|
|||||||||||
Purpose: Detect Non-Randomness |
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.
H0: the sequence was produced in a random manner Ha: the sequence was not produced in a random manner Test statistic: Z = 2.6938 Significance level: α = 0.05 Critical value (upper tail): Z1-α/2 = 1.96 Critical region: Reject H0 if |Z| > 1.96Since 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:
|
||||||||||
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. |
Learning Objectives Create and interpret frequency polygons Create and interpret cumulative frequency polygons Create and interpret overlaid frequency polygons Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms, but are especially helpful for comparing sets of data. Frequency polygons are also a good choice for displaying cumulative frequency distributions . To create a frequency polygon, start just as for histograms , by choosing a class interval. Then draw an X-axis representing the values of the scores in your data. Mark the middle of each class interval with a tick mark, and label it with the middle value represented by the class. Draw the Y-axis to indicate the frequency of each class. Place a point in the
Comments
Post a Comment