Chapter 8 – 1 Chapter 8: Bivariate Regression and Correlation Overview The Scatter Diagram Two Examples: Education & Prestige Correlation Coefficient Bivariate Linear Regression Line SPSS Output Interpretation Covariance

Chapter 8 – 2 Overview Interval Nominal Dependent Variable Independent Variables Nominal Interval Considers the distribution of one variable across the categories of another variable Considers the difference between the mean of one group on a variable with another group Considers how a change in a variable affects a discrete outcome Considers the degree to which a change in one variable results in a change in another

Chapter 8 – 3 Overview Interval Nominal Dependent Variable Independent Variables Nominal Interval Considers the difference between the mean of one group on a variable with another group Considers how a change in a variable affects a discrete outcome Considers the degree to which a change in one variable results in a change in another You already know how to deal with two nominal variables Lambda

Chapter 8 – 4 Overview Interval Nominal Dependent Variable Independent Variables Nominal Interval Considers how a change in a variable affects a discrete outcome Considers the degree to which a change in one variable results in a change in another You already know how to deal with two nominal variables Lambda Chi-square Confidence Intervals T-Test We will deal with this later in the course TODAY!

Chapter 8 – 5 Overview Interval Nominal Dependent Variable Independent Variables Nominal Interval Considers how a change in a variable affects a discrete outcome Regression Correlation You already know how to deal with two nominal variables Lambda Chi-square Confidence Intervals T-Test We will deal with this later in the course TODAY! What about this cell?

Chapter 8 – 6 Overview Interval Nominal Dependent Variable Independent Variables Nominal Interval Logistic Regression Regression Correlation You already know how to deal with two nominal variables Lambda Confidence Intervals T-Test We will deal with this later in the course TODAY! This cell is not covered in this course

Chapter 8 – 7 General Examples Does a change in one variable significantly affect another variable? Do two scores tend to co-vary positively (high on one score high on the other, low on one, low on the other)? Do two scores tend to co-vary negatively (high on one score low on the other; low on one, hi on the other)?

Chapter 8 – 8 Specific Examples Does getting older significantly influence a person’s political views? Does marital satisfaction increase with length of marriage? How does an additional year of education affect one’s earnings?

Chapter 8 – 9 Scatter Diagrams Scatter Diagram (scatterplot)—a visual method used to display a relationship between two interval-ratio variables. Typically, the independent variable is placed on the X-axis (horizontal axis), while the dependent variable is placed on the Y-axis (vertical axis.)

Chapter 8 – 10 Scatter Diagram Example % Willing to Pay Higher PricesGNP/Capital United States Ireland Netherlands Norway Sweden Russia New Zealand Canada Japan Spain Latvia Portugal Chile Switzerland Finland Mexico

Chapter 8 – 11 Scatter Diagram Example

Chapter 8 – 12 Scatter Diagram Example

Chapter 8 – 13 Linear Relationships Linear relationship – A relationship between two interval-ratio variables in which the observations displayed in a scatter diagram can be approximated with a straight line. Deterministic (perfect) linear relationship – A relationship between two interval-ratio variables in which all the observations (the dots) fall along a straight line. The line provides a predicted value of Y (the vertical axis) for any value of X (the horizontal axis.

Chapter 8 – 14 Graph the data below and examine the relationship:

Chapter 8 – 15 The Seniority-Salary Relationship

Chapter 8 – 16 Example

Chapter 8 – 17

Chapter 8 – 18 xSocial Alienation yReligiosity

Chapter 8 – 19 Example: Education & Prestige Does education predict occupational prestige? If so, then the higher the respondent’s level of education, as measured by number of years of schooling, the greater the prestige of the respondent’s occupation. Take a careful look at the scatter diagram on the next slide and see if you think that there exists a relationship between these two variables…

Chapter 8 – 20 Scatterplot of Prestige by Education

Chapter 8 – 21 Example: Education & Prestige The scatter diagram data can be represented by a straight line, therefore there does exist a relationship between these two variables. In addition, since occupational prestige becomes higher, as years of education increases, we can say also that the relationship is a positive one.

Chapter 8 – 22 Linear Regression Technique that finds a line that “ fit ” the scatter of data points in such as ways as to provide for any given value of x the best estimate of the corresponding value of y.

Chapter 8 – 23 Equation for a Straight Line Y= a + bX wherea = intercept b = slope Y = dependent variable X = independent variable X Y a rise run rise run = b

Chapter 8 – 24 Bivariate Linear Regression Equation Y = a + bX Y-intercept (a)—The point where the regression line crosses the Y-axis, or the value of Y when X=0. Slope (b)—The change in variable Y (the dependent variable) with a unit change in X (the independent variable.) The estimates of a and b will have the property that the sum of the squared differences between the observed and predicted (Y-Y) 2 is minimized using ordinary least squares (OLS). Thus the regression line represents the Best Linear and Unbiased Estimators (BLUE) of the intercept and slope. ˆ ^

Chapter 8 – 25 Back to the original scatterplot:

Chapter 8 – 26 A Representative Line

Chapter 8 – 27

Chapter 8 – 28 Ordinary Least Squares Least-squares line (best fitting line) – A line where the errors sum of squares, or e 2, is at a minimum. Least-squares method – The technique that produces the least squares line.

Chapter 8 – 29 Least Squares

Chapter 8 – 30

Chapter 8 – 31 Linear Regression

Chapter 8 – 32 Estimating the slope: b The bivariate regression coefficient or the slope of the regression line can be obtained from the observed X and Y scores.

Chapter 8 – 33 Estimating the Intercept The regression line always goes through the point corresponding to the mean of both X and Y, by definition. So we utilize this information to solve for a:

Chapter 8 – 34 The Least Squares Line!

Chapter 8 – 35 Example xSocial Alienation yReligiosity

Chapter 8 – 36 Example

Chapter 8 – 37 If a respondent had zero value of social alienation, this model predicts that his Religiosity scale score would be 14 points. For each additional unit of social alienation, our model predicts a point increase in religiosity scale. Y = (X) ˆ Interpreting the regression equation

Chapter 8 – 38 The Regression Equation Prediction Equation: Y = (X) This line represents the predicted values for Y for any and all values of X ˆ

Chapter 8 – 39 Prediction Equation: Y = (X) This line represents the predicted values for Y for any and all values of X ˆ The Regression Equation

Chapter 8 – 40 Summary: Properties of the Regression Line Represents the predicted values for Y for any and all values of X. Always goes through the point corresponding to the mean of both X and Y. It is the best fitting line in that it minimizes the sum of the squared deviations. Has a slope that can be positive or negative; null hypothesis is that the slope is zero.

Chapter 8 – 41 Covariance = Variance of X= Covariance of X and Y—a measure of how X and Y vary together. Covariance will be close to zero when X and Y are unrelated. It will be greater than zero when the relationship is positive and less than zero when the relationship is negative. Variance of X—we have talked a lot about variance in the dependent variable. This is simply the variance for the independent variable Covariance and Variance

Chapter 8 – 42 Computational Formula

Chapter 8 – 43 Coefficient of Determination Coefficient of Determination (r 2 ) – A PRE measure reflecting the proportional reduction of error that results from using the linear regression model. It reflects the proportion of the total variation in the dependent variable, Y, explained by the independent variable, X.

Chapter 8 – 44 Coefficient of Determination

Chapter 8 – 45 Coefficient of Determination

Chapter 8 – 46 Pearson’s Correlation Coefficient (r) — The square root of r 2. It is a measure of association between two interval- ratio variables. Symmetrical measure—No specification of independent or dependent variables. Ranges from –1.0 to The sign (  ) indicates direction. The closer the number is to  1.0 the stronger the association between X and Y. The Correlation Coefficient

Chapter 8 – 47 r = 0 means that there is no association between the two variables. The Correlation Coefficient Y X r = 0

Chapter 8 – 48 The Correlation Coefficient Y X r = +1 r = 0 means that there is no association between the two variables. r = +1 means a perfect positive correlation.

Chapter 8 – 49 The Correlation Coefficient Y X r = –1 r = 0 means that there is no association between the two variables. r = +1 means a perfect positive correlation. r = –1 means a perfect negative correlation.

Chapter 8 – 50 SPSS Output: % Willing to Pay Higher Price & GNP/Capital

Chapter 8 – 51 Now let’s interpret the SPSS output... SPSS Output: % Willing to Pay Higher Price & GNP/Capital

Chapter 8 – 52 Exercise A random sample of 10 elementary school students is selected, and each student is measured on a creativity score (x) using a well-defined testing instrument and on a task score (y) using a new instrument. The task score is the mean time taken to perform several hand-eye coordination tasks.

Chapter 8 – 53 Data STUDENTCREATIVITY(X)TASKS(Y) AE284.5 FR353.9 HT373.9 IO506.1 DP694.3 YR848.8 RR517.1 TG457.3 EF313.3 TJ405.2

Chapter 8 – 54 Example m262/regress/regress.html