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Dataplot Vol 1 Auxiliary Chapter

ODDS RATIO STANDARD ERROR

Name:
    ODDS RATIO STANDARD ERROR (LET)
Type:
    Let Subcommand
Purpose:
    Compute the standard error of the bias corrected odds ratio between two binary variables.
Description:
    Given two variables where each variable has exactly two possible outcomes (typically defined as success and failure), we define the odds ratio as:

      o = (N11/N12)/ (N21/N22)
          = (N11N22)/ (N12N21)

    where

      N11 = number of successes in sample 1
      N21 = number of failures in sample 1
      N12 = number of successes in sample 2
      N22 = number of failures in sample 2

    The first definition shows the meaning of the odds ratio clearly, although it is more commonly given in the literature with the second definition.

    Alternatively, the odds ratio can be given in terms of the proportions

      o = (p11/p12)/ (p21/p22)
          = (p11p22)/ (p12p21)

    where

      p11 = N11/ (N11 + N21)
            = proportion of successes in sample 1
      p21 = N21/ (N11 + N21)
            = proportion of failures in sample 1
      p12 = N12/ (N12 + N22)
            = proportion of successes in sample 2
      p22 = N22/ (N12 + N22)
            = proportion of failures in sample 2

    Success and failure can denote any binary response. Dataplot expects "success" to be coded as "1" and "failure" to be coded as "0".

    Dataplot actually returns the bias corrected version of the statistic:

      o' = {(N11+0.5) (N22+0.5)}/ {(N12+0.5) (N21+0.5)}

    In addition to reducing bias, this statistic also has the advantage that the odds ratio is still defined even when N12 or N21 is zero (the uncorrected statistic will be undefined for these cases).

    The standard error of this bias corrected odds ratio is then

      SE(o') = o'*SQRT{1/(N11+0.5) + 1/(N21+0.5) + 1/(N12+0.5) + 1/(N22+0.5)}

    where o' is the bias corrected odds ratio.

Syntax:
    LET <par> = ODDS RATIO STANDARD ERROR <y1> <y2>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> is the first response variable;
                <y2> is the second response variable;
                <par> is a parameter where the computed odds ratio standard error is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET A = ODDS RATIO STANDARD ERROR Y1 Y2
    LET A = ODDS RATIO STANDARD ERROR Y1 Y2 SUBSET TAG > 2
Note:
    The two variables need not have the same number of elements.
Note:
    There are two ways you can define the response variables:

    1. Raw data - in this case, the variables contain 0's and 1's.

      If the data is not coded as 0's and 1's, Dataplot will check for the number of distinct values. If there are two distinct values, the minimum value is converted to 0's and the maximum value is converted to 1's. If there is a single distinct value, it is converted to 0's if it is less than 0.5 and to 1's if it is greater than or equal to 0.5. If there are more than two distinct values, an error is returned.

    2. Summary data - if there are two observations, the data is assummed to be the 2x2 summary table. That is,

        Y1(1) = N11
        Y1(2) = N21
        Y2(1) = N12
        Y2(2) = N22
Note:
    The following additional commands are supported

      TABULATE ODDS RATIO STANDARD ERROR Y1 Y2 X
      CROSS TABULATE ODDS RATIO STANDARD ERROR Y1 Y2 X1 X2

      ODDS RATIO STANDARD ERROR PLOT Y1 Y2 X
      CROSS TABULATE ODDS RATIO STANDARD ERROR PLOT Y1 Y2 X1 X2

      BOOTSTRAP ODDS RATIO STANDARD ERROR PLOT Y1 Y2
      JACKNIFE ODDS RATIO STANDARD ERROR PLOT Y1 Y2

    Note that the above commands expect the variables to have the same number of observations. If the two samples are in fact of different sizes, there are two ways to address the issue:

    1. Y1 and Y2 can contain the summary data. That is,

        Y1(1) = N11
        Y1(2) = N21
        Y2(1) = N12
        Y2(2) = N22

      This is a useful option in that the data is sometimes only available in summary form. Note that this will not work for the BOOTSTRAP PLOT and JACKNIFE PLOT commands (these require raw data).

    2. You can specify a missing value for the smaller sample. For example, if Y1 has 100 observations and Y2 has 200 observations, you can do something like

        SET STATISTIC MISSING VALUE -99
        LET Y1 = -99 FOR I = 101 1 200
Default:
    None
Synonyms:
    STANDARD ERROR ODDS RATIO
Related Commands: Reference:
    Fleiss, Levin, and Paik (2003), "Statistical Methods for Rates and Proportions", Third Edition, Wiley, chapter 1.
Applications:
    Categorical Data Analysis
Implementation Date:
    2007/5
Program:
     
    let n = 1
    .
    let p = 0.2
    let y1 = binomial rand numb for i = 1 1 100
    let p = 0.1
    let y2 = binomial rand numb for i = 1 1 100
    .
    let p = 0.4
    let y1 = binomial rand numb for i = 101 1 200
    let p = 0.08
    let y2 = binomial rand numb for i = 101 1 200
    .
    let p = 0.15
    let y1 = binomial rand numb for i = 201 1 300
    let p = 0.18
    let y2 = binomial rand numb for i = 201 1 300
    .
    let p = 0.6
    let y1 = binomial rand numb for i = 301 1 400
    let p = 0.45
    let y2 = binomial rand numb for i = 301 1 400
    .
    let p = 0.3
    let y1 = binomial rand numb for i = 401 1 500
    let p = 0.1
    let y2 = binomial rand numb for i = 401 1 500
    .
    let x = sequence 1 100 1 5
    .
    let a = odds ratio standard error y1 y2 subset x = 1
    tabulate odds ratio standard error y1 y2 x
    .
    label case asis
    xlimits 1 5
    major xtic mark number 5
    minor xtic mark number 0
    xtic mark offset 0.5 0.5
    y1label Bias Corrected Odds Ratio Standard Error
    x1label Group ID
    character x blank
    line blank solid
    .
    odds ratio standard error plot y1 y2 x
        
    plot generated by sample program

Date created: 6/6/2007
Last updated: 6/6/2007
Please email comments on this WWW page to alan.heckert@nist.gov.