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CHAPTER NINE
Correlational Research Designs
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 9 | 2
Study Questions
• What are correlational research designs, and why are they used in behavioral research?
• What patterns of association can occur between two quantitative variables?
• What is the Pearson product-moment correlation coefficient? What are its uses and limitations?
• How does the chi-square statistic assess association?
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 9 | 3
Study Questions
• What is multiple regression, and what are its uses in correlational research?
• How can correlational data be used to make inferences about causal relationships among measured variables? What are the limitations of correlational designs in doing so?
• What are the best uses for correlational designs?
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Correlational Research Designs
• Correlational research designs – Used to search for and describe
relationships among measured variables
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Associations AmongQuantitative Variables
A sample data set.
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Organizing the Data
• Scatterplot – Uses a standard coordinate system where
• The horizontal axis indicates the scores on the predictor variable
• The vertical axis represents the scores on the outcome variable
– A point is plotted for each individual at the intersection of his or her scores on the two variables
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Regression Line
• Regression line – The straight line of “best fit” drawn through
the points on a scatterplot
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Scatterplot
A scatterplot with regression line.
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Linear and Nonlinear Relationships
• Linear relationship – When the association between the
variables on the scatterplot can be easily approximated with a straight line
• Nonlinear relationships – Not all relationships between variables can
be well described with a straight line
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Patterns of RelationshipsBetween Two Variables
Examples of linear relationships.
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Patterns of RelationshipsBetween Two Variables
Independent: When there is no relationship at all between the two variables
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Patterns of RelationshipsBetween Two Variables
Curvilinear relationships: Relationships that change in direction
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The Pearson Correlation Coefficient
• Pearson product-moment correlation coefficient – Used to summarize and communicate the
strength and direction of the association between two quantitative variables
– Frequently designated by the letter r– Values range from r = -1.00 to r = +1.00
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The Pearson Correlation Coefficient
• The direction of the relationship is indicated by the sign of the correlation coefficient– Positive values of r indicate positive linear
relationships– Negative values of r indicate negative linear
relationships
• The strength or effect size of the linear relationship – Indexed by the absolute value distance of the
correlation coefficient from zero
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Interpretation of r
• A significant r indicates there is a linear association between the variables.
• Coefficient of determination – The proportion of variance measure for r – It is r2
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Restriction of Range
• Restriction of range – Occurs when most participants have
similar scores on one of the variables being correlated
– The value of the correlation coefficient is reduced and does not represent an accurate picture of the true relationship between the variables
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Restriction of Range
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The Chi-Square Statistic
• Chi-square statistic (2) – Must be used to assess the relationship
between two nominal variables– Technically known as the chi-square test of
independence– Calculated by constructing a contingency
table, which displays the number of individuals in each of the combinations of the two nominal variables
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Reporting Correlations andChi-Square Statistics
• An example of reporting a correlation in a research report is
r (20) = 0.52, p < 0.01
where
20 is the sample size (N) 0.52 is the correlation coefficient 0.01 is the p-value of the observed
correlation
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Correlation Matrix
• Correlation matrix – A table showing the correlations of many
variables with each other
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A Correlation Matrix
A correlation matrix reported in APA format.
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Multiple Regression
• Multiple regression – A statistical analysis procedure using more
than one predictor variable to predict a single outcome variable
• The regression analysis provides– Multiple correlation coefficient– Regression coefficients or beta weights
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Multiple Regression
The simultaneous impact of three measured independent variables as predictors of college GPA.
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Multiple Regression
• Multiple correlation coefficient (R)– The ability of all of the predictor variables
together to predict the outcome variable
• Regression coefficients or beta weights– Indicate the relationship between each of
the predictor variables and the outcome variable
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Correlation and Causality
• Correlational research – Cannot be used to draw conclusions about
the causal relationships among the measured variables
– Although the researcher may believe the predictor variable is causing the outcome variable, the correlation between the two variables does not provide support for this hypothesis
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Alternative Explanationsof a Correlation
• Reverse causation– The causal direction is opposite what has
been hypothesized
• Reciprocal causation– The two variables cause each other
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Alternative Explanationsof a Correlation
• Common-causal variables– Variables not part of the research
hypothesis cause both the predictor and the outcome variable
• Spurious relationship– The common-causal variable produces and
“explains away” the relationship between the predictor and outcome variables
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Alternative Explanationsof a Correlation
• Extraneous variables– Variables other than the predictor cause
the outcome variable but do not cause the predictor variable
• Mediating variables– Variables caused by the predictor variable
in turn cause the outcome variable
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Longitudinal Research
• Longitudinal research design – The same individuals are measured more
than one time– The time period between the
measurements is long enough that changes in the variables of interest could occur
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Longitudinal Research
• Path analysis – An analysis technique for correlational data
from longitudinal research designs
• Path diagram – A method for displaying the results of a
path analysis – Represents the associations among a set
of variables
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Path Diagram
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Longitudinal Research
• Longitudinal research designs take a long time to conduct.
• Cross-sectional research designs– Measure people from different age groups
at the same time– Very limited in their ability to rule out
reverse causation
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Structural Equation Analysis
• Structural equation analysis – A statistical procedure that tests whether
the observed relationships among a set of variables conform to the theoretical prediction about how those variables should be causally related
• Latent variables – The conceptual variables
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Structural Equation Model