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Educational Research

Chapter 7Correlational Research

Gay, Mills, and Airasian



Topics to Be Discussed Definition, purpose, and limitation of

correlational research

Correlation coefficients and theirsignificance

Process of conducting correlational

research Relationship studies

Prediction studies



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



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



The Process Participant and instrument selection

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



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



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



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



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



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 thestrongest, the first or last?

Objective 3.3 & 3.5



Correlation Coefficients Scatterplots

Graphical presentations of correlations

Example of predicting from an attitudescale – EX 1 – to an achievement test – EX 2

Predictor variable - EX1 - is on thehorizontal axis

Criterion variable - EX 2 - is on the verticalaxis

Objective 3.4



An Example of a ScatterplotLinear Regression

30.00 40.00 50.00

ex 1

30.00

35.00

40.00

45.00

50.00

e x 2

ex2 = 11.23 + 0.72 * ex1R-Square = 0.66

Objective 3.4



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 - r 2

Examples

If r = .50 then r 2 = .25

25% of the variance in the criterion can be explainedby the predictor

If r = .70 then r 2 = .49

49% of the variance in the criterion can be explainedby the predictor




Statistical Significance 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)




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 andpredicts 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 predictsonly .09 of the variation in the criterion scores




Relationship Studies General purpose

Gain insight into variables that are related to othervariables 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 dependentvariable in experimental studies




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 eachvariable

Collect data for each instrument from each subject

Compute the appropriate correlation coefficient

Objective 6.1



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 criterionvariables

Rank in class and rank on a final exam

Phi coefficient – dichotomous predictor and criterionvariables

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 Linear relationships

Plots of the scores on two variables are bestdescribed 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



An Example of a Linear Relationship

Linear Regression

30.00 40.00 50.00

ex 1

0.7000

0.8000

0.9000

1.0000

f p

fp = 0.39 + 0.01 * ex1R-Square = 0.80

Objective 8.4



An Example of a Curvilinear Relationship

LLR Smoother

2.00 4.00 6.00 8.00 10.00

study

0.00

25.00

50.00

75.00

100.00

s c o r e

Objective 8.4



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 theyare being obscured by the analysis of the data for thetotal sample

Reduces the size of the sample

Potentially reduces variation in the scores

Objective 9.1



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




Prediction Studies Attempts to describe the predictive

relationships between or amongvariables

The predictor variable is the variable fromwhich the researcher is predicting

The criterion variable is the variable towhich the researcher is predicting




Prediction Studies Three purposes

Facilitates decisions about individuals tohelp a selection decision

Tests variables believed to be goodpredictors of a criterion

Determines the predictive validity of aninstrument

Objective 11.1



Prediction Studies Single and multiple predictors

Linear regression - one predictor and onecriterion Y’ = a + b X

r 2

Multiple regression – more than onepredictor and one criterion Y’ = a + b X1 + b X2 + … + b Xi

r 2 or the coefficient of determination

Objective 11.4



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 eachvariable 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)



Conducting a Prediction Study Issues of concern

Shrinkage – the tendency of a prediction

equation to become less accurate whenused with a group other than the one onwhich the equation was originallydeveloped

Cross validation – validation of a predictionequation with another group of subjects toidentify problematic variables

Objective 11.3



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 betweencollecting the predictor and criterion variables

Criterion variables defined in general terms (e.g.,teacher effectiveness, success in school) tend tohave lower prediction accuracy than those definedvery narrowly (e.g., overall GPA, test scores)

Objective 11.5



Differences between Types of Studies Correlational research is a general category

that is usually discussed in terms of twovariables

Relationship studies develop insight into therelationships between several variables The measurement of all variables occurs at about

the same time

Predictive studies involve the predictiverelationships between or among variables The predictor variables are collected long before

the criterion variableObjectives 11.2 & 11.3



Other Correlation Analyses Path analysis

Investigates the patterns of relationships among anumber of variables

Results in a diagram that indicates the specificmanner 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



Other Correlation Analyses Discriminant function analysis

Similar to multiple regression except thatthe criterion variable is categorical

Typically used to predict groupmembership

High or low anxiety

Achievers or non-achievers

Objective 13.2



Other Correlation Analyses Cannonical correlation

An extension of multiple regression in which morethan one predictor variable and more than onecriterion variable are used

Factor analysis

A correlational analysis used to take a largenumber of variables and group them into a smallernumber of clusters of similar variables calledfactors




A Checklist of Questions Was the correct correlation coefficient

used?

Is the validity and reliability of theinstruments acceptable?

Is there a restricted range of scores?

How large is the sample?

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