Correlation and Correlational Research Chapter 5 The Two - - PowerPoint PPT Presentation

Correlation and Correlational Research

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 22

Jacob

2 40 11

Carlos

4 20 18

Shonda

4 30 14

Alex

5 30 10

Rita

6 50 12

Mike

9 70 7

Kyoko

11 60 9

Robert

11 80 10

Deborah

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

𝜏𝑌𝑍 = 𝑌 − 𝜈𝑌)(𝑍 − 𝜈𝑍 𝑂

Standard Deviation-- Definitional Formula

𝜏𝑌 =

(𝑌−𝜈𝑌)2 𝑂

𝜏𝑍 =

(𝑍−𝜈𝑍 )2 𝑂 Pearson’s r

𝑠

𝑌𝑍 = 𝜏𝑌𝑍

𝜏𝑌𝜏𝑍

Spearman Rank-Ordered Correlation

Based on Ranks for Each of the Two Variables If no tied ranks then can use simplified formula

𝑠

𝑇𝑞𝑓𝑏𝑠𝑛𝑏𝑜 = 1 − 6 𝐸2 𝑂(𝑂2−1)

𝑠

𝑇𝑞𝑓𝑏𝑠𝑛𝑏𝑜 = 𝜏𝑌𝑍

𝜏𝑌𝜏𝑍

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 ≤ 𝑠 ≤ +1.00

• absolute value reflects strength/degree of relationship between

two variables

Interpreting Magnitude of Correlations

• square of the correlation coefficient
• 𝑠2
• aka coefficient of determination
• proportion of variability in one variable that can be accounted for through the linear

relationship with the other variable

• thus 𝑠2 = .82 = .64 as does 𝑠2 = −.82 = .64

Interpreting Magnitude of Correlations

• Is the relationship between two variables weak?

Moderate? Strong?

Cohen’s Guidelines

Guidelines from Cohen (1988) Absolute value

• f 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 𝜁2

Range Restriction

Correlation And Causation

• Bidirectionality Issue
• Third Variable Problem

Bidirectionality Problem

GPA Religiosity GPA Religiosity

Religiosity Causes GPA GPA Causes Religiosity

Correlation between Religiosity and GPA

GPA Religiosity

Third-Variable Problem

GPA Religiosity

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)

𝑠

𝑌𝑍 = 0.70

• perhaps a spurious relationship caused by a common third variable –

age (Z)

𝑠

𝑌𝑎 = 0.90

𝑠𝑍𝑎 = 0.80 𝑠

𝑌𝑍∙𝑎 = −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

𝑍 = 𝑏 + 𝑐𝑌 where 𝑍 is the predicted value of Y for a given value of X a is the Y-intercept (i.e., 𝑍 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

𝑐 = 𝐷𝑝𝑤𝑏𝑠𝑗𝑏𝑜𝑑𝑓𝑌𝑍 𝑊𝑏𝑠𝑗𝑏𝑜𝑑𝑓𝑌 = r ∗ 𝑇𝐸𝑍 𝑇𝐸𝑌

𝑏 = 𝑍 − 𝑐 𝑌

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

𝑍 = −0.84 + 1.21𝑌

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)

𝑍 = 𝑏 + 𝑐1𝑌1 + 𝑐2𝑌2 + … . +𝑐𝑙𝑌𝑙

Multiple Regression

General CAT Criterion Structured Interview Work Sample General CAT Criterion Structured Interview Work Sample

ideally want to avoid multicollinearity in order to maximize prediction

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

𝐼𝑓𝑠𝑓 𝑞𝑠𝑓𝑒𝑗𝑑𝑢𝑝𝑠𝑡 𝑏𝑠𝑓 𝑣𝑜𝑑𝑝𝑠𝑠𝑓𝑚𝑏𝑢𝑓𝑒 𝐼𝑓𝑠𝑓 𝑞𝑠𝑓𝑒𝑗𝑑𝑢𝑝𝑠𝑡 𝑏𝑠𝑓 𝑑𝑝𝑠𝑠𝑓𝑚𝑏𝑢𝑓𝑒

Benefits of Correlational Research

• prediction in everyday life
• test validation
• broad range of applications
• establishing relationship
• convergence with experiments