Quantitative Data Analysis II: Correlation and Simple Linear

Regression

SI0030Social Research Methods

Week 6

Luke Sloan

Introduction

• Last Week – Recap

• Correlation

• How To Draw A Line

• Simple Linear Regression

• Summary

Last Week - Recap

• Hypotheses

• Probability & Significance (p=<0.05)

• Chi-square test for two categorical variables

• t-test for one categorical and one interval variables

• What about a test for two interval variables?...

Correlation I

• Calculates the strength and direction of a linear relationship between two interval variables

• e.g. is there a relationship between age and income?

• Measured using the Pearson correlation coefficient (r)

• Data must be normally distributed (check with a histogram)

If not normally distributed use Spearman’s Rank Order Correlation (rho) - consult Pallant (2005:297)

Correlation II• ‘r’ can take any value from +1 to -1

• +/- indicates whether the relationship is positive or negative

• +1 or -1 is a perfect linear relationship, but usually it is not this clear cut

• Rule of thumb:– +/- 0.7 = a strong linear relationship– +/- 0.5 = a good linear relationship– +/- 0.3 = a linear relationship– Below +/- 0.3 = weak linear relationship– 0 = no linear relationship

Alternatively:- +/- 0.10 to 0.29 = weak- +/- 0.30 to 0.49 = medium- +/- 0.50 to 1.00 - strong

Correlation III

Positive relationship

Negative relationship

No relationshipPositiveRelationship

NoRelationship

NegativeRelationship

Formulate hypotheses and use scatter plots!

Correlation IV

H1 = There is a relationship between Age and the number of years a candidate has been a member of a political party

H0 = There is no relationship between Age and the number of years a candidate has been a member of a political party

What do you think?

Correlation V

Is this normal? Just to prove a point…

Correlation VI

Correlations

What was your age last birthday

Number of years a party member

What was your age last birthday Pearson Correlation 1 .425**

Sig. (2-tailed)   .000

N 4481 1874Number of years a party member Pearson Correlation .425** 1

Sig. (2-tailed) .000  

N 1874 1936

**. Correlation is significant at the 0.01 level (2-tailed).

Perfect correlation against itself (obviously!) and number of cases in analysis

Pearson’s Correlation Coefficient is r=0.43 – medium/good positive linear relationship

Significance for correlation is problematic (highly dependent on sample size) – report p-value but ignore level of significance

Correlation VII

• Don’t forget to refute or accept the null hypothesis and discuss the relationship

• Correlation is not causation!

The relationship between the number of years a candidate has been a member of a party and candidate age was explored using Pearson’s correlation

coefficient. Both variables were confirmed to have normal distributions [?] and a scatter plot revealed a linear relationship. There was a medium-strength,

positive relationship between the two variables (r=0.43, n=4481, p<0.05)... [go on to explain the relationship in detail]

How To Draw A Line I

• Correlation is indicative of a relationship, but it does not allow us to quantify it

• What if we wanted to explain how an increase in age leads to an increase in years of party membership?

• What if we wanted to predict years of party membership based only on age?

The line of best fit is a predictive – it is the regression line!

How To Draw A Line II• The regression line allows us to predict any given value of y when we

know x

• i.e. if we know the age of a candidate we can predict how long they are likely to have been a member of a political party

• Another (more useful!) example would be years in education and income

• Using a regression line we can predict someone’s income based on the number of years they have been in education

• Assumes a causal relationship – that income is ‘caused’ by years in education

How To Draw A Line III

• But… we don’t simply look very closely at the line and the axis of the scatter plot because the regression line can be written as an equation:

y = a + b x

‘y’ represents the dependent

variable (what we are trying to predict) e.g.

income

‘a’ represents the intercept(where

the regression line crosses the vertical

‘y’ axis) aka the constant

‘b’ represents the slope of the line (the association

between ‘y’ & ‘x’) e.g. how income

changes in relation to education

‘x’ represents the independent

variable (what we are using to predict ‘y’) e.g.

years in education

y = 0 + 1x

y = 0 + 2x

y = 0 + 0.5x

x axis

y ax

is

y = 0 + 0.25x

y = 1 + 1x

What about…

How To Draw A Line IV

• If we know the slope (b) and the intercept (a), for any given value of ‘x’ we can predict ‘y’

Preconditions:

Simple Linear Regression

EXAMPLE: predicting income (y) in thousands (£) from years in education (x)

Intercept (a) = 4

Slope (b) = 1.5

For someone with 10 years of

education

Equations:

y = a + bx

Income = intercept + (slope*years in education)

Income = 4 + (1.5*10) = 19 (£19,000)

Or…

Or…

Simple Linear Regression II

• Assumptions– Interval level data– Linearity between ‘x’ and ‘y’– Outliers (check scatter plot)– Sample size = 100+?

• R2 measure of ‘model fit’– Literally the Pearson’s correlation coefficient squared– R2 tells us how much of the variance in the dependent variable is

explained by the independent variable e.g. how much of the variance in income can be explained by age

– Expressed as a percentage (1.0 = 100%, 0.5 = 50% etc)

Simple Linear Regression III

H0 = There is no relationship between Age and the number of years a candidate has been a member of a political partyH1 = There is a relationship between Age and the number of years a candidate has

been a member of a political party

H2 = As the age of a candidate increases, so will the number of years that they have been a party member

‘Years as Party Member’ = intercept + (slope * ’Age’)

Simple Linear Regression IV

Model Summary

Model

R R Square Adjusted R SquareStd. Error of the

Estimate

  1 .425a .181 .180 11.995

a. Predictors: (Constant), What was your age last birthday

ANOVAb

Model

Sum of Squares df Mean Square F Sig.1 Regression 59446.085 1 59446.085 413.170 .000a

Residual 269339.696 1872 143.878    Total 328785.781 1873      

a. Predictors: (Constant), What was your age last birthday

b. Dependent Variable: Number of years a party member

Pearson’s correlation coefficient (same value!)

18% of variance in party membership (y) explained by age (x)

This tests the hypothesis that the model is a better predictor of party membership than if we simply

used the mean value of party membership

p<0.05 so the regression model is a significantly better

predictor than the mean value

Simple Linear Regression V

Coefficientsa

Model

Unstandardized Coefficients

Standardized

Coefficients

t Sig.B Std. Error Beta1 (Constant) -6.899 1.156   -5.966 .000

What was your age last birthday

.418 .021 .425 20.327 .000

a. Dependent Variable: Number of years a party member

This is the intercept (a)

y = a + b x

This is the slope (b)

p<0.05 so ‘Age’ has a significant effect on ‘Party Membership’

‘Party Membership’ = -6.9 + (0.42 * ’Age’)A one unit increase in age will result in an increase in party membership of 0.42

Or…

Simple Linear Regression VI… and this is what we saw in the original scatter plot!

• The ‘regression line’ will intercept the verticle (y) axis at -6.9

• The ‘regression line’ rises by 0.42 on the verticle axis (y) for every one unit increase on the horizontal axis (x)

• The R2 value is low because of the fanning effect (remember the histograms!)

Summary

• How to describe and quantify the relationship between two interval variables

• Correlation – the strength and direction of the association

• Regression – the causal and quantified effect of an independent on a dependent variable