Simple Linear Regression

Transcription

1 1 Excel Manual Simple Linear Regression Chapter 13 This chapter discusses statistics involving the linear regression. Excel has numerous features that work well for comparing quantitative variables both graphically and numerically. Here, we ll utilize standard functions, array functions, and scatter diagram trendlines, as well as the Data Analysis ToolPak, to measure the linear association between two variables. These methods will be used to perform hypothesis tests and create confidence intervals on the slope and y- intercept coefficients. Straight Line Regression Model There is no utility in Excel to plot the equation of a line, but you can create the plot by finding two points on a line and adding a trendline for those two points. To do this, create an area in a spreadsheet that has an area to enter the y intercept and slope. We ve entered the labels shown using the Symbols group in the Insert tab. The subscript was made by selecting the digit and lowering the font size via the Font group of the Home tab. Add an area to enter the minimum and maximum x values, with an area next to those for the resulting y values. In C7, enter the formula =$A$4 + $B$4*B7. Autofill this formula down to C8. The resulting sheet is shown below: To create the plot of the line for these parameters, select the cells B6:C8. Select the Scatter chart from the Charts group in the Insert tab. Since there are only two points, Excel graphs this as two different series or two separate lines. To fix this click Switch Rows/Columns in the Data group of the Design tab, as shown in the illustration:

2 2 Excel Manual Select Layout 3 in the Chart Layouts group in the Design tab. Finally, enter appropriate axis titles for the data being graphed. Click on the titles to select the label to change them. Scatter Diagram To create a simple scatter diagram using the data on income levels vs. food expenditures (in hundreds of dollars) from Table 13.1 on page 595 in your text, enter the data in a spreadsheet as shown below:

3 3 Excel Manual Select cells A1:B7, then choose the Scatter With Only Markers chart from the Charts group in the Insert ribbon. The following scatterplot is created by selecting Layout 3 from the Quick Layouts Group in the Design tab and filling the appropriate labels:

4 4 Excel Manual Notice that the x- and y-axes both go to 0. If you would like to zoom in on the chart a little more by eliminating the origin from the plot, right-click on the x-axis and select Format Axis. Select Fixed and enter an appropriate minimum value (we ve chosen 20 in the example below). Do the same for the y-axis. This example will use 5 for the minimum value for the y-axis. The results are shown below. The line shown is the least squares regression line, which is automatically created when Layout 3 is used:

5 5 Excel Manual The Correlation Coefficient and Linear Prediction There are four distinct methods of calculating the statistics of measurement bivariate relations. First, we ll discuss the built-in functions to be used in a worksheet. =CORREL(Array1, Array2) calculates the correlation coefficient between two measurement variables. =SLOPE(known_Y s, known_x s) calculates the slope of the line of best fit using the least squares method between two variables. Note that the response variables must be listed first and explanatory variables second. =INTERCEPT(known_y s, known_x s) calculates the y-intercept of the line of best fit using the least squares method between two variables. Again, note that the response variables must be listed first and explanatory variables second. The advantage of these functions is that if there is any text or blank cells, they will simply be ignored in the calculation. Also, the arrays used in the formula may take on the form of a column range or a defined rectangular range. By using column references, such as A:A, we can add data dynamically and don t have to manually recalculate every time a new entry is added. Using the same data as the example above from Table 13.1 on page 595 on income vs. food expenditures, add an area to display the measurement bivariate statistics as shown: The Data Analysis ToolPak offers a second method for obtaining these values along with much more statistical information. To start, select Data Analysis in the Analysis group in the Data ribbon. (If you do not see Data Analysis in the Data ribbon, see Chapter 2 for instructions on how to access this add-in feature.) Select Regression from the list.

6 6 Excel Manual Enter the response variable range of cells for the Input Y Range and the explanatory variable cells for the Input X Range. Select Labels in the input area and choose Residual Plots to let Excel create that plot for you. Click OK. The sample output is shown below. The correlation coefficient appearing in cell B4 is referred to as Multiple R. The slope is shown in cell C18 and is referred to as the x-variable coefficient. The y-intercept in cell C17 is referred to as the intercept coefficient. The advantage of this option is that it is relatively straightforward, quick, and yields a number of statistical values that may be needed. The disadvantage is that it requires all cells to be filled with numeric values (i.e. no blank or text cells are allowed) and the output is static, meaning you will need to recalculate when data is added or adjusted. Notice that numerous statistics are available, some of which (such as the Standard Error, t-stat, p-value, and the upper and lower 95%) will be discussed later in the chapter. A third method is to create a scatter diagram and right-click on any data point in the scatterplot and select Add Trendline. You can check the options for displaying r 2 and the equation on the chart. The dialog box and results are shown below:

7 7 Excel Manual The Method of Least Squares Calculating the Values of the Least Squares Regression Model Using Excel Enter the following spreadsheet: Format superscripts and subscripts using the font dialog box. Select the character that you would like to appear as a subscript, select the digit and press <shift>+<ctrl>+f or click on the Show Font menu button at the right end of the Font group label. The data shown here is from the same example as the previous one on income vs. food expenditures from Table 13.1 on page 595. Once these formulas are entered, autofill them all down to the remainder of the table. In another area of the spreadsheet, copy the header rows A1:G2 and paste them in J1:P2. Enter the following formulas (the example shown utilizes columns I through P of the same sheet): In J3, enter =Sum(A:A). Autofill this formula to the right to P3. In J4, enter =Average(A:A). Autofill this formula to the right to K4.

8 8 Excel Manual In J6, enter =Count(A:A). These formulas use only the column designation and no row designations to allow you to dynamically add data points. The formulas will recalculate as you add the data points. In J7, enter =L3-J3^2/J6 to calculate the sum squared deviations of x s. In J8, enter =M3-K3^2/J6 to calculate the sum squared deviations of y s. In J9, enter =N3-J3*K3/J6 to calculate the sum of cross-product deviations. In J10, enter =K4-J11*J4 to calculate the estimate of the y-intercept. In J11, enter =J9/J7 to calculate the estimate of the slope. In J12, enter = J8-J9^2/J7 to calculate the sum of squares due to errors. Now that the rest of the spreadsheet is completed, you can enter the formula for the list of predicted y values based on the least squares model. In F3, enter =$J$10+$J$11*A3. Autofill down to the bottom of the table. The following illustrations show the results using the income vs. food expenditure data from Table 13.1 on page 595: As you can see, this spreadsheet calculates all of the necessary statistics, such as slope (b = ), y-intercept (a = ) and more. Any number of data points may be entered in this sheet; you just need to ensure the formulas in columns C through G are completed through all

9 9 Excel Manual of the data points and use the column reference A:A instead of range formulas. You do not need to worry how far down the formula goes. The Sampling Variability of the Least Squares Estimators These formulas begin to get a little complicated using Excel, but using the sheet from the previous example, you can see how the information can build and that you can create a very dynamic sheet for any set of data points. Inference Concerning Slope b Enter the formula =SQRT(J12/(J6-2)) in J14 to calculate the standard errors of the least square estimators. This value may now be used in the spreadsheet to find test statistics and the margins of error for inferences about the slope, y-intercept, and expected response. To test the hypothesis about the slope, add the labels as shown in L6 through L13: L6: Inference about slope L7: Hypothesized b L8: SE(b) L9: T L10: α L11: T α/2 L12: Margin of Error L13: Confidence Interval L14: p-value (1-tailed) L15: p-value (2-tailed) Enter the following formulas: M8: =J13/SQRT(J7) M9: =(J11-M7)/M8 (The =STANDARDIZE function could also be used here.) M11: =TINV(M10,J2-2) M12: =M11*M8 M13: =J11-M12

10 10 Excel Manual N13: =J11+M12 M14: =TDIST(ABS(M9),J6-2,1) M15: =TDIST(ABS(M9),J6-2,2) The results are shown below first with the formulas and then with values using the same example as above with the same data from Table 13.1 on page 595 relating to income and food expenditures: Inference About the Intercept a Select cells L6:N15 and paste them to P6:R15 to create the area for the inference about the y- intercept. The labels need minor adjustments. In P6, replace slope with intercept In P7 and P8, replace the label b with a Change the formula in Q8 to =J3*SQRT(1/J6+J4^2/J7) to calculate the standard error about the intercept. In the rest of the formulas, the copying changed any reference of M to Q, which is what needs to happen. The references to J changed too, but they must be adjusted back to refer to J, as follows: Q9: =(J10-Q7)/Q8 Q11: =TINV(Q10,J6-2) Q12: =Q11*Q8 Q13: =J10-Q12 R13: =J10+Q12 Q14: =TDIST(ABS(Q9),J6-2,1) Q15: =TDIST(ABS(Q9),J6-2,2)

11 11 Excel Manual The results from inference on the intercept using the same example are shown below: Using the Regression Model for Estimating the Mean Value of y Continuing the same sheet to complete Example 13-9 in your text on page 635, we will calculate an estimate for the average food expenditure for people who have an income of $5500. We are 99% confident that the true mean food expenditure for people earning $5500 is in the interval ($ , $ ). Using the Regression Model for Predicting a Particular Value of y Using the same data and Example on page 636, we will create a confidence interval for a prediction of the food expenditure for a particular person who has an income of $5500.

12 12 Excel Manual Here, we are 99% confident that the food expenditure for a person earning $5500 is in the interval ($851.68, $ ). Linear Correlation Coefficient The correlation coefficient calculation is shown in the previous examples. The hypothesis test on the linear correlation coefficient is equivalent to the hypothesis on the slope. Performing the hypothesis test on the correlation coefficient has the advantage that it can be done even when only the r and the sample size are known. The example below uses the same example from Table 13.1 on page 595 regarding income and food expenditures. The results are shown below. The t is somewhat different than the results in the text due to the rounding of r to.95. The p-value is less than.001 for the one-tailed test, and therefore the results are significant. It appears the correlation coefficient is greater than 0.