1. Statistics for Education Research Lecture 3 Bivariate Correlations Coefficients Instructor: Dr. Tung-hsien He the@tea.ntue.edu.tw

2. Meaning: The association (relationship) between two variables Meaning: The association (relationship) between two variables Types: Types: 1. Positive Relationship (a lower-left to upper-right pattern on a scattergram) 2. Negative Relationship (a upper-left to low-right pattern on a scattergram) Index: Correlation Coefficients Index: Correlation Coefficients 1. Definition: the extent to which two sets of data are related to each other (See figure 5.1, p. 105).

3. 2. Coefficients fall between  1. 3. The absolute value of coefficients indicates the degree of the relationship, rather than the strength of the relationship. 4. When coefficient is = 0, it means no relationship whatsoever. 5. Slope indicates the general direction of relationship: See figure 5.2, p. 106.

4. Different Types of Data Call for Use of Different Bivariate Correlation Procedures Different Types of Data Call for Use of Different Bivariate Correlation Procedures 1. Pearson product-moment correlation coefficient [Pearson 積差相關係數 ]: 1. Symbol: r 2. An index of the linear relationship between two variables. 3. The two variables must be measured to produce interval or ratio scores.

5. 4. Formulas (No need to memorize them!): a. Standard Score Formula: 5.1, p. 108 b. Deviation Score Formula: 5.2, p. 109 c. Raw Score Formula: 5.3, p. 111 d. Covariance Formula: 5.5, p. 112 5. Standard Deviation, Variance, & Covariance a. SD 2 = Variance (Within one variable) b. s xy 2 (covariance: 共變數 ): 5.4, p. 112 (variances between two variables) b. s xy 2 (covariance: 共變數 ): 5.4, p. 112 (variances between two variables)

6. Spearman rho (  ) Spearman rho (  ) 1. Symbol:  or r s 2. An index of the Spearman’s rank-order correlation between two variables. 3. The two variables must be measured to produce ordinal scores with ranks but without ties (i.e., no tied ranks.) 4. Formula: 5.8, p. 122 5. Spearman  will be equal to Pearson r if no tied scores are found.

7. Kendall’s tau (τ) Kendall’s tau (τ) 1. Symbol: τ 2. An index of the rank-order correlation between two variables. 3. The two variables must be measured to produce ordinal scores with tied ranks.

8. Point Biserial Correlation (Not available in SPSS) Point Biserial Correlation (Not available in SPSS) 1. Symbol: r pb 2. An index of the bivariate correlation between two variables. 3. One of the two variables must be measured to produce interval or ratio scores, whereas the other must produce dichotomous scores, i.e., 1 or 0.

9. Biserial Correlation (Not available in SPSS) Biserial Correlation (Not available in SPSS) 1. Symbol: r bis 2. An index of the bivariate correlation between two variables. 3. One of the two variables must be measured to produce interval or ratio scores, whereas the other must produce artificial dichotomous scores.

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

11. Phi Correlation Phi Correlation 1. An index of the tetrachoric correlation between two variables. 2. The two variables must be measured to produce dichotomous scores, no matter they are artificial or not.

12. Check the assumptions before using correlation procedures: Check the assumptions before using correlation procedures: 1. Linearity: a. Definition: data are located around a straight line rather than fall exactly on it b. For a positive relationship: the increasing values on X axis will tend to entail the increasing values on Y axis and vice versa for a negative relationship. c. Check scattergrams to see whether this linearity assumption is met.

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

14. 1. Homogeneity: a. Cause negative effects on the size of correlation coefficients. b. The more homogeneous a sample is, the lower the value of coefficient will be. (Why? Think about formulas: r = s xy /s x s y s xy, = covariance If covariance decreases, that is, the amount of variances shared by the two variables become lower, what happens?

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

16. 3. Report r and r 2 : a. r 2 (r square): coefficient of determination. b. the proportion of the total variance in Y that can be associated with the variance in X. c. r tends to overestimate the strength of correlation. Even though a very low r may be found to be significant. Thus, r 2 is a better index to show this degree. (e.g., r =0.5 but r 2 = 0.25 only) 4. No causality for any type of correlation coefficient.

17. Demo of Correlation Study Demo of Correlation Study 1. Question 7 & 12 on p. 126 & 127. 2. Hypothesis for Correlation Procedures: Ho: r = 0 Ha: r  0 3. SPSS Procedures: Hands-on of Correlation Study: Check the SPSS files Hands-on of Correlation Study: Check the SPSS files

18. 3P 3P PP: Task & ego orientations are associated with self- perceived ability (self-efficacy), self-esteem, anxiety, and intrinsic motivation. IP: However, dimensions of goal orientations should be expanded into Task, Avoidance, Self-Defeating, and Self-Enhancing since: (a) the avoidance for work and avoidance for looking stupid are different, and (b) the ego orientations should be broken into self-defeating (avoiding looking stupid) & selfenhancing ( outperforming others).

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

20. Instruments: Nine instruments (see p. 76, Table 2) Instruments: Nine instruments (see p. 76, Table 2) Statistical Analysis: Statistical Analysis: a. Exploratory Factor Analysis b. Pearson r c. Multiple-Regression Results: Results: a. EFA: Table 1, p. 75 b. Pearson r: Table 2, p. 76 (What does a significant coefficient mean?)

21. c. Regression (beta): Table 3, p. 76 Interpretations: Interpretations: a. According to EFA, the four types of goal orientations can be separated; b. According to Pearson, the four speculations are confirmed, and the ego orientations could be separated into two independent orientations Meanings of Significances & One-Tailed & Two- Tailed Meanings of Significances & One-Tailed & Two- Tailed