 Dataplot Vol 1 Vol 2

# BIWEIGHT CONFIDENCE LIMITS

Name:
BIWEIGHT CONFIDENCE LIMITS
Type:
Analysis Command
Purpose:
Generates a biweight based confidence interval for the location of a variable.
Description:
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:

$$\mbox{biweight confidence limits } = \mbox{biweight location } \pm t_{\nu}\sqrt{\frac{\mbox{biweight scale}}{n}}$$

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

HELP BIWEIGHT LOCATION
HELP BIWEIGHT SCALE
Syntax:
BIWEIGHT CONFIDENCE LIMITS <y>           <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable, and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
BIWEIGHT CONFIDENCE LIMITS Y1
BIWEIGHT CONFIDENCE LIMITS Y1 SUBSET TAG > 2
Note:
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 * $$\sqrt{s_{bi}^2}$$ are printed in the table.
Default:
None
Synonyms:
None
Related Commands:
 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.
Reference:
Mosteller and Tukey (1977), "Data Analysis and Regression: A Second Course in Statistics," Addison-Wesley, pp. 203-209.
Applications:
Robust Data Analysis
Implementation Date:
2001/11
Program:

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
SET WRITE DECIMALS 4
BIWEIGHT CONFIDENCE LIMTIS Y1 TO Y4

Dataplot generates the following output:
             Confidence Limits for Biweight Location
(Two-Sided)

Response Variable: Y1

Summary Statistics:
Number of Observations:                             100
Sample Biweight Location:                       0.01272
Sample Biweight Scale                           0.78155
Standard Error:                                 0.08840
Degrees of Freedom:                            69.00000

-----------------------------------------------------------------
Confidence       t      t-Value X          Lower          Upper
Value (%)   Value         StdErr          Limit          Limit
-----------------------------------------------------------------
50.000   0.678        0.05994       -0.04722        0.07266
75.000   1.160        0.10256       -0.08983        0.11528
90.000   1.667        0.14739       -0.13466        0.16011
95.000   1.994        0.17636       -0.16364        0.18908
99.000   2.648        0.23418       -0.22146        0.24690
99.900   3.437        0.30386       -0.29114        0.31659
99.990   4.130        0.36514       -0.35242        0.37787
99.999   4.767        0.42151       -0.40879        0.43424

Confidence Limits for Biweight Location
(Two-Sided)

Response Variable: Y2

Summary Statistics:
Number of Observations:                             100
Sample Biweight Location:                       0.09524
Sample Biweight Scale                           3.52551
Standard Error:                                 0.18776
Degrees of Freedom:                            69.00000

-----------------------------------------------------------------
Confidence       t      t-Value X          Lower          Upper
Value (%)   Value         StdErr          Limit          Limit
-----------------------------------------------------------------
50.000   0.678        0.12731       -0.03207        0.22255
75.000   1.160        0.21782       -0.12258        0.31307
90.000   1.667        0.31304       -0.21780        0.40829
95.000   1.994        0.37457       -0.27933        0.46982
99.000   2.648        0.49738       -0.40213        0.59262
99.900   3.437        0.64537       -0.55013        0.74062
99.990   4.130        0.77553       -0.68028        0.87077
99.999   4.767        0.89525       -0.80000        0.99049

Confidence Limits for Biweight Location
(Two-Sided)

Response Variable: Y3

Summary Statistics:
Number of Observations:                             100
Sample Biweight Location:                       0.18511
Sample Biweight Scale                           2.86058
Standard Error:                                 0.16913
Degrees of Freedom:                            69.00000

-----------------------------------------------------------------
Confidence       t      t-Value X          Lower          Upper
Value (%)   Value         StdErr          Limit          Limit
-----------------------------------------------------------------
50.000   0.678        0.11468        0.07043        0.29980
75.000   1.160        0.19621       -0.01109        0.38133
90.000   1.667        0.28198       -0.09686        0.46710
95.000   1.994        0.33741       -0.15229        0.52252
99.000   2.648        0.44802       -0.26291        0.63314
99.900   3.437        0.58134       -0.39622        0.76645
99.990   4.130        0.69858       -0.51346        0.88369
99.999   4.767        0.80641       -0.62130        0.99153

Confidence Limits for Biweight Location
(Two-Sided)

Response Variable: Y4

Summary Statistics:
Number of Observations:                             100
Sample Biweight Location:                      -0.00512
Sample Biweight Scale                           0.93957
Standard Error:                                 0.09693
Degrees of Freedom:                            69.00000

-----------------------------------------------------------------
Confidence       t      t-Value X          Lower          Upper
Value (%)   Value         StdErr          Limit          Limit
-----------------------------------------------------------------
50.000   0.678        0.06572       -0.07085        0.06060
75.000   1.160        0.11245       -0.11757        0.10732
90.000   1.667        0.16160       -0.16673        0.15648
95.000   1.994        0.19337       -0.19849        0.18824
99.000   2.648        0.25676       -0.26189        0.25164
99.900   3.437        0.33317       -0.33829        0.32804
99.990   4.130        0.40036       -0.40548        0.39523
99.999   4.767        0.46216       -0.46729        0.45704


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Date created: 11/21/2001
Last updated: 10/09/2015