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Statistics. Correlation and regression. 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

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statistics

Statistics

Correlation and regression

introduction
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?
contents
Contents
  • Coefficients of correlation
    • meaning
    • values
    • role
    • significance
  • Regression
    • line of best fit
    • prediction
    • significance
introduction1
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?
pearson s coefficient of correlation
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
pearson s coefficient of correlation1

r = +1

r = -1

r = 0

r = 0.6

Pearson’s coefficient of correlation
scatter plot

Scatter plot

BMD

dependent variable

make inferences about

Calcium intake

independent variable

make inferences from

controlled in some cases

slide10

Calculating r

  • The value and significance of r are calculated by SPSS
slide13

Interpreting correlation

  • Large r does not necessarily imply:
    • strong correlation
      • r increases 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
slide14

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
slide16

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
slide17

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
slide18

Non-parametric correlation

  • Make no assumptions
  • Carried out on ranks
  • Spearman’s r
    • easy to calculate
  • Kendall’s t
    • has some advantages over r
    • distribution has better statistical properties
    • easier to identify concordant / discordant pairs
  • Usually both lead to same conclusions
slide20

Role of regression

  • Shows how one variable changes with another
  • By determining the line of best fit
    • linear
    • curvilinear
slide21

value of Y when X=0

change in Y when X increases by 1

Line of best fit

  • Simplest case linear
  • Line of best fit between:
    • dependent variable Y
      • BMD
    • independent variable X
      • dietary intake of Calcium

Y= a + bX

slide22

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
slide24

Assumptions

  • Only if statistical inferences are to be made
    • significance of regression
    • values of slope and intercept
slide25

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
slide26

Multivariate regression

  • More than one independent variable
    • BMD dependent on:
      • age
      • gender
      • calorific intake
      • etc
slide27

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
slide28

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