NOTE: Sometimes a linear regression attempt will fail. This can be because of insufficient rank in the matrix due to there not being enough observations or due to some of the regressors being “colinear”, that is, being linear combinations of another regressor or regressors and therefore not able to “present new data” to the regression.

17.7.1 Residual Spreadsheet

A residual spreadsheet may optionally be produced. This spreadsheet will contain the actual, predicted, and residual values for each sample, as well as the spreadsheet values for each covariate and interaction term. The residual value of a sample is defined as the difference between the sample’s actual value and its predicted value from the regresssion.

17.7.2 Linear Regression Statistical Output Viewer

As detailed in 16.2.5, a statistical output viewer will be displayed for the regression.

17.7.3 Overall Statistics

The following overall statistics are displayed for both normal and stepwise regression:

• The name of the response variable.
• The multiple correlation coefficient R. (Square root of R².)
• The coefficient of determination R².
• The adjusted-R² statistic.
• The sample size.
• The standard error of the estimate. This is computed as

  where s_e is the standard error (of the estimate), sse is the sum square of errors (sum of squares of predicted minus actual values), n is the sample size, and reg_df is the number of regressors in the full model.

• The standard deviation of the response.
• The F-statistic.
• The p-value from the regression.
• The permuted p-value, if permutation testing has been selected.
• The number of permutations, if permutation testing has been selected.
• The regression degrees of freedom.
• The residual degrees of freedom.
• The total degrees of freedom.

17.7.4 Regressor Statistics

The y-intercept is displayed. Then, the following statistics are displayed for each regressor:

• The regressor (the covariate or interaction term).
• The regression coefficient for this regressor.
• The standard error for this regressor. To compute this, a full-model-only regression is taken with all the regressors but this one as the full model, but with this regressor as a substitute dependent variable. If ssr is the sum of squares of this regressor’s actual values minus this regressor’s average, and R_r² is the R² value obtained from this regression-against-the-regressor, and the standard error of the estimate is s_e, then the standard error of the regressor s_r will be

• The value of the t-statistic for this regressor.

  where β is the regressor’s regression coefficient.

• Pr(> ∣t∣). This is the p-value from regressing using the actual full model as its full model, but using the actual full model without this regressor as its reduced model. Thus, this shows how much difference this particular regressor is making in the regression. Pr(> ∣t∣) refers to the probability that the difference made by adding this regressor is accounted for by chance, and thus that this case could be thought of as being in one of the “tails” of the t-distribution.
• Univariate Fit. This is the p-value of simply taking a regression with this regressor, all by itself, against the dependent variable.

17.7.5 Left-Out Regressors

Any potential regressors which have been left out are listed here. This list will include all regressors that were excluded from the final model of a stepwise regression.

17.7.6 Parameters

The parameters used for the regression are shown.