• Correlation

• Regression

• Multiple Regression

• Curve fitting

Correlation / regression

Correlation

• Represents the relationship between two measurements

– Examples : height and weight, education level and income, BMI and skin fold thickness, wealth and fertility

– Correlation does not represent one causing the other, usually is present if both measurements are influenced by a common factor

– The value is from -1 to 1

• 0 = no relationship

• 1 = perfect relationship

• -1 = perfect inverse relationship

Sxy

SSqxSSqyr

n

xxSSqx

22

n

yySSqy

2

2

n

yxxySxy

2

xsumx

22 xsumx ysumy

22 ysumy

xysumxy

Confidence interval for r

• Correlation is not a linear measurement

– It stretches near 0 and compresses neat 1 or -1

– It has to be

• Transformed into a normally distributed linear measurement

• Have Standard Error estimated

• Have CI estimated

• Transformed back to the original format

r

rLogz

1

15.0

3

1

n

SE

SEzF 96.1 SEzG 96.1

12exp

12exp

12exp

12exp%95

G

Gto

F

FCI

Sample size

• Iterative procedure that satisfy two equations

3

2

m

zzn ba

12211

n

rrr

Logm

WhereZa = z value for Type I errorzb = z value for Type II error

• Correlation

• Regression

• Multiple Regression

• Curve fitting

Correlation / regression

Regression

• Draw a line which best fits the relationship between x and y– The line takes the form y = a + bx

• Where a is the y value when x=0

• Where b is the slope of the line, or how much y changes for one unit of change in x

– It assumes that y is dependent on x– It explains how changes in y values are governed by changes in

x values– It allows x to predict y

• Note : x = a + by is not the mirror image of y=a +bx, as how best fit is calculated differs

Regression - example

RegressionBest fit y=a + bx

RegressionBest fit x=a + by

Sxy

SSqxb

n

xxSSqx

22

n

yySSqy

2

2

n

yxxySxy

2

xsumx

22 xsumx ysumy

22 ysumy

xysumxy

n

xb

n

ya

Confidence interval for b

2

2

).( nSSqxSxy

SSqySE xyb

tSEbtSEbCIb

t = Student’s t for sample size and Type I Error

Confidence interval for predicted y

• SE 2 components and changes with x value– SE of regression slope b

– SE of departure from residual variation

SSqx

meanxx

nSSqx

SxySSqySE xy

22

.

1

xyxyxy tSEbtSExyCI ... .

Confidence interval for predicted y

• Correlation

• Regression

• Multiple Regression

• Curve fitting

Correlation / regression

Multiple Regression

• Outcome, particularly clinical outcome

– Are subjected to multiple influences

– All of which are related to each other

• Multiple regression model is therefore commonly needed

•BMI is influenced by mother and grandparents, but

•People who married tend to have comparable BMI

•Parent’s BMI tend to be dependent influenced by grandparents’

•Multiple regression y = a + b1x1 + b2x2 + b3x3 …bixi

Multiple Regression

• Starts with a matrix of Sum/products [S]k,k from k measurements

– where Si,j is the Sxy between any pair I and j

– Where Si,i is the SSqx of variable i

• This matrix is inverted [V] = [S]-1

• The Partial Regression Coefficient bi

iiyy

iyi

VV

Vb

,,

,

• The constant an

xb

n

ya i

ki

• Correlation

• Regression

• Multiple Regression

• Curve fitting

Correlation / regression

Curve fit

• In cases where the relationship between x and y are not linear

• y = function(x)

– y = Log(x)

– y = sine(x)

• Polynomial curve fit

– A special case of multiple regression

– Will fit into any shape where y increases with x

– y = a + b1x + b2x2 + b3x3 …..bkxk

– In most biological systems fitting to the power of 3 is sufficient

Polynomial curve fit

0

2

4

6

8

10

0 1 2 3 4 5 6x

y

Data point

y = a + bx

y = a + b1x + b2x2

y = a + b1x + b2x2 + b3x3

CI of polynomial fit

• Complexity of calculating Standard Error– Summing of each individual coefficients– Residual

• Solution – 2 stage procedure– Do polynomial curve fit– Calculate error (distance between each

datapoint from the regression line)– Curve fit error

-2

0

2

4

6

8

10

12

1 2 3 4 5 6

y = 14.45 – 16.66x + 5.83x2 – 0.45x3

SD = 0.29 + 0.18x

Curve fitFemur length according to gestational age

Gestation (days)

Fem

ur

len

gth

(cm

s)