Statistics Tip of the Week: Nonparametric Methods can use Signs, Ranks, Signed Ranks, etc.

In our Tip of the Week for 12/14/2017, we said that Nonparametric Methods -- like Wilicoxon, Mann-Whitney, and Kruskal-Wallis -- can work with data that Parametric Methods -- like t-tests and ANOVA -- cannot. Nonparametric methods are often called “distribution-free,” because they are free of any assumptions about the source Distribution(s). 

They can do this because they convert the data into things like Signs, Ranks, Signed Ranks, and Ranks Sums. Here's how that works: 


Signs
We’ll be comparing Sample data to a value we specify. It could be a target, a historical value, an industry standard, etc. Let’s say that the historical Median time to complete an operation in an industrial process has been 30 seconds. We collect a Sample of 10 time measurements: 28, 31, 30, 33, 32, 28, 30, 31, 27, 32

• If a time is less than the Median of 30 seconds, we give it a negative sign.
• If it is 30 seconds, we give it a zero.
• If it is greater than 30 seconds, we give it a plus sign. ​

• Count of plusses: 5
• Count of minuses: 3

We can use the Counts of these signs – instead of the original data –
in a Nonparametric method called the Sign Test.

Ranks
Let’s take that same Sample of data, and order it from low to high. Next, assign a Rank from low to high. For ties, split the difference between the values tied. For example, there are two 28’s. These occupy two Ranks after 1 (a 2 and a 3), so we give them both a 2.5. The next Rank would be a 4, but there’s another tie, so we mark the next two as 4.5’s. 

Signed Ranks
Signed Ranks, as you might guess, combine the concepts of Signs and Ranks. But there is a change in how Signs are assigned, and one step uses absolute values, so we’ll use a different example with some negative numbers.

Let’s say we are doing an analysis of the effect of a training program on employee productivity. If we were doing a Parametric test, we’d use the Paired t-test (aka Dependent Samples t-test.) We count the number of transactions that they process in an hour. For each employee, we subtract their Before Training number from their After Training number. The information we are capturing is the difference.

Instead of plus and minus signs, we’ll use +1 and 0. We compare the data values to a specified value, as we did in our example of the historical Median of 30. Each Sample data value is their After production number minus their Before number.
We’ll be testing the Null Hypothesis that there is zero difference, so the specified value is zero. 

Step 1: Sign: For each data value, assign a Sign:

• if the data value is greater than the specified value (0 in this example), then the Sign = +1
•  if it’s less than or equal to the specified value, then:Sign=0

Step 2: Calculate the Absolute Values 

Step 3: Rank the Absolute Values to produce the Absolute Ranks
​
Step 4: Signed Rank: Multiply the Sign times the Absolute Ranks 

Signed Rank tests are the NP counterpart to the Dependent Samples (aka Paired Samples) t-test. ​

Rank Sum tests are the NP counterpart of the the Independent Samples (aka 2-Samples) t-test. Since the Rank Sum method will require a somewhat lengthy description and this Tip is getting a bit long., We'll save Rank Sums for another Statistics Tip of the Week.