Chapter 3 Lecture Notes Correlation Patterns and Correlation Coefficient PDF

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Lecture notes from Professor Brent Costleigh's class....

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Correlations ○ Can be thought of as a descriptive stat for the relationship bt two variables ○ Describes relationship between two equal-interval numeric variables ■ The correlation bt amount of time studying and amount learned ■ The correlation bt number of years of education and salary Scatter diagram - graph showing the pattern of the relationship bt two variables ○ 1. Draw the axes ■ The values of one variable go along horizontal axis and the values of the other variable go along vertical axis ○ 2. Determine the range of values to use for each variable and mark them on the axes ■ Numbers should go from low to high on each starting from where the axes meet ■ Usually your low value on each axis is 0 ■ Each axis should continue to the highest value your measure can possibly have ○ 3. Make a dot for each pair of scores Patterns of correlation ○ Linear correlation: relationship between two variables that shows up on a scatter diagram as dots roughly approximating a straight line ○ Curvilinear correlation: any association bt two variables other than a linear correlation; relationship bt two variables that shows up on a scatter diagram as dots following a systematic pattern that is not a straight line ○ No correlation: no systematic relationship between two variables Positive and negative linear correlation ○ Positive ■ High scores go with high scores ■ Low scores go with low scores ■ Medium go with medium ■ When graphed, the line goes up and to the right ● Level of education achieved and income ○ Negative ■ High scores go with low scores ● The relationship between fewer hours of sleep and higher levels of stress ○ Strength of correlation ■ How close the dots on a scatter diagram fall to a simple straight line Importance of identifying pattern of correlation ○ Use a scatter diagram to examine the pattern, direction, and strength of a correlation ■ First, determine whether it is a linear or curvilinear relationship ■ If linear, look to see if it is positive or negative ■ Then look to see if the correlation is large, small, or moderate ○ Approximating the direction and strength of a correlation allows you to double check your calculations later The Correlation coefficient ○ A number that gives the exact correlation between two variables ○ Identifies both the direction and strength

Z scores used to compare scores on different variables ■ Z scores allow you to calculate a cross-product that tells you the direction of the correlation ■ A cross-product is the result of multiplying a score on one variable by a score on the other variable ● If you multiply a high (positive) z score by a high (positive) z score, you will always get a positive cross-product ● If you multiply a low (negative) z score by a low (negative) z score, you will always get a positive cross-product ● If you multiply a high (positive) z score with a low (negative) z score or a low (negative) z score with a high (positive) z score, you will always get a negative number Formula for a correlation coefficient ® ○ r=(sum)ZxZy N ● Zx = z score for each person on the x variable ● Zy = z score for each person on the Y variable ● ZxZy = cross product of Zx and Zy ● (sum)ZxZy = sum of the cross-products of the z scores over all participants in the study Correlation coefficient ○ The sign of r(Pearson correlation coefficient) tells the general trend of a relationship between two variables ■ A + sign means the correlation is positive ■ A - sign means the correlation is negative ○ The value of r ranges from 0 to 1 ■ 1 is the highest value a correlation can have ● A correlation of 1 or -1 means that the variables are perfectly correlated ● 0 = no correlation The value of a correlation defines the strength of the correlation regardless of the sign ■ -.99 is stronger than 0.75 ■ Medium correlation is around 0.35 or -0.35 ■ -0.15 or 0.15 considered weak correlation ■ Around -0.5 or 0.5 is considered strong correlation Steps for figuring correlation coefficient ○ 1. Change all scores to z scores ■ Figure the mean and the standard deviation of each variable ■ Change each raw score to a z score ○ 2. Calculate the cross-product of the z scores for each person ■ Multiply each person’s z score on one variable by his or her z score on the other variable ○ 3. Add up the cross-products of the z scores ○ 4. Divide by the number of people in the study (number of cross products NOT number of scores) ○

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Issues in interpreting the correlation coefficient ○ Direction of causality ■ Path of causal effect (x causes y) ○ You cannot determine the direction of causality just because two variables are correlated Three possible directions of causality ○ Variable x causes variable y ■ Less sleep causes more stress ○ Variably y causes variable x ■ More stress causes people to sleep less ○ There is a third variable that causes both variable x and variable y ■ Working longer hours causes both stress and fewer hours of sleep Ruling out some possible directions of causality ○ Longitudinal study ■ A study where people are measured at two or more points in time ● Evaluating number of hours of sleep at one time point and then evaluating their levels of stress at a later time point ○ True experiment ■ A study in which participants are randomly assigned to a particular level of a variable and then measured on another variable ● Exposing individuals to varying amounts of sleep in a laboratory environment and then evaluating their stress levels The statistical significance of a correlation coefficient ○ A correlation is statistically significant if it is unlikely that you could have gotten a correlation as big as you did if in fact there was no relationship between variables ■ If the probability (p) is less than some small degree of probability (5% or 1%) the correlation is considered statistically significant Prediction (also known as regression) ○ Predictor variable (x) ■ Variable being predicted from ● Level of education achieved ○ Criterion variable (y) ■ Variable being predicted to ● Income ○ If we expect level of education to predict income, the predictor variable would be level of education and the criterion variable would be income Prediction using z scores ○ Prediction model ■ A person’s predicted z score on the criterion variable is found by multiplying the standardized regression coefficient (beta) by that person’s z score on the predictor variable ○ Formula for the prediction model using z scores: predicted Zy=(beta)(Zx) ■ Predicted Zy=predicted value of the particular person’s Z score on the criterion variable Y

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■ Zx=particular person’s Z score in the predictor variable X Steps for finding prediction using Z scores ○ 1. Determine the standardized regression coefficient (beta) ○ 2. Multiply the standardized regression coefficient (beta) by the person’s z score on the predictor variable Prediction using raw scores ○ 1. Change a person’s raw score on the predictor variable to a z score (Zx) ○ 2. Multiply the standardized regression coefficient (beta) by the person’s z score on the predictor variable. ■ Multiply beta by Zx ● This gives the predicted z score on the criterion variable (Zx) ● Predicted Zy=(beta)(Zx) ○ 3. Change the person’s predicted z score on the criterion variable to a raw score ■ Predicted y=(SDy)(predicted Zy)+My The correlation coefficient and the proportion of variance accounted for ○ Proportion of variance accounted for (r2 ) - a measure of association between variables ■ To compare correlations with each other, you have to square each correlation ■ This number represents the proportion of the total variance in one variable that can be explained by the other variable. Correlation and prediction in research articles ○ Scatter diagrams are sometimes found ○ Correlation coefficients (r) are found often in research articles ○ Correlation matrix = a table which displays the correlation of each pair of variables; each variable is listed down the side and across the top of the table Multiple correlation: the association bt a criterion variable and two or more predictor variables Multiple regression: predicting scores on a criterion variable from two or more predictor variables Multiple correlation coefficient (R) ○ Overall correlation bt the criterion variable and all of the predictor variables ■ R is usually smaller than the sum of each individual r ■ R2=the proportion of variance in the criterion variable accounted for by all of the predictor variables...