CH07

Educational Research

Chapter 7Correlational Research

Gay, Mills, and Airasian

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

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 approachObjective 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 the strongest, the first or last?

Objective 3.3 & 3.5

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

An Example of a Scatterplot

Linear Regression

30.00 40.00 50.00

ex1

30.00

35.00

40.00

45.00

50.00

ex2

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 - 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


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 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


Relationship Studies General purpose

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


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

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 7.2Objectives 7.1, 7.2, & 7.3

Linear and Curvilinear Relationships Linear 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

An Example of a Linear Relationship

Linear Regression

30.00 40.00 50.00

ex1

0.7000

0.8000

0.9000

1.0000fp

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

100.00sc

ore

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 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

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 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


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

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 determinationObjective 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 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)

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

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

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


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

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

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


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?

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