Run test

 ONE-SAMPLE TESTS

RUN TEST

One of the fundamental assumptions of the parametric test is that the observed data are random and test statistic and the subsequent analysis are based on this assumption. It is always better to check whether this assumption is true or not.

A very simple tool for checking this assumption is run test. This section is devoted to throw light on the run test. Before discussing the run test first we have to explain what we mean by a “run”.

A run in observations, is defined as a sequence of letters or symbols of one kind, immediately preceded and succeeded by letters of other kind or no letters. For example a sequence of two letters H and T as given below:

HHTHTTTHTHHHTTT                                                                                                            

In this sequence, we start with first letter H and go up to other kind of letter, that is, T. In this way, we get first run of two H’s. Then we start with this T and go up to other kind of letter, that is, H. Then we get a run of one T and so on and finally a run of three T’s. In all, we see that there are eight runs. And it is denoted as r=8. it is shown as below.

HH T H TTT H  T HHH TTT                                                                                                  1   2   3   4     5   6  7         8

Under the run test we Judged randomness of observations by using number of runs in the observed sequence. Too few runs indicates that there is some clustering or trend and too large runs indicates that there is some kind of repeated or cycles according to some patterns.

for example, the following sequence of H's and T's is obtained when tossing a coin 10 times.

HHHHHHTTTT                                                                                                                        1             2 

in the above sequence we see there is only 2 run's of 6 heads and 4 Tails, hence from this we say the similar item tend to cluster together, therefore such sequence of observation is not considered as random. now the anther sequence of 10 tosses is as following:

H T H T H T H T H T                                                                                                        1  2  3 4 5   6  7 8 9 10 .

this numbers indicates the runs for all observation  in this sequence there are 10runs of 5 runs of one head each  and 5 runs of one tail each. this sequence is cloud not be considered as random because there are too many runs they indicates the pattern.

T HHH TT H T HH T HHH TT H T H T                                                                          1    2      3   4  5  6    7   8       9 10 111213 

Here neither the number of runs too small nor too large, this type of sequence may be considered as random. 

Assumptions: 

Run test make the following assumptions:

(i) Observed data should be such that we can categorise the observations into two mutually exclusive types.

 (ii) The variable under study is continuous.                    

Procedure for RUN test

Let X1,X2,...,Xn be a set of no observations arranged in the order in which they occur. Generally, we are interested to test whether a population or a sample or a sequence is random or not. So here we consider only two-tailed case. Thus, we can take the null and alternative hypotheses as

H0 :  The observations are random

H1 :  The observations are not random [two-tailed test]

test consist following steps:

Step 1: First of all, we check the form of the given dada that the given data are in symbolical form such as sequence of H and T, A and B, etc. or in the numeric form. If the data in symbolical form then it is ok, but if data in numeric form then first we convert numeric data in symbolical form. For this, we calculate median of the given observations by using either of the following formula given below             

Median = size of [(n+1) /2 ] th observation.           

provided observations should be either in ascending or descending order of magnitude.

After that, we replace the observations which are above the median by  a symbol ‘A’ (say) and the observation which are below the median by a symbol ‘B’ (say) without altering the observed order. The observations which are equal to median are discarded form the analysis and let reduced size of the sample denoted by n.

 

Step 2: Counts number of times first symbol (A) occurs and denote it by n₁

 

Step 3: Counts number of times second symbol (B) occurs and denote it by n₂ where, n = n₁+n₂

 

Step 4: For testing the null hypothesis, the test statistic is the total number of runs so in this step we count total number of runs in the sequence of symbols and denote it by R.

 

Step 5: Obtain critical values of test statistic corresponding n₁,n₂  at α % level of significance under the condition that null hypothesis is true.  From the Table  of critical value for run test is used to obtain respectively lower (RL) and upper (RU) critical values of the number of runs for a given combination of n1 and n2 at 5% level of significance.

Note1: Generally, critical values for run test are available at 5% level of

significance so we test our hypotheses for 5% level of significance.

Step 6: Decision Rule:

To take the decision about null hypothesis, the test statistic is compared with the critical (tabulated) values.

 

If the observed number of runs(R) is either less than or equal to the lower critical value (RL) or greater than or equal to the upper critical value (RU), that is, if R < RL or R > RU then we reject the null hypothesis at 5% level of significance.

 

If R lies between  R_l and R_u , that is,  R_l < R> R_u, then we Accept

 null hypothesis at 5% level of significance