Hydrocarbon Processing - January 2021 - 47

Process Optimization
assessed: yr holds the averages for replicate observations. In the case of a single
observation, the value of the replicate
mean and the observation are identical.
In the case of multiple interrogations of
the same point in factor space, the replicate mean is their average. The column
labeled ȳ holds the grand mean-the average of all measured responses. The columns labeled 0, 1, 2 and 12 hold the data
of the X matrix in Eq. 10.
Tests for statistical significance.
To test for statistical significance, begin by constructing a general analysis
of variance (ANOVA) table (TABLE 2).
For now, note that the sum of squares
for the model (M) is SSM = 4,011, and
that of the residual error (R) is SSR =
97. SSM + SSR give a total (T), SST =
4,108. We divide SSM and SSR by their
respective degrees of freedom (DFM = 3,
DFR = 5) to arrive at their mean squares
(MSM = 1337, MSR = 19). The ratio of
MSM/MSR = F = 69.05 shows that the
variance of the model is nearly 70 times
that of the residual error. Because p =
0.0002, we have a 0.02% chance that our
model does not contain statistically significant coefficients; conversely, we have
99.98% confidence that our model is statistically significant. Generally, anything
above 95% confidence will be sufficient
(i.e., p ≤ 0.05). FIG. 8 shows the updated
spreadsheet with all the statistical tests in
place and how each calculation was derived. In addition to the General ANOVA of TABLE 2, and a more detailed one is
shown in FIG. 8 along with standard errors
(SE) for each coefficient. The degrees of
freedom are found by Eqs. 13, 14 and 15:

DFM = nc - 1
(13)
DFR = n - nc(14)
DFT = n - 1
(15)
Here, nc is the number of coefficients of
the model being fit, and n is the number
of observations used to fit the coefficients.
In the detailed ANOVA ( J11 to O18
in FIG. 8), the general ANOVA has been
expanded. For example, SSM is parsed
to reveal each individual component
comprising its sum (SSc1 + SSc2 + SSc12 =
SSM) and SSR has been parsed into components for bias (B, a.k.a. lack of fit) and
pure error (B). The coefficient c0 is not
included in the detailed ANOVA because
SSM is the squared difference from the

FIG. 8. Sample of updated spreadsheet. The updated spreadsheet now includes a more detailed
ANOVA and the calculations callout has been expanded to include the new entries.

mean (ȳ) and not the mean itself, and c0
is equivalent to the mean, i.e., c0 = ȳ.
SSB and SSE are calculated using Eqs.
16-19:
SSB = yrTyr - ŷ T ŷ(16)
SSE = y Ty - yrTyr (17)
and
DFB = nu - nc(18)
(19)
DFE = n - nu 
Here, nu corresponds to the number of
unique design points, which in this case
is 5 (p1 , and s1-s4 ). For factorial designs
of this sort, the sum of squares for each
component of the model is given Eq. 20:
SSck = ck × X T y | k(20)
where X T y | k indicates the kth element of
the X T y vector (see the calculation legend of FIG. 7).
In this analysis, c12 is obviously not
significant because it does not even exceed its standard error. This latter point
may also be seen by individual F-tests on
the coefficients, all of which are significant with the exception of the interaction term c12 ( p = 0.635 >> 0.05). Since
no significant interaction is indicated, a

FIG. 9. From p1 to p2 . The next center of
exploration (p2 ) on the path of steepest
ascent is given in factor space by the
coordinates kc*1 c 2* k as measured from
the old design center (p1 ).

purely linear model is adequate to correlate the data. The F-tests also show
that no significant bias exists ( p = 0.786
>> 0.05), thus the linear approximation
appears adequate to gauge the next step
on the path of steepest ascent. In fact, assessing the bias separately from the error
leads to a difference in the estimate of s
(±4.4 vs. ±4.9, see cells I17 vs. O17 in
FIG. 8, respectively). Because there is no
significant bias, SSR is the probably the
better basis since it has more degrees of
freedom with which to estimate s; however, the model coefficients are significant using either value.
Hydrocarbon Processing | JANUARY 2021 47
Hydrocarbon Processing - January 2021

Table of Contents for the Digital Edition of Hydrocarbon Processing - January 2021
