1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.5. Quantitative Techniques


Purpose: Detect NonRandomness 
The runs test
(Bradley, 1968)
can be used to decide if a data set is from a random process.
A run is defined as a series of increasing values or a series of decreasing values. The number of increasing, or decreasing, values is the length of the run. In a random data set, the probability that the (I+1)th value is larger or smaller than the Ith value follows a binomial distribution, which forms the basis of the runs test. 

Typical Analysis and Test Statistics 
The first step in the runs test is to count the number of runs
in the data sequence. There are several ways to define runs
in the literature, however, in all cases the formulation must
produce a dichotomous sequence of values. For example, a
series of 20 coin tosses might produce the following sequence
of heads (H) and tails (T).
 
Definition 
We will code values above the median as positive and values below
the median as negative. A run is defined as a series of consecutive
positive (or negative) values. The runs test is defined as:


Runs Test Example 
A runs test was performed for 200 measurements of beam deflection
contained in the LEW.DAT data set.
H_{0}: the sequence was produced in a random manner H_{a}: the sequence was not produced in a random mannerSince the test statistic is greater than the critical value, we conclude that the data are not random at the 0.05 significance level. 

Question 
The runs test can be used to answer the following question:


Importance 
Randomness is one of the key
assumptions in determining
if a univariate statistical process is in control. If
the assumptions of constant location and scale, randomness,
and fixed distribution are reasonable, then the univariate
process can be modeled as:
If the randomness assumption is not valid, then a different model needs to be used. This will typically be either a times series model or a nonlinear model (with time as the independent variable). 

Related Techniques 
Autocorrelation Run Sequence Plot Lag Plot 

Case Study  Heat flow meter data  
Software  Most general purpose statistical software programs support a runs test. Both Dataplot code and R code can be used to generate the analyses in this section. 