Analysis of Variance

Multiple Regression

Analysis of Variance

The Analysis of Variance table summarizes information about the sources of variation in the data. Sum of Squares represents variation present in the data. These values are calculated by summing squared deviations. In multiple regression, there are three sources of variation: Model, Error, and C Total. C Total is the total sum of squares corrected for the mean, and it is the sum of Model and Error. Degrees of Freedom, DF, are associated with each sum of squares and are related in the same way. Mean Square is the Sum of Squares divided by its associated DF (Moore and McCabe 1989).

If the data are normally distributed, the ratio of the Mean Square for the Model to the Mean Square for Error is an F statistic. This F statistic tests the null hypothesis that none of the explanatory variables has any effect (that is, that the regression coefficients ${ {\beta}_{1}}$ , ${ {\beta}_{2}}$ ,and ${ {\beta}_{3}}$ are all zero). In this case the computed F statistic (labeled F Stat) is 18.8606. You can use the p-value (labeled Pr > F) to determine whether to reject the null hypothesis. The p-value, also referred to as the probability value or observed significance level, is the probability of obtaining, by chance alone, an F statistic greater than the computed F statistic when the null hypothesis is true. The smaller the p-value, the stronger the evidence against the null hypothesis.

In this example, the p-value is so small that you can clearly reject the null hypothesis and conclude that at least one of the explanatory variables has an effect on GPA.

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