Correlation and Regression Analysis in Public Health: A Comprehensive Guide, Lecture notes of Biostatistics

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Correlation and Linear Regression

Regression and Correlation

• Many medical investigations are concerned with:
  • Establishment of relationship between two variables.
  • The strength of a relationship.
  • Predicting one variable on the basis of another.
  • – Controlling the effect of unwanted variables.Controlling the effect of unwanted variables.
• Such intentions can be addressed either by using correlation

or regression analysis.

Correlation Analysis

• Does not imply cause and effect relationship.
• The value of r ranges from -1 to +1.
• If the correlation coefficient is greater than 0, the variables are said to be positively correlated (i.e. as X increases, Y tends to increase).increase).
• If the correlation coefficient is less than 0, the variables are said to be negatively correlated (i.e. as X increases, Y tends to decrease).
• If the correlation coefficient is 0 then the variables are said to be uncorrelated.

Correlation Analysis Cont…

• The formula for computing sample correlation coefficient (r) for two variables X and Y is given as:
• Or

[ ( ) ][ ( ) ]

x x^2 y y^2

x x y y

r

• Or
• Before computing r, scattered plot between the two variables should be drawn. Why?

[  ( x  x ) ][  ( y  y ) ]

[ n( x^2 ) ( x)^2 ][n( y^2 ) ( y)^2 ]

n xy x y

r

Correlation Analysis Cont…

y y

Strong relationships Weak relationships

(continued)

y

x

x

y

x

x

Correlation Analysis Cont…

y

No relationship

(continued)

y

x

x

Correlation Analysis Cont…

Example 8.1:

• The data of a random sample of 20 countries are shown in the following table. X represents the percentage of children immunized by age one year and Y represents the under five year mortality rate.year mortality rate.
• Determine the strength of association between the two variables.

Correlation Analysis Cont…

• Country % Immunized (X) CMR/1000LB(Y) XY Y^2 X
• Bolivia
• Brazil
• Cambodia
• Canada
• China
• Czech
• Egypt
• Ethiopia
• Ethiopia
• Finland
• France
• Greece
• India
• Italy
• Japan
• Mexico
• Poland
• Russia
• Senegal
• Turkey
• UK
• Total

Correlation Analysis Cont…

• Interpretation option:
  • Rule of thumb: Size of Coefficient General Interpretation

0.8-1.0 Very strong relationship

0.6-0.8 Strong relationship

0.4-0.6 Moderate relationship

0.2-0.4 Weak relationship

0.0-0.2 Very weak or no relationship

Correlation Analysis Cont…

• Hypothesis Testing for a Correlation Coefficient
• As that of mean and percentage, it is also possible to test significance about population correlation.
• For two tailed test
  • H 0 : is 0
  • H : is different from 0



• H 1 : (^) is different from 0
• The t test statistic is given as (with n-2 df):

2 1

2

r

n t r 

 



Correlation Analysis Cont..

• The critical t value for 0.05 level of significance at 18 (n-2) degree of freedom is - 1.734. Then we calculate the test statistics.

) 5.

18 ) 0. 79 ( 1 ( 0. 79 )

20 2

1. 79 ( 1

2 (^2)   2   

  

  r

n t r

• Hence we accept the H 1 that r indicates significant negative relationship between immunization coverage and child mortality.

1  r 1 ( 0. 79 )^ 0.

Correlation Analysis Cont..

Limitations:

• Applied only to a linear relationship.
• One must not extrapolate an observed correlation beyond observed ranges of the x and y value.
• Does not differentiate dependent and independent variables.
• • Confounding by a third variable.Confounding by a third variable.

Correlation Analysis Cont..

• The formula for the Spearman Correlation Coefficient is (given that there is no tied rank):
• Where;

( 1 )

6 ( ) (^1 )

2



n n

D r (^) s

• Where;
  • 6 is a constant,
  • D is the difference between a subjects ranks on the two variables,
  • n is the number of subjects.
• Consider the following example.

Correlation Analysis Cont..

Countries (Per100,00MMR 0LB)

MMRRank

Delivery CoverageService (%)

Rank D D^2

1 315 4 55 6 -2 4 2 450 6 40 5 1 1 3 200 1 70 8 -7 49 4 250 3 79 10 -7 49

The following table presents the MMR level and delivery service coverage in 10 developing countries.

6 (  D 2 )^4 250 3 79 10 -7^49

5 243 2 75 9 -7 49 6 830 9 25 3 6 36 7 850 10 21 2 8 64 8 656 7 20 1 6 36 9 701 8 30 4 4 16 10 410 5 60 7 -2 4 308

= 1- [(6x308)/10(100-1)] = 1-[1848/990] = 1-1. = -0.

2



n n

D

rs