Why residuals?

Prism provides four ways to plot residuals with t tests, provided that you entered raw data and not averaged data as mean, n and SD or SEM.

Many scientists think of residuals as values that are obtained with regression. But the t test is really regression in disguise. It fits a model. One of the assumptions of t tests is that the residuals from that model are sampled from a Gaussian distribution. A residual plot helps you assess this assumption.

Which graph to create?

Prism can make four kinds of residual plots. For t tests, since there are only two groups, three of the four choices are not super useful. As such, the QQ plot is the most useful way to plot residuals.

•Residual plot. The X axis is the actual value of the value (unpaired tests) or difference (paired test).  The Y axis is the residual. This lets you spot residuals that are much larger or smaller than the rest.

•Homoscedasticity plot. The X axis is the actual value of the value (unpaired tests) or difference (paired test).  The Y axis is the absolute value of the residual.This lets you check whether larger values are associated with bigger residuals (larger absolute value).

•QQ plot. The X axis is the actual residual. The Y axis is the predicted residual, computed from the percentile of the residual (among all residuals) and assuming sampling from a Gaussian distribution. ANOVA assumes a Gaussian distribution of residuals, and this graph lets you check that assumption.

•Heatmap plot. The rows of this graph correspond to the rows in the original input data table. When used with an unpaired test, this graph will consist of two columns (one for each group), while there will only be one column when used with a paired test. Each cell has a color encoded from the residuals.

Diagnostics for residuals

•Are the residuals Gaussian? Prism runs four normality tests on the residuals. The residuals from both groups are pooled and entered into one set of normality tests.

How residuals are computed

Residuals with  t tests and related tests are simple to understand.

•Unpaired t test. A residual is computed for each value. Each residual is the difference between the entered value and the mean of all values for that group. A residual is positive when the corresponding value is greater than the sample mean, and is negative when the value is less than the sample mean.

•Mann-Whitney test. A residual is computed for each value. Each residual is the difference between an entered value and the median of all values for that group. A residual is positive when the corresponding value is greater than the sample median, and is negative when the value is less than the sample median.

•Paired t test. A residual is calculated for each pair. First a difference is computed for each pair. The residual   equals the actual difference between the pairs minus the mean of all such difference in the data sets.

•Ratio t test. A residual is calculated for each pair. A ratio is computed for each pair. The residual equals the logarithm of that ratio minus the mean of the logarithms of all the ratios.  

•Wilcoxon matched pairs test. A residual is calculated for each pair. First a difference is computed for each pair. The residual  equals the actual difference between the pairs minus the median of all such differences in the data sets.