An exploratory data analysis (EDA) procedure that creates an additive model to describe data in a two-way table. This additive model represents each data value in the table as a sum of four components: a common effect, a row effect, a column effect, and a residual. This format isolates the effects of the row and column factors to reveal effects that are not readily apparent in the original data, and to help you determine whether these effects justify formal statistical inference.
For example, suppose a manufacturing facility employs three shifts of welders who can choose between two different welding methods. Inspectors measure weld strength and create a two-way table of the observations, with rows corresponding to the shift during which each weld was made, and columns corresponding to the welding method. Although weld strength is influenced by both welding method and shift, the raw data itself do not indicate how each individual factor contributes to weld strength. The median polish procedure, however, helps ascertain this information.
The results of the median polish procedure are represented below. Each component of the additive model is separated from the others and labeled in bold. Interpretations of each component follow.
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|
Weld Method A |
Weld Method B |
|
|
|
Residuals: |
Row effects: | |
|
Shift 1 |
10 |
0 |
2 |
|
Shift 2 |
-1 |
1 |
0 |
|
Shift 3 |
0 |
-1 |
-2 |
|
Column effects: |
-5 |
5 |
10 = Common effect |
The common effect is 10. The common effect represents a typical measure
of weld strength against which the effects of the two factors
Median polish produces a column of row effects that indicate the influence of each level of the row factor on the observed data. The row effects above lead inspectors to conclude the following:
In other words, a pattern emerges in which weld strength decreases as shifts progress. This EDA procedure suggests that the inspectors should formally analyze the effect of work shifts on weld strength.
Median polish also produces a row of column effects that indicate the influence of each level of the column factor on the observed data. The column effects above lead the inspectors to conclude the following:
Again, the EDA reveals a pattern, and the inspectors have sufficient justification to analyze the effect of welding method on weld strength with formal statistical inference.
Lastly, the residuals remain in the body of the table. These residuals reveal extraordinary values by indicating how each observation differs from its expected value (the sum of the common, row, and column effects). For example, the only exceptional residual value is 10 for "First Shift, Weld Method A". Inspectors conclude that, although the column effects suggest that Weld Method A produces a lower median weld strength, first shift welders using Weld Method A produce welds uniquely greater than expected. This effect is worth investigating further.