1. SIMPLE LINEAR REGRESSION AND CORRELLATION
By Mpembeni RNM,
School of Public Health and Social Sciences, Dept of Epidemiology and Biostatistics
MUHAS

2. LEARNING OBJECTIVES
After successful completion of this session, you should be able to:
Describe the correlation coefficient
Describe the linear regression model
Understand and check model assumptions
Understand meaning of regression coefficients

3. ANALYSING RELATIONSHIPS BETWEEN TWO OR MORE QUANTITATIVE VARIABLES
Two commonly used Methods are:
Correlation
linear regression
Multiple Linear Regression

4. CORRELATION
The (Pearson's) correlation coefficient, r measures the closeness (strength) of the linear association
i.e. the closeness with which the points lie along the straight line
r is a bivariate correlation coefficient summarizing the magnitude and direction of the relationship between two variables

5. Characteristics of r Ranges between -1 and +1
r = 0: No linear relationship
r = 1 perfect positive relationship
r = -1 perfect negative relationship

6. Interpretation of r
If r > 0: variables are positively correlated. i.e as x increases, y tends to increase, while as x decreases, y tends to decrease
If r < 0: variables are said to be negatively correlated. i.e as x increases, y tends to decrease, while as x decreases, y tends to increase

7. Little or No Correlation: -0.3 to 0.3
Rule of thumb for r
Correlation
Strong
Weak
Positive
up and right
0.7 to 1.0
0.3 to 0.7
Negative
down and left
-1.0 to -0.7
-0.7 to -0.3
Little or No Correlation:  -0.3 to 0.3

8. SCATTER DIAGRAM
First step in investigating the relationship between two variables
Two related variables - plotted on a graph in the form of points or dots
Each point on the diagram represents a pair of values, one based on X-scale and the other based on Y-scale.
X-scale refer to the explanatory or independent variable and the Y-scale refer to the response or dependent variable.
Diagram shows visually the shape and degree of closeness of the relationship

9. Head circumference and Gestational age of 100 LBW babies

10. Scatter Plot
From the scatter plot, there is a trend of head circumference to increase with increasing gestational age

11. Strong positive correlation

12. Weak negative correlation

13. No correlation

14. CORRELATION COEFFICIENT
r=∑(X-Xˉ)(Y-ӯ)
√∑(X-X͞) 2∑(Y-Ῡ)2
= ∑XY-(∑X)(∑Y)/n
√∑x2-(∑x)2/n ∑y2(∑y)2/n

15. Example: Association between Body weight and Plasma volume

16. Calculation of r ∑xy – (∑x∑y)/n = 1615.295 – 535 x 24.02/8=8.96
∑x2 –(∑x)2/n = /8 =
∑y2-∑y2/n = – /8 = 0.678
r = 8.96
√( x 0.678)
= 0.76

17. STRENGHT OF THE ASSOCIATION BETWEEN WEIGHT AND PLASMA VOLUME
How strong is the association?

18. Simple Linear Regression
The two quantitative variables should be defined:
y refers to the dependent variable (AKA response or outcome variable)
x the independent variable (AKA explanatory or predictor variable)

19. Simple linear regression
The objective of the analysis is to see whether a change in an independent variable, x, is associated with a change in the dependent variable, y,
Be able to predict the value of the dependent variable given the value of the independent variable
Eg Age and Weight of a child under five years of age.

20. EXAMPLE Data on body weight and plasma volume of eight healthy men.
The objective of the analysis is to see whether a change in plasma volume is associated with a change in body weight.

21. ASSOCIATION BETWEEN QUANTITATIVE VARIABLES

22. .Scatter Diagram of Body weight and Plasma Volume

23. Body weight and plasma volume
There is a trend of plasma volume to increase with increasing body weight

24. LINEAR REGRESSION
When Linear relationship exists, can summarize the relationship by a line drawn through the scatter of points.
any straight line drawn on a graph can be represented by the equation: y = a + bx
where y refers to the values of the
response (dependent) variable
x to values of the explanatory
(independent) variable.

25. LINEAR REGRESSION
The constant 'a' is the intercept, the point at which the line crosses the y-axis.
That is, the value of y when x = 0.
The coefficient of x variable ('b') is the slope of the line.
It tells us the average change (increase or decrease) in y due to a unit change in x.
b is also called the regression coefficient.

26. METHOD OF LEAST SQUARES
A mathematical technique to fit a straight line to a set of points
i.e is used to estimate a and b

27. LINEAR REGRESSION
Numerator =Sxy=
Denominator =
= Sxx

28. LINEAR REGRESSION
The resultant line is called the regression line, which estimates the average value of y for a given value of x.

29. Calculating the least Square Estimates Example – data on plasma volume and body weight

30. Example

31. Example
Regression line:
Plasma volume = x body weight

32. ESTIMATION
Once you have the value of a and b, you can substitute various values of x into the equation for the line, solve for the corresponding values of y.
Eg what would be the plasma volume for an adult with 62 kgs?
77 kgs?

33. Regression line

34. INFERENCES FOR REGRESSION COEFFICIENTS
Just like in any other estimate, the standard error for the regression coefficient can be calculated.
Can test the hypothesis whether b differs significantly from b0 using a t test
The t value and the corresponding p-value are all shown in the output table.

35. Evaluation of the model
The coefficient of Determination, R2 which is the square of the Pearsons Correllation Coefficent, r, is used to assess how best the model fits the data.
This is the proportion of the variability among the observed values of y that is explained by the linear regression of y on x

36. Model Evaluation
If for example R2 is it implies that almost 61% of the variation among the observed values of y is explained by its linear relationship with the independent variable

37. EXERCISE Using the provided data set ( LBW babies)
Correlate birth weight with Gestational age
What is the Correlation Coefficient between bweight and Gestage?
Regression of Birth weight on gestational age. What is the equation of the line?
What is the estimated birth weight for a baby with 42 weeks of gestation?, 36 weeks?
What proportion of the variability of birth weight is explained by gestational age?

38. Model Unstandardized Coefficients Standardized Coefficients t Sig.
Coefficients(a)
Model Unstandardized Coefficients Standardized Coefficients t Sig.
B Std. Error Beta
1 (Constant)
gestage
a Dependent Variable: birthwt