SED navigation bar go to SED home page go to Dataplot home page go to NIST home page SED Home Page SED Staff SED Projects SED Products and Publications Search SED Pages
Dataplot Vol 2 Vol 1

BINOMIAL RATIO CONFIDENCE LIMITS

Name:
    BINOMIAL RATIO CONFIDENCE LIMITS (LET)
Type:
    Let Subcommand
Purpose:
    Compute the confidence limits for the ratio 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 ratio p1/p2 is

      (p1/p2)*EXP{+/-NORPPF(alpha/2)*SQRT((1-p1)/(n1*p1) + (1-p2)/(n2*p2))}

    with NORPPF-1 denoting the normal percent point function.

    If p1 and p2 are both 1, the resulting uncertainty is zero. If either p1 or p2 is zero, we have division by zero in the above formula. For that reason, we 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. We then use these updated estimates for p1 and p2 in the above formula.

Syntax:
    LET <ratio> <lowlim> <upplim> = BINOMIAL RATIO 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;
                <ratio> is a variable that contains the computed ratio (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 <ratio> <lowlim> and <upplim> will be parameters. Otherwise, they will be variables.

Examples:
    LET RATIO AL AU = BINOMIAL RATIO 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 RATIO AL AU = BINOMIAL RATIO 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 RATIO AL AU = BINOMIAL RATIO 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:
    2009/10
Program:
     
    LET P1 = 8/12
    LET N1 = 12
    LET P2 = 7/8
    LET N2 = 8
    LET ALPHA = 0.05
    .
    LET ARATIO LOWLIM UPPLIM = BINOMIAL RATIO CONFIDENCE LIMIT ...
        P1 N1 P2 N2 ALPHA
        
    The values of ARATIO, LOWLIM, and UPPLIM are 0.7846, 0.4687, and 1.3135, respectively.

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