Regression and Correlation Methods in Biostatistics: Linear Regression and Scatterplots, Study notes of Biostatistics

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Biostatistics

ostat st cs

Lecture 20

BIL 311

Lecturer: Dr Patricia Buendia

Lecturer: Dr. Patricia Buendia

Lecture 20 Outline „

Chapter 11 – Regression andCorrelation MethodsCorrelation Methods

Regression and Correlation

g

Methods Parameters of interest: „

Linear Regression: The Regression Line

Correlation: The Correlation Coefficient

Linear Regression

Li

i^

i^

th

fitti

f^

f^

ti

t^

t^

f

„

Li

near regression is the fitting of a function to a set of

observations

„

Usually there are two continuous variables (more are possible) divided into two types, experimentalvariables and a response variable

„

The Y variable is the observed response to the XThe

Y variable is the observed response to the X

values, so they are naturally paired data points

„

The X variable can be of two sorts:

It can be only values chosen by the experimenter

• It can be only values chosen by the experimenter– It can be observed, just like the Y values, so that the

experimenter has no control over what the X values are.

Scatterplot „^

Example

: The scatterplot below shows the data points from a

Example

:^ The scatterplot below shows the data points from a

study in which the estriol levels from pregnant women who arenear term are compared to the infant birthweight. „

If x

= estriol level and y=birthweight, then we can postulate a If x

estriol level and y birthweight, then we can postulate a

relationship between y and x that is described by:

E(y|x)=

α

β x

Linear Regression „

Definitions:–

Regression Line

: The line y=

α

βx, where

α

is the intercept

and

β

is the slope of the lineβ

p

– A

linear relationship

is one in which the relationship

between X and Y can best be represented by a straight line.

• A

curvilinear relationship

is one in which the relationship

A

curvilinear

relationship

is

one in which the relationship

between X and Y can best be represented by a curved line.(nonlinear regression)

Linear Regression „

More Definitions:–

A positive relationship

exists

when Y increases as Xincreases (i.e., when the slope

β

is positive). A negative relationship

exists

–^

A

negative relationship

exists

when Y decreases as Xincreases (i.e., when the slope

β

is negative).

g^

In the birthweight(x) and estriol(y) example we have

a positive relationship

a positive relationship

y is the dependant variable and

x the independent variable

Fitting Regression Lines „

The best fitting regression line is obtained byapplying the least squares criterion

„

The least squares line or estimated regression line

„

The least squares line, or estimated regression line,is the line y=a+bx that minimizes the sum of squareddistances of the sample points (S) from the givenline S is given by:line. S is given by:

2

1

1

2

)

(^

i

n i

i

n i

i^

bx a y

d

S^

− −

=

=^

∑

∑

=

=

Linear Regression „

Example: What is the estimated average birthweightif a pregnant woman has an estriol level of 15 mg/24hr?hr?

„

The estimated value y is denoted by ^y=a+bx

„

In our example we have

^y= 21.52+0.608(15)=30.

In our example we have

y

„

One possible use of the regression line for pregnantwomen is to identify women who are carrying a lowbi th

i ht f t

If

h

b

id

tifi d

birthweight fetus. If such women can be identifiedprior to conception, then drugs might be used toprolong the pregnancy until the fetus grows larger.

p

g

p

g

y

g

g

Goodness of fit of a regression lineGoodness

of fit of a regression line

More Definitions

Th

t t l

f^

T

t l SS

i^

(^

⎯^

„

Th

e total sum of squares or

T

otal SS

is

(y

• y)i

2

„

The regression sum of squares, or

Reg SS

, is

(^y

• i^

⎯y)

2 =bL

xy

=L

2 xy

/L

xx

y^

y

„

The residual sum of squares, or

Res SS

is

(y

-^yi

(^2) ) (^) i

=L

yy

- L

2 xy

/L

xx

„

It can be shown that

Total SS

= Reg SS + Res SS

„

It^

can be shown that

Total

SS

Reg SS + Res SS

(y

• i^

⎯y)

(^y

• ⎯i

y)

(y

-^yi

(^2) ) (^) i

= L

yy

„

Reg MS

= Reg SS/k, with k the number of predictor variables in

the model and k=1 for simple linear regressionthe model and k=1 for simple linear regression „

Res MS

= Res SS/(n-k-1)

F Test for Simple Linear RegressionF

Test for Simple Linear Regression „

To test for a good fit of a linear regression line we have theTo

test for a good fit of a linear regression line we have the hypothesis H

β

=0 versus H

β ≠

0, where

β

is the underlying

slope of the regression line. „

F

= Reg SS / [Res SS/(n-2)] with b=L

xy

/L

xx

which follows an F

1 n 2

F

Reg SS / [Res SS/(n 2)] with b L

xy

/L

xx

which follows an F

1,n-

distribution under H

„

If F> F

1,n-2,1-

α^

then reject H

0

„

If F

≤^

F

then accept H

„

If F

≤^

F

1,n-2,1-

α^

then accept H

0

„

The exact p-value is given by Pr( F

1,n-

>F)

„

Results displayed in ANOVA table

SS

df

MS

F^

P-value

Regression

a'=Reg SS

1

a'/

[a’/1]/[b’/n-2] Pr( F

1,n-

F)

Residual

b’=Res SS

n-

b’/n-

Total

a’+b’

Total

a^

b