Another Look at the Shape Parameter of the Weibull Distribution
(Beta)
It is well known that the shape parameter of the Weibull distribution,
beta (β), represents the failure rate behavior. If beta is less than 1, then the failure
rate decreases with time; if beta is greater than 1, then the failure rate increases
with time. When beta is equal to 1, the failure rate is constant. Assume
that we have two failure data sets. From one data set, the estimated beta is 5; from
another data set, the estimated beta is 3. Both have increasing failure rates. If
the two data sets were combined together, would the estimated beta for the overall
data set be a value between 3 and 5, or would it be a value outside of this range? Could it
be less than 1, which would imply that the combined data set has a failure rate
that is decreasing with time? In the article, we will use examples to show the
relationship between these betas.
In order to answer the above questions, we will look at the issue from another
perspective. Assume there is the one set of life data that is fitted using a Weibull
distribution. This data set has a shape parameter β. If we separate the data
into two groups and fit each group with a separate Weibull distribution, we
get β1 and β2 as the shape parameters of the two
groups. If we suppose that β1 < β2,
will β1 < β < β2 always be true, or is
there some other relationship between these beta values?
Suppose that a testing engineer obtained the life data shown in Table 1. According to
previous experience, the 2-parameter Weibull distribution should be used to fit the data.
The analysis using Weibull++
shows that β = 0.9812 and η = 73.6036, as shown in Figure 1.
Table
1: Failure Data
| State
(F or S) |
Time to State |
Item Index |
| F |
2 |
1 |
| S |
3 |
2 |
| F |
5 |
3 |
| S |
7 |
4 |
| F |
11 |
5 |
| S |
13 |
6 |
| S |
17 |
7 |
| S |
19 |
8 |
| F |
23 |
9 |
| F |
29 |
10 |
| S |
31 |
11 |
| F |
37 |
12 |
| S |
41 |
13 |
| F |
43 |
14 |
| S |
47 |
15 |
| S |
53 |
16 |
| F |
59 |
17 |
| S |
61 |
18 |
| S |
69 |
19 |
| F |
75 |
20 |
Figure 1: Data Analysis for
the Entire Data Set
To illustrate the subgroups parameters, we extract different subgroups
from the full data set. We could use different methods to accomplish this.
For the first method, we separate data from the middle point of the entire group. We
classify the early failure items as "Subgroup I" and the later failure
items as "Subgroup II." These two subgroups are listed in Table 2.
Table
2: Method I Subgroup Data
| |
Subgroup I (Early) |
Subgroup II (Later) |
Item
Index |
State
(F or S) |
Time to
State |
State
(F or S) |
Time to
State |
| 1 |
F |
2 |
S |
31 |
| 2 |
S |
3 |
F |
37 |
| 3 |
F |
5 |
S |
41 |
| 4 |
S |
7 |
F |
43 |
| 5 |
F |
11 |
S |
47 |
| 6 |
S |
13 |
S |
53 |
| 7 |
S |
17 |
F |
59 |
| 8 |
S |
19 |
S |
61 |
| 9 |
F |
23 |
S |
69 |
| 10 |
F |
29 |
F |
75 |
The analysis using the 2-parameter Weibull distribution shows that Subgroup
I (Early) has β1 = 1.105 and η1 = 24.0872, while
Subgroup II (Later) has β2 =3.6717 and η2 = 70.8740.
A second method to extract different subgroups is to separate the data according to the failure
index. Entries with odd index numbers are put in Subgroup I and
entries with even index numbers are
assigned to Subgroup II. These two subgroups are listed in Table 3.
Table
3: Method II Subgroup Data
| |
Subgroup I (Even) |
Subgroup II (Odd) |
Item
Index |
State
(F or S) |
Time to
State |
State
(F or S) |
Time to
State |
| 1 |
S |
3 |
F |
2 |
| 2 |
S |
7 |
F |
5 |
| 3 |
S |
13 |
F |
11 |
| 4 |
S |
19 |
S |
17 |
| 5 |
F |
29 |
F |
23 |
| 6 |
F |
37 |
S |
31 |
| 7 |
F |
43 |
S |
41 |
| 8 |
S |
53 |
S |
47 |
| 9 |
S |
61 |
F |
59 |
| 10 |
F |
75 |
S |
69 |
The analysis using the 2-parameter Weibull distribution shows that Subgroup
I (Odd) has β1' = 0.7419 and η1' = 62.4596, while
Subgroup II (Even) has β2' =2.4834
and η2' = 61.4978.
As we can see, for the first method, β < β1
< β2, and for the second method, β1' <
β < β2'. So it depends on how the subgroup was extracted
from the original data. From the above examples we also can see that there is
no fixed relationship between these betas.
Now let’s examine the definition of the standard deviation of the Weibull
distribution:
where
 is the gamma
function evaluated at the value of
 .
Table 4 lists the standard deviation values for the overall group and for each
subgroup.
Table
4: Distribution Parameters for Different Subgroups
| |
Whole
Group |
Subgroup
Early |
Subgroup
Later |
Subgroup
Odd |
Subgroup
Even |
| β |
0.981 |
1.105 |
3.6717 |
0.7419 |
2.4834 |
| η |
73.6036 |
24.0872 |
70.8740 |
62.4596 |
61.4978 |
| 1/β |
1.0941 |
0.9050 |
0.2724 |
1.3479 |
0.4027 |
| σT (standard
deviation) |
75.63642 |
21.0307 |
19.3781 |
102.75802 |
23.4846 |
We can see that the relationship between the 1/β of the whole group and
each of the subgroups are similar to the relationships of their standard deviations
σT. In other words, 1/β roughly
reflects the standard deviation of the distribution. For example, in our
first classification method I, as 1/β for the whole group is greater than either
subgroup; so is its σT is greater
than either subgroup.
It will be even clearer if we use the logarithmic transformation of the raw data to fit
a Gumbel distribution. As we know, the logarithm transform of Weibull data follows
the Gumbel distribution with μ = ln(η) and
 , where μ is the
location parameter and σ is the scale parameter of the Gumbel distribution. The
standard deviation for the Gumbel distribution is given by:
From the above equation, we can see how 1/β reflects the spread of a data
set.
Conclusions
In this article, we answered the question about the relationship between the beta
for the entire data set and the betas for subgroups of that data set. As we can
see, this relationship is not fixed, and can change. Moreover, 1/β can be
interpreted as something similar to the standard deviation of a data
set. So 1/β not only represents the behavior of the failure rate, it also
reflects the spread of a data set.
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