^{1}Department of Statistics, The University of Burdwan, Burdwan, West Bengal, India ^{2}Department of Mathematics, NSHM Knowledge Campus, Durgapur, West Bengal, India

- *Corresponding Author:
- Rabindra Nath Das

Department of Statistics

The University of Burdwan, Golapbag

Rajbati, Burdwan, West Bengal, India

**Tel:**+91-9232638970

**E-mail:**[email protected]

**Received** November 14th, 2017 - **Accepted** December 24th, 2017

**Objectives **Alkaline phosphatase is a well-known biomarker of liver disease. The article attempts to locate the determinants of alkaline phosphatase of some liver and non-liver patients. **Background **Many previous research reports consider alkaline phosphatase as a continuous, homogeneous and normally distributed response variable. In practice, alkaline phosphatase is positive, heterogeneous and non-Normally distributed response variable. There is a little study considering the real fact that the alkaline phosphatase as a non-Normal, heterogeneous and positive response variable. **Materials and Methods** The current report considers a real data set of 583 subjects with 9 continuous variables and 2 attribute characters. The patients are divided into two groups liver patients (416) and non-liver patients (167). The data set contains 441 male and 142 female patients. Source of the data set is the Machine Learning Repository's citation policy. The considered data set was obtained from the North-East of Andhra Pradesh, India. The considered response alkaline phosphatase is positive, heterogeneous and non-Normally distributed continuous variable. So, it should be analyzed using statistical joint generalized linear Gamma or Log-normal models. **Results **The mean alkaline phosphatase value is higher for the liver patients (P<0.001) than non-liver patients. The mean alkaline phosphatase value is high at senior ages (P=0.030) than the junior ages. The mean ALP decreases as the total bilirubin (P=0.006) increases, while it increases as the SGPT (P<0.001) increases. The mean alkaline phosphatase decreases as the albumin to globulin ratio (A/G) (P<0.001) increases. Interaction effect of age with total bilirubin (Age*TB) (P<0.001) is directly correlated with the mean alkaline phosphatase, while the joint interaction factor of age with total proteins (Age*TP) (P=0.010) is inversely correlated with the mean alkaline phosphatase. Again, the interaction effects of albumin to globulin ratio (A/G) with the total bilirubin (A/G*TB) (P<0.001), and with total proteins (A/G*TP) (P=0.003) are directly correlated with the mean alkaline phosphatase. The variance of alkaline phosphatase is higher for liver patients (P<0.001) than non-liver patients. The variance of alkaline phosphatase increases as the SGPT (P<0.001), or total bilirubin (P=0.088) increases. The variance of ALP increases as the albumin to globulin ratio (A/G) (P<0.001), or age (P=0.055) decreases. **Conclusions **The response alkaline phosphatase is identified as heterogeneous and Log- normally distributed. Many factors such as age, SGPT, interaction effects of albumin to globulin ratio with total bilirubin, and separately with total proteins, and also the interaction effect of age with total bilirubin are directly correlated with the mean alkaline phosphatase. The present findings, especially the interaction effects and the variance determinants of alkaline phosphatase are completely new inputs in the liver disease literature.

Alkaline Phosphatase; Bilirubin

ALP alkaline phosphatase; TB total bilirubin

A vital organ of human being is liver which has various important functions such as protein synthesis, decomposition of red blood cells, detoxification, glycogen storage, and production of biochemical and hormone. The liver has multidimensional functions, and it is also prone to many diseases. Note that there are more than 100 different forms of known liver disease caused by many risk factors [1, 2, 3]. The most common disease throughout the world is the liver disease which is caused by damaged or inflammation hepatocyte in liver [4, 5, 6, 7]. In every year, throughout the world, more than one million of new liver cancer patients are diagnosed [8]. Even more dreadful is the real situation that liver cancer is not easily identified, as the liver is able to maintain regular activity even if there are some partially damaged. The imaging diagnosis of early stage hepatic cirrhosis, hepatitis, and hepatic fibrosis is still very complicated as they lack morphological changes. However, a few hemodynamic changes of the liver diseases are generally observed [8, 9]. The most important steps in liver cancer treatment is the early diagnosis of the disease.

Many research articles have pointed that dietary
factors have great effects on liver biochemical markers [8, 10, 11]. In animals or healthy humans, fat ingestion has
been identified to elevate serum ALP levels. It is happened either if a high-fat diet is absorbed or if fat is included in a
test meal [12, 13, 14]. Intravenous infusion of glucose in
rabbits develops an increase in the hepatic excretion of
conjugated bilirubin [15]. A few medium-chain fatty acids
decrease in binding of bilirubin to albumin *in vitro* [16].
In practice, excess alcohol consumed (>80 g/day for more
than 5 years) (excluding other causes of cirrhosis) may
produce alcoholic cirrhosis [12, 17].

There is a little study on the explanatory factors of serum alkaline phosphatase (ALP), considering the real fact that it is non-normal, heteroscedastic and positive. It is identified that the response ALP is heterogeneous, and non-normally distributed. Therefore, ALP should be analyzed by joint generalized linear models (JGLMs) namely, gamma or Log-normal [18, 19, 20, 21]. In order to confirm the analysis, the present considered data set has been analyzed using both the gamma and Log-normal joint generalized linear models. It is found that joint Log-normal models give better results. Both the analysis results are given in the article.

The article attempts to know the answer of the following quarries or hypotheses. What are the explanatory factors or determinants of the serum alkaline phosphatase (ALP)? How are the explanatory factors correlated with the ALP? What are the functional activities of the explanatory factors on ALP? These quarries or hypotheses are evaluated in the article, using a real data set of 583 subjects with 9 continuous variables and 2 attribute characters.

**Materials**

The current report considers a real data set of 583
subjects with 9 continuous variables and 2 attribute
characters. The data set was collected from the North-East
of Andhra Pradesh, India. The data set can be obtained
from http://archive.ics.uci.edu/ml/machine-learningdatabases/
00225/. The statement of the explanatory
variables, and levels of the attribute characters are given in **Table 1**. The descriptive statistics such as the proportion,
mean, normal range of the biochemical parameters, and
standard deviation are displayed in **Table 1**. There are
75.82% male patients, and 24.18% female patients. Liver patients (71.5%) are more than the non-liver patients
(28.5%) in the given data set. The statement of the subject
population and the information collection method are
displayed in [22, 23]. This is not described again in the
report.

The usual regression models consider that the variance
of the response (*Y*) is constant within the whole range of
the variable. However, this assumption is not true always
[24]. Generally, biochemical data are heterogeneous. For
example, the ALP values for the liver disease patients
are heterogeneous. In order to remove the non-constant
variance of a data set, the log-transformation is generally
used, but practically, the heteroscadasticity of the data set
may not always be removed [24, **Table 2**].

Positive data from the continuous variables with
constant variance or constant coefficient of variation may
be analyzed either by the Log-normal or the gamma models
[21]. Generally, physiological data are heterogeneous, so
these two models may not give similar results [19, 20, 25, 26, 28]. In practice, generalized linear models class is
used to analyze non-normal, heteroscedastic and positive
data set. Generally, the mean and variance of the response
variable may be related in the generalized linear models, as
a result the variance of the response may be non-constant.
For the analysis of non-constant variance positive data *y _{i}* ’s, Nelder and Lee [28] have suggested to use the joint
generalized linear models (JGLMs). A detailed discussion
of JGLMs is given in [18, 19, 27, 28]. For ready reference, a
short description of the JGLMs is reproduced herein.

For the positive data *Y _{i}* ’

The log transformation Z_{i} = log (Y_{i}) is used to stabilize
the variance Var . If one needs better models,
an improved statistical tool is used. Generally, the variance
may not be stabilized always by a simple transformation
[24]. Then Nelder and Lee [28] have suggested to use
JGLMs.

For the positive dependent variable *Y _{i}*, the log
transformation

where xit and git denote respectively, the row vectors for the regression coefficients β (mean model) and γ (variance model).

For the positive dependent variable *y _{i}* ’s

where are the dispersion parameters and is the variance function. In generalized linear models, the
variance has two portions. One portion is that
depends on the mean values. The other portion is *σ*_{i}^{2} that
is free of mean values. The variance function identifies the
distribution family in GLMs. For example, the distribution
is gamma if , Poisson if , and Normal
if , etc.

The mean and variance models of JGLMs are

where *g*(⋅) and *h*(⋅) are respectively the GLM
link functions for mean and variance, and denote respectively, the row vectors for the regression
coefficients β (mean model) and γ (variance model).
Maximum likelihood (ML) method is used to estimate β
(mean model), and restricted ML (REML) method is used
to estimate γ (variance model) [18, 19].

It is well-known that the serum alkaline phosphatase
(ALP) is a liver biomarker. Higher than normal levels of
ALP (**Table 1**) in blood may indicate a problem with liver
or gall bladder. This could include hepatitis (infection),
cirrhosis (scarring), liver cancer, gallstones, or a blockage in our bile ducts. The article considers ALP as the
dependent variable to examine the hypotheses as stated
in Introduction. The response serum alkaline phosphatase
is positive with non-constant variance, and it belongs to
exponential family distribution. Therefore, such data are
generally, analyzed by the Log-normal or gamma models as
stated above. In order to confirm the derived results, both
the above models are used to perform the analysis. Now
we are interested to investigate the following using both
the models. What are the approximate true models of ALP
for the liver disease patients? What are the explanatory
factors (or determinants) of ALP? What are the effects of
the explanatory factors on the ALP? These quarries are
addressed in the next sections.

**Alkaline Phosphate (ALP) Value Analysis, Results and
Interpretations**

__Analysis:__ Alkaline phosphate (ALP) value is the
continuous positive response random variable of interest
in the article. There are remaining 8 continuous, and 2
attribute explanatory variables. In case of attribute factors,
we have considered the constraint that the effects of the
first levels are zero. Hence, for each attribute factor, the
first level is considered as the reference level by estimating
it as zero. The main effect of * A* is denoted by

Alkaline phosphate vale is considered as the response
variable in the current analysis and the rest other variables
or factors are considered as the explanatory variables. We
have modeled the response ALP using both the Log-normal
and gamma models [18, 19]. The final fitted models have
been selected on the basis of smallest Akaike information
criterion (AIC) value in each class. It is known that the AIC selects a model which minimizes the predicted additive
errors and squared error loss [28]. The Log-normal and
the gamma fitted models results of ALP are given in **Table
2**. **Table 2** shows that the Log-normal fitted models (AIC=
7042) give better results than the gamma fitted models
(AIC=7189). It is not always necessary that all the selected
effects are significant [29]. Some insignificant effects are
retained in the model in order to respect the marginality
rule, namely that when an interaction term is significant all
related lower-order interactions and main effects should
be included in the model [30]. In epidemiology, statistical
insignificant included factors or variables in the fitted
models are known as confounder. For model checking
diagnosis plots, we have plotted the absolute residuals and
normal probability plots for the fitted joint Log-normal
models (**Table 2**) as they are better fitted models.

**Figure 1(a)** presents the absolute residuals plot of
the fitted Log-normal models (**Table 2**) with respect to
fitted values. It is a straight flat diagram, implying that the
variance is constant with the running means. The normal
probability plot for the fitted mean model of Log-normal
fit (**Table 2**) is given in **Figure 1(b)**. There is no lack of fit,
or systematic departure in the **Figure 1(b)**. Therefore, the
fitted Log-normal models (**Table 2**) are approximate true
models of ALP.

The mean serum alkaline phosphatase (ALP) is negatively associated with the types of liver patients (P<0.001), or total bilirubin (TB) (P=0.006), or albumin to globulin ratio (A/G) (P<0.001), or interaction effects of albumin to globulin ratio (A/G) with total proteins (TP) (A/G*TP) (P=0.003). The mean ALP is positively associated with the age (P= 0.030), or SGPT (P<0.001), or interaction effect of age with total bilirubin (TB) (Age*TB) (P<0.001), or interaction effect of albumin to globulin ratio (A/G) with the total bilirubin (TB) (A/G*TB) (P<0.001), or interaction effect of albumin to globulin ratio (A/G) with total proteins (TP) (A/G*TP) (P=0.003).

The variance of serum alkaline phosphatase is negatively associated with the types of liver patients (P<0.001), or albumin to globulin ratio (A/G) (P<0.001), or age (P=0.055), or SGOT (P=0.186). The ALP variance is positively associated with the SGPT (P<0.001), or total bilirubin (TB) (P=0.088).

**Interpretations of Liver Disease Data Analysis**

Summarized results of the serum alkaline phosphatase
value analysis are given in **Table 2**. **Table 2** presents
results of joint gamma and Log-normal models. Note that
joint Log-normal models give better results. Therefore, the
following interpretations are drawn based on Log-normal
models (**Table 2**).

*The mean alkaline phosphates (ALP) is negatively significantly correlated with the patient types (liver patient=1, non-liver patient=2) (P<0.001). This indicates that ALP value is higher for liver disease patients than nonliver patients.

*The mean ALP is positively correlated with the subjects age (P=0.030), implying that the mean ALP value increases at senior ages.

*The mean ALP is negatively significantly correlated with the total bilirubin (TB) (P=0.006), indicating that the total bilirubin increases as the ALP value decreases.

*The mean ALP is positively significantly correlated with Alamine Aminotransferase (SGPT) (P<0.001), indicating that the mean ALP increases as the SGPT increases.

*The mean ALP is negatively significantly correlated with the Albumin to globulin ratio (A/G) (P<0.001), indicating that the mean ALP increases as the A/G ratio decreases.

*The mean ALP is positively significant associated with the interaction effect of age and the total bilirubin (TB) (Age*TB) (P<0.001), indicating that the ALP increases as the interaction effect (Age*TB) increases.

*The mean ALP is negatively significant associated with the interaction effect of age and the total protein (TP) (Age*TP) (P=0.010), indicating that the ALP increases as the interaction effect (Age*TP) decreases.

*The mean ALP is positively significant associated with the interaction effect of total bilirubin (TB) and the Albumin to globulin ratio (A/G) (TB*A/G) (P<0.001), implying that the ALP increases as the interaction effect (TB*A/G) increases.

*The mean ALP is directly significantly correlated with the interaction effect of total protein (TP) and the Albumin to globulin ratio (A/G) (TP*A/G) (P=0.003), indicating that the ALP increases as the interaction effect (TP*A/G) increases.

*The variance of Alkaline Phosphates (ALP) is inversely significantly correlated with the patient’s type (liver patient=1, non-liver patient=2) (P<0.001), implying that the ALP variance is higher for liver patients than the nonliver patients.

*The ALP variance is directly significantly correlated with Alamine Aminotransferase (SGPT) (P=0.021), implying that the ALP variance increases as the SGPT increases.

*The ALP variance is reciprocally significantly correlated with the Albumin to globulin ratio (A/G) (P<0.001), indicating that the ALP variance decreses as the A/G ratio increases.

*The ALP variance is reciprocally significantly correlated with the age (P=0.055), implying that the ALP variance is higher at junior ages.

*The ALP variance is directly significantly correlated with the total bilurubin (TB) (P= 0.088), indicating that the ALP variance increases as TB increases.

The final fitted Log-normal mean and variance models
(**Table 2**) of ALP value of the liver patients, respectively,
are

and

The article has considered Alkaline Phosphates (ALP)
as the dependent variable. The estimated ALP values can
be derived from the fitted Log-normal models. Fitted
results for the two models are given in **Table 2**. These
two fitted models show very close results, except a few
additional covariates included in the fitted Log-normal
variance model. The explanatory factors of the ALP values
have been derived in the article using the fitted Log-normal
models (**Table 2**).

It has been identified in the article that the mean alkaline phosphatase (ALP) value is higher for the liver patients than non-liver patients. It is higher at senior ages of the liver patients than the junior ages. It decreases as the total bilirubin increases, while it increases as the SGPT increases. The mean ALP decreases as the albumin to globulin ratio increases. Interaction effect such as age with total bilirubin (TB) (Age*TB), age with total proteins (TP) (Age*TP), albumin to globulin ratio (A/G) with the total bilirubin (TB) (A/G*TB), total proteins (TP) with albumin to globulin ratio (A/G) (A/G*TP) are significant effects on the mean ALP.

The article has considered 583 patients with all nonmissing
information. The current results are not similar
to earlier results. For example, SGOT is not a significant
determinant of the ALP (**Table 2**) in the present report,
but it is significant in some other articles [31, 32]. Similarly,
age, total bilirubin (TB), albumin to globulin ratio (A/G
ratio) are significant factors for ALP in the present article,
but these factors are not significant in the previous articles
[17]. In the present report, many interaction effects such
as Age*TB, Age*TP, TB*A/G, TP*A/G are significant on the
mean ALP (**Table 2**), which are not reported in the earlier
reports. Many ALP variance determinants such as patient’s
type, SGPT, age, albumin to globulin ratio, total bilirubin,
SGOT are significant in the present reports which are not
reported in the previous articles [31, 32].

The current models may provide many determinants of Alkaline Phosphates. Based on the present results, it may be possible to take the necessary action to control the Alkaline Phosphates value. Medical practitioners, researchers and the liver patients may be benefited from the current report. Every individual should care on Alkaline Phosphates values at older ages. Note that at older ages with higher values of total bilirubin, and total protein increase the Alkaline Phosphates values.

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

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