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Math Help - Frequency of testing....

  1. #1
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    Frequency of testing....

    Hi all. I'm trying to calculate how frequently an employer would need to perform random drug testing in order to achieve a desired probability of detecting a drug user within a certain time.

    Here's the relevant facts:
    There is no upper or lower limit to the frequency of testing. Subjects can be tested every day if required, or once a year.
    If a subject is using drugs it will be evident on testing for 4 days...that is, the day of use and the next 3 days.
    I need to achieve a 99.9% chance of detecting drug use within a 60 day period.

    So, here is the question....how frequently (on average) should drug testing be applied in order to achieve this 99.9% chance of detection within 60 days? I'm not talking about a regular, fixed schedule of testing - rather it will be random, with a given probability of testing each day.

    If someone could give me a formula to apply, or even the priniciples used to derive a formula this would be most helpful, because then I could experiment with the numbers a bit myself.

    Thanks in advance !
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  2. #2
    MHF Contributor
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    60 days vs 4 days

    This is only 15 periods. 1/15 = 0.06666666. This is FAR greater than your tolerance of 0.001.

    In other words, you must test very often because you cannot afford (according to your tolerance) to miss even one instance.

    All year 365 days vs 4 days is 91.25 perioods, giving 0.011, still much greater than your hoped-for tolerance.

    1) You may wish to rethink your tolerance.
    2) How many subjects?
    3) Draw a picture.

    10 days vs 4 days

    Catch-Clear-Clear-Clear-Clear-Clear-Clear-Clear-Clear-Clear
    Catch-Catch-Clear-Clear-Clear-Clear-Clear-Clear-Clear-Clear
    Catch-Catch-Catch-Clear-Clear-Clear-Clear-Clear-Clear-Clear
    Use-Catch-Catch-Catch-Clear-Clear-Clear-Clear-Clear-Clear
    Clear-Use-Catch-Catch-Catch-Clear-Clear-Clear-Clear-Clear
    Clear-Clear-Use-Catch-Catch-Catch-Clear-Clear-Clear-Clear
    Clear-Clear-Clear-Use-Catch-Catch-Catch-Clear-Clear-Clear
    Clear-Clear-Clear-Clear-Use-Catch-Catch-Catch-Clear-Clear
    Clear-Clear-Clear-Clear-Clear-Use-Catch-Catch-Catch-Clear
    Clear-Clear-Clear-Clear-Clear-Clear-Use-Catch-Catch-Catch
    Clear-Clear-Clear-Clear-Clear-Clear-Clear-Use-Catch-Catch
    Clear-Clear-Clear-Clear-Clear-Clear-Clear-Clear-Use-Catch
    Clear-Clear-Clear-Clear-Clear-Clear-Clear-Clear-Clear-Use

    On any day you test, you will catch only the users on that day and those who used 1, 2, or 3 days before the testing. This is part of your difficulty. With only a 10-day test, using can occur on any of 13 different days! It doesn't matter which day you pick, you will cover only 4 days of users if you do only one day. 4/13 = 30.77% - Again, nowhere near your 0.1%.

    Pick another day, 4 days away, of course, and increase by 4. So, 1, 5, 9 will get 12/13 = 92.31%.

    60 days, then, has 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57 - 15 testing days.

    Since 60 is a little irritating for this arrangement, you miss quite a few at the end. Picking 58 instead for 57 won't make any difference. You'll just get a different 4. In any case, you cover (15*4)/63 = 95.24%.

    However, if you extended your study period to day 61, the results certainly are improved. You add a day, but you also add a testing day, giving (16*4)/64 = 100%

    Pretty irritating, isn't it? You cannot make small adjustments.

    Note: This considers only one (1) subject. As your population grows, adjustments will get smaller and it will become more tractable. When will you be able to make an adjustment so small as 0.1%? Good thought question.
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