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Dataplot Vol 2 Vol 1

BINOMIAL PRODUCT CONFIDENCE LIMITS

Name:
    BINOMIAL PRODUCT CONFIDENCE LIMITS (LET)
Type:
    Let Subcommand
Purpose:
    Compute the standard error and confidence limits for the product of two binomial proportions.
Description:
    Given two binomial proportions, p1 and p2, with associated sample sizes, n1 and n2, the formula for the confidence interval for the product p1*p2 is

      p1*p2  +/-  NORPPF(alpha/2)*SE

    NORPPF-1 denoting the normal percent point function and SE denoting the standard error of p1*p2.

      SE = SQRT(p2**2*p1*(1 - p1)/n1 + p1**2*p2*(1 - p2)/n2)

    In the above formula, we actually use the Bayes estimators of p1 and p2:

      p1 = (V + 0.5)/(n1 + 1)
      p2 = (U + 0.5)/(n2 + 1)

    with V and U denoting the number of successes in the binomial trials.

Syntax:
    LET <se> <lowlim> <upplim> = BINOMIAL PRODUCT CONFIDENCE LIMITS
                            <p1> <n1> <p2> <n2> <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;
                <alpha> is constant or parameter that contains the significance level;
                <se> is a variable that contains the standard error of the computed product (p1*p2);
                <lowlim> is a variable that contains the computed lower confidence limit;
                <upplim> is a variable that contains the computed upper confidence limit;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    The <p1> <n1> <p2> and <n2> 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 <p1> <n1> <p2> and <n2> are all parameters, then <se>, <lowlim>, and <upplim> will be parameters. Otherwise, they will be variables.

Examples:
    LET SE AL AU = BINOMIAL PRODUCT CONFIDENCE LIMITS ...
                            P1 N1 P2 N2 ALPHA
Note:
    If you would like to use this command on raw data (i.e., you have variables containing a sequence of 0's and 1's), do something like the following

      LET YSUM1 = SUM Y1
      LET N1 = SIZE Y1
      LET P1 = YSUM1/N1
      LET YSUM2 = SUM Y2
      LET N2 = SIZE Y2
      LET P2 = YSUM2/N2
      LET SE AL AU = BINOMIAL PRODUCT CONFIDENCE LIMITS ...
                  P1 N1 P2 N2 ALPHA

    If you have group-id variables (X1 and X2), you would do something like

      SET LET CROSS TABULATE COLLAPSE
      LET YSUM1 = CROSS TABULATE SUM Y1 X1
      LET N1 = CROSS TABULATE SIZE Y1 X1
      LET P1 = YSUM1/N1
      LET YSUM2 = CROSS TABULATE SUM Y2 X2
      LET N2 = CROSS TABULATE SIZE Y2 X2
      LET P2 = YSUM2/N2
      LET SE AL AU = BINOMIAL PRODUCT CONFIDENCE LIMITS ...
                  P1 N1 P2 N2 ALPHA

    In this case, P1, N1, P2, and N2 are now variables rather than parameters.

Default:
    None
Synonyms:
    None
Related Commands: Reference:
    Private communication with Andrew Ruhkin and Bill Strawderman.
Applications:
    Statistics
Implementation Date:
    2010/07
Program:
     
    LET P1 = 8/12
    LET N1 = 12
    LET P2 = 7/8
    LET N2 = 8
    LET ALPHA = 0.95
    .
    LET BPROD = P1*P2
    LET SE LOWLIM UPPLIM = BINOMIAL PRODUCT CONFIDENCE LIMIT ...
        P1 N1 P2 N2 ALPHA
        
    The values of SE, LOWLIM, and UPPLIM are 0.14325, 0.26411, and 0.8256, respectively.

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