This statlet performs an analysis of variance for data grouped by levels of a single classification factor, where information has also been recorded about the value of a covariate.The tabs are:
The example data consists of a set of measurements made on 15 samples of fiber, 5 samples from each of 3 machines. The primary variable of interest is the strength of the fiber and any differences in strength that might exist between the 3 machines. As a covariate, the diameter of each sample of fiber was also measured.
Enter the name of the column containing the measurements, the name of the column containing the level codes for the experimental factor, and the name of the column containing the values of the covariate:
The level codes may be numeric or non-numeric. The data and covariate columns must contain numeric values.
This tab shows a plot of the data values by level code:
Note that the strength of the fibers made on machine 3 look somewhat lower than the strength of the fibers made on the other machines.
Specify the desired orientation for the plot:
Indicate the axis along which the data values will be displayed.
This tab displays a table of summary statistics for each level code:
Initially, a set of commonly computed statistics is displayed. You can request other statistics by pressing the Options button.
For a discussion of other summary statistics, refer to the One Variable Analysis statlet.
The Options button permits you to select any of 18 different statistics to compute:
For definitions of each of these statistics, refer to the Glossary.
This tab displays box-and-whisker plots by level code:
Box-and-whisker plots are useful graphs for displaying many aspects of samples of numeric data. Their construction is discussed in the One Variable Analysis statlet.
The Options button permits you to control various aspects of the plot:
These include:
Median notches - if checked, notches are added to the boxes indicating the location of uncertainty intervals for the medians. These intervals are scaled in such a way that if two intervals do not overlap, there is a statistically significant difference between the two population medians at the indicated confidence level.
Mean symbols - if checked, the location of the sample means are shown as plus signs.
Flagged outside points - if checked, the whiskers extend outward to the most extreme points which are not outside points. All outside points are marked with special symbols. If not checked, the whiskers extend outward to the minimum and maximum values in the samples.
Confidence level - specifies the confidence level for the median notches.
Direction - specifies the orientation of the plot.
This tab displays an analysis of variance table for the data:
The table takes the original variability among the 15 measured fiber strengths and divides it into three components:
The P-values in the rightmost column of the table are used to determine whether significant differences exist due to the covariate and between levels of the experimental factor. P-values below 0.05 indicate statistically significant differences at the 5% significance level. In the above example, the effect of the covariate is significant, but the differences between machines are not quite significant at the 5% significance level.
This tab displays the means for each level code together with uncertainty intervals:
The uncertainty intervals bound the estimation error in the means using one of several methods, selected by pressing the Options button. The choice of intervals is described in detail in the Oneway ANOVA statlet.
Select the desired type of uncertainty intervals:
This tab plots the level means with uncertainty intervals:
See the discussion under Means table for an explanation of the intervals.
Select the desired type of uncertainty intervals:
This tab indicates which means are significantly different from which others:
Select the desired procedure and the level of confidence:
This tab plots the residuals by level code:
If the standard deviations within each group are the same, we should see approximately the same scatter amongst the residuals for each machine. In this case, there are small differences between the spreads of the residuals in the three groups. We don’t usually start to worry, however, until the standard deviations differ by more than a factor of 3 to 1 between the largest and the smallest. The analysis of variance is known to be reasonably robust at differences of less than this magnitude, which means that the confidence levels stated are approximately correct. The differences here are much less than 3 to 1.
This tab plots the residuals versus predicted values of strength:
It is useful for detecting possible violations of the assumption of constant within group variability. Frequently, the variability of measurements increases with their mean. If so, the above plot would show points falling into a funnel-shaped pattern, increasing in spread from left to right. Such observed heteroscedasticity may often be eliminated by analyzing the logarithms of the data rather than the original data values.
This tab plots the residuals versus row number:
If the data were entered in time order, any pattern in the above plot would indicate changes over the course of the data collection.
This tab creates a normal probability plot for the residuals:
If the residuals come from a normal distribution, the points should lie approximately along a straight line as shown above.