In this simple lesson, we are going to explain in very simple and clear terms, the concept of run test in statiscs<\/p>\n
Content<\/b><\/p>\n <\/p>\n Run test is a statistical test used to determine of the data obtained from a sample is ramdom. That is why it is called Run Test for Randomness. <\/p>\n A run is a sequence of similar or like events, items or symbols that is preceded by and followed by an event, item or symbol of a different type,\u00a0 or by none at all. <\/p>\n A data scientist carrying out a research interviewed 10 persons during a survey. We denote the genders of the poeple by M for men and W for women. Scenario 1<\/i> Scenario 2 Scenario 3 Scenario 1 has only 2 runs and therefore the scenario cannot be considered random because there are to few runs<\/p>\n Scenario 2 has too many runs, 5 runs. And therefore would not be considered as random<\/p>\n Scenario 3 has 5 runs and therefore we need to perform a test to determine the randomness of the data.<\/p>\n <\/p>\n First we need to assume that the data available for the analysis consistes of a sequence of observations, recorded in order of occurence, which we can categorize into two mutually exclusive types.<\/p>\n First, you need to determine the total sample size, then the number of observation ofeach type as presented below:<\/p>\n n = total sample size Hypothesis <\/b> A. TWO-SIDED B. ONE-SIDED C. ONE-SIDED Test Statistic and Decision Rule<\/b> Critical Value<\/b> <\/p>\n <\/p>\n On a commuter train, the conductor want to see whether the passengers entering a train enter in a random manner. He observes the first 25 people, with the following sequence of males(M) and females(F).<\/p>\n F F F M M F F F F M F M M M F F F F M M F F F M M<\/p>\n Test for randomness at \u03b1 <\/span>= 0.05<\/p>\n Solution Steps<\/b><\/span> Step 2:<\/b> Find the test statistic (number of runs) FFF\u00a0 <\/span>MM \u00a0 FFFF \u00a0 <\/span>M\u00a0\u00a0 F \u00a0 <\/span>MMM\u00a0\u00a0 FFFF\u00a0\u00a0 <\/span>MM\u00a0\u00a0 FFF\u00a0\u00a0 <\/span>MM<\/p>\n Test statistic, r = 10 Step 3:<\/b> Find the critical value Step 4: <\/b>Make your decision Step 5:<\/b> Draw a Conclusion <\/p>\n <\/p>\n We have 20 people that enrolled in a drug abuse program. Test the claim that the ages of the people, according to the order in whihc they enroll occur at random, at \u03b1 <\/span>= 0.05.<\/p>\n The data are as follows: Solution Steps<\/b><\/span> The claim is the null hypothesis H0<\/sub> and the hypothesis is the alternate hypothesis H1<\/sub>. Step 2:<\/b> Find the test statistic (number of runs) Then compare the original data with the median. The replace the above median in the original sequence with an A if it is above the median and with B if it is below the median.(you can also use the mean instead of median) We can now arrange the data according to runs and we would have the output below: From the above\u00a0 we have<\/p>\n Test statistic, r = 9 Step 3:<\/b> Find the Critical Value Upper critical value = 5 Step 4:<\/b> Make the decision Step 5<\/b>: Draw your conclusion <\/p>\n <\/p>\n Table 1.0 shows the departures from normal of daily temperatures recorded at Atlanta, Georgia during February1969. We would like to know whether we may conclude that the pattern of departures above and below normal is the result of a non-random process.<\/p>\n\n
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1. What is Run Test<\/h3>\n
\nRandomness of the data is determined based on the number and nauture of runs present in the data of interest.<\/p>\n2. What is a Run?<\/h3>\n
\nRandomness of of the series\u00a0 is unlikely when there appear to be either too many or two few runs. In this case, a run test need to be carried out to determine the randomness.
\nThe Run Test when performed helps us to decide whether a sequence of events, items or symbol is the result of a random process.<\/p>\n3. Example of Runs<\/h3>\n
\nAssuming the respondents were chosen as follows:<\/p>\n
\nM M M M M F F F F F<\/p>\n
\nF M F M F M F M F M<\/p>\n
\nF F F M M F M M F F<\/p>\n4. Run Test Procedure<\/h3>\n
\nn1<\/sub> = the number of observation of one type
\nn2<\/sub> = the number of observations of the other type<\/p>\n
\nThen State the null and alternate hypothesis<\/p>\n
\nH0<\/sub>: the pattern of occurence is random
\nH1<\/sub>: the pattern of occurence is not random<\/p>\n
\nH0<\/sub>: the pattern of occurence is random
\nH1<\/sub>: the pattern of occurence is not random (because there are too few runs to be atributed as random)<\/p>\n
\nH0<\/sub>: the pattern of occurence is random
\nH1<\/sub>: the pattern of occurence is not random (because there are too few runs to be atributed as random)<\/p>\n
\nThe test statistic is r = total number of runs
\nThe decision rules is also called the acceptance or rejection criteria. It depends on the test statistic(calculated) and the value and the values of upper and lower limits(from statistical tables)<\/p>\n![](\"https:\/\/3.bp.blogspot.com\/-8U7dFbD6dMs\/WrlbemhIihI\/AAAAAAAABmQ\/SXhcRcX8HEsMDMtgz6p9b1mHdUwWIJQFACLcBGAs\/s640\/Run%2BTest.jpg\")
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\nCritical value is determined from statistical table using n1<\/sub> and n2<\/sub>
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\nWe can solve some examples to clarify this.<\/span>\u00a0<\/span><\/p>\n5. Example 1<\/h3>\n
\nStep 1<\/b>: State the null and alternate hypothesis
\nH0<\/sub>: The patter of occurence of males and females enter the train is random
\nH1<\/sub>: The pattern of occurence of males and females entering the train is not random<\/p>\n
\nYou can easily get this by grouping each run as shown below:<\/p>\n
\nn1<\/sub> = number of females = 15
\nn2<\/sub> = number of males = 10<\/p>\n
\nWe can find the lower and upper critical value from statistical run table
\nn1<\/sub> = 15, n2<\/sub> = 10
\nLower critical value = 7
\nUpper critical value = 18<\/p>\n
\nSince r = 10 which is between 7 and 18, we accept the null hypothesis (we fail to reject it)<\/p>\n
\nThere are not enough evidence to reject the claim hat the pattern of occurence of males and femals enter the train is determined by a random process<\/p>\n6. Example 2<\/h3>\n
\n18, 36, 19, 22, 25, 44, 23, 27, 27, 35, 19, 43, 37, 32, 28, 43, 46, 19, 20, 22<\/p>\n
\nStep 1:<\/b> State the hypothesis and identify the claim<\/p>\n
\nH0<\/sub>: The pattern of occurence of ages of the people enrolled in a drug abuse program is determined by a random process
\nH1<\/sub>: The pattern of occurence ofa ages of people enrolled in a drug abuse program is not random<\/p>\n
\nTo find the number of runs we first arrange the data\u00a0 in ascending order and find the median of the data set.<\/p>\n
\nI have done this using excel and the result is shown below:<\/p>\n![](\"https:\/\/1.bp.blogspot.com\/-bg3fO3xBZkQ\/WrloK2oDa-I\/AAAAAAAABmg\/VF-xs6SkqVA4hS9hiJCYUzn-Hud0DV8gwCLcBGAs\/s640\/Run%2BTest%2BExample%2B2.jpg\")
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\nB<\/span>\u00a0 A\u00a0 BBB \u00a0 <\/span>A\u00a0 B<\/span>\u00a0 A\u00a0 B<\/span>\u00a0 AAAAAA \u00a0 BBB<\/span><\/p>\n
\nn1<\/sub> = number or A runs = 9
\nn2<\/sub> = number of B runs =9<\/p>\n
\nn1<\/sub> = 9, n2<\/sub> = 9
\nFrom statistical table of Runs Test, we get the critical values<\/p>\n
\nLower critical value =15<\/p>\n
\nSince the statistic r is between the upper and lower critical values, we accept H0<\/sub><\/p>\n
\nThere is not enough evidence to reject the claim that the patter of occurence of ages of people in th program is determined by a random process<\/p>\n7. Example 3:<\/h3>\n
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