Statistics for Statistics for Education ResearchEducation Research

Lecture 3Lecture 3

Bivariate Correlations Bivariate Correlations CoefficientsCoefficients

Instructor: Dr. Tung-Instructor: Dr. Tung-hsien Hehsien He

the@tea.ntue.edu.twthe@tea.ntue.edu.tw

Meaning: The association Meaning: The association (relationship) between two variables(relationship) between two variables

Types: Types: 1. 1. Positive RelationshipPositive Relationship (a lower-left to (a lower-left to

upper-right pattern on a scattergram)upper-right pattern on a scattergram)2. 2. Negative RelationshipNegative Relationship (a upper-left (a upper-left

to low-right pattern on a scattergram)to low-right pattern on a scattergram) Index: Index: Correlation CoefficientsCorrelation Coefficients1. Definition: 1. Definition: the extent to which two the extent to which two

sets of data are related to each othersets of data are related to each other (See figure 5.1, p. 105).(See figure 5.1, p. 105).

2. 2. CoefficientsCoefficients fall between fall between 1 1..3. The absolute value of coefficients 3. The absolute value of coefficients

indicates the indicates the degree of the degree of the relationshiprelationship, , rather than the rather than the strength of the relationshipstrength of the relationship..

4. When coefficient is = 0, it means no 4. When coefficient is = 0, it means no relationship whatsoever.relationship whatsoever.

5. 5. SlopeSlope indicates the general indicates the general direction of relationship: See figure direction of relationship: See figure 5.2, p. 106. 5.2, p. 106.

Different Types of Data Call for Use of Different Types of Data Call for Use of Different Bivariate Correlation Different Bivariate Correlation ProceduresProcedures

1. 1. Pearson product-moment Pearson product-moment correlation coefficient [Pearson correlation coefficient [Pearson 積差相關積差相關係數係數 ]]::

1. Symbol: 1. Symbol: rr2. An index of the 2. An index of the linearlinear relationship relationship

between between twotwo variables.variables.3. The two variables must be measured 3. The two variables must be measured

to produce to produce intervalinterval or or ratioratio scores. scores.

4. Formulas (4. Formulas (No need to memorize themNo need to memorize them!!):):a. Standard Score Formula: 5.1, p. 108a. Standard Score Formula: 5.1, p. 108b. Deviation Score Formula: 5.2, p. 109b. Deviation Score Formula: 5.2, p. 109c. Raw Score Formula: 5.3, p. 111c. Raw Score Formula: 5.3, p. 111d. d. CovarianceCovariance Formula: 5.5, p. 112 Formula: 5.5, p. 112

5. Standard Deviation, Variance, & 5. Standard Deviation, Variance, & CovarianceCovariancea. SDa. SD2 2 = Variance (Within one variable)= Variance (Within one variable) b. b. ssxyxy

22 ((covariancecovariance:: 共變數共變數 ): 5.4, p. 112 ): 5.4, p. 112

(variances between two variables)(variances between two variables)

Spearman rho (Spearman rho ())1. Symbol: 1. Symbol: or or rrss

2. An index of the 2. An index of the Spearman’sSpearman’s rank-rank-order correlationorder correlation between between twotwo variables.variables.

3. The two variables must be measured 3. The two variables must be measured to produce to produce ordinalordinal scores scores with rankswith ranks but but without tieswithout ties (i.e., (i.e., no tied ranksno tied ranks.).)

4. Formula: 5.8, p. 1224. Formula: 5.8, p. 1225. Spearman 5. Spearman will be equal to Pearson will be equal to Pearson

r if no tied scores are found. r if no tied scores are found.

Kendall’s tau (Kendall’s tau (ττ))

1. Symbol: 1. Symbol: ττ

2. An index of the 2. An index of the rank-order rank-order correlationcorrelation between between twotwo variables.variables.

3. The two variables must be 3. The two variables must be measured to produce measured to produce ordinalordinal scores scores with tied rankswith tied ranks..

Point Biserial Correlation (Not available in Point Biserial Correlation (Not available in SPSS)SPSS)

1. Symbol: 1. Symbol: rrpbpb

2. An index of the 2. An index of the bivariate correlationbivariate correlation between between twotwo variables.variables.

3. One of the two variables must be 3. One of the two variables must be measured to produce measured to produce intervalinterval or or ratioratio scores, whereas the other must produce scores, whereas the other must produce dichotomous scoresdichotomous scores, i.e., 1 or 0. , i.e., 1 or 0.

Biserial Correlation (Not available in Biserial Correlation (Not available in SPSS)SPSS)

1. Symbol: 1. Symbol: rrbisbis

2. An index of the 2. An index of the bivariate correlationbivariate correlation between between twotwo variables.variables.

3. One of the two variables must be 3. One of the two variables must be measured to produce measured to produce interval or interval or ratio scoresratio scores, whereas the other must , whereas the other must produce produce artificial dichotomous artificial dichotomous scoresscores..

4. Artificial dichotomous values mean 4. Artificial dichotomous values mean the the original data are not keyed in original data are not keyed in with 0 or 1with 0 or 1. But, researchers decide . But, researchers decide to to transform the original data into transform the original data into dichotomous ones by assigning dichotomous ones by assigning either 0 or 1either 0 or 1 to replace original to replace original values of these data. values of these data.

Phi CorrelationPhi Correlation1. An index of the 1. An index of the tetrachoric tetrachoric

correlationcorrelation between between twotwo variables.variables.2. The two variables must be 2. The two variables must be

measured to produce measured to produce dichotomous dichotomous scores, no matter they are artificial scores, no matter they are artificial or notor not. .

Check the assumptions before using Check the assumptions before using correlation procedures:correlation procedures:

1. 1. LinearityLinearity::a. Definition: data are located around a a. Definition: data are located around a straight line rather than fall exactly on itstraight line rather than fall exactly on itb. For a positive relationship: the b. For a positive relationship: the increasing values on X axis will tend to increasing values on X axis will tend to entail the increasing values on Y axis and entail the increasing values on Y axis and vice versa for a negative relationship.vice versa for a negative relationship.c. Check c. Check scattergramsscattergrams to see whether to see whether this linearity assumption is met.this linearity assumption is met.

2. 2. Curvilinear RelationshipCurvilinear Relationship: : a. No straight line can be identified a. No straight line can be identified but some curved lines (See figure 5.3 but some curved lines (See figure 5.3 B/C).B/C).b. b. A non-linear relationshipA non-linear relationship..c. The increasing values do not entail c. The increasing values do not entail any patterned direction. any patterned direction. d. Pearson r will undermine non-d. Pearson r will undermine non-linear relationship.linear relationship.e. Check e. Check scattergramsscattergrams to see to see whether this curvilinearity appears.whether this curvilinearity appears.

1. 1. HomogeneityHomogeneity: : a. Cause negative effects on the size of a. Cause negative effects on the size of correlation coefficients.correlation coefficients.b. The more homogeneous a sample is, b. The more homogeneous a sample is, the lower the value of coefficient will be. the lower the value of coefficient will be. (Why? Think about formulas:(Why? Think about formulas:r = r = ssxyxy//ssxxssyy

ssxy, xy, = covariance = covariance If covariance decreases, that is, the If covariance decreases, that is, the amount of variances shared by the two amount of variances shared by the two variables become lower, what happens?variables become lower, what happens?

2. Sizes of Samples2. Sizes of Samples1. 1. Sizes of samples do not influence sizesSizes of samples do not influence sizes of of rr (Why? Think about a 5-person sample (Why? Think about a 5-person sample and 10-person sample? Provided that and 10-person sample? Provided that subjects in both group are relatively subjects in both group are relatively homogeneous, will the relationships of homogeneous, will the relationships of their performances on two scales be their performances on two scales be extremely different?) extremely different?) 2. 2. Sizes of samples do influence accuracySizes of samples do influence accuracy of of rr (i.e., whether r is significant or not). It (i.e., whether r is significant or not). It is because the influences of outliers will be is because the influences of outliers will be reduced considerably when they are reduced considerably when they are divided by a large divided by a large nn. .

3. Report 3. Report rr and and rr22::a. a. rr22 ( (r square)r square): coefficient of : coefficient of determination. determination. b. the proportion of the total variance in Y b. the proportion of the total variance in Y that can be associated with the variance in that can be associated with the variance in X. X. c. c. rr tends to overestimate the strength of tends to overestimate the strength of correlation. Even though a very low correlation. Even though a very low rr may may be found to be significant. Thus, be found to be significant. Thus, rr22 is a is a better index to show this degree. (e.g., better index to show this degree. (e.g., rr =0.5 but =0.5 but rr22 = 0.25 only) = 0.25 only)4. 4. No causalityNo causality for any type of correlation for any type of correlation coefficient. coefficient.

Demo of Correlation StudyDemo of Correlation Study1. Question 7 & 12 on p. 126 & 127.1. Question 7 & 12 on p. 126 & 127.2. Hypothesis for Correlation 2. Hypothesis for Correlation

Procedures:Procedures:Ho: r = 0Ho: r = 0Ha: r Ha: r 0 03. SPSS Procedures:3. SPSS Procedures: Hands-on of Correlation Study: Hands-on of Correlation Study:

Check the SPSS filesCheck the SPSS files

3P3PPP: Task & ego orientations are associated PP: Task & ego orientations are associated

with self-perceived ability (self-efficacy), with self-perceived ability (self-efficacy), self-esteem, anxiety, and intrinsic self-esteem, anxiety, and intrinsic motivation.motivation.

IP: However, dimensions of goal orientations IP: However, dimensions of goal orientations should be expanded into Task, Avoidance, should be expanded into Task, Avoidance, Self-Defeating, and Self-Enhancing since: Self-Defeating, and Self-Enhancing since: (a) the avoidance for work and avoidance (a) the avoidance for work and avoidance for looking stupid are different, and (b) the for looking stupid are different, and (b) the ego orientations should be broken into ego orientations should be broken into self-defeating (avoiding looking stupid) & self-defeating (avoiding looking stupid) & selfenhancing ( outperforming others).selfenhancing ( outperforming others).

SP: For the study 1: (a) self-defeating & SP: For the study 1: (a) self-defeating & selfenhancing should be weakly selfenhancing should be weakly correlated since they originate from the correlated since they originate from the same ego orientation, (b) self-defeating same ego orientation, (b) self-defeating orientation should be weakly related to orientation should be weakly related to avoidance orientations since they both avoidance orientations since they both contain the element of avoidance, (c) contain the element of avoidance, (c) self-enhancing orientations should be self-enhancing orientations should be weakly related to task orientations since weakly related to task orientations since both of them contain the elements of both of them contain the elements of learning efforts, and (d) task and learning efforts, and (d) task and avoidance orientations should be avoidance orientations should be negatively related since they represent negatively related since they represent contradictory learning motivation.contradictory learning motivation.

Instruments: Nine instruments (see Instruments: Nine instruments (see p. 76, Table 2)p. 76, Table 2)

Statistical Analysis:Statistical Analysis:a. Exploratory Factor Analysisa. Exploratory Factor Analysisb. Pearson rb. Pearson rc. Multiple-Regression c. Multiple-Regression Results:Results:a. EFA: Table 1, p. 75a. EFA: Table 1, p. 75b. Pearson r: Table 2, p. 76 (What b. Pearson r: Table 2, p. 76 (What

does a significant coefficient mean?)does a significant coefficient mean?)

c. Regression (beta): Table 3, p. 76c. Regression (beta): Table 3, p. 76 Interpretations: Interpretations: a. According to EFA, the four types of a. According to EFA, the four types of

goal orientations can be separated;goal orientations can be separated;b. According to Pearson, the four b. According to Pearson, the four

speculations are confirmed, and the speculations are confirmed, and the ego orientations could be separated ego orientations could be separated into two independent orientations into two independent orientations

Meanings of Significances & One-Meanings of Significances & One-Tailed & Two-TailedTailed & Two-Tailed