal effects of CRP (P = 0.0027) and DIS (P = 0.0548). This indicates that SNa levels are highly scattered for the diabetic subjects with higher CRP level than non-diabetic subjects with lower CRP level.

In the above, all the findings are discussed based on the present data analysis. It is observed herein that most of the above results are completely new findings in the literature, so these are not discussed in the earlier articles. Therefore, the present findings are not compared with the earlier published results.

5. Conclusions

The current article has developed the effects of SNa on anemia, diabetes, and heart disease subjects along with age, sex, SMH, CRP, EJF, PLC, SEC, TTF, and DEE. The fitted SNa probabilistic model has been selected herein according to the AIC rule, on comparison of joint log-normal and gamma models, graphical diagnostic checking plots (Figure 1), and standard error of the estimates. In Table 1, it is observed that both the models are identical. All the interpretations are drawn herein using the appropriate fitted SNa probabilistic model. Most of the findings focus on the real facts that are observed in practice. The article has mainly two purposes such as the development of a very complicated fitted SNa model and the interpretations of the findings. The derived outcomes regarding SNa effects herein though not completely conclusive are revealing. Scientific research should have complete faith in these derived outcomes as the fitted model has been derived with diagnostic checking and comparison of two models.

The fitted SNa models (Table 1) are obtained from the data set reported in the article [20]. For any other similar data sets, the findings will be similar to the present findings, which are not verified herein as parallel data sets are not available. The findings of the article may help medical practitioners, researchers, and patients. SNa has many complex functional roles in anemia, diabetes, and heart disease subjects and other factors, so the medical treatment process should care about SNa levels of the subjects.

Abbreviations

ANS: anemia status

BP: blood pressure

CRP: creatinine phosphokinase

DEE: death event

DIS: diabetes status

EJF: ejection fraction

HBP: high blood pressure

JGLM: joint generalized linear model

JIE: joint interaction effect

PLC: platelets count

SEC: serum creatinine

SNa: serum sodium

SMH: smoking habit

T2D: type-2 diabetes

TTF: time to follow-up

Acknowledgements

The authors are very much indebted to the editors and referees who have provided valuable comments to improve this paper, and they are also grateful to the principal data investigators, who provided the data freely for scientific study.

Funding

The authors declare no financial support for the research, authorship, or publication of this article.

Author contributions

Conceptualization, R.D. and M.D.; methodology, R.D. and S.B.; software, R.D.; validation, R.D. and S.M.; formal analysis, R.D.; investigation, R.D. and S.M.; resources, M.D.; data curation, S.B.; writing—original draft preparation, R.D.; writing—review and editing, R.D. and S.B.; visualization, R.D. and S.M.; supervision, M.D.; project administration, R.D.; funding acquisition, S.B.; All authors have read and agreed to the published version of the manuscript.

Conflict of interest

The authors confirm that this article content has no conflict of interest.

Data availability statement

The data set is available in the site https://archive.ics.uci.edu/ml/datasets/Heart+failure+clinical+records.

Institutional review board statement

Note that the current study has been performed based on a secondary data set, which was first collected by Ahmad T, Munir A, Bhatti SH, Aftab M, and Ali Raza M. The study ethics approval and the subject consents are clearly described in the original article [20], which are not reproduced herein.

Informed consent statement

Not applicable.

Sample availability

The authors declare no physical samples were used in the study.

Publisher’s note

Academia.edu stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords

anemia status (ANS), creatinine phosphokinase (CRP), ejection fraction (EJF), joint generalized linear models (JGLMs), serum sodium (SNa)