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A blank value in any of these rows just means that the corresponding variable was not already in the model and so can’t be eliminated. These p-values are calculated using the array formula The odd-numbered rows in columns L through O show the p-values which are used to determine the potential elimination of a variable from the model (corresponding to step 2b in the above procedure). Thus we see that at variable x 4 is the first variable that can be added to the model (provided its p-value is less than the alpha value of. cell Q6 contains the formula =MIN(L6:O6) and R6 contains the formula =MATCH(Q6,L6:O6,0). The values in range L8:O8 are computed using the array worksheet formula =RegRank($B$6:$E$18,$A$6:$A$18,G8:J8), which will be explained below.įor each even row in columns L through O, we determine the variable with the lowest p-value using formulas in columns Q and R. The value in cell L8 is the p-value of the x 1 coefficient for the model containing x 1 and x 3 as independent variables (since x 3 was already in the model at that stage). the value in cell L6 is the p-value of the x 1 coefficient for the model containing just x 1 as an independent variable. An “x” in one of these cells indicates that the corresponding variable is already in the model (at least at that stage) and so a p-value doesn’t need to be computed.Į.g. The even-numbered rows show the p-values for potential variables to include in the model (corresponding to steps 1a and 2a in the above procedure). We see that the model starts out with no variables (range G6:J6) and terminates with a model containing x 1 and x 4 (range G12:J12).Ĭolumns L through O show the calculations of the p-values for each of the variables. An empty cell corresponds to the corresponding variable not being part of the regression model at that stage, while a non-blank value indicates that the variable is part of the model.
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Columns G through J show the status of the four variables at each step in the process.
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The steps in the stepwise regression process are shown on the right side of Figure 1. In order to make this process clearer, let’s look at an example.Įxample 1: Carry out stepwise regression on the data in range A5:E18 of Figure 1. Note that this process will eventually stop. This leaves us with at most m+1 independent variables. Now consider the regression model of y on z 1, z 2, …, z m+ 1 and eliminate any variable z i whose regression coefficient in this model is greater than or equal to α. Then stop and conclude that the stepwise regression model contains the independent variables z 1, z 2, …, z m. Let’s call this variable z m+1 and suppose the p-value for the z m+1 coefficient in the regression of y on z 1, z 2, …, z m, z m+ 1 is p.Ģb. As in step 2a, choose the independent variable whose regression coefficient has the smallest p-value. Assuming that we have now built a stepwise regression model with independent variables z 1, z 2, …, z m (after step 1b, m = 1), we look at each of the k–m regression models in which we add one of the remaining k-m independent variables to z 1, z 2, …, z m. Then stop and conclude there is no acceptable regression model.
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z 1 is one of the independent variables x 1, …, x k) and the p-value for the z1 coefficient in the regression of y on z1 is p.ġb. Choose the independent variable whose regression coefficient has the smallest p-value in the t-test that determines whether that coefficient is significantly different from zero. Build the k linear regression models containing one of the k independent variables. 05 is chosen, here it is more common to choose a higher value such as alpha =. Whereas for most statistical tests a value of alpha =. The algorithm we use can be described as follows where x 1, …, x kare the independent variables and y is the dependent variable:Ġ. In this webpage, we describe a different approach to stepwise regression based on the p-values of the regression coefficients. The reader is once again alerted to the limitations of this approach, as described in Testing Significance of Extra Variables. This leads to the concept of stepwise regression, which was introduced in Testing Significance of Extra Variables. When there are a large number of potential independent variables that can be used to model the dependent variable, the general approach is to use the fewest number of independent variables that can do a sufficiently good job of predicting the value of the dependent variable.