correlational study.ppt

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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 correlationalresearch

! Relationship studies!

Prediction studies



Correlational Research! Definition

!

Whether and to what degree variables arerelated

!

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 thebasis 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 theresults 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 eachsubject

! Data analysis!

Compute the appropriate correlation coefficient

Objectives 2.2 & 2.3



Correlation Coefficients!

A correlation coefficient identifies thesize 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




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

ex1

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




Common variance! Definition

!

The extent to which variables vary in a systematic manner

! Interpreted as the percentage of variance in the criterionvariable 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 eventhough 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 themeasurement 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 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 describedby 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

1.0000

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 ofa high correlation

! Analysis of subgroups

! If the total sample consists of males and females eachgender 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 thehigher the likelihood of a strong correlation

! The lower the variation in scores the higherthe 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 predictiverelationships between or among

variables!

The predictor variable is the variable fromwhich 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 goodpredictors 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 + 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., theprediction equation)



Conducting a Prediction Study! Issues of concern

!

Shrinkage – the tendency of a predictionequation to become less accurate when

used with a group other than the one onwhich the equation was originallydeveloped

! Cross validation – validation of a prediction

equation 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 orreliability) diminish the accuracy of the prediction

! Intervening variables can influence the predictiveprocess 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 to

have 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 aboutthe 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 equationmodeling (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 programused 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 groupmembership

!

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 onecriterion variable are used

! Factor analysis

!

A correlational analysis used to take a largenumber of variables and group them into a smaller

number of clusters of similar variables calledfactors




A Checklist of Questions!

Was the correct correlation coefficientused?

! Is the validity and reliability of theinstruments acceptable?

! Is there a restricted range of scores?

! How large is the sample?



Statistical Assessment of

Relationships Data

Are the data quantitative or nominal?

quantitative nominal

Correlation Analysis:

r Chi-Square Analysis: !

2

Do you have more than two predictorvariables?

Do you have more than two predictorvariables?

No Yes No Yes

Regression Analysis: R Log-Linear AnalysisLogistic Regression



The Correlation Coefficient

for Association among Quantitative Variables Scatterplot

Regression Line

High SchoolGPA

CollegeGPA

4.0

3.0

2.0

1.0

1.0 2.0 3.0 4.0

A graph in which the x axis indicatesthe scores on the predictor variable

and the y axis represents the scoreson the outcome variable. A point is

plotted for each individual at theintersection of their scores.

A line in which the squared distancesof the points from the line are

minimized. (least square methods)



Linear Relationships and Nonliniar Relationships

Y Y

Y Y Y

X

X

X

X X

Positive Linear Negative Linear

Curvilinear Curvilinear Independent



The Pearson CorrelationCoefficient

Calculation

r =

[ ]! !!

""

""

])(][)([

))((

22

2

Y Y X X

Y Y X X ii

Esteem 1 Esteem 21 4 42 4 33 3 2

4 2 25 2 1

Mean 3 2.4

[(4-3)(4-2.4)]2 + ...

(4-3)2 + ... (4-2.4)2 + ...

( )

1!

" N

Z Z Y X

Sesteem1 = 0.8 Sesteem2=1.04

(4-3)/0.8 =1.67

=

=

( )( )

( ) ( )!!

"

#

$$

%

&'

!!

"

#

$$

%

&'

'

((((

( ((

N

Y Y

N

X X

N

Y X XY

2

2

2

2

4 x 4 + 4 x 3 + ...

5

4+4+3+2+2 4+3+2+2+1

4 x 4 + 4 x 4 + 3 x 3 ...

Task 1: compute r

4 x 4 + 3 x 3 + 2 x 2 ...



Interpretation of r

If the relationship between X and Y are positive:If the relationship between X and Y are negative:

-1< r <1

0 < r < 1-1 < r < 0

If p-value associated with the r is < .05

The variable X and Y are significantly correlate to each othe

Positively: 0 < r < 1, Negatively -1 < r < 0

If p-value associated with the r is >. 05

There is no significant correlation between X and Y



Reporting Correlations

“ As predicted by the research hypothesis, the variable of optimiand reported health behavior were (significantly) positively correin the sample (the data), r(20) = .52, p < .01

r(Number of Participants) = Correlation Coefficient r, p < p valu



Limitation 1. Cases in which the correlation between X and Y that havecurvilinier relationships r = 0

2. Cases in which the range of variables is restricted.Restriction of Range Example. SAT scores and college GPA

3. Cases in which the data have outliers r > |.99|



Limitation (visual)

Curviliniar Small Range Outlier



The Chi-square Statistic

for Association among nominal variables

Northerner

Southerner

Yes No

30 (.15)70 (.35)

60 (.30)40 (.20)

100 (.50)

100 (.50)

90 (.45)110 (.55)200 (1.00)

45 (.225) 55 (.275)

45 (.225) 55 (.275)

X =

5.27

)5540(

5.22

)4560(

5.27

)5570(

5.22

)4530( 2222!

+

!

+

!

+

!

! "

e

eo f

f f 2)(

=e f

!2 =

Row marginal X Column marginalN

Task 2 computation !2



Interpretation of !2

•

Go to Table E in Appendix E.

• Degree of Freedom (df):(Level of variable 1 - 1) X (Level of variable 2 -1)

•

Number of Participants

•

See the value at the intersection between Alpha p < .05 and df

If !2 is greater than the value in Table E, the contingency table

is significantly differ from the expectation.

If !2 is greater than the value in Table E, the contingency tableis not significantly differ from the expectation.



Reporting Chi-Square Statistic !2 (degree of freedom (df), Number of Participants(N)) =

Chi value, p < p value

As predicted by the research hypothesis, the southerners were morlikely to approve of a policeman striking an adult male citizen whowas being questioned as a suspect in a murder case, !2(1, N =30)34.23, p < .01

correlational study.ppt

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