Correlation and Linear Regression Peter T. Donnan Professor of Epidemiology and Biostatistics Statistics for Health Research.

Correlation and Linear Regression

Peter T. Donnan

Professor of Epidemiology and Biostatistics

Statistics for Health ResearchStatistics for Health Research

CONTENTS

• Correlation coefficients• meaning• values• role• significance

• Regression• line of best fit• prediction• significance

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INTRODUCTION

• Correlation• the strength of the linear relationship between

two variables

• Regression analysis• determines the nature of the relationship

• For example - Is there a relationship between the number of units of alcohol consumed and the likelihood of developing cirrhosis of the liver?

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PEARSON’S COEFFICIENT OF CORRELATION (r)

• Measures the strength of the linear relationship between one dependent and one independent variable• curvilinear relationships need other techniques

• Values lie between +1 and -1• perfect positive correlation r = +1 • perfect negative correlation r = -1• no linear relationship r = 0

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PEARSON’S COEFFICIENT OF CORRELATION

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r = +1

r = -1

r = 0.6

r = 0

SCATTER PLOT

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dependent variable

make inferences about

independent variable

Calcium intake

BMD

NON-NORMAL DATA

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NORMALISED WITH LOG TRANSFORMATION

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SPSS OUTPUT: SCATTER PLOT

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SPSS OUTPUT: CORRELATIONS

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Interpreting correlation

Large r does not necessarily imply: strong correlation

r tends to increase with sample size cause and effect

strong correlation between the number of televisions sold and the number of cases of paranoid schizophrenia

watching TV causes paranoid schizophrenia

may be due to indirect relationship

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Interpreting correlation

Variation in dependent variable due to: relationship with independent variable: r2 random noise: 1 - r2 r2 is the Coefficient of Determination or

Variation explained e.g. r = 0.661 r2 = = 0.44 less than half of the variation (44%) in

the dependent variable due to independent variable

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Agreement

Correlation should never be used to determine the level of agreement between repeated measures: measuring devices users techniques

It measures the degree of linear relationship You can have high correlation with poor

agreement

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Non-parametric correlation

Make no assumptions Carried out on ranks Spearman’s

easy to calculate Kendall’s

has some advantages over distribution has better statistical

properties easier to identify concordant / discordant

pairs Usually both lead to same

conclusions

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Role of regression

Shows how one variable changes with another

By determining the line of best fit Default is linear Curvilinear?

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Line of best fit

Simplest case linear Line of best fit between:

dependent variable Y BMD

independent variable X dietary intake of Calcium

value of Y when X=0

Y = a + bX

change in Y when X increases by 1

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Role of regression

Used to predict or explore associations the value of the dependent variable when value of independent variable(s)

known within the range of the known data

extrapolation is risky! relation between age and bone age

Does not imply causality

SPSS OUTPUT: REGRESSION

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Multiple regression

Later - More than one independent variable BMD may be dependent on:

agegendercalorific intakeUse of bisphosphonatesExerciseetc

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Summary

Correlation strength of linear relationship between two

variables Pearson’s - parametric Spearman’s / Kendall’s non-parametric Interpret with care!

Regression line of best fit prediction Multiple regression logistic