Excel Lab 3: Linear Regression

Transcription

1 Excel Lab 3: Linear Regression When working with data sets, the term regression (also called least squares fitting ) refers to determining the formula of a function that relates the data quantities in a way that is consistent with the given data. In this assignment, you will learn how to use some of Excel s built-in commands to perform regression for a linear model. Open a new Excel workbook, and perform the steps in this tutorial as we go. Example: (Fitting a linear function model to a data set.) Here is a data set resulting from the following experiment: We start with six trees of different heights. We measure and record the height of each tree, then measure and record the heights again after one year. Tree Initial heigh (feet) Final height (feet) x y We begin by entering this data set into an Excel spreadsheet, and producing a scatter plot of the data. (NOTE: In this table, the first column is NOT data it is just a bookkeeping index.) Here is what we get:

2 The points look fairly linear, so we try to find a regression line, i.e. a function of the form y=mx+b, where we have to figure out some good values of m and b. As noted in class, Excel has two commands, logically called SLOPE and INTERCEPT, that we can use to compute values for the slope, m, and the intercept, b, for our data set. Using the SLOPE command: To compute the slope m of the regression line, we do the following in Excel: 1. Click on an empty cell, where you want the slope value to be, and label an adjacent cell with the word slope. We have this: 2. In the selected cell, type =SLOPE( 3. At this point, you should see a little hint window (just below our selected cell in the image below).

3 This hint window tells us that the SLOPE command takes two arguments. The first is the list of dependent variable values, and the second is the list of independent variable values. (Note that this is backwards from the usual thinking, where independent comes first, and dependent comes second.) 4. The easiest way to finish inputting the SLOPE command is to first use the cursor and the left mouse button to select the column of dep. var. values. When you make the selection this way, the formula fills in automatically, like this: Notice how the blue E5:E11 (which is how Excel describes a range of values, using a colon) has been added to the SLOPE formula. 5. Next, type a comma after the blue E5:E11. Then, repeat Step 4, this time with the column of indep. var. values. This gives us this:

4 Again, this automatically enters the correct range of values ( D5:D11, in this case) into the SLOPE formula. 6. The last thing to do is type a closing parenthesis after the green D5:D11, then press the Enter key. You should now see the value of the regression line in the box where the formula was entered. Using the INTERCEPT command: To compute the intercept b of the regression line, we do exactly the same steps that we did for SLOPE, only substitute INTERCEPT for SLOPE in the formula. When finished, the formula will read =INTERCEPT(E5:E11,D5:D11).

5 (NOTE: You can always view the formula for a particular cell (assuming it has one) by just double-clicking on the cell.) After doing this, our Excel spreadsheet looks like this: So, for this data set, the regression line (or best fit linear model or least squares line ) is given by the formula y=1.01x Comparing values of the regression line model to the data: We now add a new column to our data set, so we can list the values predicted by the regression line model. After labeling the column, go to the cell next to the first data point, and enter, in Excel cell-address form, the above formula for the regression line. For the first data row (Row 6 in this case), the x-value is in cell D6, the slope-value is in cell C15, and the intercept-value is in cell D15. Thus, the formula for the regression line is written =C$15*D6+D$15 So, this is what you enter in the cell. Then, use the select and drag method to extend the calculation to the other data points. Here is what you should see:

6 Question: Why did we put a $ in the middle of the cell address for the slope and intercept? Answer: By default, when we select and drag a formula, the cells used by that formula are automatically updated, as we have seen in previous Excel assignments, and in class. This is fine for the x-values in our regression line Excel formula. However, we do NOT want it to use the SAME slope and intercept values in each of the cells that we drag the formula to. By putting a $ between the letter and number in a cell address, we are telling Excel to NOT shift their cell addresses. That way, when we drag the formula down to the second data point, the resulting formula will be =C$15*D7+D$15 (which is good). If, on the other hand, we did NOT include the dollar signs, the formula in cell F7 would read =C16*D7+D16 which would be bad, because there is no value at all in cells C16 or D16. Plotting the regression curve: Next, we want to view the graph of the regression line (this graph is called the regression curve), along with a scatter plot of the data. Now that we have a column of values of the regression line, this can be done in a cool and easy way, like this: 1. First, select the entire column of regression line values (including the column label line ). 2. Move the cursor to an edge of the selected range of cells, until the cursor changes from its usual white cross shape, to a thin black cross with accompanying white arrow shape. While you have this cursor, press and hold the left mouse button, and drag the cursor into the scatter plot we made earlier for the data. Once the cursor is in the plot area, release the mouse button. This will add the new points to the graph, and give us a plot like this:

7 The new points are added as pink squares, as indicated by the legend on the plot. Usually, when we show a data set and regression curve together, we keep the data as a scatter plot, but show the regression curve as a regular curve. To do this: 1. Place the cursor on one of the pink squares in the plot, then click the right mouse button. From the resulting menu, select Format Data Series. You will get a window that looks like this: 2. This window is divided into a Line part and a Marker part. (Here, marker refers to the pink squares.) Note that, currently, the None radio button is selected under Line, while Automatic is selected under Marker. This is why the current plot shows the pink squares, but no curve. To change this, select Automatic under Line, and select None under Marker. Then, click OK. Our plot now looks like this:

8 This way of displaying the plots makes it easier to compare the model (i.e. the regression line) to the data. Exercises. 1. The table below gives data from the following experiment: Ten cells (living cells, not Excel cells), each having different mass (measured in micrograms), was treated with a specific type of toxin. The amount of toxin was increased, little by little, until the cell died. The amount of toxin that killed the cell is recorded as data (under the heading, Toxin Tolerance ). Mass Toxin Tolerance a. Use regression to determine the best fit linear model for this data, and produce a plot showing the data and regression curve together, as was done in the example above. b. Would you consider a linear model to be a reasonable choice for this data? Explain your answer.

9 2. Print your Excel worksheet, put your name on it, and turn it in on the due date. (NOTE: You do NOT need to print this Word document and turn it in.)