Educational Research
Chapter 7
Correlational Research
Gay, Mills, and Airasian
Topics to Be Discussed
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Definition, purpose, and limitation of
correlational research
Correlation coefficients and their
significance
Process of conducting correlational
research
Relationship studies
Prediction studies
Correlational Research
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Definition
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Purpose
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Whether and to what degree variables are
related
Determine relationships
Make predictions
Limitation
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Cannot indicate cause and effect
Objectives 1.1, 1.2, & 1.3
The Process
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Problem selection
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Variables to be correlated are selected on the
basis of some rationale
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Math attitudes and math achievement
Teachers’ sense of efficacy and their effectiveness
Increases the ability to meaningfully interpret
results
Inefficiency and difficulty interpreting the
results from a shotgun approach
Objective 2.1
The Process
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Participant and instrument selection
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Minimum of 30 subjects
Instruments must be valid and reliable
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Design and procedures
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Higher validity and reliability requires smaller samples
Lower validity and reliability requires larger samples
Collect data on two or more variables for each
subject
Data analysis
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Compute the appropriate correlation coefficient
Objectives 2.2 & 2.3
Correlation Coefficients
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A correlation coefficient identifies the
size and direction of a relationship
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Size/magnitude
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Ranges from 0.00 – 1.00
Direction
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Positive or negative
Objectives 3.1, 3.2, & 3.3
Correlation Coefficients
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Interpreting the size of correlations
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General rule
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Less than .35 is a low correlation
Between .36 and .65 is a moderate correlation
Above .66 is a high correlation
Predictions
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Between .60 and .70 are adequate for group
predictions
Above .80 is adequate for individual predictions
Objective 3.5
Correlation Coefficients
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Interpreting the size of correlations (cont.)
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Criterion-related validity
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Above .60 for affective scales is adequate
Above .80 for tests is minimally acceptable
Inter-rater reliability
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Above .90 is very good
Between .80 and .89 is acceptable
Between .70 and .79 is minimally acceptable
Lower than .69 is problematic
Objective 3.5
Correlation Coefficients
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Interpreting the direction of correlations
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Direction
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Positive
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Negative
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High scores on the predictor are associated with high
scores on the criterion
Low scores on the predictor are associated with low
scores on the criterion
High scores on the predictor are associated with low
scores on the criterion
Low scores on the predictor are associated with high
scores on the criterion
Positive or negative does not mean good or bad
Objective 3.3
Correlation Coefficients
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Interpreting the size and direction of
correlations using the general rule
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+.95 is a strong positive correlation
+.50 is a moderate positive correlation
+.20 is a low positive correlation
-.26 is a low negative correlation
-.49 is a moderate negative correlation
-.95 is a strong negative correlation
Which of the correlations above is the
strongest, the first or last?
Objective 3.3 & 3.5
Correlation Coefficients
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Scatterplots
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Graphical presentations of correlations
Example of predicting from an attitude
scale – EX 1 – to an achievement test –
EX 2
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Predictor variable - EX1 - is on the
horizontal axis
Criterion variable - EX 2 - is on the vertical
axis
Objective 3.4
An Example of a Scatterplot
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50.00
ex2 = 11.23 + 0.72 * ex1
R-Square = 0.66
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Linear Regression
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45.00
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ex2
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40.00
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35.00
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30.00
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30.00
40.00
ex1
50.00
Objective 3.4
Correlation Coefficients
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Common variance
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Definition
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The extent to which variables vary in a systematic manner
Interpreted as the percentage of variance in the criterion
variable explained by the predictor variable
Computation
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The squared correlation coefficient - r2
Examples
2
 If r = .50 then r = .25
 25% of the variance in the criterion can be explained
by the predictor
2
 If r = .70 then r = .49
 49% of the variance in the criterion can be explained
by the predictor
Objectives 3.6 & 3.7
Statistical Significance
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Statistical significance
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Is the observed coefficient different from 0.00?
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Determining statistical significance
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Does the correlation represent a true relationship?
Is the correlation only the result of chance?
Consult a table of the critical values of r
See Table A.2 in Appendix A
Three common levels of significance
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.01 (1 chance out of 100)
.05 (5 chances out of 100)
.10 (10 chances out of 100)
Objectives 4.1 & 4.3
Statistical Significance
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Sample size and statistical significance
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Small samples require higher correlations for significance
Large samples require lower correlations for significance
Practical significance and statistical significance
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Small correlation coefficients can be statistically significant even
though they have little practical significance
 +.20
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Statistically significant at the .05 level if the sample is about 100
Little or no practical significance because it is very low and
predicts only .04 of the variation in the criterion scores
-.30
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Statistically significant at the .05 level if the sample is about 40
Little or no practical significance because it is low and predicts
only .09 of the variation in the criterion scores
Objectives 4.2 & 4.4
Relationship Studies
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General purpose
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Gain insight into variables that are related to other
variables relevant to educators
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Achievement
Self-esteem
Self-concept
Two specific purposes
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Suggest subsequent interest in establishing cause
and effect between variables found to be related
Control for variables related to the dependent
variable in experimental studies
Objectives 5.1 & 5.2
Conducting Relationship Studies
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Identify a set of variables
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Limit to those variables logically related to the criterion
Avoid the shotgun approach
 Possibility of erroneous relationships
 Issues related to determining statistical significance
Identify a population and select a sample
Identify appropriate instruments for measuring each
variable
Collect data for each instrument from each subject
Compute the appropriate correlation coefficient
Objective 6.1
Types of Correlation Coefficients
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The type of correlation coefficient depends on the
measurement level of the variables
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Pearson r - continuous predictor and criterion variables
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Spearman rho – ranked or ordinal predictor and criterion
variables
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Rank in class and rank on a final exam
Phi coefficient – dichotomous predictor and criterion
variables
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Math attitude and math achievement
Gender and pass/fail status on a high stakes test
See Table 7.2
Objectives 7.1, 7.2, & 7.3
Linear and Curvilinear Relationships
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Linear relationships
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Plots of the scores on two variables are best
described by a straight line
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Math scores and science scores
Teacher efficacy and teacher effectiveness
Curvilinear relationships
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Plots of scores on two variables are best described
by functions
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Age and athletic ability
Anxiety and achievement
Estimated by the eta correlation
Objectives 8.1, 8.2, & 8.3
An Example of a Linear Relationship
1.0000
fp = 0.39 + 0.01 * ex1
R-Square = 0.80
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0.9000
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fp
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0.7000
Linear Regression
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0.8000
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30.00
40.00
50.00
ex1
Objective 8.4
An Example of a Curvilinear Relationship
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100.00
LLR Smoother
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75.00
score
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50.00
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25.00
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0.00
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2.00
4.00
6.00
8.00
10.00
study
Objective 8.4
Factors that Influence Correlations
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Sample size
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The larger the sample the higher the likelihood of
a high correlation
Analysis of subgroups
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If the total sample consists of males and females each
gender represents a subgroup
Results across subgroups can be different because they
are being obscured by the analysis of the data for the
total sample
Reduces the size of the sample
Potentially reduces variation in the scores
Objective 9.1
Factors that Influence Correlations
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Variation
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The greater the variation in scores the
higher the likelihood of a strong correlation
The lower the variation in scores the higher
the likelihood of a weak correlation
Attenuation
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Correlation coefficients are lower when the
instruments being used have low reliability
A correction for attenuation is available
Objectives 9.2 & 9.3
Prediction Studies
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Attempts to describe the predictive
relationships between or among
variables
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The predictor variable is the variable from
which the researcher is predicting
The criterion variable is the variable to
which the researcher is predicting
Objectives 10.1 & 10.2
Prediction Studies
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Three purposes
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Facilitates decisions about individuals to
help a selection decision
Tests variables believed to be good
predictors of a criterion
Determines the predictive validity of an
instrument
Objective 11.1
Prediction Studies
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Single and multiple predictors
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Linear regression - one predictor and one
criterion
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Y’ = a + bX
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r2
Multiple regression – more than one
predictor and one criterion
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Y’ = a + bX1 + bX2 + … + bXi
r2 or the coefficient of determination
Objective 11.4
Conducting a Prediction Study
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Identify a set of variables
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Identify a population and select a sample
Identify appropriate instruments for measuring each
variable
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Ensure appropriate levels of validity and reliability
Collect data for each instrument from each subject
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Limit to those variables logically related to the criterion
Typically data is collected at different points in time
Compute the results
 The multiple regression coefficient
 The multiple regression equation (i.e., the
prediction equation)
Conducting a Prediction Study
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Issues of concern
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Shrinkage – the tendency of a prediction
equation to become less accurate when
used with a group other than the one on
which the equation was originally
developed
Cross validation – validation of a prediction
equation with another group of subjects to
identify problematic variables
Objective 11.3
Conducting a Prediction Study
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Issues of concern (cont.)
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Errors of measurement (e.g., low validity or
reliability) diminish the accuracy of the prediction
Intervening variables can influence the predictive
process if there is too much time between
collecting the predictor and criterion variables
Criterion variables defined in general terms (e.g.,
teacher effectiveness, success in school) tend to
have lower prediction accuracy than those defined
very narrowly (e.g., overall GPA, test scores)
Objective 11.5
Differences between Types of Studies
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Correlational research is a general category
that is usually discussed in terms of two
variables
Relationship studies develop insight into the
relationships between several variables
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The measurement of all variables occurs at about
the same time
Predictive studies involve the predictive
relationships between or among variables
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The predictor variables are collected long before
the criterion variable
Objectives 11.2 & 11.3
Other Correlation Analyses
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Path analysis
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Investigates the patterns of relationships among a
number of variables
Results in a diagram that indicates the specific
manner by which variables are related (i.e., paths)
and the strength of those relationships
An extension of this analysis is structural equation
modeling (SEM)
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Clarifies the direct and indirect relationships among
variables based on underlying theoretical constructs
More precise than path analysis
Often known as LISREL for the first computer program
used to conduct this analysis
Objective 13.1
Other Correlation Analyses
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Discriminant function analysis
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Similar to multiple regression except that
the criterion variable is categorical
Typically used to predict group
membership
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High or low anxiety
Achievers or non-achievers
Objective 13.2
Other Correlation Analyses
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Cannonical correlation
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An extension of multiple regression in which more
than one predictor variable and more than one
criterion variable are used
Factor analysis
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A correlational analysis used to take a large
number of variables and group them into a smaller
number of clusters of similar variables called
factors
Objectives 13.3 & 13.4
A Checklist of Questions
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Was the correct correlation coefficient
used?
Is the validity and reliability of the
instruments acceptable?
Is there a restricted range of scores?
How large is the sample?