5
Screening Formulation Components

"Look before you leap."

 “The Fox and the Goat,” Aesop's Fables

Overview

In Chapter 2 we introduced the idea of experimental strategy that calls for screening experiments to be run prior to optimization experiments. In this chapter we introduce the screening concepts and designs for formulations. As the Aesop fable suggests, it is important to take a broad view of the possible components before getting into formulation optimization. We discuss screening experiments when the components can be varied over the full range of the components, 0 to 100%. We also discuss screening experiments when the components can be varied over equal ranges. Looking at both approaches enables one to clearly see the concepts and objectives that are associated with screening formulations. In Chapter 7 we will return to screening and discuss screening formulations when the components have lower and upper bounds but cannot be varied over equal ranges.

CHAPTER CONTENTS

Overview

5.1 Purpose of Screening Experiments

5.2 Screening Concepts for Formulations

5.3 Simplex Screening Designs

5.4 Graphical Analysis of Simplex-Screening Designs

5.5 After the Screening Design

5.6 Estimation of the Experimental Variation

5.7 Summary and Looking Forward

5.8 References

5.1 Purpose of Screening Experiments

It is not unusual, particularly at the beginning of a formulations development program, to have several components that could be included in the formulation. We suspect that not all of the components need to be in the formulation but we don’t know which components would be most useful. Component cost, ease of blending, and size of effect are always considerations as well as other issues.

Screening experiments are particularly useful in formulation studies that have these characteristics:

•   A literature review and related discussions with colleagues have led to a long list of components and perhaps process variables that could have important effects on the product characteristics.

•   You don’t know much about the magnitude and nature of the effects of the components and process variables.

•   The list of components is long, but you want to use a small number of components to simplify the problem. However, you don’t know which components are the best to use.

The first step in a good experimental strategy for such situations is to run screening experiments to find out which of the potentially important variables have the largest effects, positive and negative. Screening experiments will sometimes solve the problem. In most instances, however, it is necessary to study the effects of the most important variables (typically three to five) in greater detail. Response surface (optimization) designs are often useful in such experiments. These designs enable one to accurately quantify the effects of the important variables and to develop prediction equations that can be used to estimate the response within the region of experimentation.

In the case of independent (non-formulation) variables, the results of screening experiments are analyzed by fitting a linear model to the data. Here is the form for this model:

E(y) = b₀ + b₁x₁ + b₂x₂ + ….  + b_qx_q

In the model, x_i, an independent variable, is the standardized form of variable i and the b's are coefficients that describe the linear effects of the variables. Each x is typically studied at two levels, coded -1 for the low level and +1 for the high level.

The relative effects of the variables are assessed by ranking the estimated coefficients (b_i) in order by absolute value. A variable is assumed to be unimportant if the associated coefficient is sufficiently small. The sign of the coefficient shows the direction of the effect of the variable. A positive coefficient indicates that an increase in that x value caused the y value to increase; a negative coefficient shows that the y value decreased when the corresponding x value was increased.

The conclusion that a variable, x, is not important means that, in the low-to-high direction for x in the factor space, the y value does not change significantly from a practical viewpoint. In effect, estimating and comparing factor effects determined from screening designs is a search for directions through the factor space for which the response is constant, or nearly so. The following section discusses how these concepts are used in the formulation studies.

5.2 Screening Concepts for Formulations

Part of the philosophy of screening independent (non-formulation) variables is applicable to the screening of component variables in formulations. There are, however, some important differences in the concept of screening as applied to formulations, the experimental designs used, and the graphical displays. The differences arise because of the formulation constraint discussed earlier. To repeat, the x values (component levels) in formulation experiments must all be between 0 and 1, and the total of all x values in any formulation must be 1.0. The resulting experimental region is a simplex.

The screening design procedures to be discussed are recommended for any formulation problems involving six or more components. For five or fewer components, there is little advantage in running screening designs unless it is of interest only to determine the magnitude and direction of the component effects. (See Section 5.3 for an example.) In most instances, it is better to collect enough data to fit a quadratic model, particularly if finding the optimum formulation is the objective.  For example, in a five-component formulation problem, a good design for a quadratic model will usually require that only 20 to 25 formulations be made up and tested. In most cases, this is a reasonable number of experiments to run.

For six or more components, however, the number of experiments necessary to fit a quadratic model is very high, as shown in Chapter 3. Also, in early stages of experimentation, we are often uncertain about whether we are working with the right components and whether they are at the right levels. Therefore, for formulation problems involving six or more components, small screening experiments are strongly recommended early in an experimental program.

In screening components, we follow the principle of looking for directions through the component space for which the response is constant, or nearly so. We again use linear models and assume that important components will have large linear effects. There are exceptions, of course, and the recommended designs include at least one center point to detect any large curvature that may be present in the response surface.

The purposes of running a screening design are to increase understanding of the component effects and to reduce the candidate components to a reasonable number of essential components to be studied in subsequent experiments. An essential component may be a single component (x) or the sum of two or more components. Nonessential components can be found by identifying those components that, either singly or in groups, have no effect or equal effects. In these situations the response (y) will be constant in certain directions. In mathematical terms, a component is said to have no effect if the associated coefficient (b_i) in the mixture model is equal to the average of the other coefficients (b_j, j not equal to i) in the model. Two or more coefficients are said to have equal effects if the associated coefficients are approximately equal.

An example of a response surface in which one component has no effect is shown in Figure 5.1. Here is the model for this surface:

E(y) = 80x₁ + 90x₂ + 100x₃

Figure 5.1 – Linear Blending Model: One Component Has No Effect

We see that b₂ is equal to the average of b₁ and b₃. In such a situation, the E(y) value does not change along any line perpendicular to the x₂=0 base. In particular, E(y) does not change along the perpendicular that passes through the x₂ vertex with coordinates (0, 1, 0).  This line is called the x₂ axis of the simplex. Consider the following, where the coefficient b_i is equal to the average of the remaining coefficients in the model (that is, E_i=0):

E_i = b_i- (1/(q- 1)) (b₁ + b₂ + ... + b_j + ... + b_q),  i not equal to j = (1/(q-1)) (qb_i - (b₁ + b₂ + ... + b_q))

If this is true, then there is no variation in E(y) along any line perpendicular to the x_i = 0 base of the simplex. In this case, the following is true:

E₂ = b₂ - 1/2(b₁ + b₃) = 90 - 1/2(80 + 100) = 0

We call E_i the linear effect of x_i over the full simplex. The situation in which there is no variation along lines parallel to the x_i axis (i.e., in the x_i direction) results in a reduction of the number of essential components because component x₂ has no effect. In practice, of course, E_i will rarely equal exactly zero. However, there are often E_i that are approximately zero, or at least much smaller than the other E_i.

The response surface shown in Figure 5.2 has two components with no effect. All of the variation is due to component x₁.

Figure 5.2 – Linear Blending Model: Two Components Have No Effect

The response surface in Figure 5.2 is described by this equation:

E(y) = 90x₁ + 80x₂ + 80x₃

In the equation, two coefficients are equal (b₂ =b₃). In this case, the E(y) value is constant along the x₂, x₃ base (x₁ = 0) of the simplex and is also constant along any line parallel to the base--that is, along any contour where x₂ + x₃ is constant. In this situation, x₂ and x₃ have equal effects within the experimental region and their sum x₂ + x₃ can be considered as one component, thus reducing the number of essential components.

It is also possible for the components of a mixture system to have both equal effects and no effects. Consider the following model:

E(y) = 80x₁ + 90x₂ + 90x₃ + 100x₄

b₂ = b₃

b₂ = (b₁ + b₃ + b₄)/3

b₃ = (b₁ + b₂ + b₄)/3

In the model, there are two essential components, x₁ and x₄. In this example, the components with equal effects also have no effects.

An important result from a screening experiment is the estimation of the component effects and the determination of which are the largest effects on the responses.

We determine this by computing the linear effects, Ei, and ranking the linear effects of the components.

5.3 Simplex Screening Designs

The composition of the blends in the screening design depends on the shape of the experimental region. If the region is the entire simplex or can be expressed as a simplex in terms of pseudo-components when all the components have lower bounds, we recommend a 2q + 1 blend design that contains the q pure component blends, q interior points, and the overall centroid. The use of pseudo-components, which is handled automatically by JMP, is discussed in detail in Chapter 6. The responses at the pure component blends have a large effect on the estimates of the b_i coefficients. The interior and center responses show whether severe curvature is present. The interior points are the same as the simplex checkpoints discussed in Chapter 3.

In situations in which it is suspected that complete elimination of one component will have a large effect on the response, it is recommended that the q end effect blends also be evaluated. The end effect blends consist of all components in equal concentrations except for the component in question, which is set to zero. Hence, we use the term end effect.  Mathematically, this implies that x_i = 0, x_j = 1/(q - 1), for all j unequal to i, for the end effect blend for x_i. The addition of end effect blends increases the number of blends in the design to 3q + 1. The four classes of points in the simplex screening design identified by Snee and Marquardt (1976) are summarized in Tables 5.1 and 5.1a.

Table 5.1 – Simplex Screening Design

Blends	No. Blends	Composition
A	q	Pure components
B	q	Interior points midway between the vertices and the simplex centroid
C	1	Centroid of the simplex
D	q	End effect blends

Table 5.1a – Simplex Screening Design Blend Composition

Code	Point Type	No.	x1	x2	.	.	.	xq
A	Vertices	1	1	0	.	.	.	0
		2	0	1	.	.	.	0
		.	.	.	.			.
		.	.	.		.		.
		.	.	.			.	.
		q	0	0	.	.	.	1
B	Interior	1	(q+1)/2q	1/2q				1/2q
		2	1/2q	(q+1)/2q				1/2q
		.	.	.	.			.
		.	.	.		.		.
		.	.	.			.	.
		q	1/2q	1/2q				(q+1)/2q
C	Centroid	1	1/q	1/q	.	.	.	1/q
D	End Effects	1	0	1/(q-1)	.	.	.	1/(q-1)
		2	1/(q-1)	0	.	.	.	1/(q-1)
		.	.	.	.			.
		.	.	.		.		.
		.	.	.			.	.
		q	1/(q-1)	1/(q-1)				0

The distribution of points in the simplex screening design is easy to see in the case of three components. Such an example is shown in Table 5.2, which shows the blends for a three-component rocket propellant study published by Kurotori (1966).

Table 5.2 – Rocket Propellant Study: Blend Compositions

Point Type	Blend	x1	x2	x3	Y
Vertices	1	1	0	0	350
	2	0	1	0	450
	3	0	0	1	650
Interior	4	2/3	1/6	1/6	690
	5	1/6	2/3	1/6	700
	6	1/6	1/6	2/3	980
Centroid	7	1/3	1/3	1/3	1000
End Effect	8	0	1/2	1/2	750
	9	1/2	0	1/2	750
	10	1/2	1/2	0	400

The blends are shown graphically in Figure 5.3. Here we see that the blends line up along the component axes as follows:

Component Axis	Vertex Blend No.	Interior Blend No.	Centroid Blend No.	End Effect Blend No.
X1	1	4	7	8
X2	2	5	7	9
X3	3	6	7	10

We note that the centroid blend is common to all the component axes. Discussing a three-component example is for illustrative purposes only. As discussed earlier, we do not think of screening experiments involving three components. This is a unique situation in which the simplex centroid design, including checkpoints, is identical to the simplex screening design.

Figure 5.3 – Rocket Propellant Study: Graphical Display of Blend Compositions

The six-component simplex screening design shown in Table 5.3 is a specific example of the general design from Table 5.1; in this case, it is a 3q + 1 design that includes end effect blends. Such designs are often designated as ABCD designs because they contain all four types of points. A simplex screening design without end effect blends is often referred to as an ABC design. Two applications of simplex screening designs are discussed below.

The ABCD screening design for six components is shown in Table 5.3. This six- component screening design follows the simplex screening design definition in Tables 5.1 and 5.1a.

Table 5.3 – Six-Component Simplex Screening Design

Point Type	Blend	X1	X2	X3	X4	X5	X6
Vertices	1	1	0	0	0	0	0
	2	0	1	0	0	0	0
	3	0	0	1	0	0	0
	4	0	0	0	1	0	0
	5	0	0	0	0	1	0
	6	0	0	0	0	0	1
Interior	7	7/12	1/12	1/12	1/12	1/12	1/12
	8	1/12	7/12	1/12	1/12	1/12	1/12
	9	1/12	1/12	7/12	1/12	1/12	1/12
	10	1/12	1/12	1/12	7/12	1/12	1/12
	11	1/12	1/12	1/12	1/12	7/12	1/12
	12	1/12	1/12	1/12	1/12	1/12	7/12
Centroid	13	1/6	1/6	1/6	1/6	1/6	1/6
End Effect	14	0	1/5	1/5	1/5	1/5	1/5
	15	1/5	0	1/5	1/5	1/5	1/5
	16	1/5	1/5	0	1/5	1/5	1/5
	17	1/5	1/5	1/5	0	1/5	1/5
	18	1/5	1/5	1/5	1/5	0	1/5
	19	1/5	1/5	1/5	1/5	1/5	0

Using JMP to Generate Simplex Screening Designs. JMP can be used to generate the simplex screening designs described in Table 5.1 using the ABCD Design option. To do this, use the following commands: DOE ► Classical ► Mixture Design ► ABCD Design. The ABCD design produced by JMP contains the pure component blends, 50/50 binary blends, interior blends, and the overall centroid blend. This design can be large-- particularly for four or more components--because the number of 50/50 blends increases rapidly as the number of components increases.

The designs that are described in Table 5.1 are considerably smaller and are produced in the following way:

1.   Generate the ABCD design using JMP.

2.   Remove the binary (50/50) blends from the JMP ABCD design.

3.   Add the end effect blends as described in Table 5.1a.

4.   The resulting design will be the recommended screening design described in Table 5.1a.

The difference in design size can be seen by comparing the number of blends in the JMP designs and the designs described in Table 5.1. For seven, eight, and nine components, the Table 5.1 simplex screening designs have 22, 25, and 28 blends, respectively. The JMP designs contain 36, 45, and 55 blends, resulting in an increased design size of 64%, 80%, and 96%, respectively.

Five-Component, Placebo-Tablet Screening Design. Formulation scientists want to understand the effects of five commonly used diluents in the formulations of pharmaceutical tablets (Lewis et al. 1999). A diluent is an ingredient that lacks pharmacologic activity but is pharmaceutically necessary or desirable. It may be lactose or starch, and it is particularly useful in increasing the bulk of potent drug substances with a mass too small for dosage to allow manufacture or administration. The objective in this experiment is to understand the magnitude and direction of the effect of each of the diluents. With this objective it is appropriate to consider a screening design.

A five-component ABCD design was selected. The hardness values for each in the 3x5 +1 = 16 blends in the simplex screening design are shown in Table 5.4.

Table 5.4 – Five-Component, Placebo-Tablet Screening Design

Point Type	Blend	Lactose	MCCellulose	Corn Starch	Calcium Phosphate	Mannitol	Tablet Hardness
Vertices	1	1	0	0	0	0	13.3
	2	0	1	0	0	0	3.0
	3	0	0	1	0	0	0.0
	4	0	0	0	1	0	1.7
	5	0	0	0	0	1	17.8
Interior	6	0.6	0.1	0.1	0.1	0.1	13.3
	7	0.1	0.6	0.1	0.1	0.1	5.7
	8	0.1	0.1	0.6	0.1	0.1	5.0
	9	0.1	0.1	0.1	0.6	0.1	10.9
	10	0.1	0.1	0.1	0.1	0.6	18.5
Centroid	11	0.2	0.2	0.2	0.2	0.2	10.3
End Effect	12	0	0.25	0.25	0.25	0.25	6.6
	13	0.25	0	0.25	0.25	0.25	7.6
	14	0.25	0.25	0	0.25	0.25	12.7
	15	0.25	0.25	0.25	0	0.25	11.9
	16	0.25	0.25	0.25	0.25	0	8.0

The next step in the analysis of a simplex screening design is to construct the component effects plot, which shows how the response varies as the level of each component is varied along the component axis. The predicted response plot was introduced by Snee (1975) to study the effects of components. Snee (2011) later changed the name to component effects plot, which is more descriptive of the value and use of the plot. For further discussion of the component effects plot, see Snee and Piepel (2013).

Table 5.5 shows how the component effects plot is constructed. A separate curve is plotted for each component by plotting the average response versus the level of the component in the blend. For the five-component example there are five curves. The result is a plot of the average response values with the centroid response being common to the curve for each component.

The placebo-tablet component effects are shown graphically in Figure 5.4. The plot was constructed using the overlay plot command in JMP. (Use the following commands: Graph ► Overlay Plot.) The component effects plot (Figure 5.4) enables us to easily see the following effects:

Effect on Tablet Hardness	Components
Positive	Lactose, Mannitol
Negative	MCC, Corn Starch, Calcium Phosphate

Table 5.5 – Placebo-Tablet Screening: Response (Tablet Hardness) along Component Axes

Point Type	Component Level	Lactose	MCCellulose	Corn Starch	Calcium Phosphate	Mannitol
Vertices	1.0	13.3	3.0	0.0	1.7	17.8
Interior	0.6	13.3	5.7	5.0	10.9	18.5
Centroid	0.2	10.3	10.3	10.3	10.3	10.3
End Effect	0.0	6.6	7.6	7.6	11.9	8.0

Figure 5.4 – Placebo-Tablet Screening Study: Component Effects Plot

The effects of corn starch and MCC are essentially linear. Lactose, calcium phosphate, and mannitol show some curvature but the linear model captures the major portion of the blending response variation. As discussed in Chapter 2, the goal of screening experiments is to identify the components with the largest effects. These components will be studied further in an optimization experiment. In this case, further experimentation was deemed unnecessary.

In our experience and that of others, the majority of the blending relationship is the linear effect, which is larger than the curvilinear (e.g., quadratic) effect. The linear effect then captures the majority of the variation in the blending response. The strength of the linear blending effects can be assessed by including the overall centroid in the design and assessing the fit of the linear model by examining the adjusted R² and RMSE statistics.

The results of simplex screening designs are also more formally analyzed using regression analysis, which was discussed in Chapter 4. The regression analysis produces estimates of the coefficients in the linear blending model and the coefficient standard errors. Component effects can be calculated using the formula defined above. A better approach that produces standard errors and tests of significance for the effects is to use the Cox model estimation approach (Cox 1971). This is accomplished in JMP by fitting the linear blending model (Analyze ► Fit Model), clicking the Red Triangle by Response, clicking Estimates, and selecting Cox Mixtures. The Cox model will be discussed in detail in Chapter 7.

The effects of the components in the placebo-tablet study were confirmed when computed using regression analysis as shown in Table 5.6. All of the effects are statistically significant (p < 0.05). The linear blending effects model has an adjusted R² = 0.78, indicating that the curvature in the blending response surface is not large and that the linear model captures a major portion of the blending response.

The standard errors for the component effects, shown in Table 5.6, can be used to construct 95% confidence limits for the effects. The confidence limits provide a range in which the true component effect is expected to lie.

The standard errors of the component effects are calculated by fitting the Cox (1971) model to the data using standard least squares regression such as that used by JMP software. The Cox model will be discussed further in Chapter 7.

Table 5.6 – Five-Component Placebo-Tablet Experiment: Component Effects

Component	Coefficient	Component Effect	Effect Std Error	t-Ratio	p-Value	Component Effect %
Lactose	15.35	7.75	2.5	3.10	0.010	84.8
Microcrystalline Cellulose (MCC)	4.41	-5.91	2.5	-2.37	0.037	-64.7
Corn Starch	0.89	-10.32	2.5	-4.13	0.002	-113.0
Calcium Phosphate	4.58	-5.70	2.5	-2.28	0.043	-62.4
Mannitol	20.49	14.18	2.5	5.68	0.000	155.1

In the case of the lactose component, these statistics indicate that the true effect of lactose is between 2.75 and 12.75 hardness units (with 95% confidence). The component effects (E_i) and associated 95% confidence limits are shown in Figure 5.5. Figure 5.5 was constructed in JMP using the plot in the Quality and Process platform (Analyze ► Quality and Process ► Variability/Attribute Gauge Chart). This is not actually an attribute gauge chart, of course, but is a simple way to produce this plot.

In Figure 5.5, we see that all the component effects are statistically significant, as none of the confidence intervals include zero. The interpretation of Figure 5.5 regarding statistical significance will be identical to the interpretation of the p-values shown in Table 5.6. A plot of component effects and the associated confidence intervals such as that shown in Figure 5.5 is useful as a summary graphic, particularly when a large number of components is involved.

Practical Significance of Component Effects. The last column in Table 5.6 expresses the component effect as a percentage of the average value of the data collected in the experiment, which is tablet hardness = 9.14 in the case of the placebo-tablet study.  Negative percentages are interpreted in the usual way: as the level of the component is increased, the response decreases. Every case is different, depending on the subject matter. A rule of thumb is that a component effect may be of practical importance if it is larger than 5 to 10% of the average value.

Tests of significance and the associated p-values provide a measure of statistical significance, regardless of whether the effects are larger than can be attributed to experimental variation. Statistical significance does not imply practical significance (whether the effect is large enough to warrant action be taken). Practical significance is determined by subject matter considerations and the goals of the study.

Figure 5.5 – Placebo-Tablet Screening Study: Plot of Component Effects

5.4 Graphical Analysis of Simplex-Screening Designs

One important advantage of the simplex-screening designs is that the resulting data can be easily analyzed graphically. All of the points in these designs lie on the component axes of the simplex. The graphical analysis consists of plotting the response along each component axis, thus producing the component effects plot (Snee 1975). The centroid is common to all of the component axes. Figure 5.4 previously presented the component effects plot for the Placebo-Tablet screening study. As discussed above, it’s easy to see the effects of the five components—in both magnitude and direction.

Figure 5.3 previously showed the distribution of blends across the three-component simplex. We noted how the blends in the design lie on the component axes. The component effects plot of the rocket propellant data is shown in Figure 5.6. In the figure,  we see that all three components have large effects and that there is a high degree of curvature in the response surface. The centroid seems to produce the largest, or near largest, response value. High response levels also occur at low x₁ and at low x₂ levels.

Figure 5.6 – Rocket Propellant Study: Plot of Component Effects

The component effects plot (Figure 5.6) illustrates the gross characteristics of the contour surface, which is a principal objective of screening designs. This does not, however, mean that component effects plots can be considered as replacements for response surface contour plots. Contour plots are used to understand the nature of the response surface--formulations that produce maximum or minimum responses, desired levels of the formulation involving several responses, etc. Screening designs and the associated component effects plots generally do not contain sufficient data to reach such conclusions. Response surface designs, such as those discussed in Chapters 4 and 6, are needed to better understand the response surface.

As noted earlier, in the case of three components, the blends in the ABCD simplex screening design are identical to those in the simplex response surface design. The rocket propellant experiment is such an example. This is a characteristic of the q = 3 design only and does not hold for designs involving four or more components.

Motor Octane Study. A more complex example is the ABCD screening design for the ten-component motor octane data shown in Table 5.7 (Snee and Marquardt 1976).  The component effects plot is given in Figure 5.7. Evaluating only 31 blends (3x10 + 1) produced a great deal of useful information about a complicated formulation system. Using the screening design seems reasonable in this case because at least 66 blends (10 pure components, 45 binary blends, 10 interior checkpoints, and the overall centroid) would have been required to estimate all the coefficients in the 55-term quadratic model. Approximately one third (21/66) of these blends are included in the screening design; hence, if it were decided to develop the full model later, a fair percentage of the experimental work will already have been done in the screening design.

The conclusions reached from an examination of the component effects plot of the motor octane data (Figure 5.7) are summarized in Table 5.8.

Table 5.7 – Motor Octane Study: Blend Octane Responses (y)

Figure 5.7 – Motor Octane Study: Component Effects Plot

Table 5.8 – Motor Octane Study: Conclusions Reached from the Component Effects Plot

Components	Effects
Lt Straight Run	Large negative effect
Lt. Straight Run, LH Cat Cracked A, LH Cat Cracked B, HHCC A and HHCC B	Negative Effects; All other components have positive effects
LH Cat Cracked A and LH Cat Cracked B; HHCC A and HHCC B; LL Cat Cracked and LHCC; Reformate A and Poly.	Four pairs of components with equal or nearly equal blending behavior
Reformate A, LL Cat Cracked, LHCC, HHCC A and HHCC B	Have the Smallest effects
All – Response surface curvature is small	Nonlinear blending is not large

These observations should be confirmed by a multiple regression analysis of the data. In this analysis, the linear blending model is fit to the data by least squares:

E(y) = b₁x₁ + b₂x₂ + ….  + b_qx_q

These coefficients are used to calculate effects (E_i) of the components:

$${\text{E}}_{\text{i}}=({\text{b}}_{\text{i}}-{\overline{b}}_{\text{i}})$$

In the equation, $\overline{b_{\text{i}}}$ is the average of all coefficients other than for x_i. Here is an example:

$$\overline{b}=({\displaystyle {\sum }_{j\ne i}^q{b}_j)/(q-1})$$

The resulting coefficients and effects for the ten-component motor octane example are shown in Table 5.9. Clearly, these results confirm what was seen in Figure 5.7.

Table 5.9 – Motor Octane Study: Component Effects

Component	Regression Coefficient	Component Effect	Effect Std Error	t-Ratio	P-Value	Component Effect %
Lt Straight Run	67.50	-13.06	0.552	-23.64	<.0001	-16.5
Reformate A	82.49	3.59	0.552	6.51	<.0001	4.5
Reformate B	85.00	6.36	0.552	11.52	<.0001	8.0
LL Cat Cracked	81.54	2.55	0.552	4.61	0.0001	3.2
LH Cat Cracked A	76.80	-2.73	0.552	-4.94	<.0001	-3.4
LH Cat Cracked B	76.88	-2.64	0.552	-4.78	0.0001	-3.3
LHCC	81.90	2.94	0.552	5.33	<.0001	3.7
HHCC A	78.43	-0.91	0.552	-1.66	0.113	-1.1
HHCC B	78.67	-0.65	0.552	-1.18	0.252	-0.8
POLY	83.34	4.54	0.552	8.23	<.0001	5.7

The standard errors for the component effects, shown in Table 5.9, can be used to construct 95% confidence limits for the effects. The confidence limits provide a range in which the true component effect is expected to lie. Table 5.10 shows this calculation for component HHCC A.

Table 5.10 – Motor Octane Study: Confidence Limits for Effect of HHCC A

Statistic	Effect	Effect Standard Error	Lower Confidence Limit	Upper Confidence Limit
HHCC A Effect	-0.9	0.552	-2.1	0.3

These statistics indicate that the true effect of HHCC A, if we had an unlimited amount of data, would be between -2.1 and 0.3 units (with 95% confidence). Hence, HHCC A has no detectable effect because zero is included in the confidence interval. A similar conclusion is reached for HHCC B (component 9). Evaluating the confidence limits generally supports the conclusions reached from the graphical analysis.

The component effects and associated 95% confidence limits are shown in Figure 5.8. Confidence limits that do not include zero are considered statistically significant. The interpretation of Figure 5.8 regarding statistical significance will be identical to the interpretation of the p-values shown in Table 5.8.

Figure 5.8 – Motor Octane Study: Component Effects with 95% Confidence Limits

The last column in Table 5.9 expresses the component effect as a percentage of the average value of the data collected in the experiment, which in this case is Octane = 79.3.

The results of the Motor Octane Study are summarized in Table 5.11. These findings are essentially the same as developed from the graphical analysis reported earlier in Table 5.8.

Table 5.11 – Motor Octane Study: Summary of Component Effects Findings

Component	Effect	Findings
Light Straight Run	-13.1	Large negative effect
LH Cat Cracked A, LH Cat Cracked B, HHCC A, HHCC B,	-2.7, -2.6, -0.9, -0.6	Small negative effects
Reformate A, Reformate B, LL Cat Cracked, LH CC, POLY	3.6, 6.4, 2.6, 2.9, 4.5	Positive effects
Reformate A, LL Cat Cracked, LHCC, HHCC A, HHCC B	3.6, 2.6, 2.9, -0.9, -0.6	Smallest effects
LH Cat Cracked A, LH Cat Cracked B	-2.7, -2.6	Equal negative effects
LL Cat Cracked, LHCC	2.6, 2.9	Equal positive effects
HHCC A, HHCC B	-0.9, -0.6	Equal negative effects, not statistically significant
Reformate A, POLY	3.6, 4.5	Equal positive effects
All Components	Response Surface Curvature is small	Nonlinear blending is not large

One additional finding regarding the Motor Octane Study involves the degree of linear blending. In Figure 5.7 it appears that the response surface is essentially linear. This is confirmed by examining the fit of the linear blending model. In this case, the adjusted R² was 0.96 for the linear blending model, confirming the conclusions reached from the simplex-screening design plot regarding the linearity of the blending response surface.  Having learned this information, what should we do with it? How should this guide our future experimentation and modeling? We discuss this in the next section.

5.5 After the Screening Design

The most appropriate next steps in experimentation after the screening design has been completed depend on the objectives of the study. If high octane numbers were the primary consideration, for example, it would be reasonable to do the following experimentation:

Components	Next Steps
HHCC A and HHCC B have small effects.	Fix at some appropriate levels.
LH Cat Cracked A and LH Cat Cracked B: LL Cat Cracked and LHCC; Reformate A and Poly are pairs with equal effects.	Combine pairs of components into single components.
Reformate A, Reformate B and Poly have positive effects.	Study further to understand effects.
Lt Straight Run has strong negative effect,	Set at zero or some small level.

To summarize: Fix components HHCC A and HHCC B at appropriate levels: treat the sums of LH Cat Cracked A and LH Cat Cracked B and LL Cat Cracked and LHCC as two new components. Component Lt Straight Run could be either dropped or included in further studies at a low level if that were economically interesting. Components Reformate A, Reformate B, and Poly would be retained for further study for their positive effects. Components Reformate A and Poly have equal effects and could also be considered for combination as a single component.

It seems reasonable to consider a five-component system consisting of components Reformate A, Reformate B, Poly, and combinations of LH Cat Cracked A and LH Cat Cracked B and LL Cat Cracked and LHCC. In such a system, components Lt Straight Run, HHCC A, and HHCC B would be fixed at some appropriate levels. The resulting data could be fit with a quadratic model. Twenty-one blends would be a reasonable number: five pure components, ten 50/50 blends, five interior checkpoints, and the overall centroid. The quadratic model with five components would include 15 terms; 5 linear terms and 5x4/2 = 10 cross products.

The five pure components have already been evaluated in the screening design. The octanes of the component blends--LL Cat Cracked and LHCC, and HHCC A and HHCC B--can be estimated from a weighted average of the octanes of components LL Cat Cracked and LHCC, and HHCC A and HHCC B, respectively. We need, therefore, to test only 16 additional blends: the 50/50's, the check blends, and the new centroid.

The net result would be evaluation of 47 blends, 31 from the screening experiment and these 16 additional. Conversely, if one chose not to use a sequential approach, 66 blends would be needed to develop the ten-component quadratic model (55 coefficients). The sequential approach results in a 29% savings in experimental work in this case, plus a more thorough understanding of the system. A further benefit is the availability of important results after the first 31 blends have been evaluated.

Screening designs become more difficult to construct when there are both upper and lower limits on the concentrations of some or all of the components. In many instances, the factor space cannot be described as a simplex, and it becomes more challenging to set up a good screening design. Extreme vertices designs are effective in these situations. The construction of these designs is discussed in Chapters 6, 7, and 8.

5.6 Estimation of the Experimental Variation

In the previous section, and in Tables 5.6, 5.9, and 5.10, we saw standard errors for the component effects. These standard errors measure the uncertainty in the estimates of the component effects and are used to construct confidence limits and tests of significance for the component effects. The effect standard errors are a function of the standard deviation of the experimental variation--i.e., experimental error or noise. In screening experiments, we are interested in detecting large effects—those effects that are perhaps five or six times the experimental error. Precise estimates of component effects and an accurate estimate of experimental error are, therefore, of less importance in screening experiments than in response surface experiments as discussed in Chapter 4. We noted earlier that in most instances the results of a screening experiment can be determined by simply ranking the absolute values of the component effects. An estimate of experimental error will be needed, however, if it is of interest to compute confidence limits for the component coefficients and effects. To obtain an estimate of the experimental error, use one of the following options:

•   Replicate five to ten blends in the design. It is good experimental practice to replicate the overall centroid since it lies on each component axis.

•   Use the estimate of experimental variation obtained in previous studies of the system.

•   Use the regression analysis residual mean square error.

One should keep in mind that this last approach will overestimate the experimental variation if the response surface contains much curvature. The graphical analysis should indicate whether a linear blending model will give an adequate approximation to the response surface or if there is noteworthy curvature.

5.7 Summary and Looking Forward

Following the experimental strategy introduced in Chapter 2, we used screening experiments to identify the components that have the largest effects. These components are subsequently studied in optimization experiments to identify the best formulations. In this chapter, we introduced the concepts used to screen formulation components and discussed screening designs for the situation in which the experimental region is the full simplex (all components can be varied over the range 0 to 100%). Formulation experiments in which the components have constraints or are varied over different ranges will be discussed in Chapters 6, 7, and 8. In the next chapter we discuss the general problem of constrained formulations, including the identification of the usable experimental region, and the potential usage of pseudo-components.

5.8 References

Cox, D. R. (1971) “A Note on Polynomial Response Functions for Mixtures.” Biometrika, 58 (1), 155-159.

Kurotori, I. S. (1966) “Experiments with Mixtures of Components Having Lower Bounds.” Industrial Quality Control, 22 (11), May 1966, 592-596.

Lewis, G. A., D. Mathieu and R. Phan-Tan-Luu. (1999) Pharmaceutical Experimental Design, Marcel Dekker, New York, NY.

Snee, R. D. (1975) “Discussion of: ‘The use of gradients to aid in the interpretation of mixture response surfaces’.” Technometrics. 17 (4), 425-430.

Snee, R. D. and D. W. Marquardt. (1976) “Screening Concepts and Designs for Experiments with Mixtures.” Technometrics, 18 (1), 19-29.

Snee, R.D. (2011) “Understanding Formulation Systems-A Six Sigma Approach.” Quality Engineering, 23 (3), 278-286.

Snee, R.D. and G.F. Piepel. (2013) “Assessing Component Effects in Formulation Systems.” Quality Engineering, 25 (1), 46-53.