Joint Distributions AND CORRELATION Coefficients (Part 3)

Textbook Credits Textbook Shavelson, R.J. (1996). Statistical reasoning for the behavioral sciences (3rd Ed.). Boston: Allyn & Bacon. Supplemental Material Ruiz-Primo, M.A., Mitchell, M., & Shavelson, R.J. (1996). Student guide for Shavelson statistical reasoning for the behavioral sciences (3rd Ed.). Boston: Allyn & Bacon.

Overview Joint Distributions Correlation Coefficients

Joint Distributions and Correlation Coefficients Correlational studies answer the question “What is the relationship of variable X and variable Y?” or “How are scores on one measure (X) associated with scores on another measure(Y)?” First, we want to summarize the scores, and Second, examine the relationship between the scores on the two measures First step: Arrange the scores to represent them in the form of a joint distribution (the representation of a pair of scores for each subject) Second step: Summarize the relationship represented by the JD with a single number we call correlation coefficient(a descriptive statistic that represents the magnitude of the relation, 0 to |1|, and the direction of the relation, + or -).

The Psychological Belief Scale and Student Achievement Research Example The Psychological Belief Scale and Student Achievement Intuition and prior experience suggest that it is easier to learn from teachers who have the same beliefs as the students Prediction(intuition): Students with similar beliefs as their instructors will earn the highest scores on exams Exam scores should decrease as the difference in the students’ and instructors’ beliefs increases Study: 3 introductory Psych. classes at 3 different colleges with 7 students each, with variable X representing a Belief Score and Y representing an Exam Score What method to use? General: Combine the data and for all 3 classes examine one overall average X with Y? More Specific: Examine X and Y in each class separately? Are the data consistent with predictions?

The Psychological Belief Scale and Student Achievement Research Example The Psychological Belief Scale and Student Achievement Test 2 types of belief approaches: Humanistic(H) & Behavioristic(B) Example: The central focus of the study of human behavior should be The specific principles that apply to unique individuals(H) The general principles that apply to all individuals(B) Instructors and students received the belief scale beginning of course Behavioristic orientation on the belief scale indicative by high scores Humanistic orientation on the belief scale indicative by low scores

Joint Distribution: Tabular Representation Behavioristic orientation on the belief scale indicative by high scores Humanistic orientation on the belief scale indicative by low scores Achievement(exam) score: students’ total scores earned in all class exams

Joint Distribution: Tabular Representation Divided into 3 classes with 3 columns each. Take Class 1 as example: Low belief scores are associated with moderately high exam scores(subjects 1 & 2) Moderate belief scores are associated with high exam scores (subjects 3, 4, & 7) High belief scores are associated with low exam scores (subjects 5 & 6)

Relationship of Student’s Belief & Exam Scores Lines represent relationship between belief scale scores & exam scores The magnitude of students’ scores differ from one class to the next as each instructor gave a different exam So all exam scores were converted to standard scores showing how far above (+) or below (-) the class average a particular exam falls

Scatterplot of Student’s Belief & Exam Scores A graphical representation of a JD showing pairs of each subject’s scores

Scatterplots for 3 classes & Instructor’s Belief Score Comparison of Scatterplots for each of the 3 classes in the study Curvilinear Relationship Linear Relationship Suspect Outlier

Correlation Coefficients: Linear Relationships

Properties of Linear Correlation Coefficients The coefficient can take values from -1.00 to + 1.00 - A correlations of -0.95 indicates a very strong negative relationship between X & Y - A correlation of +0.95 indicates a very strong positive relationship between X & Y - A correlation of 0 indicates that there is no linear relationship between X & Y The sign indicates the direction of the relationship between 2 variables A positive relationship means: - Low scores on X go with low scores on Y - High scores on X go with high scores on Y(As X scores , Y scores ) A negative relationship means: - Low scores on X go with high scores on Y - High scores on X go with low scores on Y(As X scores , Y scores )

Determining the Correlation Coefficient Magnitude Scatterplot characteristics are indicative of slope and data clustering: - Correlation is 0 if slope is horizontal & vertical slope is undefined - The clustering of data points determines the magnitude of correlation - Tight clustering means the magnitude of the correlation coefficient is high - Lose clustering means the magnitude of the correlation coefficient is low

SAT & GPA Relationships Scatterplot characteristics are indicative of slope and data clustering: - Correlation is 0 if slope is vertical or horizontal

SAT & GPA Relationships Developing Statistics - Student’s #1 deviation score on the SAT is: - Student’s #1 deviation score on the GPA is: - Student 1 earned scores below the mean for both SAT and GPA 𝒙=𝑿− 𝑿 =𝟒𝟓𝟎−𝟓𝟕𝟕=−𝟏𝟐𝟕 𝒚=𝒀− 𝒀 =𝟐.𝟒𝟎−𝟑.𝟏𝟎=−𝟎.𝟕𝟎

SAT & GPA Minitab Results Descriptive Statistics: SAT(X), GPA(Y) Total Variable Count Mean StDev Variance Sum SAT(X) 5 577.0 126.1 15895.0 2885.0 GPA(Y) 5 3.100 0.477 0.228 15.500

Covariance of SAT & GPA Scores Measuring how two sets of deviation go together or covary - Student’s #1 covariance (cross product)is: - Note: When |x| and |y| are large  xy is large (students 1 & 5) - Note: When |x| and |y| are small  xy is small (students 2, 3, &4) - Covariance: - Pearson product-moment correlation coefficient measures the strength with X and Y - Correlation coefficient: 𝑪𝒐𝒗𝒙𝒚= 𝒙𝒚 𝑵−𝟏 = 𝟐𝟏𝟑.𝟔𝟓 𝟒 =𝟓𝟑.𝟒𝟏 𝒄𝒐𝒓𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏 𝑿, 𝒀 =𝒓𝒙𝒚= 𝑪𝒐𝒗𝒙𝒚 𝒔𝒙𝒔𝒚 = 𝟓𝟑.𝟒𝟏 𝟏𝟐𝟔.𝟎𝟖 𝟎.𝟒𝟖 =𝟎.𝟖𝟗 𝒙𝒚= −𝟏𝟐𝟕 −𝟎.𝟕𝟎 =𝟖𝟖.𝟗𝟎

Covariance of SAT & GPA Scores Measuring how two sets of deviation go together or covary - Student’s #1 covariance (cross product)is: - Note: When |x| and |y| are large  xy is large (students 1 & 5) - Note: When |x| and |y| are small  xy is small (students 2, 3, &4) - Covariance: - Pearson product-moment correlation coefficient measures the strength with X and Y - Correlation coefficient: 𝒙𝒚= −𝟏𝟐𝟕 −𝟎.𝟕𝟎 =𝟖𝟖.𝟗𝟎 𝑪𝒐𝒗𝒙𝒚= 𝒙𝒚 𝑵−𝟏 = 𝟐𝟏𝟑.𝟔𝟓 𝟒 =𝟓𝟑.𝟒𝟏 𝒄𝒐𝒓𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏 𝑿, 𝒀 =𝒓𝒙𝒚= 𝑪𝒐𝒗𝒙𝒚 𝒔𝒙𝒔𝒚 = 𝟓𝟑.𝟒𝟏 𝟏𝟐𝟔.𝟎𝟖 𝟎.𝟒𝟖 =𝟎.𝟖𝟗 Minitab Results Covariances: SAT(X), GPA(Y) SAT(X) GPA(Y) SAT(X) 15895.000 GPA(Y) 53.413 0.228 Correlations: SAT(X), GPA(Y) Pearson correlation of SAT(X) and GPA(Y) = 0.888

Correlation Between SAT & GPA Scores Looking at the scatterplot to validate the correlation findings - A linear relationship with a positive slope indicates a positive correlation - The absolute magnitude 0.89 provides an index of the relationship strength(-1to +1) - Points cluster closely about an imaginary line validating the relationship magnitude

Minitab Output: SAT & GPA Scores Looking at the scatterplot to validate the correlation findings - A linear relationship with a positive slope indicates a positive correlation - The absolute magnitude 0.89 provides an index of the relationship strength(-1to +1) - Points cluster closely about an imaginary line validating the relationship magnitude

Excel Output: SAT & GPA Scores

Excel Output SAT & GPA Scores

The Squared Correlation Coefficient The squared correlation coefficient is the coefficient of determination - It is the amount of variability that can be explained between X & Y Recall: The larger |rxy| is, the stronger the relationship between X & Y We previously found that: So Now we want to convert to percentage of variance - Tells us the percentage that X shares with Y in terms of variability to one another - The % of variance in Y and X that can be explained is: 𝒓𝒙𝒚= 𝑪𝒐𝒗𝒙𝒚 𝒔𝒙𝒔𝒚 = 𝟓𝟑.𝟒𝟏 𝟏𝟐𝟔.𝟎𝟖 𝟎.𝟒𝟖 =𝟎.𝟖𝟗 𝒓 𝟐 𝒙𝒚= 𝟎.𝟖𝟗 𝟐 =𝟎.𝟕𝟗𝟐𝟏 𝒓 𝟐 𝒙𝒚 ×𝟏𝟎𝟎= 𝟎.𝟕𝟗𝟐𝟏 ×𝟏𝟎𝟎=𝟕𝟗.𝟐𝟏%

Percentage of Variance Pictorial representation of the % of variance in exam scores accounted for by the variability in belief scores (computed from class 3 data) Variability in X Variability in Y

Spearman Rank Correlation Coefficient Non-linear (curvilinear) monotonic increasing or decreasing functions Monotonically decreasing f Monotonically increasing f

Spearman Rank Correlation Coefficient Example: Y is a monotonically increasing function of X

Spearman Rank Correlation Coefficient Rank ordering the data for both X & Y and graph - The converted ordered graph is now linear - We can now compute the Pearson correlation coefficient for ranks between X & Y

Spearman Rank Correlation Coefficient

Sources of Misleading Correlation Coefficients Too much confidence can lead to misleading interpretations - Restriction of the range of values on one of the variables may reduce the magnitude of the correlation coefficient

Sources of Misleading Correlation Coefficients Too much confidence can lead to misleading interpretations - Use of extreme groups may inflate the correlation coefficient

Sources of Misleading Correlation Coefficients Too much confidence can lead to misleading interpretations - Combining groups with different means on one or both variables may have an unpredictable effect on the correlation coefficient

Sources of Misleading Correlation Coefficients Too much confidence can lead to misleading interpretations - Extreme scores (Outliers) may have a marked effect on the correlation coefficient, especially if the sample size is small

Sources of Misleading Correlation Coefficients Too much confidence can lead to misleading interpretations - A curvilinear relationship between X and Y may account for a near-zero correlation coefficient No systematic relationship Curvilinearly related: Use the eta (h) ratio coefficient measurement instead of the Pearson correlation coefficient

Correlation and Causality Correlation does not imply causality Many possible interpretations of a correlation coefficient: Most common problem inferring causality from correlation: Selectivity! X: beliefs Y: Achievement Z: Knowledge gained from related courses

Practice Exercises Part 3 Practice Exercises Select a hypothetical product or a process and create some test data of your choice (plausible, no more than 10) as shown in textbook/class Show your type of experimental approach Create a detailed table of frequency distributions Display your data with different types of graphs Calculate the measures of central tendency and variability Calculate the Z-score(s) and indicate the relative position in the normal distribution. Provide any other pertinent information as a result Part 3 Practice Exercises Represent your joint distribution data in a tabular form Create a scatterplot of your data Create a covariance table (as table 6-4) and calculate the covariance Calculate the correlation of the two variables Calculate the R squared value and explain your findings as a result

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