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A linear regression fits the lineor as Excel prefers to call itto the existing data set. It does so through a technique known as minimizing the sum of the squares of the error terms. To get the complete result of a regression analysis, select a range 5 rows by 2 columns and array-enter the LINEST function as shown in Figure 4.
The first row contains the 2 coefficients a1 and a0 respectively. The rest of the information is important in understanding how well the regression line fits the data, how significant the individual coefficients are, as well as the significance of the regression as a whole. It also contains key elements needed to build confidence intervals for interpolated or extrapolated estimates, a subject covered in the section titled Confidence Intervals.
But, first, we start with some nomenclature. The number of data points is given by n. The number of independent variables is given by k. If a constant is included in the regression, it increases k by 1.
Each of the recorded observations is denoted by the pair of values. For eachthe value predicted by the regression is given by. The value that one gets from the regression is. For reasons that will soon be apparent, we start with the last row. There are two values and.
These are aggregate measures of something we have already looked at the level of an individual data point. Recall that for each individual data point, the measure of how much the regression explains is and how much remains unexplained is.
The first is the sum of the squared values of how well the regression fits the data or. The second is the sum of the squared values of how much remains unexplained or. Row 4 contains two values: The degrees of freedom is given by the expression n-k, where n and k are explained earlier in this section.
The F statistic, or the observed F-value, is a measure of the significance of the regression as a whole. For the technically minded it tests the null hypothesis that all of the coefficients are insignificant against the alternative hypothesis that at least one of the coefficients is significant.Story.
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