The block plot is a series of k basic block plots with each
basic block plot for a main effect. Each basic block plot asks the
question as to whether that particular factor is important:
The i-th basic block plot, which targets factor i and
asks whether factor Xi is important, is formed by:
- The first block plot asks the question:
"Is factor X1 important?
- The second block plot asks the question:
"Is factor X2 important?
- Continue for the remaining factors.
For a particular combination of robustness factor settings
(horizontally), there will be two points plotted above it
- Vertical Axis: Response
- Horizontal Axis: All 2k-1 possible
combinations of the (k-1) non-target factors (that
is, "robustness" factors). For example, for the block plot
focusing on factor X1 from a 23 full factorial
experiment, the horizontal axis will consist of all
23-1 = 4 distinct combinations of factors X2
and X3. We create this robustness factors axis because
we are interested in determining if X1 is important
robustly. That is, we are interested in whether X1 is
important not only in a general/summary kind of way, but also
whether the importance of X is universally and
consistently valid over each of the 23-1 = 4
combinations of factors X2 and X3. These four
combinations are (X2, X3) = (+, +), (+, -), (-, +),
and (-, -). The robustness factors on the horizontal axis change
from one block plot to the next. For example, for the
k = 3 factor case:
- the block plot targeting X1 will have robustness
factors X2 and X3;
- the block plot targeting X2 will have robustness
factors X1 and X3;
- the block plot targeting X3 will have robustness
factors X1 and X2.
- Plot Character: The setting (- or +) for the target factor
Xi. Each point in a block plot has an
associated setting for the target factor
Xi. If Xi = "-", the
corresponding plot point will be "-"; if
Xi = "+", the corresponding plot
point will be "+".
In a block plot, these two plot points are surrounded by a box
(a block) to focus the eye on the
internal within-block differences as opposed to the distraction
of the external block-to-block differences. Internal block
differences reflect on the importance of the target factor (as
desired). External block-to-block differences reflect on the
importance of various robustness factors, which is not of primary
- one plot point for Xi = "-"; and
- the other plot point for Xi = "+".
Large within-block differences (that is, tall blocks) indicate a
large local effect on the response which, since all robustness
factors are fixed for a given block, can only be attributed to the
target factor. This identifies an "important" target factor.
Small within-block differences (small blocks) indicate that the
target factor Xi is unimportant.
For a given block plot, the specific question of interest is thus
Is the target factor Xi important? That is,
as we move within a block from the target factor setting of "-" to
the target factor setting of "+", does the response variable
value change by a large amount?
The height of the block reflects the "local" (that is, for that
particular combination of robustness factor settings) effect on
the response due to a change in the target factor settings. The
"localized" estimate for the target factor effect for
Xi is in fact identical to the difference in the
response between the target factor Xi at the "+"
setting and at the "-" setting. Each block height of a robustness
plot is thus a localized estimate of the target factor effect.
In summary, important factors will have both
where the "consistency" is over all settings of robustness factors.
Less important factors will have only one of these two properties.
Unimportant factors will have neither property.
- consistently large block heights; and
- consistent +/- sign arrangements
How to interpret
From the block plot, we are looking for the following:
We will discuss each of these in turn.
- Important factors (including 2-factor interactions);
- Best settings for these factors.
Important factors (including 2-factor interactions):
Look at each of the k block plots. Within a given block plot,
Are the corresponding block heights
consistently large as we scan across the
within-plot robustness factor
settings--yes/no; and are the within-block
sign patterns (+ above -, or - above +)
consistent across all robustness factors
To facilitate intercomparisons, all block plots have the same
vertical axis scale. Across such block plots,
This scanning and comparing of the k block plots easily
leads to the identification of the most important factors. This
identification has the additional virtue over previous steps in
that it is robust. For a given important factor, the consistency
of block heights and sign arrangement across robustness factors
gives additional credence to the robust importance of that factor.
The factor is important (the change in the response will be large)
irrespective of what settings the robustness factors have. Having
such information is both important and comforting.
- Which plot has the consistently largest block heights,
along with consistent arrangement of within-block +'s
and -'s? This defines the "most important factor".
- Which plot has the consistently next-largest block heights,
along with consistent arrangement of within-block
+'s and -'s? This defines the "second most important
- Continue for the remaining factors.
Important Special Case; Large but Inconsistent:
What happens if the block heights are large but not consistent?
Suppose, for example, a 23 factorial experiment is being
analyzed and the block plot focusing on factor X1 is being
examined and interpreted so as to address the usual question of
whether factor X1 is important.
Let us consider in some detail how such a block plot might
appear. This X1 block plot will have 23-1 = 4
combinations of the robustness factors X2 and X3 along
the horizontal axis in the following order:
(X2, X3) = (+, +); (X2, X3) = (+, -);
(X2, X3) = (-, +); (X2, X3) = (-, -).
If the block heights are consistently large (with "+" above "-" in
each block) over the four combinations of settings for X2 and
X3, as in
block height (= local X1 effect)
then from binomial considerations there is one chance in
24-1 = 1/8 \( \approx \)
12.5 % of the the four local X1 effects having the same
sign (i.e., all positive or all negative). The usual statistical
cutoff of 5 % has not been achieved here, but the 12.5 % is suggestive.
Further, the consistency of the four X1 effects
(all near 30) is evidence of a robustness of the X effect
over the settings of the other two factors. In summary, the
On the other hand, suppose the 4 block heights for
factor 1 vary in the following cyclic way:
- Factor 1 is probably important (the issue of how large the
effect has to be in order to be considered important will
be discussed in more detail in a later section); and
- The estimated factor 1 effect is about 30 units.
block height (= local X1 effect)
then how is this to be interpreted?
The key here to such interpretation is that the block plot is
telling us that the estimated X1 effect is in fact at
least 20 units, but not consistent. The effect is
changing, but it is changing in a structured way. The "trick"
is to scan the X2 and X3 settings and deduce what that
substructure is. Doing so from the above table, we see that the
estimated X1 effect is 30
and then the estimated X1 effect drops 10 units to 20
- for point 1 (X2, X3) = (+, +) and
- for point 3 (X2, X3) = (-, +)
We thus deduce that the estimated X1 effect is
- for point 2 (X2, X3) = (+, -) and
- for point 4 (X2, X3) = (-, -)
When the factor X1 effect is not consistent, but in fact changes
depending on the setting of factor X3, then definitionally that
is said to be an "X1*X3 interaction". That is precisely
the case here, and so our conclusions would be:
- 30 whenever X3 = "+"
- 20 whenever X3 = "-"
Note that we are using the term important in a qualitative sense here.
More precise determinations of importance in terms of statistical
or engineering significance are discussed in later sections.
- factor X1 is probably
- the estimated factor X1 effect is 25
(the average of 30, 20, 30, and 20);
- the X1*X3 interaction is probably important;
- the estimated X1*X3 interaction is about 10
(the change in the factor X1 effect as X3
changes = 30 - 20 = 10);
- hence the X1*X3 interaction is less
important than the X1 effect.
The block plot gives us the structure and the detail to allow
such conclusions to be drawn and to be understood. It is a
valuable adjunct to the previous analysis steps.
After identifying important factors, it is also of use to
determine the best settings for these factors. As usual, best
settings are determined for main effects only (since main
effects are all that the engineer can control). Best settings
for interactions are not done because the engineer has
no direct way of controlling them.
In the block plot context, this determination of best factor
settings is done simply by noting which factor setting (+ or -)
within each block is closest to that which the engineer is
ultimately trying to achieve. In the defective springs case,
since the response variable is percent acceptable springs, we are clearly
trying to maximize (as opposed to minimize, or hit a target) the
response and the ideal optimum point is 100 %. Given this, we would
look at the block plot of a given important factor and note within
each block which factor setting (+ or -) yields a data value
closest to 100 % and then select that setting as the best for that
From the defective springs block plots, we would thus conclude that
- the best setting for factor 1 is +;
- the best setting for factor 2 is -;
- the best setting for factor 3 cannot be easily determined.