DOE++ also allows
for the analysis of single factor R-DOE experiments. This analysis is
similar to the analysis of single factor designed experiments mentioned
in Chapter
6. In single factor R-DOE analysis, the focus is on discovering whether
change in the level of a factor affects reliability and how each of the
factor levels are different from the other levels. The analysis models
and calculations are similar to multi-factor R-DOE analysis.
This section is divided into the following subsections:
To illustrate single factor R-DOE analysis, consider the data in Table
11.1 where life data readings for a product are taken at three levels
of a certain factor, . Factor may either be a stress that is thought
to affect life or three different designs of the same product or the same
product manufactured by three different machines or operators, etc. The
goal of the experiment is to see if there is a change in life due to change
in the levels of the factor. The design for this experiment is shown in
Figure 11.10. The life of the product is assumed
to follow the Weibull distribution. Therefore, the life characteristic
to be used in the R-DOE analysis is the scale parameter, . Since factor has three levels, the model for the
life characteristic, , is:(29)
where is the intercept, is the effect coefficient for the
first level of the factor ( is represented as "A[1]"
in DOE++) and is the effect coefficient for the
second level of the factor ( is represented as "A[2]"
in DOE++). Two indicator variables, and are the used to represent the three
levels of factor such that:
Table 11.1: Data obtained from a single factor R-DOE experiment.
The following hypothesis test needs to be carried out in this example:
where . The statistic for this test is:
where is the value of the likelihood function
corresponding to the full model, and is the likelihood value for the reduced
model. To calculate the statistic for this test, the MLE estimates of
the parameters must be obtained.
Following the procedure used in the analysis of multi-factor R-DOE experiments,
MLE estimates of the parameters are obtained by differentiating the log-likelihood
function :
Substituting from Eqn. (29)
and setting the partial derivatives to zero, the parameter estimates are
obtained as , , and . These parameters are shown in Figure
11.11 in the MLE Information table.
Figure 11.11:
MLE results for the experiment in Example 11.5.
Knowing the MLE estimates, the likelihood ratio test for the significance
of factor can be carried out. The likelihood
value for the full model, , is the value of the likelihood function
corresponding to the model :
The likelihood value for the reduced model, , is the value of the likelihood function
corresponding to the model :
Then the likelihood ratio is:
If the null hypothesis, , is true then the likelihood ratio
will follow the Chi-Squared distribution. The number of degrees of freedom
for this distribution is equal to the difference in the number of parameters
between the full and the reduced model. In this case, this difference
is 2. The value corresponding to the likelihood
ratio on the Chi-Squared distribution with two degrees of freedom is:
Assuming that the desired significance is 0.1, since , is rejected it is concluded that,
at a significance of 0.1, at least one of the parameters, or , is non-zero. Therefore, factor affects the life of the product. This
result is shown in the Likelihood Ratio Test table in Figure 11.11.
Additional results for single factor R-DOE analysis obtained from DOE
++ include information on the life characteristic and comparison of life
characteristics at different levels of the factor.
Results in the Life Characteristic Summary table, include information
about the life characteristic corresponding to each treatment level of
the factor. If is represented as , then Eqn. (29)
can be written as:
The respective equations for all three treatment levels for a single
replicate of the experiment can be expressed in matrix notation as:
where:
Knowing , and , the predicted value of the life characteristic
at any level can be obtained. For example, for the second level:
Thus:
The variance for the predicted values of life characteristic can be
calculated using the following equation:
where is the variance-covariance matrix
for , and . [Note]
Substituting the required values:
From the previous matrix, . Therefore, the 90% confidence interval
() on is:
Since the 90% confidence interval on is:
Results for other levels can be calculated in a similar manner and are
shown in Figure 11.12.
Figure 11.12:
Life characteristic results for the experiment in Example 11.5.
Results under Life Comparisons include information on how life is different
at a level in comparison to any other level of the factor. For example,
the difference between the predicted values of life at levels 1 and 2
is (in terms of the logarithmic transformation):
The pooled standard error for this difference can be obtained as:
If the covariance between and is taken into account, then the pooled
standard error is:
This is the value displayed by DOE++. Knowing the pooled standard error
the confidence interval on the difference can be calculated. The 90% confidence
interval on the difference in (logarithmic) life between levels 1 and
2 of factor is:
Since the confidence interval does not include zero it can be concluded
that the two levels are significantly different at . Another way to test for the significance
of the difference in levels is to observe the value. The statistic corresponding to this difference
is:
The value corresponding to this statistic,
based on the standard normal distribution, is:
Since it can be concluded that the levels
are significantly different at . The results for other levels can
be calculated in a similar manner and are shown in Figure 11.12.
. Factor 
may either be a stress that is thought
to affect life or three different designs of the same product or the same
product manufactured by three different machines or operators, etc. The
goal of the experiment is to see if there is a change in life due to change
in the levels of the factor. The design for this experiment is shown in
Figure 11.10. The life of the product is assumed
to follow the Weibull distribution. Therefore, the life characteristic
to be used in the R-DOE analysis is the scale parameter, 
. Since factor 
has three levels, the model for the
life characteristic, 
, is:
(29)
