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We occasionally nd that the straight-line regression model Y is inappropri0 1x ate because the true regression function is nonlinear. Sometimes nonlinearity is visually determined from the scatter diagram, and sometimes, because of prior experience or underlying theory, we know in advance that the model is nonlinear. Occasionally, a scatter diagram will exhibit an apparent nonlinear relationship between Y and x. In some of these situations, a nonlinear function can be expressed as a straight line by using a suitable transformation. Such nonlinear models are called intrinsically linear. As an example of a nonlinear model that is intrinsically linear, consider the exponential function Y

This function is intrinsically linear, since it can be transformed to a straight line by a logarithmic transformation ln Y ln

Dim ti as TextImaging.TextToImage Private Sub Page_Load( _ ByVal sender As System.Object, _ ByVal e As System.EventArgs) Handles MyBase.Load ti = New TextImaging.TextToImage() Image1.ImageUrl = ti.RenderText( Regular , _ 36 , Impact , My Home Page , LightBlue , Blue ) End Sub

0 1x

To edit a function, click the cell containing the function that you want to edit, click the Formulas tab and click the Insert Function button. Excel displays the function s Function Arguments dialog box, where you can change the cell references or values as needed.

This transformation requires that the transformed error terms ln are normally and independently distributed with mean 0 and variance 2. Another intrinsically linear function is Y By using the reciprocal transformation z Y

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The code to create an instance of the TextToImage class and execute the RenderText method is actually quite simple. The browser output is shown in Figure 16.10. The call to RenderText returned a URL to the image, which was assigned to the ImageURL property. Viewing the source HTML from the browser reveals the result of the RenderText call.

0 1z

Sometimes several transformations can be employed jointly to linearize a function. For example, consider the function Y letting Y* exp 1 1

dr.CancelEdit() dr.BeginEdit() End If dt.DefaultView.RowFilter = emp_id= & newId & BindTable() End Sub

1 Y , we have the linearized form ln Y*

0 1x

This code sets a RowFilter to the added row. This ensures that the correct row will be in edit mode and no other rows are available to distract the user. Finally, the Cancel method is modified to delete the new row if the Cancel button has been clicked. This code follows:

For examples of tting these models, refer to Montgomery, Peck, and Vining (2001) or Myers (1990).

11-10 11-11

Our development of regression analysis has assumed that x is a mathematical variable, measured with negligible error, and that Y is a random variable. Many applications of regression analysis involve situations in which both X and Y are random variables. In these situations, it

Private Sub dgEmployee_CancelCommand( _ ByVal source As Object, _ ByVal e As System.Web.UI.WebControls.DataGridCommandEventArgs) _ Handles dgEmployee.CancelCommand dgEmployee.EditItemIndex = -1 Get the DataTable from the Session. Dim dt As DataTable = CType(Session( Employee ), DataTable) Get the DataRow to be canceled. Dim PrimaryKey As String = dgEmployee.DataKeys(e.Item.DataSetIndex) would be better if this was an autonumber If PrimaryKey = *NEW ID* Then Dim dr As DataRow = dt.Rows.Find(PrimaryKey) dt.Rows.Remove(dr) End If dt.DefaultView.RowFilter = BindTable() End Sub

is usually assumed that the observations (Xi, Yi), i 1, 2, p , n are jointly distributed random variables obtained from the distribution f (x, y). For example, suppose we wish to develop a regression model relating the shear strength of spot welds to the weld diameter. In this example, weld diameter cannot be controlled. We would randomly select n spot welds and observe a diameter (Xi) and a shear strength (Yi) for each. Therefore (Xi, Yi) are jointly distributed random variables. We assume that the joint distribution of Xi and Yi is the bivariate normal distribution pre2 2 sented in 5, and Y and Y are the mean and variance of Y, X and X are the mean and variance of X, and is the correlation coef cient between Y and X. Recall that the correlation coef cient is de ned as

(11-35)

Figure 8.10 shows the Add method in action. Notice that when the Add button is clicked, the DataGrid automatically resizes to a single row, which is the row that is being added. The only thing left is to allow the deletion of DataRows.

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