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

Overview Independent Variables Interval Nominal Dependent Variable 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

You already know how to deal with two nominal variables Overview You already know how to deal with two nominal variables Independent Variables Nominal Interval Considers how a change in a variable affects a discrete outcome Lambda Dependent Variable Interval Nominal Considers the difference between the mean of one group on a variable with another group 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 Overview You already know how to deal with two nominal variables TODAY! Independent Variables Nominal Interval Considers how a change in a variable affects a discrete outcome Lambda Dependent Variable Interval Nominal Considers the degree to which a change in one variable results in a change in another Confidence Intervals T-Test We will deal with this later in the course

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

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

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)?

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?

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.)

Scatter Diagram Example The data…

Scatter Diagram Example

A Scatter Diagram Example of a Negative Relationship

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.

Graph the data below and examine the relationship:

The Seniority-Salary Relationship

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…

Scatterplot of Prestige by Education

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.

The mean age for U.S. residents. Take your best guess? If you know nothing else about a person, except that he or she lives in United States and I asked you to his or her age, what would you guess? The mean age for U.S. residents. Now if I tell you that this person owns a skateboard, would you change your guess? (Of course!) With quantitative analyses we are generally trying to predict or take our best guess at value of the dependent variable. One way to assess the relationship between two variables is to consider the degree to which the extra information of the second variable makes your guess better. If someone owns a skateboard, that is likely to indicate to us that s/he is younger and we may be able to guess closer to the actual value.

Take your best guess? Similar to the example of age and the skateboard, we can take a much better guess at someone’s occupational prestige, if we have information about her/his years or level of education.

Equation for a Straight Line Y= a + bX where a = intercept b = slope Y = dependent variable X = independent variable X Y a rise run = b

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

SPSS Regression Output (GSS) Education & Prestige

SPSS Regression Output (GSS) Education & Prestige Now let’s interpret the SPSS output...

The Regression Equation Prediction Equation: Y = 6.120 + 2.762(X) This line represents the predicted values for Y for any and all values of X ˆ

The Regression Equation Prediction Equation: Y = 6.120 + 2.762(X) This line represents the predicted values for Y for any and all values of X ˆ

Interpreting the regression equation Y = 6.120 + 2.762(X) ˆ If a respondent had zero years of schooling, this model predicts that his occupational prestige score would be 6.120 points. For each additional year of education, our model predicts a 2.762 point increase in occupational prestige.

Ordinary Least Squares Least-squares line (best fitting line) – A line where the errors sum of squares, or e2, is at a minimum. Least-squares method – The technique that produces the least squares line.

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.

Covariance and Variance 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

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:

Back to the original scatterplot:

A Representative Line

Other Representative Lines

Calculating the Regression Equation

Calculating the Regression Equation

The Least Squares Line!

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.

Coefficient of Determination Coefficient of Determination (r2) – 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.

Coefficient of Determination

Coefficient of Determination

The Correlation Coefficient Pearson’s Correlation Coefficient (r) — The square root of r2. 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 +1.0. The sign () indicates direction. The closer the number is to 1.0 the stronger the association between X and Y.

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

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

The Correlation Coefficient 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. Y r = –1 X