Introduction to Mixture Designs

Abstract

In previous chapters, we discussed situations where our factors, or independent variables (X’s), were categorical or continuous, and there were no constraints which limited our choice of combinations of levels which these variables can assume. In this chapter, we introduce a different type of design called a mixture design, where factors (X’s) are components of a blend or mixture. For instance, if we want to optimize a recipe for a given food product (say bread), our X’s might be flour, baking powder, salt, and eggs. However, the proportions of these ingredients must add up to 100% (or 1, if written as fractions), which complicates our design and analysis if we were to use only the techniques covered up to now.

Appendix

Example 17.7 Vitamin Mixture using R

To analyze the same vitamin-mixture example in R, we can import the data as previously, or we can create our own data. The second option can be implemented in two ways: using the SLD() function (mixexp package) or the mixDesign() function (qualityTools package). The steps are as follows:

# Option 1: using the SLD() function

> vitamin <- SLD(3, 3)

# The first and second arguments of the SLD() function refer to the number of factors and levels, respectively

> score <- c(32, 38, 40, 43, 33, 37, 43, 31, 42, 36) > vitamin <- data.frame(vitamin, score) > vitamin

	x1	x2	x3	score
1	1.0000000	0.0000000	0.0000000	32
2	0.6666667	0.3333333	0.0000000	38
(…)
10	0.0000000	0.0000000	1.0000000	36

# Option 2: using the mixDesign() function

> vitamin <- mixDesign(p=3, n=3, type="lattice", axial=FALSE, +randomize=FALSE)

# p and n refer to the number of factors and levels, respectively

Warning messages: 1: In `[<-`(`*tmp*`, i, value = <S4 object of class +"doeFactor">) : implicit list embedding of S4 objects is deprecated 2: In `[<-`(`*tmp*`, i, value = <S4 object of class +"doeFactor">) : implicit list embedding of S4 objects is deprecated 3: In `[<-`(`*tmp*`, i, value = <S4 object of class +"doeFactor">) : implicit list embedding of S4 objects is deprecated > score <- c(32, 38, 33, 40, 31, 43, 43, 42, 36, 37) > response(vitamin) <- score [1] "score" > vitamin

	StandOrder	RunOrder	Type	A	B	C	score
1	1	1	1-blend	1.0000000	0.0000000	0.0000000	32
2	2	2	2-blend	0.6666667	0.3333333	0.0000000	38
(…)
10	10	10	<NA>	0.3333333	0.3333333	0.3333333	37

The analysis of this design can also be performed in two ways, with similar (fortunately!!) results:

# Option 1: using the lm() function

> vitamin_model <- lm(score~-1+x1+x2+x3+x1:x2+x1:x3+x2:x3, +data=vitamin) > summary(vitamin_model) Call: lm(formula = score ~ -1 + x1 + x2 + x3 + x1:x2 + x1:x3 + x2:x3, +data = vitamin) Residuals:

1	2	3	4	5	6	7
-0.8571	0.8571	-0.4286	0.2857	1.7143	-0.8571	-0.4286
8	9	10
-1.2857	0.8571	0.1429

Coefficients:

	Estimate	Std. Error	t value	Pr(>|t|)
x1	32.857	1.331	24.687	1.60e-05	***
x2	42.714	1.331	32.093	5.62e-06	***
x3	35.857	1.331	26.941	1.13e-05	***
x1:x2	4.500	5.892	0.764	0.4876
x1:x3	-11.571	5.892	-1.964	0.1210
x2:x3	13.500	5.892	2.291	0.0837	.

--- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘’ 1 Residual standard error: 1.414 on 4 degrees of freedom Multiple R-squared: 0.9994, Adjusted R-squared: 0.9986 F-statistic: 1186 on 6 and 4 DF, p-value: 1.891e-06

# Option 2: using the MixModel() function

> MixModel(frame=vitamin, "score", mixcomps=c("A", "B", +"C"), model=2)

	coefficient	Std. err.	t.value	Prob
A	32.85714	1.330950	24.6869805	1.597878e-05
B	42.71429	1.330950	32.0930746	5.619531e-06
C	35.85714	1.330950	26.9410091	1.128541e-05
B:A	4.50000	5.891883	0.7637626	4.875729e-01
C:A	-11.57143	5.891883	-1.9639610	1.210039e-01
B:C	13.50000	5.891883	2.2912878	8.373821e-02

Residual standard error: 1.414214 on 4 degrees of freedom Corrected Multiple R-squared: 0.9561644 Call: lm(formula = mixmodnI, data = frame) Coefficients:

A	B	C	A:B	A:C	B:C
32.86	42.71	35.86	4.50	-11.57	13.50

There are also two ways we can create a triangular surface in R (shown in Fig. 17.16) using the contourPlot3() function, which will result in the same (again, fortunately) graph.

Fig. 17.16

Triangular surface in R

# Option 1:

> contourPlot3 (A, B, C, score, data=vitamin, form="score~+-1+A+B+C+A:B+A:C+B:C")

# Option 2:

> contourPlot3 (A, B, C, score, data=vitamin, +form=vitamin_model)

Example 17.8 A New Alloy Mixture using R

We can use the same software packages to design and analyze a simplex-centroid design. The options are shown below:

# Option 1: using the SCD() function

> alloy <- SCD(4) > strength <- c(102, 110, 164, 140, 116, 113, 170, 156, +138, 153, 134, 141, 140, +143, 142) > alloy <- data.frame(alloy, strength) > alloy

	x1	x2	x3	x4	strength
1	1.0000000	0.0000000	0.0000000	0.0000000	102
2	0.0000000	1.0000000	0.0000000	0.0000000	110
(…)
15	0.2500000	0.2500000	0.2500000	0.2500000	36

# Option 2: using the mixDesign() function

> alloy <- mixDesign(p=4, type="centroid", axial=FALSE, +randomize=FALSE) Warning messages: 1: In `[<-`(`*tmp*`, i, value = <S4 object of class +"doeFactor">) : implicit list embedding of S4 objects is deprecated 2: In `[<-`(`*tmp*`, i, value = <S4 object of class +"doeFactor">) : implicit list embedding of S4 objects is deprecated 3: In `[<-`(`*tmp*`, i, value = <S4 object of class +"doeFactor">) : implicit list embedding of S4 objects is deprecated 4: In `[<-`(`*tmp*`, i, value = <S4 object of class +"doeFactor">) : implicit list embedding of S4 objects is deprecated > strength <- c(140, 164, 110, 102, 153, 138, 156, 170, +113, 116, 143, 140, 134, +141, 142) > response(alloy) <- strength [1] "strength" > alloy

	Stand Order	Run Order	Type	A	B	C	D	score
1	1	1	1-blend	0.0000000	0.0000000	0.0000000	1.0000000	140
2	2	2	1-blend	0.0000000	0.0000000	1.0000000	0.0000000	164
(⋮)
15	15	15	center	0.2500000	0.2500000	0.2500000	0.2500000	142

The analysis of this design can also be performed in two ways, with similar (‘nuff said) results:

# Option 1: using the lm() function

> alloy_model <- lm(strength~-1+x1+x2+x3+x4+x1:x2+x1:x3+x1:+x4+x2:x3+x2:x4+x3:x4, data=alloy) > summary(alloy_model) Call: lm(formula = strength ~ -1 + x1 + x2 + x3 + x4 + x1:x2 + x1:+x3 + x1:x4 +x2:x3 + x2:x4 + x3:x4, data = alloy) Residuals:

1	2	3	4	5	6	7
-0.55271	-0.63166	-0.65797	-1.68429	-0.06304	0.04222	4.14748
8	9	10	11	12	13	15
0.35801	4.46327	4.56854	5.02123	-4.21561	-4.45245	-5.16298
15
-1.18005

Coefficients:

	Estimate	Std. Error	t value	Pr(>|t|)
x1	102.553	5.467	18.757	7.93e-06	***
x2	110.632	5.467	20.235	5.45e-06	***
x3	164.658	5.467	30.116	7.57e-07	***
x4	141.684	5.467	25.914	1.60e-06	***
x1:x2	37.883	23.559	1.608	0.168742
x1:x3	-82.590	23.559	-3.506	0.017180	*
x1:x4	174.936	23.559	7.425	0.000698	***
x2:x3	71.989	23.559	3.056	0.028241	*
x2:x4	29.515	23.559	1.253	0.265674
x3:x4	-18.959	23.559	-0.805	0.457517

--- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘’ 1 Residual standard error: 5.531 on 5 degrees of freedom Multiple R-squared: 0.9995, Adjusted R-squared: 0.9984 F-statistic: 944.1 on 10 and 5 DF, p-value: 1.51e-07

# Option 2: using the MixModel() function

> MixModel(frame=alloy, "strength", mixcomps=c("A", "B", +"C", "D"), model=2)

	coefficient	Std. err.	t.value	Prob
A	102.55271	8.292953	12.3662476	6.125730e-05
B	110.63166	8.292953	13.3404418	4.233074e-05
C	164.10534	8.292953	19.7885291	6.087322e-06
D	142.23692	8.292953	17.1515406	1.233387e-05
B:A	37.88344	35.734106	1.0601479	3.375799e-01
C:A	-72.64288	35.734106	-2.0328724	9.775260e-02
D:A	164.98870	35.734106	4.6171212	5.750377e-03
B:C	81.93607	35.734106	2.2929374	7.039056e-02
B:D	19.56765	35.734106	0.5475902	6.075199e-01
C:D	-18.95867	35.734106	-0.5305483	6.184412e-01

Residual standard error: 8.38936 on 5 degrees of freedom Corrected Multiple R-squared: 0.9361068 Call: lm(formula = mixmodnI, data = frame) Coefficients:

A	B	C	D	A:B	A:C	A:D
102.55	110.63	164.11	142.24	37.88	-72.64	164.99
B:C	B:D	C:D
81.94	19.57	-18.96