COMMON WEIBULL SHAPE PARAMETER

COMMON WEIBULL SHAPE PARAMETER

Analysis Command

Test whether k samples that follow a 2-parameter Weibull distribution have a common shape parameter.

In a number of Weibull modeling applications, it is desired to test whether different groups of the data follow 2-parameter Weibull distributions having a common shape parameter. For example, in testing fiber strength, it is commonly assumed that fibers of different gauge length will have a common shape shape parameter but differing scale parameters. This assumption can be tested using a test given by McCool (2012).

Before applying the test described here, each of the samples should be tested to ensure that they can in fact be adequately modeled with a 2-parameter Weibull distribution. If not, then applying this test is not meaningful.

Given k samples, the test procedure is as follows.

1. For each sample, fit a 2-parameter Weibull distribution. Identify the shape parameters as \( \hat{\beta}_{1}, \hat{\beta}_{2}, \ldots , \hat{\beta}_{k} \).

2. Determine the minimum and maximum values of the shape parameter.

3. The test statistic is
   \( \frac{\hat{\beta}_{\max}} {\hat{\beta}_{\min}} \)

   If all of the shape parameters are exactly equal, then this test statistic would be equal to one. Values of this test statistic that are significantly larger than one indicate that the shape parameters are not equal.

4. Critical values are determined via simulation. Specifically, generate k 2-parameter Weibull samples with a common shape parameter (Dataplot uses a value of 2 for the shape parameter and a value of 1 for the scale parameter). The ratio above is computed for 10,000 simulations.

COMMON WEIBULL SHAPE PARAMETER <y> <tag>
                        <SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable;
            <tag> is a group-id variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

The response variable and the group-id variables should have the same number of observations.

MULTIPLE COMMON WEIBULL SHAPE PARAMETER <y1> ... <yk>
                        <SUBSET/EXCEPT/FOR qualification>
where <y1> ... <yk> is a list of 2 to 30 response variables;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

The response variables do not need to have equal sample sizes.

REPLICATED COMMON WEIBULL SHAPE PARAMETER <y> <x1> ... <xk>
                        <SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable;
            <x1> ... <xk> is a list of 1 to 6 group-id variables;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

All variables should have the same number of observations.

COMMON WEIBULL SHAPE PARAMETER Y X
REPLICATED COMMON WEIBULL SHAPE PARAMETER Y X1 X2
MULTIPLE COMMON WEIBULL SHAPE PARAMETER Y1 TO Y5
COMMON WEIBULL SHAPE PARAMETER Y X SUBSET X > 1

The following parameters are saved by this command.

STATVAL	-	value of the test statistic
STATCDF	-	CDF for the test statistic
PVALUE	-	p-value for the test statistic
CUTOFF0	-	0% critical value
CUTOFF50	-	50% critical value
CUTOFF75	-	75% critical value
CUTOFF90	-	90% critical value
CUTOFF95	-	95% critical value
CUTOF975	-	97.5% critical value
CUTOFF99	-	99% critical value
CUTOF999	-	99.9% critical value

The following statistics are also supported
LET A = COMMON WEIBULL SHAPE TEST Y X
LET A = COMMON WEIBULL SHAPE TEST CDF Y X
LET A = COMMON WEIBULL SHAPE TEST PVALUE Y X
LET A = COMMON WEIBULL SHAPE TEST CV90 Y X
LET A = COMMON WEIBULL SHAPE TEST CV95 Y X
LET A = COMMON WEIBULL SHAPE TEST CV99 Y X


Dataplot statistics can be used in a number of commands. For details, enter

HELP STATISTICS

If the shape parameters are deemed to be statistically equivalent, then a common shape parameter will be estimated. You can define the cut-off for statistical equivalence by entering the command
LET ALPHA = <value>

where <value> is 0.90, 0.95, or 0.99. If this command is not given, the default value for alpha will be 0.95.

The equation used to estimate the common shape parameter is given on page 238 of McCool. If a common shape parameter is estimated, then the corresponding scale parameters for each of the groups based on this common shape parameter are written to the file "dpst1f.dat".

McCool defines this test for either uncensored data or for type II censored data (i.e., there are a fixed number of failures). McCool provides tables of critical values for various values of n (number of observations per group), r (number of failures), and k (number of groups). His tables assume that n and r are the same for all k groups. However, the requirement for equal n and r values is for the purpose of creating manageable tables and is not a theoretical restriction.

Dataplot currently only supports the uncensored data case. Dataplot performs the simulations for the critical values dynamically, so it does not require equal sample sizes for the groups.

Note the similarity of this test to the Hartley test. For k samples that are normally distributed and have equal sample size, Hartley's test computes the ratio of the largest variance to the smallest variance. Hartley's test is quite sensitive to departures from normality. Likewise, the common Weibull shape parameter test is only meaningful if the individual samples follow 2-parameter Weibull distributions.

None

None

MAXIMUM LIKELIHOOD	= Perform a distributional maximum likelihood estimation.
GOODNESS OF FIT	= Perform a distributional goodness of fit test.

McCool (2012), "Using the Weibull Distribution: Reliability, Modeling, and Inference", Wiley, pp. 236-238.

Distributional Modeling

2014/4

. Step 1:   Read the data
.
skip 25
read mccool82.dat y1 y2 y3 y4 y5
let y x = stack y1 y2 y3 y4 y5
set write decimals 5
set minmax 1
.
. Step 2:   Perform the command
.
common weibull shape test y x
    

The following output is generated.

             Test for Common Weibull Shape Parameter
             (For the 2-Parameter Weibull Distribution)
  
 Response Variable: Y
 Group-ID Variable: X
  
 H0: Homogeneous Shape Parameters
 Ha: Shape Parameters Are Not Homogeneous
  
 Summary Statistics:
 Total Number of Observations:                        50
 Number of Groups:                                     5
 Number of Groups Used in Test:                        5
  
 Minimum Value of Shape Parameter:               1.93917
 Maximum Value of Shape Parameter:               3.65176
 Test Statistic Value:                           1.88316
 CDF of Test Statistic:                          0.53140
 P-Value:                                        0.46860
  
 Estimate of Common Shape Parameter:             2.48034
  
  
 Percent Points of the Reference Distribution
 -----------------------------------
   Percent Point               Value
 -----------------------------------
             0.0    =          0.000
            50.0    =          1.845
            75.0    =          2.200
            90.0    =          2.640
            95.0    =          2.986
            97.5    =          3.265
            99.0    =          3.726
            99.9    =          5.094
  
 Conclusions (Upper 1-Tailed Test)
 ----------------------------------------------
   Alpha    CDF   Critical Value     Conclusion
 ----------------------------------------------
     10%    90%            2.640      Accept H0
      5%    95%            2.986      Accept H0
    2.5%  97.5%            3.265      Accept H0
      1%    99%            3.726      Accept H0
  
  
             Confidence Interval for Common Shape Parameter
  
 ------------------------------------------
   Confidence          Lower          Upper
  Coefficient          Limit          Limit
 ------------------------------------------
        50.00        2.12965        2.49122
        80.00        1.97370        2.67145
        90.00        1.88467        2.77648
        95.00        1.80180        2.88262
        99.00        1.65531        3.09005
        99.90        1.50608        3.32405
 ------------------------------------------
    

.
. Step 3:   Compute the test statistic
.
let statval = common weibull shape test y x
let statcdf = common weibull shape test cdf y x
let pval    = common weibull shape test pvalue y x
let cv90    = common weibull shape test cv90 y x
.
print statval statcdf pval cv90
    

The following output is generated.

 PARAMETERS AND CONSTANTS--

    STATVAL --        1.88316
    STATCDF --        0.53120
    PVAL    --        0.46260
    CV90    --        2.63340