Chapter 9: Correlational Research

Chapter 9. Correlational Research Chapter Objectives

Distinguish between positive and negative bivariate correlations, create scatterplots to illustrate them, and recognize factors that can influence the size of correlation coefficients

Calculate a coefficient of determination and interpret its meaning

Understand how a regression analysis accomplishes the goal of prediction

Chapter Objectives

Describe the research situations in which correlational procedures are likely to be used

Describe the logic of the multivariate procedures of multiple regression and factor analysis, and understand how to interpret the results of these procedures

Correlation and Regression: The Basics

Finding the relationship between two variables without being able to infer causal relationships

Correlation is a statistical technique used to determine the degree to which two variables are related

Three types of [linear] correlations: Positive correlation Negative correlation No correlation

Correlation and Regression: The Basics

Positive correlation Higher scores on one

variable associated with higher scores on a second variable

Correlation and Regression: The Basics

Negative correlation Higher scores on one

variable associated with lower scores on a second variable

Correlation and Regression: The Basics

Correlation coefficient Pearson’s r Statistical tests include:

• Pearson’s r, Spearman’s rho Ranges from –1.00 to +1.00 Numerical value = strength of correlation

• Closer to -1.00 or +1.00, the stronger the correlation Sign = direction of correlation

• Positive or Negative

Correlation and Regression: The Basics

Scatterplots Graphic representations of data from your two variables One variable on X-axis, one on Y-axis Examples:

Correlation and Regression: The Basics

Scatterplots Creating a scatterplot from data

• Each point represents an individual subject

Correlation and Regression: The Basics

Scatterplots from the hypothetical GPA data for positive (top) and negative (bottom) correlations

Correlation and Regression: The Basics

Scatterplots Correlation assumes a

linear relationship, but scatterplot may show otherwise

Curvilinear correlation coefficient will be close to zero

• Left half strong positive

• Right half strong negative

Correlation and Regression: The Basics

Coefficient of determination Equals value of Pearson’s r2

• Proportion of variability in one variable that can be accounted for (or explained) by variability in the other variable

• The remaining proportion can be explained by factors other than your variables

r = .60 r2 = .36

• 36% of the variability of one variable can be explained by the other variable

• 64% of the variability can be explained by other factors

Correlation and Regression: The Basics

Regression Analysis – Making Predictions The process of predicting individual scores AND estimating the

accuracy of those predictions Regression line – straight line on a scatterplot that best

summarizes a correlation• Y = bX + a

• Y = dependent variable—the variable that is being predicted• Predicting GPA from study hours Y = GPA

• X = independent variable—the variable doing the predicting• Predicting GPA from study hours X = study hours

• a = point where regression line crosses Y axis• b = the slope of the line

• Use the independent variable (X) to predict the dependent variable (Y)

Correlation and Regression: The Basics

Regression lines for the GPA scatterplots

Study time (X) of 40 predicts GPA (Y) of 3.5

Goof-off time (X) of 40 predicts GPA (Y) of 2.1

Interpreting Correlations

Correlations and causality Directionality problem

• Given correlation between A and B, A could cause B, or B could cause A

Third variable problem • Given correlation

between A and B• uncontrolled third

variable could cause both A and B to occur

• Partial correlations “partial out” possible third variable

Interpreting Correlations

Caution: correlational statistics vs. correlational research

Not identical• Correlational research could involve t tests• Experimental research could examine relationship between IV

and DV

Using correlations The need for correlational research

• Some IVs cannot be manipulated• Subject variables• Practical/ethical reasons

• e.g., brain damage

Combining Correlational and Experimental Research

Research example 27: Loneliness and anthropomorphism

Study 1: correlation between loneliness and tendency to anthropomorphize

• r = .53 Studies 2 & 3: manipulated loneliness to tests its effects

on likelihood to anthropomorphize• IVstudy1 = [false] personality feedback (will be lonely, will have many

connections with others)• DVstudy1 = degree of belief in supernatural beings (e.g., God, Devil,

ghosts)• IVstudy2 = induce feeling of connection or disconnection

• DVstudy1 = anthropomorphic ratings of own pets and others’ pets

• Results feelings of disconnection (loneliness) increased Ss likelihood to anthropomorphize

Multivariate Analysis

Bivariate vs. multivariate analyses Multiple regression

One dependent variable More than one independent variable Relative influence of each predictor variable can be

weighted• Examples:• predicting school success (GPA) from (a) SAT scores and (b)

high school grades• predicting susceptibility to colds from (a) negative life

events, (b) perceived stress, and (c) negative affect

Multivariate Analysis

Factor analysis After correlating all possible scores, factor analysis

identifies clusters of intercorrelated scores• First cluster factor could be called verbal fluency• Second cluster factor could be called spatial skill

Often used in psychological test development