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.

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            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 to 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 tcal < ttab, 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 = 
ij
 
      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

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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. 2nd 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); 3rd Edition, New York, Schaum’s outline series: McGraw-Hill.
APPENDIX A
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

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