Chapter 7 Correlational Research Gay, Mills, and Airasian Educational Research Chapter 7 Correlational 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 approach Objective 2.1

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

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 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 Objectives 3.6 & 3.7

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

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

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

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 7.2 Objectives 7.1, 7.2, & 7.3

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

An Example of a Linear Relationship Objective 8.4

An Example of a Curvilinear Relationship 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 Objectives 9.2 & 9.3

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

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 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 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 Objectives 11.2 & 11.3

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 Objectives 13.3 & 13.4

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?