REGRESSION MODEL TO RELATIONSHIP BETWEEN THE BLOOD PRESSURE AND THE WEIGHT, AGE AND HEIGHT

MULTIPLE REGRESSION MODELS AND USE THE MODEL TO THE RELATIONSHIP BETWEEN THE BLOOD PRESSURE AND THE WEIGHT, AGE AND HEIGHT

CHAPTER ONE

1.0   INTRODUCTION

                A common factor of many scientific investigations is that variation in the value of one variable is caused to a great extent, by variation in the values of other related variables. For instance, variation in crop yield can largely be explained in terms of variation in the amount of rainfall and the quantity of fertilizer applied. The amount of fuel consumed by a certain brand of car over a given distance varies according to the age and the speed of the car and so forth. Therefore, a primary goal of many statistical investigations is to establish relationships which make it possible to predict one variable in terms of others.

               Regression analysis is a statistical investigation of the relationship between a dependent variable Y and one or more independent variable(s) X or X’s, and the use of the modeled relationship to predict, control or optimize the value of the dependent variable Y. The relationship is formulated in an equation that express the values of Y in terms of the corresponding values of X or X’s and enables future values of Y to be predicted in terms of the observed values of X, or to be controlled or optimized by calculating the values of X or X’s. The independent variables X’s are also called explanatory variables or controlled variables, while the dependent variable Y is also called the response variable.

            Regression models are of various kinds. A regression study involving only two variables, a dependent variable Y and one independent variable X, is called a simple linear regression or univariate regression while a study involving a Y-variable and two or more X-variable is called a multiple regression. The term bivariate regression and multivariate regression are often used to distinguish between multiple regression involving two X-variables and those involving more than two X-variable. If a regression is linear in the X’s and the parameters, we refer to it as a simple linear regression or a multiple linear regression depending on whether it involves one X-variable or more than X-variables. An example of a simple linear regression model is:

……………………………………(1) 

While an example of a multiple linear regression model is:

        …………………………………………………………….. (2)

                  

Regression being linear in the X’s and the parameters means that no term in the model involves second and higher powers of the X’s or the parameters, or a product or quotient of two X’s or two parameters.

1.1       AIMS AND OBJECTIVES OF THE STUDY

The main aim of this project is t
o derive the multiple regression models and use the model to the relationship between the blood pressure and the weight, age and height of 100 individuals

1.2       SCOPE OF THE STUDY

This project is restricted to the data gotten from the Federal Medical Centre Owo, Ondo State which covers the blood pressure, weight, age, height of 100 individuals in 2010.

1.3   IMPORTANCE OF STUDY

   In Statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis help one to understand  how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates the conditional expectation of the dependent variables given the independent variables i.e. the average value of the dependent variable when the independent variables are fixed. Less commonly, the focus is on a quantile, or other location parameter of the conditional distribution of the dependent variable given the independent variables called the regression function.

      Regression analysis is widely used for forecasting, where its use has substantial overlap with the field of machine learning. Regression analysis is also used to understand which among the independent variables are related to the dependent variable, and to explore the forms of these relationships.

1.4   DEFINITION OF TERMS

During the course of this research, so many terminologies and abbreviations were encountered which are precisely defined below:

DATA: data is a fact or piece of information, especially when examined and used to find out things or to make decision.

PARAMETERS: This is something that decides or limits the way in which something can be done.

ERROR: Error is a random variable with a mean of zero conditional on the explanatory variables.

REGRESSION ANALYSIS: This is a statistical tool, which helps to predict one variable from another variables or variables on the basis of assumed nature of the relationship between the variables.

DBP: Diastolic Blood Pressure

SBP: Systolic Blood Pressure

TSS: Total Sum of Squares

SSE: Sum of Squares due to Error

DF: Degree of Freedom

CHAPTER TWO

2.1   LITERATURE REVIEW

            Regression analysis is a statistical methodology that utilizes the relationship between two or more quantitative variables so that one variable can be predicted from the other(s). This methodology is widely used in business, the social and behavioral sciences, biological sciences and many other disciplines.

    The term “regression” was coined by Sir Francis Galton in the nineteenth century to describe a biological phenomenon. The phenomenon was that the heights of descendants of tall ancestors tend to regress down towards a normal average. For Galton, regression had only this biological meaning, but his was later extended by Udny Yule and Karl Pearson in 1913 to a more general statistical context. In the work of Yule and Pearson, the joint distribution of the response and explanatory variables is assumed to be Gaussian. This assumption was weakened by R. A. Fisher in his work of 1922 and 1925. Fisher assumed that the conditional distribution of the response variable is Gaussian, but the joint distributions need not to be. In this respect, Fisher’s assumption is closer to Gauss’ formulation of 1821.

        The earliest form of regression was the method of least squares which was published by Legendre in 1805 and by Gauss in 1809. Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the sun (mostly comets, but also later the then newly discovered minor planets). Gauss published a further development of the theory of least squares in 1821, including a version of the Gauss-Markov theorem.

          Blair [1962] described regression analysis as a mathematical measure of the average relationship between two or more variables in terms of the original units of the data.

        Hamburg [1970] said regression analysis refers to the methods by which estimates are made of the values of a variable from knowledge of the values of one or more other variables and to the measurement of the errors involved in this estimation process.

        Yamane [1974] and Karylowski [1985] both said in a presentation that one of the most frequently used techniques in economics and business research to find a relation between two or more variables that are related casually is regression.

        Chou [1978] says regression analysis attempts to establish the ‘nature of the relationship’ between variables i.e. to study the functional relationship between the variables and thereby provide a mechanism for prediction or forecasting.

        Multiple regression methods continue to be an area of research. In recent decades, various definition and derivation have been made, and the derivations have been to various life problems.

      Koutsoyiannis [1973] in his book ‘Theory of Econometrics’ used co-factor approach to derive the multiple regression model and then used the derivation to show that the economic theory postulates that the quantity demanded for a given commodity depends on the price and on consumers’ income.

      Schaeffer and McClave [1982] in their book ‘Statistics for Engineer’ used the inverse matrix method to derive the multiple regression and then went further using the derivation to show that the average amount of energy required to heat a house depends not only on the air temperature, but also on the size of the house, the amount of insulation, and the type of heating unit.

       Okeke [2009] in his book ‘Fundamentals of Analysis of variance in statistics designed Experiment’ used the crammer’s rule to show the multiple regression and then used it to show that a chemical process may depend on temperature, pressure and concentration of the catalyst.

CHAPTER THREE

3.0   SOURCE OF DATA

The data used in this project work was collected from the Federal Medical Center Owo Ondo State.

3.1   METHOD OF DATA COLLECTION

        The data used in this project work is a secondary source.

3.2   METHOD OF ANALYSIS

        MULTIPLE REGRESSION ANALYSIS

The aim of multiple regression is to examine the nature of the relationship between a given dependent variable and two or more independent variable. The model describing the relationship between the dependent variable Y and a set of k independent
variable  can be expressed as

       

Here, n is the number of observation on both the dependent and the independent variable. is the ith observation on the dependent variable Y.  are known constants representing respectively the ith observation on the independent and normally distributed

In this project work, we are going to use multiple regression to examine the nature of the relationship between blood pressure and weight, age and height.

3.3   PARAMETER ESTIMATION IN MULTIPLE PEGRESSION

 can be writer in matrix notation as     

n x 1                    n x (k+1)        (k+1)x1  nx1

Which can still be written as

The sum of squared residuals is

                      

Differentiating with respect to B and equation to zero we have

Dividing both sides by two

Where   

3.4   HYPOTHESIS TO BE TESTED

We are going to test if weight, age, or height cannot single handedly cause a change in blood pressure (H₀) or otherwise (H₁).

DECISION: If t_cal < t_tab, we accept H₀ otherwise reject H₀ and accept H₁.

3.5   ANOVA TABLE FOR MULTIPLE REGRESSION

Source	DF	SS	Ms	F-ratio
Regression	K	SSR
Error	N-K-1	SSE	SSE
Total	n-1	SST

Where SSR = 

      SST =
 3.6   COEFFICIENT OF MULTIPLE CORRELATION

          The coefficient measures the proportion of the total variation in the dependent variables y that is ascribed or attributed to y on the independent variables that are include in the regression if is given as:

CHAPTER FOUR

DATA ANALYSIS

4.0   INTRODUCTION

This chapter presents the summary of data to be studied and analyzed. Since the procedure under listed in solving multiple regression problem have been known in the previous chapter , we shall now emphasize on real life problem using the data then from Federal Medical center Owo Ondo State as case study.

          Furthermore, the steps and formula stated in chapter three can also be applied here, but we shall be focusing only on the results extracted from statistical software called MINITAB used in analyzing this data.

4.1   THE MINITAB RESULTS SHOWING THE RELATIONSHIP BETWEEN THE SYSTOLIC BLOOD PRESSURE AND THE WEIGHT, AGE, AND HEIGHT OF 100 INDIVIDUALS

The regression equation is:

 4.2   ADEQUACY OF THE MODEL

To test the significance of each parameter

P-value = 0.624

CRITICAL VALUE FROM THE T-TABLE

Decision:- Since  = 4.8899  critical value =1.980 we reject  and accept  and conclude  that weight contribute majorly to change in SBP i.e any increase in weight, lead to change in SBP and it is significant  at  = 5%

HYPOTHESIS FOR  

      

P-value = 0.000

Decision:- Since  = 5.3774  critical value

We reject  and conclude that age is a major factor for change in SBP i.e. the older we become, the nearer we are to high blood pressure and it is significant at   = 5%

HYPOTHESIS FOR

    

P-value = 0.884

Decision: Since  we accept  and conclude that height cannot single handedly contribute to a change in SBP and it is not significant at

4.3           ANALYSIS OF VARIANCE

Source	DF	SS	MS	F	P
Regression	3	2675.57	891.86	10.52	0.000
Error	96	8135.47	84.74
Total	99	10811.04

TEST OF HYPOTHESIS

     

DECISION

Since the P-value = 0.000 falls in the acceptance region, we conclude that weight, age, height are factors to be considered for a change in SBP

4.4   COEFFICIENT OF MULTIPLE CORRELATION

From the ANOVA table, we have

               = 0.2472 X 100% = 24.74%

This implies that 24.74% of the total variation are normal in terms of change in blood pressure when the factors i.e. weight () Age () and height () were considered.

          In other words, 75.26% are suffering from either high blood pressure or low blood pressure

CHAPTER FIVE

5.1   SUMMARY

           Regression analysis is a statistical tool used to establish linear relationship between predetermined variables and dependent variable and thereby be able predicts the future variable using the other variables.

         In this project, multiple regression was exhaustively discussed and was used in analyzing the effect of weight, age, height on blood pressure and conclusions were drawn.

5.2   CONCLUSION

             From the result of the statistical software MINITAB used in analyzing the data, and from the interpretation of the result in chapter four, it was observed that weight contribute majorly to change in blood pressure when age and height are held fixed. Age also contribute majorly to change in blood pressure when weight and height are held fixed but height does no contribute to change in blood pressure when weight and age are held fixed. But the three (weight, age and height) jointly contribute immensely to a change in blood pressure.

5.3   RECOMMENDATION

        Since it has been shown that any unit change in our weight in conjunction with the advancement of our age can cause a change in our blood pressure, it is advised that we concentrate on our weight to avoid being over weighted and as we advance in age, we should always go for regular check of our blood pressure.

        The health sector should also endeavour to always give adequate advice to the public on the effect of the food we eat and engaging in excessive thought (too much of thinking) on our blood pressure.

REFERENCES

 Arua, A.I. and Okafor, F.C. (1997); Fundamentals of Statistics for Higher Education: Fijac Academic Press.

Dixon, W.J. and Massey, F.J. (1969); Introduction to Statistical Analysis: New York McGraw-Hill book company.

Draper, N.R., and Smith, H. (1988); Applied Regression Analysis, New York: John wiles & sons.

Francis, A. (1986); Business Mathematics and Statistics: DP Publications, Aldine house, Aldine place 142/144, Uxbridge road, London.

Gupta, S.P. (1969); Statistical Methods: Sultan Chand and Sons, 23, Daryaganj, New Delhi.

Kleinbaum, D.G., Kupper, L.L., and Muller, K.E. (1988); Applied Regression Analysis and other Multivariable Methods. 2^nd Edition, Boston: PWS-Kent Publishing Company.

Koutsoyiannis, A. (1973); Theory of Econometrics: Palgrave Houndmills, Basingstoke and Hampshire New York.

Okeke, A.O. (2009); Fundamentals of Analysis of Variances in Statistical Designed Experiments: Macro Academic Publishers; 1, Anigbogu close Achara layout Enugu.

Schaeffer R.L. and McClave, J.T. (1982); Statistics for Engineers: PWS Publishers, a division of Wadsworth, Inc. USA.

Spiegel, M.R., and Stephens, L.J. (1999); 3^rd Edition, New York, Schaum’s outline series: McGraw-Hill.

APPENDIX A

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S/N	Weight (kg)	SBP	DBP	AGE	Height (m)	BMI (kg/m2)
1	72	164	82	72	1.76	23.24380
2	65	108	70	24	1.71	22.22906
3	67	128	75	17	1.62	25.52964
4	70	124	70	23	1.67	25.09950
5	75	139	74	42	1.48	34.24032
6	67	144	65	51	1.83	20.00657
7	73	127	75	22	1.89	20.43616
8	75	136	78	41	2.1	17.0068
9	78	138	76	64	1.55	32.46618
10	78	112	68	28	1.71	26.67487
11	71	129	80	24	1.62	27.0538
12	71	140	90	63	1.22	48.3741
13	72	115	63	22	1.71	18.80921
14	55	120	80	24	1.89	19.03642
15	68	123	85	27	2.0	17.75000
16	71	135	64	23	2.01	16.08871
17	84	134	97	57	1.62	32.00732
18	60	125	88	23	1.68	21.2585
19	77	122	80	26	1.65	28.28283
20	56	115	70	19	1.92	15.19097
21	52	129	80	22	1.55	21.71166
22	70	130	80	26	1.65	21.64412
23	78	126	73	26	1.82	23.54788
24	75	112	65	24	1.98	19.1307
25	66	129	80	53	1.52	28.56648
26	70	125	76	23	2.04	16.82045
27	67	126	79	60	1.82	20.22703
28	59	113	74	24	1.55	24.55775
29	65	123	72	23	1.80	20.06173
30	55	128	72	33	1.77	17.55562
31	60	136	67	42	1.74	19.81768
32	88	140	90	76	1.77	28.08899
33	80	119	76	25	1.55	33.29865
34	70	124	76	25	1.83	20.90239
35	78	116	66	27	1.83	23.29123
36	84	140	80	55	1.98	21.42639
37	82	130	80	29	1.86	23.70216
38	63	125	70	20	1.74	20.80856
39	74	134	92	65	1.58	27.6469
40	55	147	84	69	1.07	48.03913
41	63	133	87	52	1.77	20.10916
42	70	131	95	75	1.70	24.22145
43	73	117	70	22	2.01	18.66886
44	75	125	68	24	1.86	21.67881
45	59	108	63	28	1.49	26.57538
46	76	130	77	26	1.92	20.61632
47	67	117	74	40	1.83	20.00657
48	60	124	77	23	1.71	20.51913
49	80	128	75	60	1.86	23.12326
50	70	132	80	55	1.98	17.85532
51	75	132	73	71	1.79	23.10751
52	62	160	89	75	1.21	42.34683
53	73	123	63	23	1.89	20.43616
54	75	136	80	60	1.34	41.76877
55	60	117	71	25	1.49	27.02581
56	82	134	83	44	1.89	22.95568
57	72	127	74	28	1.52	31.16343
58	64	130	72	80	1.55	26.63892
59	75	139	81	49	1.98	19.1307
60	68	131	76	63	1.74	22.46003
61	84	127	92	27	1.74	27.744745
62	75	128	82	21	1.56	30.81854
63	70	149	96	68	1.74	23.12062
64	78	128	77	24	1.71	26.67487
65	80	116	77	26	1.71	27.35885
66	52	120	80	25	1.74	17.17532
67	64	105	60	23	1.71	22.22906
68	60	129	74	21	1.65	22.03857
69	70	122	76	27	1.39	36.23001
70	60	118	84	22	1.59	24.97399
71	69	129	82	33	1.23	45.60777
72	74	132	81	40	1.83	22.09681
73	70	121	80	23	1.77	22.34352
74	65	121	73	35	1.70	20.74755
75	80	122	77	26	1.89	22.39579
76	74	123	79	34	1.79	23.09541
77	56	124	75	31	1.72	18.92915
78	70	128	80	43	1.68	24.80159
79	70	124	77	23	1.74	23.12062
80	69	115	69	59	1.68	24.44728
81	70	121	79	38	1.18	50.27291
82	72	126	76	23	1.98	18.36547
83	79	116	76	30	1.77	25.22625
84	68	125	81	24	1.77	21.70513
85	71	112	68	21	1.55	29.55255
86	75	112	72	35	1.77	23.93948
87	66	112	70	25	1.71	22.57105
88	67	125	85	77	1.63	25.21736
89	75	120	70	24	1.89	20.99605
90	72	121	74	23	1.77	22.9819
91	68	118	65	81	1.83	20.30517
92	72	113	75	22	1.89	20.15621
93	75	127	70	63	1.07	65.50790
94	77	113	76	24	1.55	32.04995
95	80	102	68	26	1.71	27.35885
96	75	115	65	24	1.98	19.1307
97	70	118	80	40	1.62	26.67276
98	78	110	76	63	1.22	42.40527
99	60	120	63	72	1.71	20.51913
100	59	115	80	90	1.69	20.65751

Abbreviations

WT = Weight

SBP = Systolic Blood pressure

DBP = Diastolic Blood pressure

BMI = Body mass index =

HT = Height.

APPENDIX B

Regression Analysis

The regression equation is

SBP = 110 + 0.60 WT(Kg) + 0.270 AGE + 0.68 HT(M)

Predictor      Coef      StDev         T       P

Constant   109.63       10.89      10.06   0.000

WT(Kg)        0.0603      0.1227       0.49    0.624

AGE              0.27008       0.05021       5.38    0.000

HT(M)              0.683       4.678           0.15    0.884

5 = 9.206      R-Sq = 24.7%     R-Sq(adj) = 22.4%

Analysis of Variance

Source       DF                SS                  MS         F       P

Regression    3     2675.57     891.86     10.52    0.000

Error 96 8135.47   84.74
Total        99    10811.04

Source        DF Seq SS
WT(Kg)        1       62.03

AGE              1     2611.74

HT(M)          1        1.81

Unusual Observations