Step-up/step-down analysis
Overall, there are two different types of variable rule analysis: binominal one-step and binominal step-up/step down (cf. Tagliamonte 2006: 140).
The binomial one-step analyzes all cells at once, allowing the analyst “to examine each of the cells and see how much each combination differs from the expected” (Tagliamonte 2006: 139). This enables the user to pinpoint certain cells which fit the model least well, and to exclude the respective tokens. However, as this type of analysis is rarely applied by most variationists, because it neither assesses statistical significance nor the relative strength of the factor groups, we will now focus our attention on the generally preferred binominal step-up/step down method (cf. Tagliamonte 2006: 140).
The main difference between these two types of analysis is that unlike the binominal one-step procedure, the binominal step up/step down method analyzes each cell one step at a time. The main aim is to find groups which cause the model to change significantly when being added or subtracted. Therefore, the program tests each factor group and retains the most significant ones, by continually adding further groups until no further additions result in a significant change. The ultimately remaining groups are also referred to as the step-up solution (cf. Tagliamonte 2006: 140).
This method is called step-up/step down, because it consists of two parts, which are essentially based on the same principle but proceed in reversed directions. While the step-up analysis begins at Level '0', which includes no factors, and progresses by adding new groups in the respective levels, the step-down analysis “starts by calculating the likelihood of the model when all the factor groups are included in the regression simultaneously” (Tagliamonte 2006: 143) and goes on by abstracting the least significant groups one after another. However, the results in regards to finding the run which is the best fit of the model for the data should be identical in both steps (cf. Tagliamonte 2006: 226ff.).
If this is not the case or if, for example, the factor weights for each factor at each level fluctuate considerably, then there might be a glitch in the data. Then the analyst must cross-tabulate the various factor groups as noted in the section entitled The result file, in order to ascertain whether factor groups are overlapping or interacting, or whether certain cells remain empty or are widely diverging across factor groups.
Some of the terms which can be seen in Figure 7 might require some information, in order to understand the facts given in such a file. For this information go to Important terms.
After having rerun the data several times, the analyst will ultimately come to the point where the factors' weights fluctuate only slightly at the various levels, convergence is reached for each run and the results for both the step-up and step-down analysis match. When this is the case and the best fit of the model has been found, the analyst must begin interpreting the data (cf. Tagliamonte 2006: 226ff.).
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