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Chapter 15 (Ch. 13 in 2nd Can.) Association Between Variables Measured at the Interval-Ratio Level: Bivariate Correlation and Regression

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**Introduction: Scattergrams / Scatterplots**

Graphs that display relationships between two interval-ratio variables. The Regression Line, Slope, and Intercept. The regression line, y=a+bX, summarizes the linear relationship between X and Y. Predicts the score of Y from a score of X. b represents the slope of the line. a, called the intercept, is the point on the Y-axis where the regression line crosses it. Pearson’s r and the Coefficient of Determination (r2) r is a measure of association for two I-R variables. r2 tells you how much variation in the dependent variable is explained by the independent variable.

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**Scattergram / Scatterplot**

Has two dimensions: The X (independent) variable is arrayed along the horizontal axis. The Y (dependent) variable is arrayed along the vertical axis. Each dot on a scattergram is a case in the data set. The dot is placed at the intersection of the case’s scores on X and Y.

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**Example of a Hypothetical Scattergram Showing the Relationship Between X and Y**

Shows the relationship between % College Educated (X) and Voter Turnout (Y) on election day for the 50 cities. Turnout

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**Scattergram Example (cont.)**

Horizontal X axis - % of population of a city with a college education. Scores range from 15.3% to 34.6% and increase from left to right. Turnout

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**Scattergram Example (cont.)**

Vertical (Y) axis is voter turnout. Scores range from 44.1% to 70.4% and increase from bottom to top Turnout

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**The Regression Line on a Scattergram**

A single straight line that comes as close as possible to all data points. “least squares regression line” Indicates strength and direction of the relationship. Turnout

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**Strength of Regression Line**

The greater the extent to which dots are clustered around the regression line, the stronger the relationship. This relationship is weak to moderate in strength. Turnout

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**Direction of Regression Line**

Positive: regression line rises left to right. Negative: regression line falls left to right. This a positive relationship: As % college educated increases, % turnout increases. Turnout

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**Scattergrams and Linearity**

Inspection of the scattergram should always be the first step in assessing the correlation between two interval-ratio variables. In addition to assessing the strength and direction, the relationship must also be linear.

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**The Regression Line: Formula**

This formula defines the regression line: y = a + bx Where: Y = score on the dependent variable a = the Y intercept or the point where the regression line crosses the Y axis. b = the slope of the regression line or the amount of change produced in Y by a unit change in X X = score on the independent variable

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**Regression and Prediction**

We can use the regression line to find the predicted value of y (symbolized as y’) for values of x. Once we know the values of the coefficients b and a, we can use the following prediction formula by substituting any value for x to predict y. The predicted level of y can be calculated by: We can also use the regression formula to accurately plot the regression line on our scattergram.

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Regression Analysis: Healey’s definitional formula for calculating the slope of the line (Formula 15.2 or 13.2 in 2nd Can.) Note: The numerator is the covariation of x and y (how x and y vary together). The denominator is the sum of the squared deviations around the mean of x

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**Regression Analysis:. computational formula. for b (Formula 13**

Regression Analysis: *computational formula* for b (Formula 13.3 in 2nd) Below is the computational (working) formula to calculate b. It is a re-arrangement of the theoretical formula and is much easier to calculate! The slope tells you what the change in Y is, for every unit of X. The sign of the slope coefficient (+/- b) tells you whether the relationship is positive or negative.

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Regression Analysis The Y intercept (a) is computed from Healey, Formula 15.3 (or 13.4 in 2nd): The intercept (a) is the point where the regression line crosses the Y-axis, when X=0.

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**Results of a Hypothetical Regression Analysis of the Relationship Shown in the Scattergrams Above:**

For the relationship between % college educated and % turnout: Assume b (slope) = .42 Assume a (Y intercept)= 50.03 A slope of .42 means that % turnout increases by .42 (less than half a percent) for every unit increase of 1 in % college educated. The Y intercept means that the regression line crosses the Y axis at Y =

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**An example of prediction:**

We can use the regression equation y’=a+bx for prediction. For instance, we could ask, what % turnout would be expected in a city where only 10% of the population was college educated? What % turnout would be expected in a city where 70% of the population was college educated? This is a positive relationship so the value for Y increases as X increases. Our prediction: For X =10, Y = 54.5 For X =70, Y = 79.7

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**Calculating the Correlation Coefficient: Formula for Pearson’s r**

Definitional formula for Pearson’s r: *Use the computational formula to calculate*:

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**Pearson’s r Like Gamma, r varies from -1.00 to +1.00**

Pearson’s r is a measure of association for Interval-Ratio variables. For the hypothetical relationship between % college educated and turnout, assume r =.32 This relationship would be positive and weak to moderate. As level of education increases, % turnout increases.

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**The Coefficient of Determination: r2**

Total variation in y ( ) is the sum of the explained variation ( ) and the unexplained variation ( ) The explained variation (the portion explained by x) is represented by the formula: Or, alternatively: r2 = (r)2

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**Practical Example using Healey Problem 15.1 (Problem 13.1 in 2nd Can.)**

The computation and interpretation of a, b, Pearson’s r and r2 will be illustrated using a similar example from Healey Problem 15.1 ( % Turnout by Education (Years of Schooling) but with only 5 cases) The variables are: Voter turnout (Y) is the dependent variable. Average years of school (X) is the independent variable. The sample is 5 cities. This is only to simplify the calculation. A sample of 5 is actually very small.

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Data from Problem 15.1: The scores on each variable are displayed in table format: Y = % Turnout X = Years of Education City X Y A 11.9 55 B 12.1 60 C 12.7 65 D 12.8 68 E 13.0 70

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**1. Draw and Interpret the Scattergram:**

The relationship between X and Y is linear. Estimate regression line. Relationship is positive and strong.

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**2. Make a Computational Table:**

X Y X2 Y2 XY 11.9 55 141.61 3025 654.5 12.1 60 146.41 3600 726 12.7 65 161.29 4225 825.5 12.8 68 163.84 4624 870.4 13.0 70 169 4900 910 ∑X = 62.5 ∑Y = 318 ∑X2 =782.15 ∑Y2 = 20374 ∑XY = Sums (Σ) are needed to compute b, a, and Pearson’s r. As well, the mean of X and Y are needed:

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3. Next, calculate b and a…. Calculate slope: Calculate y-intercept:

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Interpret Slope (b), the Intercept (a) For every unit increase in X, Y increases by This means that for 1 additional year of schooling, voter turnout goes up by 12.67%. This is the point at which the regression line crosses the Y-axis (when X is equal to 0, Y is equal to )

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**Find the Regression Line. :**

Find the Regression Line*: *Note: you can now substitute two values for X and solve for Y to find points to plot the actual regression line on your scattergram. For prediction: Suppose years of schooling = 10 years… Then, Y = (10) = We would predict that when average years of education is 10 years, the voter turnout would be 31.92%

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4. Pearson’s r Calculate the correlation coefficient r

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**Interpret Pearson’s r An r of 0**

Interpret Pearson’s r An r of 0.98 indicates an extremely strong relationship between years of education and voter turnout for these five cities (use the table given in Ch. 14 to estimate strength)

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5. Find the Coefficient of Determination (r2) and Interpret: The coefficient of determination is r2 = Education, by itself, explains 96.8% of the variation in voter turnout.

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**6. Testing r for significance:**

We can test the relationship between % turnout and years of education (represented by Pearson’s r) for significance using the 5 step model and the following formula: Degrees of Freedom = N-2

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**Step 1: Assumptions There are 3 main assumptions…**

1. The dependent and independent are normally distributed. We can test this by looking at the histograms for the two variables. 2. The relationship between X and Y is linear. We can check this by looking at the scattergram. 3. The relationship is homoscedastic. We can test homoscedasticity by looking at the scattergram and observing that the data points form a “roughly symmetrical, cigar-shaped pattern” about the regression line. If the above 3 assumptions have been met, then we can use linear regression and correlation and test r for significance.

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**Step 2: Null and Alternate Hypotheses:**

Ho: ρ = 0.0 H1: ρ ≠ 0.0 (Note that ρ (rho) is the population parameter, while r is the sample statistic.) Step 3: Sampling Distribution and Critical Region: S.D. = t-distribution Alpha = .05 DF = n - 2 = = 3 tcritical = 3.182

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**Step 4. Computing the Test Statistic:**

Use Formula 15.6 in Healey (13.6 in 2nd Can.) Step 5. Decision and Interpretation: Tobtained = 9.53 > tcritical = 3.182 Reject Ho. The relationship between % turnout and years of schooling is significant.

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**Always include a brief summary of your results:**

There is a very strong, positive relationship between % voter turnout and years of schooling for the five cities. As years of schooling increase, the % of voter turnout goes up. The relationship is significant (t=9.53, df=3, α = .05) . Years of schooling explain 96.8% of the variation in % voter turnout.

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Practice Problem Working with a partner, calculate, interpret and summarize the results for Healey 1st Can. #15.1 (2nd Can. 13.1) for “% Turnout” and “Unemployment” and for “% Turnout” and “Negative Campaigning”.

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