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Dataplot Vol 2 Vol 1

RUKHIN 3 TEST

Name:
    RUKHIN 3 TEST (LET)
Type:
    Let Subcommand
Purpose:
    For four binomial proportions, compute the p-value and lower and upper confidence limits for

      H0: p1*p2 = p3*p4

    with p1, p2, p3, and p4 denoting the four binomial proportions.

Description:
    For each of the binomial proportions, we actually use the correction terms

      pi = (xi + 0.5)/(ni + 1)

    with pi xi, and ni denoting the binomial proportion, the number of successes and the number of trials for the i-th binomial proportion.

    The computed test statistic is

      p1*p2 - p3*p4

    and the associated standard error is

      SE = SQRT(term1 + term2 + term3 + term4)  where
      term1 = (p2**2)*p1*(1 - p1)/n1
      term2 = (p1**2)*p2*(1 - p2)/n2
      term3 = (p4**2)*p3*(1 - p3)/n3
      term4 = (p3**2)*p4*(1 - p4)/n4

    The p-value for a lower tailed test is

      1 - NORCDF((p3*p4 - p1*p2)/SE)

    The p-value for an upper tailed test is

      1 - NORCDF((p1*p2 - p3*p4)/SE)

    The p-value for a two tailed test is

      2*(1 - NORCDF((p3*p4 - p1*p2)/SE)

    The confidence interval is

      (p1*p2 - p3*p4) +/- NORPPF(ALPHA/2)*SE

    Although this is a fairly specialized test, it does have applicability in the following test scenario. Suppose we are testing scanning devices to see if they can detect certain radionuclides. Further suppose we have a "primary" screening device and then a "secondary" screening device. For example, the "primary" device may be an alarm which then sends the object to a more accurate (but more costly or more time consuming) device to perform an id. In this case, the binomial success for the primary device is that it alarms when the radionuclide is present and the binomial success for the secondary device is that it returns the correct id. The RUKHIN 3 test could then be used to compare the performance of two different sets of primary/secondary devices.

Syntax 1:
    LET <pval> <lowlim> <upplim> = RUKHIN 3 TEST
                            <p1> <n1> <p2> <n2> <p3> <n3> <p4> <n4> <alpha>
                            <SUBSET/EXCEPT/FOR qualification>
    where <p1> is constant, parameter, or variable that contains the proportion of successes for the first sample;
                <n1> is constant, parameter, or variable that contains the number of trials for the first sample;
                <p2> is constant, parameter, or variable that contains the proportion of successes for the second sample;
                <n2> is constant, parameter, or variable that contains the number of trials for the second sample;
                <p3> is constant, parameter, or variable that contains the proportion of successes for the third sample;
                <n3> is constant, parameter, or variable that contains the number of trials for the third sample;
                <p4> is constant, parameter, or variable that contains the proportion of successes for the fourth sample;
                <n4> is constant, parameter, or variable that contains the number of trials for the fourth sample;
                <alpha> is constant or parameter that contains the significance level;
                <pval> contains the returned p-value;
                <lowlim> contains the computed lower confidence limit;
                <upplim> contains the computed upper confidence limit;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs a two-tailed hypothesis test.

    The <p1> <n1>, <p2>, <n2>, <p3>, <n3>, <p4>, and <n4> arguments can be either parameters or variables. If they are variables, then the variables must have the same number of elements. The <alpha> argument is always assumed to be either a constant or a parameter.

    If the arguments are all parameters, then <pval>, <lowlim>, and <upplim> will be parameters. Otherwise, they will be variables.

Syntax 2:
    LET <pval> <lowlim> <upplim> = RUKHIN 3 LOWER TAIL TEST
                            <p1> <n1> <p2> <n2> <p3> <n3> <p4> <n4> <alpha>
                            <SUBSET/EXCEPT/FOR qualification>
    where <p1> is constant, parameter, or variable that contains the proportion of successes for the first sample;
                <n1> is constant, parameter, or variable that contains the number of trials for the first sample;
                <p2> is constant, parameter, or variable that contains the proportion of successes for the second sample;
                <n2> is constant, parameter, or variable that contains the number of trials for the second sample;
                <p3> is constant, parameter, or variable that contains the proportion of successes for the third sample;
                <n3> is constant, parameter, or variable that contains the number of trials for the third sample;
                <p4> is constant, parameter, or variable that contains the proportion of successes for the fourth sample;
                <n4> is constant, parameter, or variable that contains the number of trials for the fourth sample;
                <alpha> is constant or parameter that contains the significance level;
                <pval> contains the returned p-value;
                <lowlim> contains the computed lower confidence limit;
                <upplim> contains the computed upper confidence limit;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs a lower-tailed hypothesis test. However, the confidence limits correspond to the two-tailed test.

    The <p1> <n1>, <p2>, <n2>, <p3>, <n3>, <p4>, and <n4> arguments can be either parameters or variables. If they are variables, then the variables must have the same number of elements. The <alpha> argument is always assumed to be either a constant or a parameter.

    If the arguments are all parameters, then <pval>, <lowlim>, and <upplim> will be parameters. Otherwise, they will be variables.

Syntax 3:
    LET <pval> <lowlim> <upplim> = RUKHIN 3 UPPER TAIL TEST
                            <p1> <n1> <p2> <n2> <p3> <n3> <p4> <n4> <alpha>
                            <SUBSET/EXCEPT/FOR qualification>
    where <p1> is constant, parameter, or variable that contains the proportion of successes for the first sample;
                <n1> is constant, parameter, or variable that contains the number of trials for the first sample;
                <p2> is constant, parameter, or variable that contains the proportion of successes for the second sample;
                <n2> is constant, parameter, or variable that contains the number of trials for the second sample;
                <p3> is constant, parameter, or variable that contains the proportion of successes for the third sample;
                <n3> is constant, parameter, or variable that contains the number of trials for the third sample;
                <p4> is constant, parameter, or variable that contains the proportion of successes for the fourth sample;
                <n4> is constant, parameter, or variable that contains the number of trials for the fourth sample;
                <alpha> is constant or parameter that contains the significance level;
                <pval> contains the returned p-value;
                <lowlim> contains the computed lower confidence limit;
                <upplim> contains the computed upper confidence limit;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs an upper-tailed hypothesis test. However, the confidence limits correspond to the two-tailed test.

    The <p1> <n1>, <p2>, <n2>, <p3>, <n3>, <p4>, and <n4> arguments can be either parameters or variables. If they are variables, then the variables must have the same number of elements. The <alpha> argument is always assumed to be either a constant or a parameter.

    If the arguments are all parameters, then <pval>, <lowlim>, and <upplim> will be parameters. Otherwise, they will be variables.

Examples:
    LET PVAL AL AU = RUKHIN 3 TEST P1 N1 P2 N2 P3 N3 P4 N4 ALPHA
    LET PVAL AL AU = RUKHIN 3 LOWER TAILED TEST ...
                P1 N1 P2 N2 P3 N3 P4 N4 ALPHA
    LET PVAL AL AU = RUKHIN 3 UPPER TAILED TEST ...
                P1 N1 P2 N2 P3 N3 P4 N4 ALPHA
Default:
    None
Synonyms:
    RUKHIN THREE is a synonym for RUKHIN 3
Related Commands: Reference:
    Private communication with Andrew Rukhin and Bill Strawderman.
Applications:
    Statistics
Implementation Date:
    2010/06
Program:
     
    LET P1 = 0.95
    LET N1 = 40
    LET P2 = 0.8
    LET N2 = 40
    LET P3 = 0.95
    LET N3 = 40
    LET P4 = 0.6
    LET N4 = 40
    LET ALPHA = 0.90
    .
    LET PVAL AL AU = RUKHIN 3 TEST P1 N1 P2 N2 P3 N3 P4 N4 ALPHA
        
    The returned values of PVAL, AL, and AU are 0.0715, 0.0160, and 0.3505, respectively.

Date created: 10/5/2010
Last updated: 10/5/2010
Please email comments on this WWW page to alan.heckert@nist.gov.