Examining paired t test results to detect assumption violations

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All the following results are provided as part of a two-sample paired t test analysis.

Results for paired differences:

Results for residuals:


  • Normality test for paired differences:
  • If the assumptions for the t testhold, the paired differences should come from a normal distribution. Departures from normality can suggest the presence of outliers in the data, or of a nonnormal distribution in one or more of the samples. The normality test will give an indication of whether the population from which the paired differences was drawn appears to be normally distributed, but will not indicate the cause(s) of the nonnormality. The smaller the sample size, the less likely the normality test will be able to detect nonnormality.
  • Histogram for paired differences:
  • The histogram for the paired differences has a reference normal distribution curve for a normal distribution with the same mean and variance as the sample. This provides a reference for detecting gross nonnormality when the sample sizes are large.
  • Boxplot for paired differences:
  • Suspected outliers appear in a boxplot as individual points o or x outside the box. If these appear on both sides of the box, they also suggest the possibility of a heavy-tailed distribution. If they appear on only one side, they also suggest the possibility of a skewed distribution. Skewness is also suggested if the mean (+) does not lie on or near the central line of the boxplot, or if the central line of the boxplot does not evenly divide the box. Examples of these plots will help illustrate the various situations.
  • Normal probability plot for paired differences:
  • For values sampled from a normal distribution, the normal probability plot, (normal Q-Q plot) has the points all lying on or near the straight line drawn through the middle half of the points. Scattered points lying away from the line are suspected outliers. Examples of these plots will help illustrate the various situations.
  • Plot of observed values vs sample number:
  • A plot of observed values against sample number may be helpful in detecting interaction between pairs and samples (treatments). If there is no interaction, the line segments (one for each pair) should be parallel or nearly so. If the line segments are not at all close to parallel, then it's likely that there is an interaction between pairs and samples (treatments) in their joint effect on the observed measurement.
  • Normality test for residuals:
  • If the assumptions for the t test hold, all the residuals should come from the same normal distribution with mean 0. Departures from normality can suggest the presence of outliers in the data, or of a nonnormal distribution in one or more of the populations from which the samples were drawn. The normality test will give an indication of whether the population from which the paired differences were drawn appears to be normally distributed, but will not indicate the cause(s) of the nonnormality. The smaller the sample size, the less likely the normality test will be able to detect nonnormality.
  • Histogram for residuals:
  • The histogram for residuals has a reference normal distribution curve for a normal distribution with the same mean and variance as the residuals. This provides a reference for detecting gross nonnormality when the sample sizes are large.
  • Boxplot for residuals:
  • Suspected outliers appear in a boxplot as individual points o or x outside the box. If these appear on both sides of the box, they also suggest the possibility of a heavy-tailed distribution. If they appear on only one side, they also suggest the possibility of a skewed distribution. Skewness is also suggested if the mean (+) does not lie on or near the central line of the boxplot, or if the central line of the boxplot does not evenly divide the box. Examples of these plots will help illustrate the various situations.
  • Normal probability plot for residuals:
  • For data sampled from a normal distribution, the normal probability plot, (normal Q-Q plot) has the points all lying on or near the straight line drawn through the middle half of the points. Scattered points lying away from the line are suspected outliers. Examples of these plots will help illustrate the various situations.
  • Residuals plotted against fitted values:
  • For the paired differences themselves, the residual paired difference is simply the paired difference minus the mean of all paired differences. A graph of these residuals against their fitted values would consist of a vertical "stack" of residuals, since there is only a single mean for the paired differences. Except for the subtraction of a constant value (the mean) from all the paired differences, this graph is identical in appearance to a distribution graph of the paired differences, or the graph of the paired differences against the mean of the paired differences. Outliers may appear as anomalous points in the graph (although an outlier may not turn up in the residuals plot by virtue of affecting the mean so that its fitted value lies near it). The graph of the residuals for individual values from each sample plotted against the corresponding fitted values should suggest a horizontal band across the graph, because the variation (vertical range) of the residuals should be independent of the size of the fitted values. A wedge-shaped fan pattern like the profile of a megaphone, with a noticeable flare either to the right or to the left as shown in the picture suggests that the variance in the values increases in the direction the fan pattern widens (usually as the fitted value increases), and this in turn suggests that a transformation of the paired differences may be appropriate. If the pattern to the residuals plot is curvilinear, then there may be interaction between pairs and samples (treatments). The plot of observed values vs samples (treatments) may also help detect the present of such interactions.

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