BIWEIGHT CONFIDENCE LIMITS

BIWEIGHT CONFIDENCE LIMITS

Analysis Command

Generates a biweight based confidence interval for the location of a variable.

Mosteller and Tukey (see Reference section below) define two types of robustness:
1. resistance means that changing a small part, even by a large amount, of the data does not cause a large change in the estimate

2. robustness of efficiency means that the statistic has high efficiency in a variety of situations rather than in any one situation. Efficiency means that the estimate is close to optimal estimate given that we distribution that the data comes from. A useful measure of efficiency is:
   Efficiency = (lowest variance feasible)/(actual variance)

Many statistics have one of these properties. However, it can be difficult to find statistics that are both resistant and have robustness of efficiency.

Standard confidence intervals are base in the mean and variance. These are the optimal estimators if the data are in fact from a Gaussian population. However, they lack both resistance and robustness of efficiency. The biweight confidence interval is based on estimates of of location and scale that are both resistant and have robustness of efficiency. Therefore it should provide a reasonable confidence interval when the normality assumption cannot be validated. Note that it is still a symmetric confidence interval. However, symmetry is a much looser assumption than normality.

The biweight confidence interval for the population biweight location is defined by:

where the biweight location and biweight scale are location and scale estimators based on the biweight and = 0.7*(n-1). The definitions for the biweight location and biweight scale estimators are given in:

HELP BIWEIGHT LOCATION
HELP BIWEIGHT SCALE

BIWEIGHT CONFIDENCE LIMITS <y>           <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable, and where the <SUBSET/EXCEPT/FOR qualification> is optional.

BIWEIGHT CONFIDENCE LIMITS Y1
BIWEIGHT CONFIDENCE LIMITS Y1 SUBSET TAG > 2

A table of confidence intervals is printed for alpha levels of 50.0, 75.0, 90.0, 95.0, 99.0, 99.9, 99.99, and 99.999. The sample biweight location estiamte and sample biweight scale estimate are also printed. The t-value and t-value * are printed in the table.

None

None

CONFIDENCE LIMITS	= Compute a Gaussian based confidence limit.
T-TEST	= Perform a t-test.
BIWEIGHT LOCATION	= Compute a biweight location estimate.
BIWEIGHT SCALE	= Compute a biweight scale estimate.

"Data Analysis and Regression: A Second Course in Statistics", Mosteller and Tukey, Addison-Wesley, 1977, pp. 203-209.

Robust Data Analysis

2001/11

 
    LET Y1 = NORMAL RANDOM NUMBERS FOR I = 1 1 100
    LET Y2 = LOGISTIC RANDOM NUMBERS FOR I = 1 1 100
    LET Y3 = CAUCHY RANDOM NUMBERS FOR I = 1 1 100
    LET Y4 = DOUBLE EXPONENTIAL RANDOM NUMBERS FOR I = 1 1 100
    BIWEIGHT CONFIDENCE LIMTIS Y1
    BIWEIGHT CONFIDENCE LIMTIS Y2
    BIWEIGHT CONFIDENCE LIMTIS Y3
    BIWEIGHT CONFIDENCE LIMTIS Y4
    

Dataplot generates the following output:

           *************************************
           **  BIWEIGHT CONFIDENCE LIMITS Y1  **
           *************************************
      
      
                        CONFIDENCE LIMITS FOR BIWEIGHT LOCATION
                                (2-SIDED)
      
               NUMBER OF OBSERVATIONS     =      100
               BIWEIGHT LOCATION          =   0.1272405E-01
               BIWEIGHT SCALE             =   0.7815597
               STANDARD ERROR             =   0.8840586E-01
               DEGREES OF FREEDOM         =              69
      
        CONFIDENCE   T     T X STDERR       LOWER         UPPER
        VALUE (%)  VALUE                    LIMIT         LIMIT
     ---------------------------------------------------------------
          50.000   0.678  0.599447E-01  -.472206E-01  0.726687E-01
          75.000   1.160  0.102561      -.898370E-01  0.115285
          90.000   1.667  0.147394      -.134670      0.160118
          95.000   1.995  0.176365      -.163641      0.189089
          99.000   2.649  0.234185      -.221461      0.246909
          99.900   3.437  0.303870      -.291146      0.316594
          99.990   4.130  0.365145      -.352421      0.377869
          99.999   4.768  0.421484      -.408760      0.434208
      
      
           *************************************
           **  BIWEIGHT CONFIDENCE LIMITS Y2  **
           *************************************
      
      
                        CONFIDENCE LIMITS FOR BIWEIGHT LOCATION
                                (2-SIDED)
      
               NUMBER OF OBSERVATIONS     =      100
               BIWEIGHT LOCATION          =   0.9524417E-01
               BIWEIGHT SCALE             =    3.525512
               STANDARD ERROR             =   0.1877635
               DEGREES OF FREEDOM         =              69
      
        CONFIDENCE   T     T X STDERR       LOWER         UPPER
        VALUE (%)  VALUE                    LIMIT         LIMIT
     ---------------------------------------------------------------
          50.000   0.678  0.127315      -.320711E-01  0.222559
          75.000   1.160  0.217827      -.122583      0.313072
          90.000   1.667  0.313047      -.217802      0.408291
          95.000   1.995  0.374578      -.279334      0.469822
          99.000   2.649  0.497381      -.402137      0.592625
          99.900   3.437  0.645383      -.550139      0.740628
          99.990   4.130  0.775525      -.680280      0.870769
          99.999   4.768  0.895182      -.799938      0.990426
      
      
           *************************************
           **  BIWEIGHT CONFIDENCE LIMITS Y3  **
           *************************************
      
      
                        CONFIDENCE LIMITS FOR BIWEIGHT LOCATION
                                (2-SIDED)
      
               NUMBER OF OBSERVATIONS     =      100
               BIWEIGHT LOCATION          =   0.1851178
               BIWEIGHT SCALE             =    2.860590
               STANDARD ERROR             =   0.1691328
               DEGREES OF FREEDOM         =              69
      
        CONFIDENCE   T     T X STDERR       LOWER         UPPER
        VALUE (%)  VALUE                    LIMIT         LIMIT
     ---------------------------------------------------------------
          50.000   0.678  0.114683      0.704353E-01  0.299800
          75.000   1.160  0.196214      -.110959E-01  0.381331
          90.000   1.667  0.281985      -.968669E-01  0.467103
          95.000   1.995  0.337411      -.152293      0.522529
          99.000   2.649  0.448029      -.262911      0.633147
          99.900   3.437  0.581346      -.396228      0.766464
          99.990   4.130  0.698574      -.513456      0.883692
          99.999   4.768  0.806358      -.621240      0.991476
      
      
           *************************************
           **  BIWEIGHT CONFIDENCE LIMITS Y4  **
           *************************************
      
      
                        CONFIDENCE LIMITS FOR BIWEIGHT LOCATION
                                (2-SIDED)
      
               NUMBER OF OBSERVATIONS     =      100
               BIWEIGHT LOCATION          =  -0.5124480E-02
               BIWEIGHT SCALE             =   0.9395723
               STANDARD ERROR             =   0.9693154E-01
               DEGREES OF FREEDOM         =              69
      
        CONFIDENCE   T     T X STDERR       LOWER         UPPER
        VALUE (%)  VALUE                    LIMIT         LIMIT
     ---------------------------------------------------------------
          50.000   0.678  0.657256E-01  -.708501E-01  0.606011E-01
          75.000   1.160  0.112452      -.117576      0.107327
          90.000   1.667  0.161608      -.166732      0.156484
          95.000   1.995  0.193373      -.198498      0.188249
          99.000   2.649  0.256769      -.261894      0.251645
          99.900   3.437  0.333174      -.338299      0.328050
          99.990   4.130  0.400359      -.405483      0.395234
          99.999   4.768  0.462131      -.467256      0.457007