1. Chapter 7 Correlational Research Gay, Mills, and Airasian
Educational Research
Chapter 7
Correlational Research
Gay, Mills, and Airasian

2. Topics to Be Discussed
Definition, purpose, and limitation of correlational research
Correlation coefficients and their significance
Process of conducting correlational research
Relationship studies
Prediction studies

3. Correlational Research
Definition
Whether and to what degree variables are related
Purpose
Determine relationships
Make predictions
Limitation
Cannot indicate cause and effect
Objectives 1.1, 1.2, & 1.3

4. The Process Problem selection
Variables to be correlated are selected on the basis of some rationale
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

5. The Process Participant and instrument selection Design and procedures
Minimum of 30 subjects
Instruments must be valid and reliable
Higher validity and reliability requires smaller samples
Lower validity and reliability requires larger samples
Design and procedures
Collect data on two or more variables for each subject
Data analysis
Compute the appropriate correlation coefficient
Objectives 2.2 & 2.3

6. Correlation Coefficients
A correlation coefficient identifies the size and direction of a relationship
Size/magnitude
Ranges from 0.00 – 1.00
Direction
Positive or negative
Objectives 3.1, 3.2, & 3.3

7. Correlation Coefficients
Interpreting the size of correlations
General rule
Less than .35 is a low correlation
Between .36 and .65 is a moderate correlation
Above .66 is a high correlation
Predictions
Between .60 and .70 are adequate for group predictions
Above .80 is adequate for individual predictions
Objective 3.5

8. Correlation Coefficients
Interpreting the size of correlations (cont.)
Criterion-related validity
Above .60 for affective scales is adequate
Above .80 for tests is minimally acceptable
Inter-rater reliability
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

9. Correlation Coefficients
Interpreting the direction of correlations
Direction
Positive
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
Negative
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

10. Correlation Coefficients
Interpreting the size and direction of correlations using the general rule
+.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

11. Correlation Coefficients
Scatterplots
Graphical presentations of correlations
Example of predicting from an attitude scale – EX 1 – to an achievement test – EX 2
Predictor variable - EX1 - is on the horizontal axis
Criterion variable - EX 2 - is on the vertical axis
Objective 3.4

12. An Example of a Scatterplot
Objective 3.4

13. Correlation Coefficients
Common variance
Definition
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
The squared correlation coefficient - r2
Examples
If r = .50 then r2 = .25
25% of the variance in the criterion can be explained by the predictor
If r = .70 then r2 = .49
49% of the variance in the criterion can be explained by the predictor
Objectives 3.6 & 3.7

14. Statistical Significance
Is the observed coefficient different from 0.00?
Does the correlation represent a true relationship?
Is the correlation only the result of chance?
Determining statistical significance
Consult a table of the critical values of r
See Table A.2 in Appendix A
Three common levels of significance
.01 (1 chance out of 100)
.05 (5 chances out of 100)
.10 (10 chances out of 100)
Objectives 4.1 & 4.3

15. Statistical Significance
Sample size and statistical significance
Small samples require higher correlations for significance
Large samples require lower correlations for significance
Practical significance and statistical significance
Small correlation coefficients can be statistically significant even though they have little practical significance
+.20
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
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

16. Relationship Studies General purpose Two specific purposes
Gain insight into variables that are related to other variables relevant to educators
Achievement
Self-esteem
Self-concept
Two specific purposes
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

17. Conducting Relationship Studies
Identify a set of variables
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

18. Types of Correlation Coefficients
The type of correlation coefficient depends on the measurement level of the variables
Pearson r - continuous predictor and criterion variables
Math attitude and math achievement
Spearman rho – ranked or ordinal predictor and criterion variables
Rank in class and rank on a final exam
Phi coefficient – dichotomous predictor and criterion variables
Gender and pass/fail status on a high stakes test
See Table 7.2
Objectives 7.1, 7.2, & 7.3

19. Linear and Curvilinear Relationships
Plots of the scores on two variables are best described by a straight line
Math scores and science scores
Teacher efficacy and teacher effectiveness
Curvilinear relationships
Plots of scores on two variables are best described by functions
Age and athletic ability
Anxiety and achievement
Estimated by the eta correlation
Objectives 8.1, 8.2, & 8.3

20. An Example of a Linear Relationship
Objective 8.4

21. An Example of a Curvilinear Relationship
Objective 8.4

22. Factors that Influence Correlations
Sample size
The larger the sample the higher the likelihood of a high correlation
Analysis of subgroups
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

23. Factors that Influence Correlations
Variation
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
Correlation coefficients are lower when the instruments being used have low reliability
A correction for attenuation is available
Objectives 9.2 & 9.3

24. Prediction Studies
Attempts to describe the predictive relationships between or among variables
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

25. Prediction Studies Three purposes
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

26. Prediction Studies Single and multiple predictors
Linear regression - one predictor and one criterion
Y’ = a + bX r2
Multiple regression – more than one predictor and one criterion
Y’ = a + bX1 + bX2 + … + bXi
r2 or the coefficient of determination
Objective 11.4

27. Conducting a Prediction Study
Identify a set of variables
Limit to those variables logically related to the criterion
Identify a population and select a sample
Identify appropriate instruments for measuring each variable
Ensure appropriate levels of validity and reliability
Collect data for each instrument from each subject
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)

28. Conducting a Prediction Study
Issues of concern
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

29. Conducting a Prediction Study
Issues of concern (cont.)
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

30. Differences between Types of Studies
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
The measurement of all variables occurs at about the same time
Predictive studies involve the predictive relationships between or among variables
The predictor variables are collected long before the criterion variable
Objectives 11.2 & 11.3

31. Other Correlation Analyses
Path analysis
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)
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

32. Other Correlation Analyses
Discriminant function analysis
Similar to multiple regression except that the criterion variable is categorical
Typically used to predict group membership
High or low anxiety
Achievers or non-achievers
Objective 13.2

33. Other Correlation Analyses
Cannonical correlation
An extension of multiple regression in which more than one predictor variable and more than one criterion variable are used
Factor analysis
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

34. A Checklist of Questions
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?