Correlation

Lecture 4 Survey Research & Design in Psychology James Neill, 2009 Correlations

Purpose of correlation
Covariation
Correlational analyses

Types of answers
Types of correlation
Interpretation
Assumptions / Limitations

Dealing with several correlations

Overview

Purpose of correlation The purpose of correlation is to help address the question: “ What is the

relationship or
degree of association or
amount of shared variance

between two variables? ” Direction Strength Statistical sig.

Purpose of correlation Another way of putting the correlational question is: “ To what extent are two variables

dependent or
independent

of one another? ”

The world is made of covariation.

We observe covariations in the psycho-social world. We can measure our observations. Depictions of violence in the environment. Psychological states such as stress and depression.

Correlation example: World population and total carbon emissions

Correlation is not causation e.g.,: Stop global warming: Become a pirate

Causation may be in the eye of the beholder : It's a rather interesting phenomenon

Covariations are the basis of more complex models.

Correlational analyses Correlational analyses are commonly used to examine the extent to which two variables have a simple linear relationship . Correlations provide the building blocks for:

Factor analysis
Reliability
Regression etc.

Correlational analyses Of interest in examining the linear relationship between 2 variables is the:

direction,
strength, and

ranges from -1 to +1

Sign indicates direction
Size indicates strength

statistical significance

Correlational analyses Measures the extent to which:

differences in one variable can be predicted from differences in the other variable
one variable varies with another variable

Types of answers

No relationship (independence)‏
Linear relationship:

As one variable ↑ s, so does the other (+ve)‏
As one variable ↑ s, the other ↓ s (-ve)‏

Non-linear
Restricted range
Heterogeneous samples

Types of correlation To decide which of the following types of correlation to use, we need to consider the levels of measurement for each variable:

Phi ( Φ ) / Cramer’s V
Spearman’s rank / Kendall’s Tau b
Point bi-serial r pb
Product-moment or Pearson’s r

Types of correlation and LOM Scatterplot Product-moment correlation ( r )‏ Int/Ratio   Recode Scatterplot Point bi-serial or Spearman/Kendall Scatterplot or clustered bar chart Spearman's Rho or Kendall's Tau Ordinal Scatterplot, bar chart or error-bar chart Point bi-serial correlation ( r pb )‏ Clustered bar-graph Chi-squared Phi (φ) or Cramer's V Clustered bar-graph Chi-squared Phi (φ) or Cramer's V Nominal Int/Ratio Ordinal Nominal

Tufte: Graphics reveal data 6.89 8.0 5.73 5.0 4.74 5.0 5.68 5.0 7.91 8.0 6.42 7.0 7.26 7.0 4.82 7.0 5.56 8.0 8.15 12.0 9.13 12.0 10.84 12.0 12.50 19.0 5.39 4.0 3.10 4.0 4.26 4.0 5.25 8.0 6.08 6.0 6.13 6.0 7.24 6.0 7.04 8.0 8.84 14.0 8.10 14.0 9.96 14.0 8.47 8.0 7.81 11.0 9.26 11.0 8.33 11.0 8.84 8.0 7.11 9.0 8.77 9.0 8.81 9.0 7.71 8.0 12.74 13.0 8.74 13.0 7.58 13.0 5.76 8.0 6.77 8.0 8.14 8.0 6.95 8.0 6.58 8.0 7.46 10.0 9.14 10.0 8.04 10.0 Y4 X4 Y3 X3 Y2 X2 Y1 X1

Tufte: Graphics reveal data.

Nominal by Nominal

Contingency tables

Bivariate frequency tables
Marginal totals
Can include %

Contingency tables

Contingency table: Example

Example Contingency table: Example

Clustered bar graph

Bar graph of frequencies or percentages with the category axis clustered by coloured bars to indicate the two variable’s categories

Clustered bar graph

Chi-squared

Chi-squared: Example

Phi (  ) & Cramer’s V Phi (  )‏

Two dichotomous variables (2x2, 2x3, 3x2)‏
e.g., Gender & Pass/Fail

Cramer’s V

Two dichotomous variables (3x3 or greater)‏
e.g., Favourite Season x Favourite Sense

Phi (  ) & Cramer’s V : Example

Ordinal by Ordinal

Spearman’s rho ( r s )‏

For ranked (or recoded to ordinal) data
Uses product-moment correlation, but interpretations must be adjusted to consider the underlying ranked scales

e.g. Olympic Placing vs. World Ranking

Kendall’s Tau-b

For ordinal/ranked data
Takes joint ranks into account
Ranges -1 to +1, but only for square tables

Dichotomous by Interval/Ratio

Point-biserial correlation ( r pb )‏

One dichotomous & one continuous variable
Calculate as for Pearson’s r, but adjust to consider the underlying scales

e.g., gender and self-esteem

Point-biserial correlation ( r pb )‏: Example

Product-moment correlation ( r )

For two interval and/or ratio variables
r = cov xy

s x s y

Interval/Ratio by Interval/Ratio

Scatterplots

Plot each pair of observations (X, Y)‏
x = predictor variable (independent)‏
y = criterion variable (dependent)‏
Check for:

outliers
linearity

‘ Line of best fit’

y = a + bx

The correlation between 2 variables is a measure of the degree to which pairs of numbers (points) cluster together around a best-fitting straight line
By convention, the independent variable (IV) should be plotted on the x (horizontal) axis, and the dependent variable (DV) on the y (vertical) axis.

Scatterplots

Scatterplot showing relationship between age & blood cholesterol

What's wrong with this scatterplot? IV should treated as X and DV as Y, although this is not always distinct.

Why is infant mortality positively linearly associated with the number of physicians (with the effects of GDP removed)? – because more doctors tend to be deployed to areas with infant mortality (socio-economic status aside). S catterplot example: Strong positive (.81)‏

Scatterplot example: Weak positive (.14)‏

Scatterplot example: Moderately strong negative (-.76)‏

Covariance

Variance shared by 2 variables
Covariance reflects the direction of the relationship:

+ve cov indicates + relationship -ve cov indicates - relationship. Cross products

Covariance: Cross-products -ve cross products -ve dev. products -ve dev. products +ve dev. products +ve dev. products

Covariance

Covariance is dependent on the scale of measurement used
Can’t compare magnitude of cov across different scales of measurement (e.g., age by weight in kilos versus age by weight in grams).
Therefore, standardise covariance -> > correlation

Correlation formulae

For a given set of data the covariance between X and Y is .80. The SD of X is 2 and the SD of Y is 3. The resulting correlation is closest to: a. .00 b. .15 c. .80 d. -.30 Covariance, SD , and correlation: Quiz Answer: .80 / 2 x 3 = .15

The correlation between 2 variables is:

an effect size – i.e., standardised measure of amount of covariation.

Correlation & effect size

Correlation - SPSS

Hypothesis testing Almost all correlations are not 0, therefore: “ What is the likelihood that a relationship between variables is a ‘true’ relationship, or could it simply be a result of random sampling variability or ‘chance’?”

Significance of correlation

Null hypothesis (H 0 ): assumes that there is no ‘true’ relationship
Alternative hypothesis (H 1 ): assumes that the relationship is real
We initially assume the null hypothesis, and evaluate whether the data support the alternative hypothesis

Null hypothesis: H 0 :  = 0
Alternative hypothesis H 0 :  0

 rho  = population product-moment correlation coefficient

How do we test the null hypothesis?

Use statistical tests of probability which produce a p value
Select a criterion (or critical value) for statistical significance (alpha ( α ) level); commonly .05‏

How do we test the null hypothesis?

Calcute a p value and compare it to the critical value.
If p is less than the critical value, this indicates statistical significance, i.e., that there is less than a X% chance that the relationship being tested is due to random sampling variability.

Imprecision in hypothesis testing

Type I error : rejecting the null hypothesis when it is true
Type II error : Accepting the null hypothesis when it is false
Statistical significance is a function of:

effect size,
sample size
alpha level

A 1- or 2-tailed significance test can be done in an effort to infer to a population
Result will depend on the power of study (i.e., higher N, p , and r , more likely to be sig.)‏
Alternatively look up r tables with df = N - 2

Scatterplot showing a confidence interval for a line of best fit

US States 4 th Academic Achievement by SES

Significance of correlation df critical ( N -2) p = .05 5 .67 10 .50 15 .41 20 .36 25 .32 30 .30 50 .23 200 .11 500 .07 1000 .05

If the correlation between Age and test Performance is statistically significant, it means that: a. there is an important relationship between Age and test Performance b. the true correlation between Age and Performance in the population is equal to 0 c. the true correlation between Age and Performance in the population is not equal to 0 d. getting older causes you to do poorly on tests Significance of correlation

Descriptive assumptions

Levels of measurement ≥ interval
Linear relations
No outliers

Inferential assumptions

Homoscedasticity
Similar, normal underlying distributions
Correction for attenuation (unreliability)‏
Minimal and normally distributed measurement error

Homoscedasticity

Factors that affect correlation

Covariation
Restricted range
Heterogenous samples
Scale has no effect

Coefficient of Determination ( r 2 )‏

CoD = The proportion of variance or change in one variable that can be accounted for by another variable.
e.g., r = .60, r 2 = .36

Interpreting correlation (Cohen, 1988)‏ A correlation is an effect size, so guidelines re strength can be suggested. Strength r r 2 weak: .1 to .3 (1 to 10%)‏ moderate: .3 to .5 (10 to 25%)‏ strong: >.5 (> 25%)‏

Size of correlation (Cohen, 1988)‏ WEAK (.1 - .3)‏ MODERATE (.3-.5)‏ STRONG (>.5)‏

Interpreting correlation Strength r r 2 very weak 0 - .19 (0 to 4%)‏ weak .20 - .39 (4 to 16%)‏ moderate .40 - .59 (16 to 36%)‏ strong .60 - .79 (36% to 64%)‏ very strong .80 - 1.00 (64% to 100%)‏

Interpreting correlation

Interpreting correlation 4

Correlation of this scatterplot = .9

What do you estimate the correlation of this scatterplot of height and weight to be? a. -.5 b. -1 c. 0 d. .5 e. 1

What do you estimate the correlation of this scatterplot to be? a. -.5 b. -1 c. 0 d. .5 e. 1

Linearity & non-linearity
Range restriction
Types of answers
Types of correlation
Interpretation

Assumptions / Limitations: Overview

Non-linear relationships Check scatterplot Can a linear relationship ‘capture’ the lion’s share of the variance? If so,use r .

Non-linear relationships

If non-linear, consider transforming variables to ‘create’ linear relationship = equivalent to finding a non-linear mathematical function to describe the relationship between the variables

Range restriction

Range restriction is when sample contains restricted (or truncated) range of scores

e.g., fluid intelligence and age < 18 might have linear relationship

If range restriction, be cautious in generalising beyond the range for which data is available

E.g., fluid intelligence does not continue to increase linearly with age after age 18

Range restriction

Sub-samples (e.g., males & females) may artificially increase or decrease overall r.
Solution - calculate r separately for sub-samples & overall, look for differences

Scatterplot of Same-sex & Opposite-sex Relations by Gender ♂ r = .67 ♀ r = .52

Scatterplot of Weight and Self-esteem by Gender ♂ r = .50 ♀ r = -.48

Effect of outliers

Outliers can disproportionately increase or decrease r .
Options

compute r with & without outliers
get more data for outlying values
recode outliers as having more conservative scores
transformation
recode variable into lower level of measurement

Age & self-esteem ( r = .63)‏

Age & self-esteem (outliers removed) r = .23

Dealing with Several Correlations

Dealing with several correlations

Scatterplot matrices and correlation matrices organise correlations amongst several variables at once.
Example (Kliewer et al, 1998)‏

99 young children
Measured level of

Witnessed violence, Intrusive thoughts, Social support, and Internalizing symptoms

Correlation matrix: Example of an APA Style Correlation Table

Scatterplot matrix

Checklist

Graphs & Scatterplots

Outliers?
Linear?
Does each variable have a reasonable range?
Are there sub-samples to consider?

Choose appropriate measure of association
Conduct inferential test (if needed)‏
Interpret/Discuss

Reporting

Relate back to research hypothesis
Describe & interpret correlation

direction of relationship
size/strength
significance

If many correlations, report in a table
Acknowledge limitations e.g.,

Heterogeneity (sub-samples)‏
Range restriction
Causality?

Writing-up example “ Number of children and marital satisfaction were inversely related ( r (48) = -.35, p < .05), such that contentment in marriage tended to be lower for couples with more children. Number of children explained approximately 10% of the variance in marital satisfaction, a small-moderate effect.”

Key points

Covariations are the building blocks of more complex analysies, e.g., reliability analysis, factor analysis, multiple regression
Correlation does not prove causation – may be in opposite direction, co-causal, or due to other variables.

Key points

Check scatterplots to see whether a correlation makes sense
Choose measure of correlation based on levels of measurement .
Report measures of effect size (e.g., Φ , Cramer's V , r , r 2 ), direction of relationship , and inferential statistical significance .