The 2 designs are a special category of the
factorial experiments where all the factors are at two levels. The fact
that these designs contain factors at only two levels and are orthogonal
greatly simplifies their analysis even when the number of factors is large.
The use of 2 designs in investigating a large number
of factors calls for a revision of the notation used previously for the
ANOVA models. The case for revised notation is made stronger by the fact
that the ANOVA and multiple linear regression models are identical for
2 designs because all factors are only
at two levels. Therefore, the notation of the regression models is applied
to the ANOVA models for these designs, as explained below.
Based on the notation of Chapter
6, the ANOVA model for a two level factorial experiment with three
factors would be as follows:(1)where:
represents the overall mean.
represents the independent effect
of the first factor (factor ) out of the two effects and .
represents the independent effect
of the second factor (factor ) out of the two effects and .
represents the independent effect
of the interaction out of the other interaction effects.
represents the effect of the third
factor (factor ) out of the two effects and .
represents the effect of the interaction
out of the other interaction effects.
represents the effect of the interaction
out of the other interaction effects.
represents the effect of the interaction
out of the other interaction effects
.
is the random error term.
The notation for a linear regression model having three predictor variables
with interactions is:(2)
The notation for the regression model is much more convenient, especially
for the case when a large number of higher order interactions are present.
[Note]
In two level experiments, the ANOVA model requires only one indicator
variable to represent each factor for both qualitative and quantitative
factors. Therefore, the notation for the multiple linear regression model
can be applied to the ANOVA model of the experiment that has all the factors
at two levels. For example, for the experiment of Eqn. (1),
can represent the overall mean instead
of , and can represent the independent effect,
, of factor . Other main effects can be represented
in a similar manner. The notation for the interaction effects is much
more simplified, e.g. can be used to represent the three
factor interaction effect, .
As mentioned earlier, it is important to note that the coding for the
indicator variables for the ANOVA models of two level factorial experiments
is reversed from the coding followed in the Chapter 6, Analysis
of Experiments. Here represents the first level of the
factor while represents the second level. This
is because for a two level factor a single variable is needed to represent
the factor for both qualitative and quantitative factors. [Note]
For quantitative factors, using for the first level (which is the
low level) and 1 for the second level (which is the high level) keeps
the coding consistent with the numerical value of the factors. [Note]
The change in coding between the two coding schemes does not affect the
analysis except that signs of the estimated effect coefficients will be
reversed (i.e. numerical values of , obtained based on the coding of Chapter 6, and , obtained based on the new coding,
will be the same but their signs would be opposite).
In summary, the ANOVA model for the experiments with all factors at
two levels is different from the ANOVA models for other experiments in
terms of the notation in the following two ways:
The notation of the regression models is used for
the effect coefficients.
The coding of the indicator variables is reversed.
Consider the design matrix, , for the 2 design shown in Figure 7.3
(b). The () matrix is:
Notice that, due to the orthogonal design of the matrix, the has been simplified to a diagonal
matrix which can be written as:
where represents the identity matrix of the
same order as the design matrix, . Since there are eight observations
per replicate of the 2 design, the ' matrix for replicates of this design can be written
as:
The matrix for any 2 design can now be written as:(3)
Then the variance-covariance matrix for the 2 design is:(4, 5)
Note that the variance-covariance matrix for the 2 design is also a diagonal matrix. Therefore,
the estimated effect coefficients (, , etc.) for these designs are uncorrelated.
This implies that the terms in the 2 design (main effects, interactions)
are independent of each other. Consequently, the extra sum of squares
for each of the terms in these designs is independent of the sequence
of terms in the model, and also independent of the presence of other terms
in the model. As a result the sequential and partial sum of squares for
the terms are identical for these designs and will always add up to the
model sum of squares. Multicollinearity is also not an issue for these
designs. [Note]
It can also be noted from Eqn. (5), that in addition
to the matrix being diagonal, all diagonal
elements of the matrix are identical. This means that
the variance (or its square root, the standard error) of all estimated
effect coefficients are the same. The standard error, , for all the coefficients is: (6)
This property is used to construct the normal probability plot of effects
in 2 designs and identify significant effects
using graphical techniques. [Note]
For details on the normal probability plot of effects in DOE++, refer
to Chapter 7, Normal
Probability Plot of Effects.
To illustrate the analysis of a full factorial 2 design, consider a three factor experiment
to investigate the effect of honing pressure, number of strokes and cycle
time on the surface finish of automobile brake drums. Each of these factors
is investigated at two levels. The honing pressure is investigated at
levels of 200 and 400 , the number of strokes used is 3 and
5 and the two levels of the cycle time are 3 and 5 seconds. The design
for this experiment is set up in DOE++ as shown in Figures 7.5
and 7.6 . It is decided to run two replicates
for this experiment. The surface finish data collected from each run (using
randomization) and the complete design is shown in Figure 7.7.
The analysis of the experiment data is explained next.
Figure 7.5: Design properties for the experiment in Example
7.1.
Figure 7.6: Factor properties for the experiment in Example
7.1.
Figure: 7.7: Experiment design for Example 7.1 to investigate
the surface finish of automobile brake drums.
The applicable ANOVA model using the notation for 2 designs is:(7)where the indicator variable, represents factor (honing pressure), represents the low level of 200 and represents the high level of 400 . Similarly, and represent factors (number of strokes) and (cycle time), respectively. is the overall mean, while , and are the effect coefficients for the
main effects of factors , and , respectively. , and are the effect coefficients for the
, and interactions, while represents the interaction.
If the subscripts for the run (; 1 to 8) and replicates (; 1,2) are included, then the model
can be written as:(8)
To investigate how the given factors affect the response, the following
hypothesis tests need to be carried:
This test investigates the main effect of factor (honing pressure). The statistic for
this test is:where is the mean square for factor and is the error mean square. Hypotheses
for the other main effects, and , can be written in a similar manner.
This test investigates the two factor interaction . The statistic for this test is:where is the mean square for the interaction
and is the error mean square. Hypotheses
for the other two factor interactions, and , can be written in a similar manner.
This test investigates the three factor interaction . The statistic for this test is:where is the mean square for the interaction
and is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA
model of Eqn. (7) in the form .
Knowing the matrices , and , the extra sum of squares for the
factors can be calculated. [Note]
These are used to calculate the mean squares that are used to obtain the
test statistics. Since the experiment design is orthogonal, the partial
and sequential extra sum of squares are identical. The extra sum of squares
for each effect can be calculated as shown next. As an example, the extra
sum of squares for the main effect of factor is:
where is the hat matrix and is the matrix of ones. The matrix
can be calculated using where is the design matrix, , excluding the second column that
represents the main effect of factor . Thus, the sum of squares for the
main effect of factor is:
Similarly, the extra sum of squares for the interaction effect is:
The extra sum of squares for other effects can be obtained in a similar
manner.
Knowing the extra sum of squares, the test statistic for the effects
can be calculated. For example, the test statistic for the interaction
is:
where is the mean square for the interaction and is the error mean square. [Note]
The value corresponding to the statistic,
, based on the distribution with one degree of freedom
in the numerator and eight degrees of freedom in the denominator is: [Note]
Assuming that the desired significance is 0.1, since value > 0.1, it can be concluded
that the interaction between honing pressure and number of strokes does
not affect the surface finish of the brake drums. Tests for other effects
can be carried out in a similar manner. The results are shown in the ANOVA
Table in Figure 7.8. The values S, R-sq and R-sq(adj)
in the figure indicate how well the model fits the data. The value of
S represents the standard error of the model, R-sq represents the coefficient
of multiple determination and R-sq(adj) represents the adjusted coefficient
of multiple determination. For details on these values refer to Chapter
5, Multiple Linear Regression
Analysis.
Figure 7.8: ANOVA table for the experiment in Example
7.1.
The estimate of effect coefficients can also be obtained:
The coefficients and related results are shown in the Regression Information
Table in Figure 7.9. In the table, the Effect column
displays the effects, which are simply twice the coefficients. The Standard
Error column displays the standard error, . The Low CI and High CI columns display
the confidence interval on the coefficients. The interval shown is the
90% interval as the significance is chosen as 0.1. The T Value column
displays the statistic, , corresponding to the coefficients.
The P Value column displays the value corresponding to the statistic. (For details on how these
results are calculated, refer to Chapter 6, Analysis
of Experiments.) Plots of residuals can also be obtained from DOE++
to ensure that the assumptions related to the ANOVA model of Eqn. (7) are not violated.
Figure 7.9: Regression Information table for the experiment
in Example 7.1.
Model Equation
From the analysis results in Figure 7.8, it is
seen that effects , and are significant. In DOE++, the values for the significant effects
are displayed in red in the ANOVA Table for easy identification. Using
the values of the estimated effect coefficients, the model for the present
2 design in terms of the coded values
can be written as:
To make the model hierarchical, the main effect, , needs to be included in the model
(because the interaction is included in the model). [Note]
The resulting model is:
This equation can be viewed in DOE++, as shown in Figure 7.10,
using the Show Analysis Summary icon in the Control Panel. The equation
shown in the figure will match the hierarchical model once the required
terms are selected using the Select Effects icon.
Figure 7.10: The model equation for the experiment of
Example 7.1.
designs are a special category of the
factorial experiments where all the factors are at two levels. The fact
that these designs contain factors at only two levels and are orthogonal
greatly simplifies their analysis even when the number of factors is large.
The use of 2
designs in investigating a large number
of factors calls for a revision of the notation used previously for the
ANOVA models. The case for revised notation is made stronger by the fact
that the ANOVA and multiple linear regression models are identical for
2
designs because all factors are only
at two levels. Therefore, the notation of the regression models is applied
to the ANOVA models for these designs, as explained below.
(1)
.
.
.
(2)
(3)
(4, 5)
(6)
(7)
(8)
