statistics, regression, correlation, test of significance, regression equation, medical statistics

1. Regression and correlation analysis
Prof. Amany R.Abo-El-Seoud
Community Medicine ,Zagazig
University,EGYPT

2.  The term regression analysis refers to the methods used to
estimate the values of a variable from a knowledge of the
values of another variable
Regression
Regression and correlation analysis
 correlation analysis refers to the methods used to measure the
strength of the association (correlation) among these variables.
Our study here will concentrate on the relationship between
two variables only.
Correlation analysis

3.  Regression analysis is classified into many types according to
the type of relationship between the two variables such as:
 linear,
 exponential,
 logarithmic and
 power regression analysis.
In the next paragraphs will concentrate on the linear regression
analysis, which is our objective.
 The term linear means that an equation of a straight line is
used to describe the relationship between the two variables i.
e. the relationship between the two variables is represented by
a straight line, which is usually called the regression line.
Linear regression analysis:
Regression and correlation analysis

4.  In studying the relationship between two variables it is
advisable to plot the data on a graph as a first step. This allows
visual examination of the extent of association between the
variables.
 The chart used for this purpose is known as a scatter diagram
which is a graph on which each plotted points represents an
observed pair of values of the dependent (Y) and independent
(X) variables.
Scatter diagram:
Regression and correlation analysis
0 1 2 3 4 5 6 7
0
1
2
3
4
5
6
7
Dose (mg/kg)
Numberofdeadanimals

5.  The relationship between the two variables X and Y should be
represented by a straight line that is the regression line. The
regression line is the best or the ideal line or its equation
which represents a scatter diagram which may describe a
possible correlation between two variable.
 The equation of the linear regression line is calculated by the
following equation: Y = aX + b
Where Y is the dependent variable, X is the independent
variable, a is the slope of the linear regression line and b is
the intercept of the regression line with the Y-axis.
 the values of a and b of the equation of straight line can be
calculated:
Regression Equation:
Regression and correlation analysis
X  Y - N XY
a =
(X)2 - N X2
X  XY - X2 Y
b =
(X)2 - N  X2

6. Example:
The relationship between the toxic dose of a drug in mg/kg and
number of dead animals induced by its toxic effects was plotted
diagramatically using the X-axis for the dose of the drug and the Y-
axis for the number of dead animals. The following relationship was
obtained:
Dose (mg/kg) Number of dead animals
(X) (out of six: Y)
3 0
4 1
5 2
6 4
7 5
8 6
Calculate the equation of the linear regression line?

7. Example:
Solution:
X Y X2 XY
3
4
5
6
7
8
0
1
2
4
5
6
9
16
25
36
49
64
0
4
10
24
35
48
X = 33 Y = 18 X2 = 199 XY = 121
X  Y - N XY (33 X 18) - 6 X 121
a = = = 1.886
(X)2 - N X2 (33)2 - (6 X 199)
X  XY - X Y (33 X 121) - (199 X 18)
b = = = - 3.89
(X)2 - N  X2 (33 )2 - (6 X 199)
The equation is Y = 1.886X - 3.89

8.  When two series of observations are made, it is often found
that the observations in one series (dependent series) vary
correspondingly with those in the other (independent series).
The correlation between the two series of observations can be
tested by determination of the correlation coefficient (r).
 The correlation coefficient is measured on a scale that varies
from +1 through zero to -1. 1 expresses complete correlation
between the variables. When one variable increases with the
increase of the other, the correlation is positive while when
one variable decreases with the increase of the other variable,
the correlation is negative. Complete absence of correlation is
represented by r = 0.
The correlation coefficient:
Regression and correlation analysis
r = + 1 r = - 1 r = 0

9. The significance of the correlation between two variables can
be tested by comparing the value of t calculated according to
the following formula:
The correlation coefficient equation:
Regression and correlation analysis
_ _
 ( X - X )  ( Y - Y )  (dx . dy)
r = =
_ _
 (X - X)2.  (Y - Y)2  dx
2 . dy
2
n-2
t = r
1-r2

10. Example:
From the previous example mentioned in regression line , calculate
the correlation coefficient and test the significance of the
correlation ? ( t = 2.57 at P<0.05 & D. F. = 5 ) .
Regression and correlation analysis

11. Solution:
The value of the calculated t is higher than that of the tabulated t , therefore
there is a positive significant correlation between the dose of the drug and its
toxic effect
X dx dx
2 Y dy dy
2 dx . dy
3
4
5
6
7
8
-2.5
-1.5
-0.5
+0.5
+1.5
+2.5
6.25
2.25
0.25
0.25
2.25
6.25
0
1
2
4
5
6
-3
-2
-1
+1
+2
+3
9
4
1
1
4
9
+7.5
+3.0
+0.5
+0.5
+3.0
+7.5
 (dx . dy) 22
r = = = 0.994
 dx
2 . dy
2 17.5 X 25
n-2 6 - 2
t = r = 0.994 = 18.15
1-r2 1 - (0.994)2
Regression and correlation analysis

12. Another solution using Instat® software:
Regression and correlation analysis

13. Another solution using Instat® software:
Regression and correlation analysis

14. Another solution using Instat® software:
Regression and correlation analysis

15. Another solution using Instat® software:
Regression and correlation analysis

16. Another solution using Instat® software:
Regression and correlation analysis

17. Another solution using Instat® software:
Regression and correlation analysis

18. Example:
The dose response curve of a new antihypertensive agent was
performed using five rats, the following results were obtained:
Dose ( mg/kg ): 2 4 6 8 10
Decrease in B. P.(mmHg): 15 20 25 30 35
A) Find the equation of the linear regression line?
B) Calculate the correlation coefficient?

19. Example:
Solution:
X Y X2 XY
2
4
6
8
10
15
20
25
30
35
4
16
64
84
100
30
80
150
240
350
X = 30 Y = 125 X2 = 268 XY = 850
X  Y - N XY (30 X 125) - 5 X 850
a = = = 1.134
(X)2 - N X2 (30)2 - (5 X 268)
X  XY - X Y (30X 850) - (30 X 125)
b = = = - 49.4
(X)2 - N  X2 (30 )2 - (5 X 268)
The equation is Y = 1.134X – 49.4