Correlation andCorrelationalResearch

Chapter 5

The Two Disciplines of Scientific Psychology

• Lee Cronbach• APA Presidential Address

Fundamentals of Correlation • correlations reveal the degree of statistical association

between two variables

• used in both experimental and non-experimental research designs

• Correlational/Non-Experimental research • establishes whether naturally occurring variables are

statistically related

Correlational Research

• in correlational research, variables are measured rather than manipulated

• manipulation is the hallmark of experimentation which enables researchers to draw causal inferences

• distinction between measurement and manipulation drives the oft-cited mantra “correlation does not equal causation”

Direction of Relationship

Positive

• two variables tend to increase or decrease together

• higher scores on one variable on average are associated with higher scores on the other variable

• lower scores on one variable on average are associated with lower scores on the other variable

• e.g., relationship between job satisfaction and income

Direction of Relationship

Negative

• two variables tend to move in opposite directions

• higher scores on one variable are on average associated with lower scores on the other variable

• lower scores on one variable are on average associated with higher scores on the other variable

• e.g., relationship between hours video game playing and hours reading

Hypothetical Data

Participant Weekly Hours of

TV Watched

(X)

Perceived Crime

Risk (%)

(Y1)

Trust in

Other People

(Y2)Wilma 2 10 22Jacob 2 40 11Carlos 4 20 18Shonda 4 30 14Alex 5 30 10Rita 6 50 12Mike 9 70 7Kyoko 11 60 9Robert 11 80 10Deborah 19 70 6

Graphing Bivariate Relationships

a two-dimensional graph

• values of one of the variables are plotted on the horizontal axis (labelled as X and known as the abscissa)

• values of the other observations are plotted on the vertical axis (often labelled as Y and known as the ordinate)

Scatterplots/Scattergram

Positive (Direct) Relationship Negative (Inverse) Relationship

Calculating Correlations

Pearson product-moment correlation coefficient• Pearson’s r

• Variables measured on interval or ratio scale

Spearman’s rank-order correlation coefficient• Spearman’s rho

• One or both variables measured on ordinal scale

Depends on scale of measurement

Pearson’s r

• based on a ratio that involve the covariance and standard deviations of the two variables (X and Y)

• the covariance is a number that reflects degree to which two variables vary together

• as with variance, covariance calculation differs for populations and samples

• deal with population calculations

Ordinal or Ratio Scales

Pearson’s r

Covariance -- Definitional Formula

𝜎𝑋𝑌 = 𝑋 − 𝜇𝑋)(𝑌 − 𝜇𝑌

𝑁

Standard Deviation-- Definitional Formula

𝜎𝑋 = (𝑋−𝜇𝑋)2

𝑁𝜎𝑌 =

(𝑌−𝜇𝑌 )2

𝑁Pearson’s r

𝑟𝑋𝑌 =𝜎𝑋𝑌

𝜎𝑋𝜎𝑌

Spearman Rank-Ordered Correlation

Based on Ranks for Each of the Two Variables

If no tied ranks then can use simplified formula

𝑟𝑆𝑝𝑒𝑎𝑟𝑚𝑎𝑛 = 1 −6 𝐷2

𝑁(𝑁2−1)

𝑟𝑆𝑝𝑒𝑎𝑟𝑚𝑎𝑛 =𝜎𝑋𝑌

𝜎𝑋𝜎𝑌

Interpreting Magnitude of Correlations

• In addition to considering the direction of the relationship (i.e., positive or negative), we need to attend to the strength of the relationship.

• correlation only takes on limited range of values

−1.00 ≤ 𝑟 ≤ +1.00

• absolute value reflects strength/degree of relationship between two variables

Interpreting Magnitude of Correlations

• square of the correlation coefficient

• 𝑟2

• aka coefficient of determination

• proportion of variability in one variable that can be accounted for through the linear relationship with the other variable

• thus 𝑟2 = .82 = .64 as does 𝑟2 = −.82 = .64

Interpreting Magnitude of Correlations

• Is the relationship between two variables weak? Moderate? Strong?

Cohen’s Guidelines

Guidelines from Cohen (1988)

Absolute value of r

Weak .10 - .29

Moderate .30 - .49

Strong > .50

Interpreting Magnitude of Correlation

• If a psychological researcher reports a correlation of .33 between integrity and job performance, can one say that the two variables are 33% related?

• No• r2 (coefficient of determination) reveals how much of the

differences in Y scores are attributable to differences in X scores

• .332 = .1089

• so only about 11% of the variability is accounted for

Coefficient of determination

Nonlinear Relationships

• magnitude of the correlation coefficient influenced by degree on non-linearity

test

performance

Alertness

sleepy alert panic

r = 0

• can assess the strength of non-linear relationships with alternative

statistical procedures such as 𝜀2

Range Restriction

Correlation And Causation

• Bidirectionality Issue

• Third Variable Problem

Bidirectionality Problem

GPAReligiosity

GPAReligiosity

Religiosity Causes GPA

GPA Causes Religiosity

Correlation between Religiosity and GPA GPAReligiosity

Third-Variable Problem

GPAReligiosity

Correlation between Religiosity and GPA

Parenting Style

• spurious relationship

Strategies to Reduce Causal Ambiguity in Correlational Research

Statistical approaches• measure and statistically control for a third variable • partial correlation analysis

• e.g., relationship between right-hand palm size (X) & verbal ability (Y)𝑟𝑋𝑌 = 0.70

• perhaps a spurious relationship caused by a common third variable –age (Z)

𝑟𝑋𝑍 = 0.90 𝑟𝑌𝑍 = 0.80

𝑟𝑋𝑌∙𝑍 = −0.076

Research Designs • Cross-Sectional Designs

• bidirectionality potential problem

• Prospective Longitudinal design• X measured at Time 1• Y measured at Time 2 • Rules out bidirectionality problem

• Cross-lagged panel design • Measure X and Y at Time 1• Repeat X and Y measurement at Time 2• Examine pattern of relationships (i.e., cross-lagged

correlations) across variables and time

Cross-Lagged Panel Design Eron et al., 1972

Drawing Causal Conclusions

• How do we rule out all plausible third variables

(confounds) using correlational research designs?

• We can’ t – only the control afforded by rigorous

experimentation provides strong tests of causation

• as noted by some recent researchers employing such designs:

“longitudinal correlational research can be used to compare

the relative plausibility of alternative causal perspectives” but

they “do not provide a strong test of causation”

Correlation/Regression and Prediction

• A goal of science is to forecast future events

• In simple linear regression, scores on X can be used to predict scores on Y assuming a meaningful relationship (r) has been established between X and Y in past research

Linear Regression

• interest in predicting scores on one variable (Y) based upon linear relationship with another variable (X)

• X is the predictor; Y is the criterion

Regression Equation

• based on formula for straight line

𝑌 = 𝑎 + 𝑏𝑋

where 𝑌 is the predicted value of Y for a given value of X

a is the Y-intercept (i.e., 𝑌 for X = 0)

b is the slope of the regression line

• can be plotted on scatterplot

Regression Equation - Calculation

• need to calculate values for • a – the y-intercept and

• b – the slope

𝑏 =𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝑋𝑌

𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝑋= r ∗

𝑆𝐷𝑌

𝑆𝐷𝑋

𝑎 = 𝑌 − 𝑏 𝑋

Interpreting Regression Equation

For example assume were looking at the relationship between how many children a couple has (Y) and the number of years they’ve been married. From a sample we calculate the following:

thus, if a couple is married for 0 years we would predict that they would have -0.84 of a child

for each year they’re married we’d expect couple to have an additional 1.21 children

𝑌 = −0.84 + 1.21𝑋

Multiple (Linear) Regression

• Multiple predictors are used to predict a criterion measure

• ideally want as little overlap as possible between predictors (X’s)

• i.e., want each predictor to account for unique variance in criterion (Y)

𝑌 = 𝑎 + 𝑏1𝑋1 + 𝑏2𝑋2 + … .+𝑏𝑘𝑋𝑘

Multiple Regression

GeneralCAT

Criterion

Structured Interview

WorkSample

GeneralCAT

Criterion

Structured Interview

WorkSample

ideally want to avoid multicollinearity in order to maximize prediction

Example - One Criterion (Y) and Three Predictors (s)

𝐻𝑒𝑟𝑒 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 𝑢𝑛𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑒𝑑 𝐻𝑒𝑟𝑒 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑒𝑑

Benefits of Correlational Research

• prediction in everyday life

• test validation

• broad range of applications

• establishing relationship

• convergence with experiments