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Another way to achieve approximate linearity is to add explanatory variables to the model. These could be entirely new variables or variables such as X I2 or Xl X 2 , which are constructed from other variables.
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If the plots indicate unequal variances of the errors, a transformation of Y may help. If Y is always positive and its range spans several orders of magnitude, such as from 0.01 to to, a frequently used transformation is the logarithm function. If Y is a percentage, a commonly used transformation for stabilizing its variance is the arcsine square-root function.
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Unequal Variances. Outliers. If a data point stands apart from the main body of points in a plot, it should be investigated to see what may have caused this. Perhaps a number has been incorrectly recorded. Or the experimenter who collected the data may remember some special circumstance for that data point. If special factors have influenced an outlier, one could remove it from the data set and use a linear regression model only for the remaining data. (The information that certain factors may produce outliers should be retained and could even be the most interesting result of the analysis.) However, an outlier may legitimately belong in the same model with the other data; it may be that the process that generated the data is one that occasionally produces extreme values. When an outlier is kept in the data set, its effect on the analysis can be checked by running two analyses, one with and one without the outlier. The outlier's effect can be constrained by using least-absolute-deviations, M-, or nonparametric regression.
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Since there is always a lot of subjectivity involved in planning and interpreting a statistical analysis, each of the following examples should be viewed as only one of many possible analyses.
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Example 1. Consider the turnip green data described in Section 1.2 and displayed in Table 1.1. Plots of the response variable, Y = vitamin B2 , versus each of the three explanatory variables, Xl = sunlight, X 2 = soil moisture, and X3 = air temperature, are shown in Figure 2.1. The plot of Y versus X 2 looks nonlinear. A corrective measure for this is to add to the model as a fourth explanatory variable. That is, let us modify the model to be
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Now we want to estimate the coefficients in this model. Suppose we decide to use the method of least squares. Following the procedure described in 3, we obtain Y = 119.6 - 0.03367X I + 5.425X2 - 0.5026XJ 0.1209Xi as an estimate of the regression equation. The least-squares estimate of the standard deviation of the errors is a = 6.104. Next we calculate the residuals i = Yi - Yi and plot their standardized values versus the fitted values Yi and the three explanatory variables Xii' X i2 , and X iJ These four plots are shown in Figure 2.2. No curvature is apparent in these plots. The plot versus X 2 , soil moisture, emphasizes that this variable takes on only three different values-2.0, 7.0, and 47.4. We would have to be cautious about the validity of the estimated regression equation for data with soil-moisture measurements of 20 or 30. This plot also suggests that the variance of the errors for X 2 = 7.0 may be smaller than for the other two values of X 2 But the observed difference in variance could simply be due to chance since the difference is caused by only two or three points with large residuals. A standardized least-squares residual can be regarded as a possible outlier, to be investigated, if its absolute value is greater than 2.0. (Some statisticians would not worry unless the absolute value was greater than 2.5.) The plots show one possible outlier, plant number 19, which has a standardized residual of 2.286. In the absence of a specific reason for omitting it, we retain it in the model. Moreover, its residual is not too extreme, and among more than 20 observations we expect to see at least one observation whose standardized residual is larger than 2 in absolute value. Thus we see that linear regression model (2.3) seems to adequately fit these data. Now we check whether the model can be simplified by dropping one or more explanatory variables. Using the procedure presented in 3, we can test the four hypotheses that f31 = 0, f32 = 0, f3J = 0, and f34 = O. The test of f31 = 0 indicates that perhaps f31 = 0, and hence that explanatory variable XI' the amount of sunlight, does not contain much information about the response variable beyond the information already contained in the other explanatory variables. Let us simplify the model to
(2.4 ) The least-squares estimate of the regression equation is Y = 120.6 + 4.904X 2 - 0.5716X3 - 0.1108Xi and the estimate of the standard deviation of the errors is fT = 6.223. Testing the three hypotheses f32 = 0, f3J = 0, and f34 = 0 in this model, we find that all three explanatory variables are necessary. The residual plots for this model look similar to the corresponding plots in Figure 2.2 for model (2.3), with the slight difference that, in the simplified