Regression Analysis Chapter 10

Regression and Correlation Techniques that are used to establish whether there is a mathematical relationship between two or more variables, so that the behavior of one variable can be used to predict the behavior of others. Applicable to “Variables” data only. “Regression” provides a functional relationship (Y=f(x)) between the variables; the function represents the “average” relationship. “Correlation” tells us the direction and the strength of the relationship. The analysis starts with a Scatter Plot of Y vs X. The analysis starts with a Scatter Plot of Y vs X

Simple Linear Regression What is it? Determines if Y depends on X and provides a math equation for the relationship (continuous data) y Examples: Process conditions and product properties Sales and advertising budget x Does Y depend on X? Which line is correct?

Simple Linear Regression rise run m = slope = Y b = Y intercept = the Y value at point that the line intersects Y axis. rise run Simple linear regression will help you to understand the relationship between the process output (Y) and any factor that may affect it (X). Understanding this relationship will allow you to predict the Y, given a value of X. This is especially useful when the Y variable of interest is difficult or expensive to measure. For example, measuring the strength of a hardened steel part would require a destructive test. A model of the relationship between annealing temperature and strength would be useful in predicting the material's strength. b X A simple linear relationship can be described mathematically by Y = mX + b

Simple Linear Regression slope = rise run = (6 - 3) (10 - 4) 1 2 Y rise run 5 intercept = 1 X 5 10 Y = 0.5X + 1

Simple regression example An agent for a residential real estate company in a large city would like to predict the monthly rental cost for apartments based on the size of the apartment as defined by square footage. A sample of 25 apartments in a particular residential neighborhood was selected to gather the information

Size Rent 850 950 1450 1600 1085 1200 1232 1500 718 1485 1700 1136 1650 726 935 700 875 956 1150 1100 1400 1285 1985 2300 1369 1800 1175 1225 1245 1259 896 1361 1040 755 1000 800 1750 The data on size and rent for the 25 apartments will be analyzed in EXCEL.

Scatter plot Scatter plot suggests that there is a ‘linear’ relationship between Rent and Size

Interpreting EXCEL output Regression Equation Rent = 177.121+1.065*Size

Interpretation of the regression coefficient What does the coefficient of Size mean? For every additional square feet, Rent goes up by $1.065

Using regression for prediction Predict monthly rent when apartment size is 1000 square feet: Regression Equation: Rent = 177.121+1.065*Size Thus, when Size=1000 Rent=177.121+1.065*1000=$1242 (rounded)

Using regression for prediction – Caution! Regression equation is valid only over the range over which it was estimated! We should interpolate Do not use the equation in predicting Y when X values are not within the range of data used to develop the equation. Extrapolation can be risky Thus, we should not use the equation to predict rent for an apartment whose size is 500 square feet, since this value is not in the range of size values used to create the regression equation.

Why extrapolation is risky Extrapolated relationship In this figure, we fit our regression model using sample data – but the linear relation implicit in our regression model does not hold outside our sample! By extrapolating, we are making erroneous estimates!

Correlation (r) “Correlation coefficient”, r, is a measure of the strength and the direction of the relationship between two variables. Values of r range from +1 (very strong direct relationship), through “0” (no relationship), to –1 (very strong inverse relationship). It measures the degree of scatter of the points around the “Least Squares” regression line

Coefficient of correlation from EXCEL The sign of r is the same as that of the coefficient of X (Size) in the regression equation (in our case the sign is positive). Also, if you look at the scatter plot, you will note that the sign should be positive. R=0.85 suggests a fairly ‘strong’ correlation between size and rent.

Coefficient of determination (r2) “Coefficient of Determination”, r-squared, (sometimes R- squared), defines the amount of the variation in Y that is attributable to variation in X

Getting r2 from EXCEL It is important to remember that r-squared is always positive. It is the square of the coefficient of correlation r. In our case, r2=0.72 suggests that 72% of variation in Rent is explained by the variation in Size. The higher the value of r2, the better is the simple regression model.

Standard error (SE) Standard error measures the variability or scatter of the observed values around the regression line.

Getting the standard error (SE) from EXCEL In our example, the standard error associated with estimating rent is $194.60.

Is the simple regression model statistically valid? It is important to test whether the regression model developed from sample data is statistically valid. For simple regression, we can use 2 approaches to test whether the coefficient of X is equal to zero using t-test using ANOVA

Is the coefficient of X equal to zero? In both cases, the hypothesis we test is: What could we say about the linear relationship between X and Y if the slope were zero?

Using coefficient information for testing if slope=0 P-value 7.52E-08 =7.52*10-8 =0.0000000752 t-stat=7.740 and P-value=7.52E-08. P-value is very small. If it is smaller than our a level, then, we reject null; not otherwise. If a=0.05, we would reject null and conclude that slope is not zero. Same result holds at a=0.01 because the P-value is smaller than 0.01. Thus, at 0.05 (or 0.01) level, we conclude that the slope is NOT zero implying that our model is statistically valid.

Using ANOVA for testing if slope=0 in EXCEL F=59.91376 and P-value=7.51833E-08. P-value is again very small. If it is smaller than our a level, then, we reject null; not otherwise. Thus, at 0.05 (or 0.01) level, slope is NOT zero implying that our model is statistically valid. This is the same conclusion we reached using the t-test.

Confidence interval for the slope of Size The 95% CI tells us that for every 1 square feet increase in apartment Size, Rent will increase by $0.78 to $1.35.

Summary Simple regression is a statistical tool that attempts to fit a straight line relationship between X (independent variable) and Y (dependent variable) The scatter plot gives us a visual clue about the nature of the relationship between X and Y EXCEL, or other statistical software is used to ‘fit’ the model; a good model will be statistically valid, and will have a reasonably high R-squared value A good model is then used to make predictions; when making predictions, be sure to confine them within the domain of X’s used to fit the model (i.e. interpolate); we should avoid extrapolation