Definition

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 first block plot asks the question:
"Is factor X_{1} important?
 The second block plot asks the question:
"Is factor X_{2} important?
 Continue for the remaining factors.
The ith basic block plot, which targets factor i and
asks whether factor X_{i} is important, is formed by:
 Vertical Axis: Response
 Horizontal Axis: All 2^{k1} possible
combinations of the (k1) nontarget factors (that
is, "robustness" factors). For example, for the block plot
focusing on factor X_{1} from a 2^{3} full factorial
experiment, the horizontal axis will consist of all
2^{31} = 4 distinct combinations of factors X_{2}
and X_{3}. We create this robustness factors axis because
we are interested in determining if X_{1} is important
robustly. That is, we are interested in whether X_{1} 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 2^{31} = 4
combinations of factors X_{2} and X_{3}. These four
combinations are (X_{2}, X_{3}) = (+, +), (+, ), (, +),
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 X_{1} will have robustness
factors X_{2} and X_{3};
 the block plot targeting X_{2} will have robustness
factors X_{1} and X_{3};
 the block plot targeting X_{3} will have robustness
factors X_{1} and X_{2}.
 Plot Character: The setting ( or +) for the target factor
X_{i}. Each point in a block plot has an
associated setting for the target factor
X_{i}. If X_{i} = "", the
corresponding plot point will be ""; if
X_{i} = "+", the corresponding plot
point will be "+".
For a particular combination of robustness factor settings
(horizontally), there will be two points plotted above it
(vertically):
 one plot point for X_{i} = ""; and
 the other plot point for X_{i} = "+".
In a block plot, these two plot points are surrounded by a box
(a block) to focus the eye on the
internal withinblock differences as opposed to the distraction
of the external blocktoblock differences. Internal block
differences reflect on the importance of the target factor (as
desired). External blocktoblock differences reflect on the
importance of various robustness factors, which is not of primary
interest.
Large withinblock 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 withinblock differences (small blocks) indicate that the
target factor X_{i} is unimportant.
For a given block plot, the specific question of interest is thus
Is the target factor X_{i} 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
X_{i} is in fact identical to the difference in the
response between the target factor X_{i} 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
 consistently large block heights; and
 consistent +/ sign arrangements
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.

How to interpret

From the block plot, we are looking for the following:
 Important factors (including 2factor interactions);
 Best settings for these factors.
We will discuss each of these in turn.
Important factors (including 2factor 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
withinplot robustness factor
settingsyes/no; and are the withinblock
sign patterns (+ above , or  above +)
consistent across all robustness factors
settingsyes/no?
To facilitate intercomparisons, all block plots have the same
vertical axis scale. Across such block plots,
 Which plot has the consistently largest block heights,
along with consistent arrangement of withinblock +'s
and 's? This defines the "most important factor".
 Which plot has the consistently nextlargest block heights,
along with consistent arrangement of withinblock
+'s and 's? This defines the "second most important
factor".
 Continue for the remaining factors.
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.
Important Special Case; Large but Inconsistent:
What happens if the block heights are large but not consistent?
Suppose, for example, a 2^{3} factorial experiment is being
analyzed and the block plot focusing on factor X_{1} is being
examined and interpreted so as to address the usual question of
whether factor X_{1} is important.
Let us consider in some detail how such a block plot might
appear. This X_{1} block plot will have 2^{31} = 4
combinations of the robustness factors X_{2} and X_{3} along
the horizontal axis in the following order:
(X_{2}, X_{3}) = (+, +); (X_{2}, X_{3}) = (+, );
(X_{2}, X_{3}) = (, +); (X_{2}, X_{3}) = (, ).
If the block heights are consistently large (with "+" above "" in
each block) over the four combinations of settings for X_{2} and
X_{3}, as in
(X_{2}, X_{3})

block height (= local X_{1} effect)

(+, +)

30

(+, )

29

(, +)

29

(, )

31

then from binomial considerations there is one chance in
2^{41} = 1/8 \( \approx \)
12.5 % of the the four local X_{1} 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 X_{1} effects
(all near 30) is evidence of a robustness of the X effect
over the settings of the other two factors. In summary, the
above suggests:
 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.
On the other hand, suppose the 4 block heights for
factor 1 vary in the following cyclic way:
(X_{2}, X_{3})

block height (= local X_{1} effect)

(+, +)

30

(+, )

20

(, +)

30

(, )

20

then how is this to be interpreted?
The key here to such interpretation is that the block plot is
telling us that the estimated X_{1} 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 X_{2} and X_{3} settings and deduce what that
substructure is. Doing so from the above table, we see that the
estimated X_{1} effect is 30
 for point 1 (X_{2}, X_{3}) = (+, +) and
 for point 3 (X_{2}, X_{3}) = (, +)
and then the estimated X_{1} effect drops 10 units to 20
 for point 2 (X_{2}, X_{3}) = (+, ) and
 for point 4 (X_{2}, X_{3}) = (, )
We thus deduce that the estimated X_{1} effect is
 30 whenever X_{3} = "+"
 20 whenever X_{3} = ""
When the factor X_{1} effect is not consistent, but in fact changes
depending on the setting of factor X_{3}, then definitionally that
is said to be an "X_{1}*X_{3} interaction". That is precisely
the case here, and so our conclusions would be:
 factor X_{1} is probably
important;
 the estimated factor X_{1} effect is 25
(the average of 30, 20, 30, and 20);
 the X_{1}*X_{3} interaction is probably important;
 the estimated X_{1}*X_{3} interaction is about 10
(the change in the factor X_{1} effect as X_{3}
changes = 30  20 = 10);
 hence the X_{1}*X_{3} interaction is less
important than the X_{1} effect.
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.
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.
Best settings:
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
factor.
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.
