Multivariate Time Series Analysis Using Heart Disease Patent in Tamilnadu

doi:10.21203/rs.3.rs-3945284/v1

Download PDF

Article

Multivariate Time Series Analysis Using Heart Disease Patent in Tamilnadu

https://doi.org/10.21203/rs.3.rs-3945284/v1

This work is licensed under a CC BY 4.0 License

Version 1

posted

You are reading this latest preprint version

This study looked into the incidence of heart disease in Tamilnadu between January 1, 2016, and December 31, 2020. Heart disease cases have been found to be positively correlated with one another. The Granger causality tests used in the Breusch-Godfrey LM approach for testing, residual serial auto correlation assumes a VAR model for the error vector _t, while the contrary was not true. can be determined using common model selection criteria such as the Final Prediction Error, Hannan-Quinn Information Criterion, Schwarz Information Criterion, and Akaike Information Criterion (FPE) time series plots, the Augmented Dickey Fuller (ADF) test, the Philip Perrons (PP) test, and the Kwiatkowski-Phillip-Schmidt-Shin test, determine the series' stationarity (KPSS) test. For both the ADF test and the PP test. The OLS-CUSUM test for stability.

Health sciences/Cardiology

Health sciences/Diseases

Health sciences/Health care

Health sciences/Medical research

Physical sciences/Mathematics and computing

Heart disease

VAR model

Breusch Godfrey LM methodology

Granger causality tests

OLS-CUSUM test

stability test

The phrase "heart disease," which is often referred to as "cardiac disease," is used to denote "any incapacity of the heart to function properly," which includes (a) The blood vessels that carry blood to the heart muscles. which can result in irregular heart rhythms, a heart attack, or chest aches (arrhythmias). (b) High blood pressure and an enlarged left heart make the heart work harder than necessary to” pump blood to the rest of the body.” The result is a thickening and stiffening of the left ventricle (left ventricular hypertrophy).

The ability of the ventricles is hampered by these alterations. Heart attack, heart failure, and sudden cardiac death are all risks that are increased by this illness. Heart failure (c) The stress on the heart brought on by high blood pressure over time may cause the heart's muscles to weaken and function less effectively. The overloaded heart eventually just wears out and stops working.

Unhealthy arteries or blood vessels are the root cause of heart problems. When an artery is constricted or obstructed, it is harmful. Strong, elastic, and flexible arteries indicate good health. Their inner lining is smooth, allowing blood to flow easily and adequately nourish and oxygenate the tissues and critical organs. High blood pressure can eventually damage the arteries by totally rupturing it due to the increased pressure of blood flowing through them. Blood clots can completely stop blood flow when dietary fats from eaten meals gather at damaged spots and enter the bloodstream. In situations where the blood vessel is narrowed but not completely closed off, just a little amount of blood can pass through. The arteries all over the body may be damaged by these problems. Heart disease develops if these events cause damage to the heart's arteries or blood vessels.

“Instead of being a disease or condition, high blood pressure is a risk factor for diseases that people would prefer to stay away from. These ailments include heart attacks, strokes, kidney issues, and other circulatory system-related conditions (blood circulation). A prevalent and significant for renal and cardiovascular disease is high blood pressure.” As people get older, hypertension is more common, especially isolated systolic hypertension. (Fahey et al. 2004)

The types of high blood pressure: primary high blood pressure is accounts for 90–95 percent of cases. The other condition is secondary high blood pressure, which has a known aetiology. Neither syndrome has a known cause. This particular case is a complication of another illness. Alcohol abuse can cause to rise. The chance of having high blood pressure is also increased by a number of chronic illnesses, such as diabetes, renal disease, excessive cholesterol, and sleep apnea. Pregnancy can occasionally cause high as well. (1988, Pickering). There is still a great deal of mystery around the origins of high blood pressure. Over 95% of people do not have an underlying aetiology for their condition. It's possible that a number of variables contribute to most people's high blood pressure. The main suspects are lifestyle choices and hereditary factors. Age, body type, and family history are all genetic influences. Drinking, smoking, exercise frequency, stress, obesity, and a high salt intake are examples of lifestyle and habits. (2004) Fahey et. al.

Without using drugs, changing one's lifestyle is the greatest way to cure high blood pressure. Altering one's diet, engaging in regular exercise, and quitting smoking will reduce the numerous dangers that might result in high blood pressure or raise cardiovascular risk. According to statistics, increasing physical activity, reducing weight, abstaining from alcohol, and modifying one's diet to include more fruits and vegetables and less salt will, on average, drop systolic blood pressure by 4 mmHg. (2004) Fahey et al.

The data will be modeled with the Vector Autoregressive (VAR) model. “The VAR model is the multivariate form of the univariate. Autoregressive (AR) model shows that there is a relationship between present and past value”. In response one way to incorporate the influence of one variable on other variables over time is to use a model which was popularized by Box and Tiao (1981). It is mainly used for structural analysis and forecasting. “The specific modeling technique used in this study is the Vector Autoregressive (VAR) modelling approach. The use of the VAR model for time series was proposed by Sims (1980)”. The appropriate lag length (p), which should be specified long enough for the residuals not to be serially correlated, can be determined using standard model selection criteria.

2.1. The Vector Autoregressive model (VAR)

Assume ${Y}_{t}$ is a K by 1 vector stochastic process. A ${p}^{th}$ order VAR model of inter-related time series, written as VAR(p), is a process evolves according to

$${Y}_{t}= {\varPhi }_{0}+{\varPhi }_{0}{Y}_{t-1}+{\varPhi }_{0}{Y}_{t-2}\cdots {\varPhi }_{0}{Y}_{t-p}+{ϵ}_{t} \cdots \left(1\right)$$

Where, ${\varPhi }_{0}$ is a 𝑘 by 1 vector of intercept parameters, ${\varPhi }_{j}$ is a 𝑘 by 𝑘 parameter matrices, with $j=\text{1,2},\dots .,p$ and ${ϵ}_{t}$ is a vector.The general approach to VAR model estimation is to fit the $VAR \left(p\right)$ with the order $p = 0,..., {p}_{max}$ and the value of $p$ should minimize some model selection criteria. Model selection criteria for VAR (𝑝) models have the form

$$IC\left(p\right)=ln\left|\sum \left(p\right)\right|+{c}_{T}.\phi (n,p)$$

Where, $\sum \left(p\right)={T}^{-1} {\widehat{\epsilon }}_{t}{\widehat{\epsilon }}_{t}^{{\prime }}$ is the residual covariance matrix without a degree of freedom correction from the $VAR \left(p\right)$ model, ${c}_{T}$ is a sequence indexed by the sample size 𝑇, and (𝑛,) is a penalty function which penalizes large VAR (𝑝) models. The three most common information criteria are the Akaike (AIC), Schwarz-Bayesian (BIC) and Hannan-Quinn (HQ) derived as,

$$AIC\left(p\right)=ln\left|\sum \left(p\right)\right|+\frac{2}{T}p{n}^{2} \cdots \left(2\right)$$

$$BIC\left(p\right)=ln\left|\sum \left(p\right)\right|+\frac{ln\left[T\right]}{T}p{n}^{2} \cdots \left(3\right)$$

$$HQ\left(p\right)=ln\left|\sum \left(p\right)\right|+\frac{2ln\left[lnT\right]}{T}p{n}^{2} \cdots \left(4\right)$$

A VAR-model has been estimated, it is of pivotal interest to see whether the residuals obey the model’s assumptions. That is, one should check for the absence of serial correlation and heteroscedasticity and also see if the error process is normally distributed. For testing the lack of serial correlation in the residuals of a VAR (p)-model, the LM test proposed by Breusch [1978] and Godfrey [1978] .

2.2. Estimation of the VAR model

The order of a VAR model is “ number of lags included in the model. The appropriate lag length (𝑝), should be estimated long enough for the residuals not to be serially correlated. The lag length plays a crucial role in diagnostic tests as well as in the estimation of VAR models for impulse response analysis and variance decomposition (Bhasin, 2004). Four of the selection criteria i.e. the Final Prediction Error (FPE)”, Akaike Information Criterion (AIC), the Schwarz Information Criterion(SC), and the Hannan-Quinn Information Criterion(HQ) support the inclusion of 2 lags as seen from Table 4.9. Hence, lag 2 is considered the optimal choice for the VAR order.

2.3. The Breush-Godfrey LM Test

The Breusch-Godfrey LM methodology for testing residual serial auto correlation assumes a VAR model for the error vector ${\epsilon }_{t}= {D}_{1}{\epsilon }_{t-1}+\cdots + {D}_{h}{\epsilon }_{t-h}+{v}_{t}$ where ${v}_{t}$ is white noise. ${\epsilon }_{t}$ is equal to ${v}_{t}$ if there is no residual serial correlation. Therefore we test for the hypotheses

$${ H}_{0}: {D}_{1}={D}_{2}\cdots {D}_{h}=0 against$$

$${H}_{1}: {D}_{j}\ne 0 for j=\text{1,2}, \cdots ,h$$

are any 𝐾 dimensional VAR, the Breusch-Godfrey LM test statistic is given by

$${LM}_{h}=T\left(K-tr\left({\stackrel{-}{\varSigma }}_{R}^{-1} {\widehat{\varSigma }}_{e}\right)\right) \cdots \left(5\right)$$

2.4. Granger Causality test

One of the main uses of VAR models is forecasting. “The structure of the VAR model provides information about a variable’s or a group of variables’ forecasting ability for other variables. The following intuitive notion of a variable’s forecasting ability is due to Granger (1969). Granger causality is a good indicator that a VAR, rather than a univariate model, is needed”. A scalar random variable $\left\{{x}_{t}\right\}$ is said to not Granger cause $\left\{{y}_{t}\right\}$ if

$$E\left[{y}_{t}/{x}_{t-1,}{y}_{t-1, }{x}_{t-2,}{y}_{t-2, }\right]=E\left[{y}_{t}/{y}_{t-1, }{y}_{t-2, }\cdots \right]$$

That is, $\left\{{x}_{t}\right\}$ does not Granger cause $\left\{{y}_{t}\right\}$ if the forecast of ${y}_{t}$ is the same whether conditioned on past values of ${x}_{t}$ or not. In such a case we say $\left\{{x}_{t}\right\}$does not Granger cause $\left\{{y}_{t}\right\},$or $\left\{{x}_{t}\right\}$is exogenous to $\left\{{y}_{t}\right\}.$It just says that 𝑥 is helpful in forecasting 𝑦, which might happen for other reasons than direct causality. A bivariate VAR (2) can generally be illustrated as:

$$\left[\begin{array}{c}{x}_{t}\\ {y}_{t}\end{array}\right]=\left[\begin{array}{cc}{\varphi }_{11.1}& {\varphi }_{12.1}\\ {\varphi }_{21.1}& {\varphi }_{22.1}\end{array}\right]\left[\begin{array}{c}{x}_{t-1}\\ {y}_{t-1}\end{array}\right]+\left[\begin{array}{cc}{\varphi }_{11.2}& {\varphi }_{12.2}\\ {\varphi }_{21.2}& {\varphi }_{22.2}\end{array}\right]\left[\begin{array}{c}{x}_{t-2}\\ {y}_{t-2}\end{array}\right]+\left[\begin{array}{c}{ϵ}_{1,t}\\ {ϵ}_{2,t}\end{array}\right]$$

“In general if ${\varphi }_{21.1}={\varphi }_{21.2}=0$, then $\left\{{x}_{t}\right\}$ does not Granger cause$\left\{{y}_{t}\right\}$. If it happens to be the case that $\left\{{x}_{t}\right\}$does not Granger cause $\left\{{y}_{t}\right\}$and ${ϵ}_{1,t}$and ${ϵ}_{2,t}$ have no contemporary correlation, then 𝒚𝒕 is said to be weakly exogenous, and ${y}_{t}$ can be modeled completely independently of ${x}_{t}$. This test can be conducted irrespective of whether ${x}_{t}$ Granger causes ${y}_{t}$ or not. An F-test is then used to determine whether the coefficients of the past values of 𝑋 are jointly zero. The null hypothesis for such a test is 𝑋 does not Granger cause 𝑌”. Now, considering a bivariate VAR (𝑝) model written out in scalar form as,

$${x}_{t}={\varphi }_{1}+\sum _{i=1}^{p}{\varphi }_{11}^{\left(i\right)}{x}_{i,t-i}+\sum _{i=1}^{p}{\varphi }_{12}^{\left(i\right)}{x}_{i,t-i}+{ϵ}_{1t}$$

$${y}_{t}={\varphi }_{1}+\sum _{i=1}^{p}{\varphi }_{21}^{\left(i\right)}{x}_{i,t-i}+\sum _{i=1}^{p}{\varphi }_{22}^{\left(i\right)}{x}_{i,t-i}+{ϵ}_{2t}$$

Then test for Granger causality from 𝑥 to 𝑦 is an F-test for the joint significance of OLS regression of F-statistic for testing the hypothesis is,

$${H}_{0}: {\beta }_{k-p+1}={\beta }_{k-2+p}=\cdots {\beta }_{k}=0 against$$

$${H}_{1}: {\beta }_{k-i}\ne 0 for i=\text{1,2}, \cdots ,q-1$$

in any regression model

$$y={\beta }_{0}+{\beta }_{1x1}+\cdots {\beta }_{k}{x}_{k}+\mu \cdots \left(6\right)$$

$$F=\frac{\left({SSE}_{r}-{SSE}_{ur}\right)/q}{{SSE}_{ur}/\left(n-k-1\right)} \cdots \left(7\right)$$

where ${SSE}_{r}$ is the residual sum of squares from the model under ${H}_{0}$ and ${SSE}_{u}$ is the residual sum of squares for the model in Eq. (6). Under the null hypothesis the test statistic in Eq. (7) is F-distributed with 𝑞 = 𝑑𝑓_𝑟 − 𝑑𝑓_𝑢𝑟 and − 𝑘 − 1 degrees of freedom.

Granger causality in a bivariate $VAR \left(p\right)$ model we drop $q = p$ variables in a model with $n = T$ observations and $k = 2p$ variables beyond the constant.

Hence,

$$F=\frac{\left({SSE}_{r}-{SSE}_{ur}\right)/p}{{SSE}_{ur}/\left(T-2p-1\right)} \sim F(p,T-2p-1) \cdots \left(8\right)$$

Two aggregate series of heart disease cases are employed in the empirical study. Table 1 below includes some descriptive data, such as the mean, median, lowest and maximum values of cases. According to the table's outcome, there were 3 heart disease cases at a minimum every month during the study period. In Tamilnadu, there were 9.935 heart disease cases on average every month. The median number of instances of hypertension and heart disease is 65 and 6, respectively, according to the sample data for the five years, which consists of 60 data points.

Table – 1: Descriptive Analysis of Heart Disease

Disease	Observation	Mean	Median	Min.	Max.
Heart	60	9.9330	8	0	56

When looking at the average number of instances each month, the month of March had the greatest average number (16.8) while the month of November had the lowest average number of cases (ie 4.4 average number of cases). The largest number of heart disease cases—57 cases—occurred in the month of March, while the lowest number—0 cases—occurred in the month of June.

Table-2:

Month

Mean

Min.

Max.

Median

January

February

March

April

May

June

July

August

September

October

November

December

10.4

11.6

16.8

14.4

5.4

9.2

12.4

11.6

5.2

4.4

5.8

115 2 6

3.1. Unit root properties of individual series

Time series plots, the Augmented Dickey Fuller (ADF) test, the Philip Perrons (PP) test, and the Kwiatkowski-Phillip-Schmidt-Shin (KPSS) test is the series stationary. Unit Root Test Results for this appear to indicate that the series is stationary for both the ADF test and the PP test. The ACF plots do not gradually decay, despite the fact that there is a sizable spike at the initial lag of the PACF, which is a careful observation to support this finding. But until the numerous unit is verified these observations, they can only be considered hypotheses. The ADF test, PP test, and KPSS test were among the stationarity tests used in this study.

Figure − 1: A time series display of the heart disease cases at levels in Tamilnadu

Table − 3: ADF unit root test results for heart disease cases

HEART DISEASE

TEST STATISTIC

P-VALUE

CONSTANT

CONSTANT + TREND

NONE

-4.2458

-4.4031

-3.0821

0.0013

0.0045

0.0026

Table − 4: PP unit root test results for heart disease cases

HEART DISEASE

TEST STATISTIC

P-VALUE

CONSTANT

CONSTANT + TREND

NONE

-4.2153

-4.2973

-2.9628

0.0014

0.0061

0.0037

Table − 5: KPSS unit root test results for heart disease cases

TEST	TEST STATISTIC	P-VALUE
KPSS	0.3962	0.0788

Results from Tables 3, 4, and 5, indicate that the heart cases just like the hypertension cases are stationary at levels.

3.2. Stability Test

The OLS-CUSUM test was used to assess the parameters. Figure 3.2.1 shows that the model's cumulative residuals are within the 95% confidence interval. Therefore, it may be said that the model's parameters are structurally stable.

:Forecast for heart disease and hypertension cases. The model was used to forecast heart disease and hypertension because it passed nearly all of the pertinent diagnostic tests. The tables below display the findings as follows:

Table 6

Monthly forecasts of heart disease cases.
Month	Forecasts	LCL	UCL
January February March April May June July August September October November December	7 5 7 8 9 10 10 10 10 10 10 10	-11.0340 -14.0400 -15.4183 -15.51935 -14.8445 -14.4445 -14.3964 -14.4355 -14.4422 -14.4190 -14.3935 -14.3792	24.95995 25.05432 28.67774 32.43230 34.19446 34.70181 34.75353 34.71623 34.71005 34.73432 34.76046 34.77487

3.3. Estimation of the VAR order

The order of a VAR model is the number of lags included in the model. The appropriate lag length (𝑝), should be estimated long enough for the residuals not to be serially correlated. The lag length plays a crucial role in diagnostic tests as well as in the estimation of VAR models for impulse response analysis and variance decomposition (Bhasin, 2004). Four of the selection criteria i.e. the Final Prediction Error (FPE), Akaike Information Criterion (AIC), the Schwarz Information Criterion(SC), and the Hannan-Quinn Information Criterion(HQ) support the inclusion of 2 lags for the VAR order. The fact that the majority of the criteria support the inclusion of two lags, the AIC which was the main guideline also supports that. Therefore the estimated VAR is VAR(2) model is as presented Table − 11.

Table − 7: The long run estimation results for heart disease cases

variable

Lag

Estimate

Std. error

t-value

p-value

Heart

Constant

0.4084

-0.0923

2.9840

0.0931

0.1871

1.9300

3.0983

-0.0983

1.2300

0.0023**

0.2331

0.3891

3.4. Granger Causality Test

An effective method for identifying whether a time series is suitable for forecasting. It is used when the coefficient of the lag of the other variable is greater than zero. At the usual significance level of 5%, the result demonstrates that hypertension Granger-causes heart illnesses. This indicates that future values of heart disease can be predicted using the past data on hypertension. But according to this study, the contrary is not true.

Table − 8: presents the result from the Granger causality tests.

NULL HYPOTHESIS	F-STATISTIC	P-VALUE
Hypertension does not Granger-cause Heart Disease	-28.938	0.0005
Heart Disease does not Granger-cause Hypertension	-20.3293	0.7209

Granger At the usual significance level of 5%, the result demonstrates that hypertension Granger-causes heart illnesses. This indicates that future values of the past data on hypertension. But according to this study, the contrary is not true. The aforementioned test clearly showed that the claim that "heart disease does not Granger cause Hypertension" was untrue.

In the VAR model, the dependent variables are subject to shocks from each of the variables. Knowing how the variables interact with one standard deviation of positive shock applied to the error terms allows for the identification of the variable's response. Thus, a unit shock is calculated for each variable from each equation separately. applied to the erroneous terms, and the long-term impacts on the VAR system are indicated. The endogenous variables in the shocks on level are identified using a conventional decomposition.

3.5. Impulse Response Function.

Impulse responses show how well the dependant variables in the VAR model are able to respond to shocks from each of the variables. Knowing how the variables interact with one standard deviation of positive shock applied to the error terms allows for the identification of the variable's response. Therefore, a unit shock is administered to the erroneous terms for each variable from each equation independently, and the consequences on the VARsystem over time are observed. The endogenous variables on level are identified using a conventional decomposition.

The impulsive response is graphically depicted. The y-axis displays the direction and magnitude of the impulse, or the percentage deviation in the dependent variable from its baseline level, while the x-axis displays the time horizon, or the duration of the shock. The black lines inside the red dashed lines show the % variance, while the red dashed lines show the lower and upper bounds of the confidence interval. Figure 4.4 depicts how heart illnesses react to hypertension standard deviation innovations and to their own shocks (Heart Disease shocks).

3.6. Analysis of the Response from Heart Disease cases

Figure 3.2 shows that when a shock of one standard deviation is applied to the cases of heart disease, the response to the shock (by the cases of heart disease) is a gradual decrease in the number of cases from period one to period three, after which it maintains uniformity until the conclusion. The response of cardiac instances to a one standard deviation shock to hypertension cases is also an instant decrease followed by a gradual increase from period one to period four, following which it remains constant the entire time until the end.

A small increase in heart disease cases in Tamilnadu is anticipated for the year 2020. Even when the trend or rise is minor, stakeholders in the area shouldn't let their guard down. Instead, the “Ministry of Health” should coordinate with local healthcare providers to give locals comprehensive education about the risk factors for heart diseases and the importance of visiting medical facilities to check their blood pressure.

The highest monthly incidence of heart disease cases reported in Tamilnadu was 57. Heart disease cases occurred on average 9.934 times per month. The months of March had the greatest monthly average for heart disease cases (16.9 cases), and the months of November had the lowest monthly average for heart disease cases (4.9 cases). For the study period, the months of January were related with the highest monthly average of hypertension cases.

Author Contribution

Co-author of the article

Mackay J, Mensah G. Atlas of “heart disease and stroke.” Geneva: World Health Organization; 2004
Kaur M. “Blood pressure trends and hypertension among rural and urban Jat women of Haryana, India”. Coll Antropol 2012;36:139 – 44.
Dutta A, Ray MR. “Prevalence of hypertension and pre-hypertension in rural women: A report from the villages of West Bengal, a state in the eastern part of India.” Aust J Rural Health 2012;20:219–25.
“World Health Organization” (2014). Non communicable diseases country profile. Retrieved 18.5.2015.
Gupta R. Recent trends in coronary heart disease epidemiology in India. Indian Heart J. 2008;60:B4–B18.
“Hypertension Study Group Prevalence, awareness, treatment and control of hypertension among the elderly in Bangladesh and India: a multicentre study.” Bull World Health Organ 2001; 79:490–500
Malhotra P, Kumari S, Kumar R, Jain S, Sharma BK.” Prevalence and determinants of hypertension in an un-industrialised rural population of North India.” J Hum Hypertens 1999; 13:467–472
Gupta PC, Gupta R,” Pednekar MS. Hypertension prevalence and blood pressure trends in 88 653 subjects in Mumbai, India.” J Hum Hypertens 2004; 18:907–910
Mohan V, Deepa M, Farooq S, Datta M, Deepa R. Prevalence,” awareness and control of hypertension in Chennai: the Chennai Urban Rural Epidemiology Study (CURES-52)”. J Assoc Physicians India 2007; 55:326–332
J. P & Marks J, W. 2011.” High blood pressure (hypertension)”. Available at: http://www.medicinenet.com/high_blood_pressure/article.htm. [Accessed on January 2015]
Dickey, D. A., and Fuller, W. A., (1979). “Distribution of the Estimators for Autoregressive Time Series with a Unit-root” Journal of the American Statistical Association, 74: 427–431.

IN TAMILNADU

No competing interests reported.

Download PDF

Version 1

posted

You are reading this latest preprint version

Multivariate Time Series Analysis Using Heart Disease Patent in Tamilnadu

Status:

Version 1

Abstract

Figures

1. Introduction

2. Model description

2.1. The Vector Autoregressive model (VAR)

2.2. Estimation of the VAR model

2.3. The Breush-Godfrey LM Test

2.4. Granger Causality test

3. Results and Discussion

3.1. Unit root properties of individual series

3.2. Stability Test

3.3. Estimation of the VAR order

3.4. Granger Causality Test

3.5. Impulse Response Function.

3.6. Analysis of the Response from Heart Disease cases

4. Conclusion

Declarations

Author Contribution

References

Unsectioned Paragraphs

Additional Declarations

Status:

Version 1