Johnson transformations are used in a way similar to Box-Cox transformations. First, apply a transformation to the response, and then use the transformed data with a normal distribution to find capability.
As with using other distributions to fit to nonnormal data, we should investigate the reasons for our data being in the shape it is before attempting Johnson transformations. For more notes on what to look out for, see the Capability analysis for nonnormal distributions recipe.
The main benefit of Johnson transformations over Box-Cox transformations is the ability of the former to transform data with negative values or 0 values. They can also be useful in situations where a process or data set has an extreme boundary condition that makes other distributions difficult to fit to. One example of where this may be used is for breaking stresses.
As with the previous examples on nonnormal distributions and Box-Cox transformations, we will use the data on patient waiting times at an accident and emergency ward. To do this, let's compare the output with the lognormal distribution and the Box-Cox transformation.
We need to check that the transformation is appropriate with the Johnson transformation from the Quality Tools menu before running a normal capability analysis on the transformed results.
The following steps will help us use the Johnson transformation for our data:
- Open the worksheet
Wait time.mtwby using Open Worksheet from the File menu. - Navigate to Stat | Quality Tools and select Johnson Transformation.
- Enter
Wait time (1 Mins)in the Single Column field and click on OK. - Check the results in the chart to see if the transformation and the function used to transform the data will work.
- Navigate to Stat | Quality Tools | Capability and select Normal.
- Insert the wait time column in the Single column field.
- Enter the Subgroup sizes as
1. - In the Upper spec field, enter
240. - Click on the Transform button.
- Select the option Johnson Transformation.
- Click on OK in each dialog box.
Steps 1 to 3 are used to check if the Johnson transformation will work on the data and if we ran the distribution ID plots as in the Capability analysis for nonnormal distributions recipe. It would not be essential to check the transformation using the Johnson Transformation tool. This tool is used to show if the transformation will work and if this is the optimum transformation function. It also allows the transformed data to be stored directly in the worksheet. The graphical page displayed shows that Minitab searches for the highest P-value to find the transformation.
The Johnson transformation in Minitab considers three transformation functions. These are for bounded, lognormal, and unbounded functions. The parameters of the transformation function are found from the function that has the highest P-value that is greater than the decision level. The default value used here is 0.1.
By selecting the Johnson transformation from the transformation functions within the normal capability analysis, we will automatically find the best transformation. If no transformation is possible, it will return an untransformed result.