Linear Regression and Correlation

GOALS Understand and interpret the terms dependent and independent variable. Calculate and interpret the coefficient of correlation, the coefficient of determination, and the standard error of estimate. Conduct a test of hypothesis to determine whether the coefficient of correlation in the population is zero. Calculate the least squares regression line. Construct and interpret confidence and prediction intervals for the dependent variable.

Regression Analysis - Uses Some examples. Is there a relationship between the amount Healthtex spends per month on advertising and its sales in the month? Can we base an estimate of the cost to heat a home in January on the number of square feet in the home? Is there a relationship between the miles per gallon achieved by large pickup trucks and the size of the engine? Is there a relationship between the number of hours that students studied for an exam and the score earned?

Correlation Analysis and Scatter Diagram Correlation Analysis is the study of the relationship between variables. It is also defined as group of techniques to measure the association between two variables. A Scatter Diagram is a chart that portrays the relationship between the two variables. It is the usual first step in correlations analysis

Dependent vs. Independent Variable DEPENDENT VARIABLE The variable that is being predicted or estimated. It is scaled on the Y-axis. INDEPENDENT VARIABLE The variable that provides the basis for estimation. It is the predictor variable. It is scaled on the X-axis.

Regression Example The sales manager of Copier Sales of America, which has a large sales force throughout the United States and Canada, wants to determine whether there is a relationship between the number of sales calls made in a month and the number of copiers sold that month. The manager selects a random sample of 10 representatives and determines the number of sales calls each representative made last month and the number of copiers sold.

Scatter Diagram

The Coefficient of Correlation, r The Coefficient of Correlation (r) is a measure of the strength of the relationship between two variables. It requires interval or ratio-scaled data. It can range from -1.00 to 1.00. Values of -1.00 or 1.00 indicate perfect and strong correlation. Values close to 0.0 indicate weak correlation. Negative values indicate an inverse relationship and positive values indicate a direct relationship.

Perfect Correlation

Correlation Coefficient - Interpretation

Correlation Coefficient - Formula

Coefficient of Determination The coefficient of determination (r2) is the proportion of the total variation in the dependent variable (Y) that is explained or accounted for by the variation in the independent variable (X). It is the square of the coefficient of correlation. It ranges from 0 to 1. It does not give any information on the direction of the relationship between the variables.

Correlation Coefficient - Example Using the Copier Sales of America data which a scatter plot was developed earlier, compute the correlation coefficient and coefficient of determination.

Correlation Coefficient - Example

Correlation Coefficient - Example How do we interpret a correlation of 0.759? First, it is positive, so we see there is a direct relationship between the number of sales calls and the number of copiers sold. The value of 0.759 is fairly close to 1.00, so we conclude that the association is strong. However, does this mean that more sales calls cause more sales? No, we have not demonstrated cause and effect here, only that the two variables—sales calls and copiers sold—are related.

Coefficient of Determination (r2) - Example The coefficient of determination, r2 ,is 0.576, found by (0.759)2 This is a proportion or a percent; we can say that 57.6 percent of the variation in the number of copiers sold is explained, or accounted for, by the variation in the number of sales calls.

Testing the Significance of the Correlation Coefficient H0:  = 0 (the correlation in the population is 0) H1:  ≠ 0 (the correlation in the population is not 0) Reject H0 if: (in SPSS) P-value <.05

Correlation and Cause High correlation does not mean cause and effect For example, it can be shown that the consumption of Georgia peanuts and the consumption of aspirin have a strong correlation. However, this does not indicate that an increase in the consumption of peanuts caused the consumption of aspirin to increase. Likewise, the incomes of professors and the number of inmates in mental institutions have increased proportionately. Further, as the population of donkeys has decreased, there has been an increase in the number of doctoral degrees granted. Relationships such as these are called spurious correlations.

Practice Home sales.sav Correlation AnalyzeCorrelatebivariate Select Appraised Land Value Appraised value of improvements Total appraised value Sale price

Regression Analysis In regression analysis we use the independent variable (X) to estimate the dependent variable (Y). The relationship between the variables is linear. Both variables must be at least interval scale. The least squares criterion is used to determine the equation.

Linear Regression Model

Regression Analysis – Least Squares Principle The least squares principle is used to obtain a and b. The equations to determine a and b are:

Illustration of the Least Squares Regression Principle

Computing the Slope of the Line

Computing the Y-Intercept

Regression Equation - Example Recall the example involving Copier Sales of America. The sales manager gathered information on the number of sales calls made and the number of copiers sold for a random sample of 10 sales representatives. Use the least squares method to determine a linear equation to express the relationship between the two variables. What is the expected number of copiers sold by a representative who made 20 calls?

Finding the Regression Equation - Example

Computing the Estimates of Y Step 1 – Using the regression equation, substitute the value of each X to solve for the estimated sales

Plotting the Estimated and the Actual Y’s

The Standard Error of Estimate The standard error of estimate (sy.x) measures the scatter, or dispersion, of the observed values around the line of regression A formula that can be used to compute the standard error:

Standard Error of the Estimate - Example Recall the example involving Copier Sales of America. The sales manager determined the least squares regression equation is given below. Determine the standard error of estimate as a measure of how well the values fit the regression line.

Graphical Illustration of the Differences between Actual Y – Estimated Y

Assumptions Underlying Linear Regression For each value of X, there is a group of Y values, and these Y values are normally distributed. The means of these normal distributions of Y values all lie on the straight line of regression. The standard deviations of these normal distributions are equal. The Y values are statistically independent. This means that in the selection of a sample, the Y values chosen for a particular X value do not depend on the Y values for any other X values.

Practice Home sales.sav Regression AnalyzeRegressionlinear Select Independent variable Appraised Land Value Appraised value of improvements Total appraised value Dependent variable Sale price

Multiple Linear Regression and Correlation Analysis

GOALS Describe the relationship between several independent variables and a dependent variable using multiple regression analysis. Compute and interpret the multiple standard error of estimate, the coefficient of multiple determination, and the adjusted coefficient of multiple determination. Conduct a test of hypothesis on each of the regression coefficients. Use and understand qualitative independent variables.

Multiple Regression Analysis The general multiple regression with k independent variables is given by: The least squares criterion is used to develop this equation. Because determining b1, b2, etc. is very tedious, a software package such as Excel or MINITAB is recommended.

Multiple Regression Analysis For two independent variables, the general form of the multiple regression equation is: X1 and X2 are the independent variables. a is the Y-intercept b1 is the net change in Y for each unit change in X1 holding X2 constant. It is called a partial regression coefficient, a net regression coefficient, or just a regression coefficient.

Regression Plane for a 2-Independent Variable Linear Regression Equation

Multiple Linear Regression - Example Salsberry Realty sells homes along the east coast of the United States. One of the questions most frequently asked by prospective buyers is: If we purchase this home, how much can we expect to pay to heat it during the winter? The research department at Salsberry has been asked to develop some guidelines regarding heating costs for single-family homes.

Multiple Linear Regression - Example Three variables are thought to relate to the heating costs: (1) the mean daily outside temperature, (2) the number of inches of insulation in the attic, and (3) the age in years of the furnace (a device used for heating). To investigate, Salsberry’s research department selected a random sample of 20 recently sold homes. It determined the cost to heat each home last January, as well (data in next slide) Attic

Multiple Linear Regression - Example

The Multiple Regression Equation – Interpreting the Regression Coefficients The regression coefficient for mean outside temperature (X1) is 4.583. The coefficient is negative and shows an inverse relationship between heating cost and temperature. As the outside temperature increases, the cost to heat the home decreases. The numeric value of the regression coefficient provides more information. If we increase temperature by 1 degree and hold the other two independent variables constant, we can estimate a decrease of $4.583 in monthly heating cost.

The Multiple Regression Equation – Interpreting the Regression Coefficients The attic insulation variable (X2) also shows an inverse relationship: the more insulation in the attic, the less the cost to heat the home. So the negative sign for this coefficient is logical. For each additional inch of insulation, we expect the cost to heat the home to decline $14.83 per month, regardless of the outside temperature or the age of the furnace.

The Multiple Regression Equation – Interpreting the Regression Coefficients The age of the furnace variable (X3) shows a direct relationship. With an older furnace, the cost to heat the home increases. Specifically, for each additional year older the furnace is, we expect the cost to increase $6.10 per month.

Applying the Model for Estimation What is the estimated heating cost for a home if the mean outside temperature is 30 degrees, there are 5 inches of insulation in the attic, and the furnace is 10 years old?

Multiple Standard Error of Estimate The multiple standard error of estimate is a measure of the effectiveness of the regression equation. It is measured in the same units as the dependent variable. It is difficult to determine what is a large value and what is a small value of the standard error. The formula is:

Multiple Regression and Correlation Assumptions The independent variables and the dependent variable have a linear relationship. The dependent variable must be continuous and at least interval-scale. The residual must be the same for all values of Y. When this is the case, we say the difference exhibits homoscedasticity. The residuals should follow the normal distribution with mean 0. Successive values of the dependent variable must be uncorrelated.

The ANOVA Table The ANOVA table reports the variation in the dependent variable. The variation is divided into two components. The Explained Variation is that accounted for by the set of independent variable. The Unexplained or Random Variation is not accounted for by the independent variables.

Minitab – the ANOVA Table

Coefficient of Multiple Determination (r2) Characteristics of the coefficient of multiple determination: 1. It is symbolized by a capital R squared. In other words, it is written as because it behaves like the square of a correlation coefficient. 2. It can range from 0 to 1. A value near 0 indicates little association between the set of independent variables and the dependent variable. A value near 1 means a strong association. 3. It cannot assume negative values. Any number that is squared or raised to the second power cannot be negative. 4. It is easy to interpret. Because R2 is a value between 0 and 1 it is easy to interpret, compare, and understand.

Coefficient of Multiple Determination (r2) - Formula

Minitab – the ANOVA Table

Adjusted Coefficient of Determination The number of independent variables in a multiple regression equation makes the coefficient of determination larger. If the number of variables, k, and the sample size, n, are equal, the coefficient of determination is 1.0. To balance the effect that the number of independent variables has on the coefficient of multiple determination, statistical software packages use an adjusted coefficient of multiple determination.

Adjusted Coefficient of Determination - Example

Correlation Matrix A correlation matrix is used to show all possible simple correlation coefficients among the variables. The matrix is useful for locating correlated independent variables. It shows how strongly each independent variable is correlated with the dependent variable.

Global Test: Testing the Multiple Regression Model The global test is used to investigate whether any of the independent variables have significant coefficients. The hypotheses are:

Global Test continued The test statistic is the F distribution with k (number of independent variables) and n-(k+1) degrees of freedom, where n is the sample size. Decision Rule: Reject H0 if F > F,k,n-k-1 Or in SPSS when in ANOVA table; p-value<.05

Interpretation The computed F is 21.90, or When p-value<.05, so we can reject H0 The null hypothesis that all the multiple regression coefficients are zero is therefore rejected. Interpretation: some of the independent variables (amount of insulation, etc.) do have the ability to explain the variation in the dependent variable (heating cost).

Evaluating the Assumptions of Multiple Regression 1. There is a linear relationship. That is, there is a straight-line relationship between the dependent variable and the set of independent variables. 2. The variation in the residuals is the same for both large and small values of the estimated Y To put it another way, the residual is unrelated whether the estimated Y is large or small. 3. The residuals follow the normal probability distribution. 4. The independent variables should not be correlated. That is, we would like to select a set of independent variables that are not themselves correlated. 5. The residuals are independent. This means that successive observations of the dependent variable are not correlated. This assumption is often violated when time is involved with the sampled observations.

Analysis of Residuals A residual is the difference between the actual value of Y and the predicted value of Y. Residuals should be approximately normally distributed. Histograms and are useful in checking this requirement. A plot of the residuals and their corresponding Y’ values is used for showing that there are no trends or patterns in the residuals.

Multicollinearity Multicollinearity exists when independent variables (X’s) are correlated. Correlated independent variables make it difficult to make inferences about the individual regression coefficients (slopes) and their individual effects on the dependent variable (Y). However, correlated independent variables do not affect a multiple regression equation’s ability to predict the dependent variable (Y).

Effect of Multicollinearity in Not a Problem: Multicollinearity does not affect a multiple regression equation’s ability to predict the dependent variable A Problem: Multicollinearity may show unexpected results in evaluating the relationship between each independent variable and the dependent variable (a.k.a. partial correlation analysis),

Variance Inflation Factor A general rule is if the correlation between two independent variables is between -0.70 and 0.70 there likely is not a problem using both of the independent variables. A more precise test is to use the variance inflation factor (VIF). The value of VIF is found as follows: The term R2j refers to the coefficient of determination, where the selected independent variable is used as a dependent variable and the remaining independent variables are used as independent variables. A VIF greater than 10 is considered unsatisfactory, indicating that independent variable should be removed from the analysis.

Independence Assumption The fifth assumption about regression and correlation analysis is that successive residuals should be independent. When successive residuals are correlated we refer to this condition as autocorrelation. Autocorrelation frequently occurs when the data are collected over a period of time.

Residual Plot versus Fitted Values The graph shows the residuals plotted on the vertical axis and the fitted values on the horizontal axis. Note the run of residuals above the mean of the residuals, followed by a run below the mean. A scatter plot such as this would indicate possible autocorrelation.

Durbin–Watson statistic The Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation in the residuals from a regression analysis. It is named after James Durbin and Geoffrey Watson. If et is the residual associated with the observation at time t, then the test statistic is

Durbin–Watson statistic Since d is approximately equal to 2(1-r), where r is the sample autocorrelation of the residuals, d = 2 indicates that appears to be no autocorrelation, its value always lies between 0 and 4. If the Durbin–Watson statistic is substantially less than 2, there is evidence of positive serial correlation. As a rough rule of thumb, if Durbin–Watson is less than 1.0, there may be cause for alarm. Small values of d indicate successive error terms are, on average, close in value to one another, or positively correlated. If d > 2 successive error terms are, on average, much different in value to one another, i.e., negatively correlated. In regressions, this can imply an underestimation of the level of statistical significance.

Qualitative Independent Variables Frequently we wish to use nominal-scale variables—such as gender, whether the home has a swimming pool, or whether the sports team was the home or the visiting team—in our analysis. These are called qualitative variables. To use a qualitative variable in regression analysis, we use a scheme of dummy variables in which one of the two possible conditions is coded 0 and the other 1.

Qualitative Variable - Example Suppose in the Salsberry Realty example that the independent variable “garage” is added. For those homes without an attached garage, 0 is used; for homes with an attached garage, a 1 is used. We will refer to the “garage” variable as X4.The data shown on the table are entered into the MINITAB system.

Qualitative Variable - Minitab

Using the Model for Estimation What is the effect of the garage variable? Suppose we have two houses exactly alike next to each other in Buffalo, New York; one has an attached garage, and the other does not. Both homes have 3 inches of insulation, and the mean January temperature in Buffalo is 20 degrees. For the house without an attached garage, a 0 is substituted for in the regression equation. The estimated heating cost is $280.90, found by: For the house with an attached garage, a 1 is substituted for in the regression equation. The estimated heating cost is $358.30, found by: Without garage With garage

Testing the Model for Significance We have shown the difference between the two types (with/without garbage) of homes to be $77.40, but is the difference significant? We conduct the following test of hypothesis. H0: βi = 0 H1: βi ≠ 0 Reject H0 if p-value<.05 Conclusion: if the regression coefficient is not zero, the independent variable garage should be included in the analysis.

Practice Use employee data.sav Analyzeregressionlinearcrosstab Dependent variable Current salary Independent variablesbeginning salary; Months since hired; Minority classification (Qualitative data) In “statistics” box; choose “collinearity diagnostics” In “residuals”, choose “Durbin-Watson”

End of Chapter