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significant predictor of demand

Outliers and Influential Observations

After a regression line has been computed for a group of data, a point which lies far from the line (and thus has a large residual value) is known as an outlier. Such points may represent erroneous data, or may indicate a poorly fitting regression line. If a point lies far from the other data in the horizontal direction, it is known as an influential observation. The reason for this distinction is that these points have may have a significant impact on the slope of the regression line. Notice, in the above example, the effect of removing the observation in the upper right corner of the plot:

With this influential observation removed, the regression equation is now

 People.Phys = 1650 + 21.3 People.Tel. 

The correlation between the two variables has dropped to 0.427, which reduces the  value to 0.182. With this influential observation removed, less that 20% of the variation in number of people per physician may be explained by the number of people per television. Influential observations are also visible in the new model, and their impact should also be investigated.


Residuals

Once a regression model has been fit to a group of data, examination of the residuals (the deviations from the fitted line to the observed values) allows the modeler to investigate the validity of his or her assumption that a linear relationship exists. Plotting the residuals on the y-axis against the explanatory variable on the x-axis reveals any possible non-linear relationship among the variables, or might alert the modeler to investigate lurking variables. In our example, the residual plot amplifies the presence of outliers. 


Lurking Variables

If non-linear trends are visible in the relationship between an explanatory and dependent variable, there may be other influential variables to consider. A lurking variable exists when the relationship between two variables is significantly affected by the presence of a third variable which has not been included in the modeling effort. Since such a variable might be a factor of time (for example, the effect of political or economic cycles), a time series plot of the data is often a useful tool in identifying the presence of lurking variables.


Extrapolation

Whenever a linear regression model is fit to a group of data, the range of the data should be carefully observed. Attempting to use a regression equation to predict values outside of this range is often inappropriate, and may yield incredible answers. This practice is known as extrapolation. Consider, for example, a linear model which relates weight gain to age for young children. Applying such a model to adults, or even teenagers, would be absurd, since the relationship between age and weight gain is not consistent for all age groups.