Lecture 4:Correlation and Regression

Laura McAvinue

School of Psychology

Trinity College Dublin

Correlation

• Relationship between two variables

– Do two variables co-vary / co-relate?• Is mathematical ability related to IQ?• Are depression and anxiety related?

– Does variable Y vary as a function of variable X?• Does error awareness vary as a function of ability to sustain

attention?• Does accuracy of memory decline with age?

Correlation

• Direction– Do both variables move in the same direction?– Do they move in opposite directions?

• Degree– What is the degree or strength of the relationship?

• Analysis– Scatterplot– Correlation Coefficient

• Statistical significance

Scatterplot

Describe the relationship between the two variables using a scatterplotVisual representation of the relationship between the

variables

Plot each observation in the study, displaying its value on variable X and variable YPlace the predictor variable on the X axis

The independent variable, which is making the prediction

Place the criterion variable on the Y axisThe dependent variable, which is being predicted

Participant Anxiety Depression1 1 12 3 33 6 6

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Dep

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No Relationship

Random Scatter

Positive Relationship

Direction in scatter

Negative Relationship

Direction in Scatter

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No. of Pints

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What is the relationship between verbal coherence and the number of pints of beer consumed?

Sometimes, the direction of the relationship might not be as obvious…

Regression Line

• Useful to add a regression line

– Model of the relationship– Straight line that best represents the relationship

between the two variables• ‘The line of best fit’

– Helps us to understand the direction of the relationship

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No. of Pints

Verb

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Adding the regression line helps us see the direction ofthe relationship

Direction of Relationship

• Positive– Two variables tend to move in the same direction

• As X increases, Y also increases• As X decreases, Y also decreases

• Negative– Two variables tend to move in opposite directions

• As X increases, Y decreases• As X decreases, Y increases

A Positive Relationship

A Negative Relationship

Degree of Relationship

• Degree or strength of relationship

– Calculate a correlation coefficient• Pearson Product-Moment Correlation Coefficient (r)

– Statistic that varies between -1 and 1• r = 0, no relationship between the variables

– Change in X is not associated with systematic change in Y

• r = 1, perfect positive correlation– Increase in X associated with systematic increase in Y

• r = -1, perfect negative correlation– Increase in X associated with systematic decrease in Y

Perfect

Negative

relationship

Interpretation of r

Perfect Positive relationshi

p-1 0 +1

AbsolutelyNo

relationship

Closer Pearson r is to one of the extremes,the stronger the relationship between the variables

Calculation of Pearson r

• Based on the covariance– A statistic representing the degree to which two

variables vary together– Based on how an observation deviates from the mean

on each variable

Calculation of Pearson r

• Covariance is not suitable as measure of degree of relationship– Absolute value is a function of standard deviations– Scale the covariance by the standard deviations

• Pearson r

Assessing Magnitude of r

• Cohen’s (1988) standards• Small Medium Large

.1 - .29 .3 - .49 .5 - 1

• Statistical Significance– Test the null hypothesis that the true correlation in the population

(rho) is zero

• Ho: ρ = 0

– Calculate the probability of obtaining a correlation of this size if the true correlation is zero

– If p < .05, reject Ho and conclude that it is unlikely that the results are due to chance, the correlation obtained represents a true correlation in the population

Summary

• Interested in the relationship between two variables

• Direction and degree of relationship– Scatterplot & regression line

• Direction

– Correlation Coefficient• Magnitude• Statistical significance

As temperature increases, ice-cream consumption increases

r = .73 (large)

n = 12

p = .007

As temperature increases, hot whiskey consumption decreases

r = -.908 (large)

n = 12

p <.001

Issues to consider

• Assumption of linearity

– Pearson correlation assumes there is a linear relationship between the two variables

– Assumes the relationship can be represented by a straight line

– It is possible that the relationship might be better represented by a curved line

• Examine scatterplot

– Curve-fitting procedures

VAR00003

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Linear?

VAR00003

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Non-linear?

STRESS

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Non-linear

Issues to consider

• Correlation can be affected by– Range restrictions– Heterogeneous subsamples– Extreme observations

• Correlation does not mean causation

Regression

• The regression line– A straight line that represents the relationship

between two variables– Useful to add to the scatterplot to help us see the

direction of the relationship– But it’s much more than this…

• Prediction– Regression line enables us to predict Variable Y on

the basis of Variable X

Regression

• If you have an equation of the line that represents the relationship between Variables X & Y, you can use it to predict a value of Y given a certain value of X.

X = 63

Y’ = 45

Regression Equation

ˆ Y bX a

Predicted

value of Y

Predicting

value of X

Regression Coefficients

The basic equation of a line

Regression Equation

ˆ Y bX ab

The slope of the regression line

The amount of change in Y associated with a one-unit change in X

a

The intercept

The point where the regression line crosses the Y axis

The predicted value of Y when X = 0

Regression Equation

X

Y’

a

b ˆ Y bX a

Same intercept,

different slopes

Same slope,

different intercepts

Summary

• The relationship between two variables, X & Y

• Correlation– Degree and direction of relationship

• Regression– Predict Y, given X– More on regression next lecture…