Statistics

Correlation and regression

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Introduction Some methods involve one variable

is Treatment A as effective in relieving arthritic pain as Treatment B?

Correlation and regression used to investigate relationships between variables most commonly linear relationships between two variables

is BMD related to dietary calcium level?

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Contents

Coefficients of correlation 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

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

r = +1

r = -1

r = 0.6

r = 0

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Scatter plot

dependent variablemake inferences

about

independent variable

controlled in some cases

Calcium intake

BMD

make inferences from

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Non-Normal data

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Normalised

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Calculating r The value and significance of r are calculated by

SPSS

SPSS output: scatter plot

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SPSS output: correlations

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Interpreting correlationLarge r does not necessarily imply:

strong correlation r increases with sample size

cause and effectstrong 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 factors: 1 - r2 r2 is the Coefficient of Determination e.g. r = 0.661 r2 = = 0.44 less than half of the variation 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 1, 2, 3 and 2, 4, 6 are perfectly positively

correlated

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Assumptions Errors are differences of predicted values of Y

from actual values To ascribe significance to r:

distribution of errors is Normal variance is same for all values of independent

variable X

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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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Calculation of value and significance

Computer does it!

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Role of regression Shows how one variable changes with

another By determining the line of best fit

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

the value of the dependent variable when value of independent variable(s)

known within the range of the known data

extrapolation risky! relation between age and bone age

Does not imply causality

SPSS output: regression

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Assumptions

Only if statistical inferences are to be made significance of regression values of slope and intercept

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Assumptions If values of independent variable are randomly

chosen then no further assumptions necessary Otherwise

as in correlation, assumptions based on errors balance out (mean=0) variances equal for all values of independent variable not related to magnitude of independent variable

seek advice / help

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Multivariate regressionMore than one independent

variable BMD dependent on:

agegendercalorific intakeetc

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Logistic regression The dependent variable is binary

yes / no predict whether a patient with Type 1

diabetes will undergo limb amputation given history of prior ulcer, time diabetic etc result is a probability

Can be extended to more than two categories Outcome after treatment

recovered, in remission, died

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Summary Correlation

strength of linear relationship between two variables

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

Regression line of best fit prediction multivariate logistic