Difference between revisions of "Design and analysis of experiments (2013)"

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Class date(s):

08 March 2013 to 09 April 2013

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This page is out of date. Please see the latest version of these notes.

Course notes and slides

Course textbook (print chapter 5)
Slides for class

Class materials

08 Mar 2013 (Class 08C): Audio and video and guest speaker's slides (Emily Nichols)
12 Mar 2013 (Class 09A): Audio and video
13 Mar 2013 (Class 09B): Audio and video
15 Mar 2013 (Class 09C): Audio and video
19 Mar 2013 (Class 10A): Audio and video
20 Mar 2013 (Class 10B): Audio and video
26 Mar 2013 (Class 11A): Audio and video
27 Mar 2013 (Class 11B): Audio and video
02 Apr 2013 (Class 12A): Audio and video
03 Apr 2013 (Class 12B): Audio and video
05 Apr 2013 (Class 12C): Audio and video
09 Apr 2013 (Class 13A-part1): Audio and video

Software source code

Source code to estimate the DOE model (scroll down if the code is not visible)

R

MATLAB

T <- c(-1, +1, -1, +1)  # centered and scaled temperature
S <- c(-1, -1, +1, +1)  # centered and scaled substrate concentration
y <- c(69, 60, 64, 53)  # conversion is the response, y
mod <- lm(y ~ T + S + T * S)
summary(mod)

Call:
lm(formula = y ~ T + S + T * S)

Residuals:
ALL 4 residuals are 0: no residual degrees of freedom!

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)     61.5         NA      NA       NA
T               -5.0         NA      NA       NA
S               -3.0         NA      NA       NA
T:S             -0.5         NA      NA       NA

Residual standard error: NaN on 0 degrees of freedom
Multiple R-squared:     1,	Adjusted R-squared:   NaN 
F-statistic:   NaN on 3 and 0 DF,  p-value: NA

% Set up the X matrix:

n = 4;
temperature = [-1, +1, -1, +1];
substrate = [-1, -1, +1, +1];

X = [ ones(n, 1), temperature', substrate', (temperature .* substrate)'];

% or you can type it in directly (but this is error prone)
X = [+1 -1 -1 +1; ...
     +1 +1 -1 -1; ...
     +1 -1 +1 -1; ...
     +1 +1 +1 +1];

% Conversion is your y-variable
y = [69, 60, 64, 53]';

% Regression coefficients = b = [Intercept, b_T, b_S, b_{TS}]
b = inv(X'*X)*X'*y

3-factor example <rst> <rst-options: 'toc' = False/> The data are from a plastics molding factory which must treat its waste before discharge. The :math:`y` variable represents the average amount of pollutant discharged [lb per day], while the 3 factors that were varied were:

	-	:math:`C`: the chemical compound added [A or B]

- :math:`T`: the treatment temperature [72°F or 100°F] - :math:`S`: the stirring speed [200 rpm or 400 rpm] - :math:`y`: the amount of pollutant discharged [lb per day]

.. tabularcolumns:: |l|l||c|c|c||c|

+-----------+-------+---------------+-----------------+-----------------+-----------------+ | Experiment| Order | :math:`C` | :math:`T` [°F] | :math:`S` [rpm] | :math:`y` [lb] | +===========+=======+===============+=================+=================+=================+ | 1 | 5 | A | 72 | 200 | 5 | +-----------+-------+---------------+-----------------+-----------------+-----------------+ | 2 | 6 | B | 72 | 200 | 30 | +-----------+-------+---------------+-----------------+-----------------+-----------------+ | 3 | 1 | A | 100 | 200 | 6 | +-----------+-------+---------------+-----------------+-----------------+-----------------+ | 4 | 4 | B | 100 | 200 | 33 | +-----------+-------+---------------+-----------------+-----------------+-----------------+ | 5 | 2 | A | 72 | 400 | 4 | +-----------+-------+---------------+-----------------+-----------------+-----------------+ | 6 | 7 | B | 72 | 400 | 3 | +-----------+-------+---------------+-----------------+-----------------+-----------------+ | 7 | 3 | A | 100 | 400 | 5 | +-----------+-------+---------------+-----------------+-----------------+-----------------+ | 8 | 8 | B | 100 | 400 | 4 | +-----------+-------+---------------+-----------------+-----------------+-----------------+

We showed the cube plot for this system in the class on 10 March. From the cube plot we could already see the main factors, and even the **CS** interaction was noticeable.

**C effect**: There are 4 estimates of :math:`C = \displaystyle \frac{(+25) + (+27) + (-1) + (-1)}{4} = \frac{50}{4} = \bf{12.5}`
**T effect**: There are 4 estimates of :math:`T = \displaystyle \frac{(+1) + (+3) + (+1) + (+1)}{4} = \frac{6}{4} = \bf{1.5}`
**S effect**: There are 4 estimates of :math:`S = \displaystyle \frac{(-27) + (-1) + (-29) + (-1)}{4} = \frac{-58}{4} = \bf{-14.5}`
**CT interaction**: There are 2 estimates of :math:`CT`. Recall that interactions are calculated as the half difference going from high to low. Consider the change in :math:`C` when

- :math:`T_\text{high}` (at :math:`S` high) = 4 - 5 = -1 - :math:`T_\text{low}` (at :math:`S` high) = 3 - 4 = -1 - First estimate = [(-1) - (-1)]/2 = 0 - :math:`T_\text{high}` (at :math:`S` low) = 33 - 6 = +27 - :math:`T_\text{low}` (at :math:`S` low) = 30 - 5 = +25 - Second estimate = [(+27) - (+25)]/2 = +1

- Average **CT** interaction = (0 + 1)/2 = **0.5** - You can interchange :math:`C` and :math:`T` and still get the same result.

**CS interaction**: There are 2 estimates of :math:`CS`. Consider the change in :math:`C` when

- :math:`S_\text{high}` (at :math:`T` high) = 4 - 5 = -1 - :math:`S_\text{low}` (at :math:`T` high) = 33 - 6 = +27 - First estimate = [(-1) - (+27)]/2 = -14 - :math:`S_\text{high}` (at :math:`T` low) = 3 - 4 = -1 - :math:`S_\text{low}` (at :math:`T` low) = 30 - 5 = +25 - Second estimate = [(-1) - (+25)]/2 = -13

- Average **CS** interaction = (-13 - 14)/2 = **-13.5** - You can interchange :math:`C` and :math:`S` and still get the same result.

**ST interaction**: There are 2 estimates of :math:`ST`: (-1 + 0)/2 = **-0.5**, calculate in the same way as above.

**CTS interaction**: There is only a single estimate of :math:`CTS`:

- :math:`CT` effect at high :math:`S` = 0 - :math:`CT` effect at low :math:`S` = +1 - :math:`CTS` interaction = [(0) - (+1)] / 2 = **-0.5**

- You can calculate this also by considering the :math:`CS` effect at the two levels of :math:`T` - Or, you can calculate this by considering the :math:`ST` effect at the two levels of :math:`C`. - All 3 approaches give the same result.

Next, use computer software to verify that

.. math::

y = 11.25 + 6.25x_C + 0.75x_T -7.25x_S + 0.25 x_C x_T -6.75 x_C x_S -0.25 x_T x_S - 0.25 x_C x_T x_S

The :math:`\mathbf{X}` matrix and :math:`\mathbf{y}` vector used to calculate the least squares model:

.. math::

\begin{bmatrix} 5\\30\\6\\33\\4\\3\\5\\4 \end{bmatrix} &= \begin{bmatrix} +1 & -1 & -1 & -1 & +1 & +1 & +1 & -1\\ +1 & +1 & -1 & -1 & -1 & -1 & +1 & +1\\ +1 & -1 & +1 & -1 & -1 & +1 & -1 & +1\\ +1 & +1 & +1 & -1 & +1 & -1 & -1 & -1\\ +1 & -1 & -1 & +1 & +1 & -1 & -1 & +1\\ +1 & +1 & -1 & +1 & -1 & +1 & -1 & -1\\ +1 & -1 & +1 & +1 & -1 & -1 & +1 & -1\\ +1 & +1 & +1 & +1 & +1 & +1 & +1 & +1\\ \end{bmatrix} \begin{bmatrix} b_0 \\ b_C \\ b_T \\ b_{S} \\ b_{CT} \\ b_{CS} \\ b_{TS} \\ b_{CTS} \end{bmatrix} \\ \mathbf{y} &= \mathbf{X} \mathbf{b} </rst>

# Create the design matrix in a quick way in R
C <- T <- S <- c(-1, +1)
design <- expand.grid(C=C, T=T, S=S)
design
   C  T  S
1 -1 -1 -1
2  1 -1 -1
3 -1  1 -1
4  1  1 -1
5 -1 -1  1
6  1 -1  1
7 -1  1  1
8  1  1  1

C <- design$C
T <- design$T
S <- design$S
y <- c(5, 30, 6, 33, 4, 3, 5, 4)

# Full factorial model
# mod.full <- lm(y ~ C + T + S + C*T + C*S + S*T + C*T*S)

# This powerful notation will expand all terms up to the 3rd order interactions
mod.full <- lm( y ~ (C + T + S)^3 ) 

summary(mod.full)
Call:
lm(formula = y ~ C + T + S + C * T + C * S + S * T + C * T * S)

Residuals:
ALL 8 residuals are 0: no residual degrees of freedom!

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)    11.25         NA      NA       NA
C               6.25         NA      NA       NA
T               0.75         NA      NA       NA
S              -7.25         NA      NA       NA
C:T             0.25         NA      NA       NA
C:S            -6.75         NA      NA       NA
T:S            -0.25         NA      NA       NA
C:T:S          -0.25         NA      NA       NA

# Guide to estimating significant coefficients, ignoring the intercept (1st coefficient)
coeff.full <- coef(mod.full)[2:length(coef(mod.full))]

# Pareto plot of the absolute coefficients
library(lattice)
coeff <- sort(abs(coeff.full), index.return=TRUE)
barchart(coeff$x, 
         xlim=c(0, max(abs(coeff.full))+0.1),
         xlab=list("Magnitude of effect", cex=1.5), 
         ylab = list("Effect", cex=1.5),
         groups=(coeff.full>0)[coeff$ix], col=c("lightblue", "orange"),
         scales=list(cex=1.5)
)

# Eliminate some factors: use the regular linear model notation
# Eliminate C*T*S, C*T and S*T
mod.sub <- lm(y ~ C + T + S + C*S)
summary(mod.sub)
confint(mod.sub)
coeff.sub <- coef(mod.sub)[2:length(coef(mod.sub))]
coeff <- sort(abs(coeff.sub), index.return=TRUE)
barchart(coeff$x, 
         xlim=c(0, max(abs(coeff.sub))+0.1),
         xlab=list("Magnitude of effect", cex=1.5), 
         ylab = list("Effect", cex=1.5),
         groups=(coeff.sub>0)[coeff$ix], col=c("lightblue", "orange"),
         scales=list(cex=1.5)
)

Fractional factorial code in R (see Saturated Design example in the slides)

A <- B <- C <- c(-1, +1)
design <- expand.grid(A=A, B=B, C=C)
design
#    A  B  C
# 1 -1 -1 -1
# 2  1 -1 -1
# 3 -1  1 -1
# 4  1  1 -1
# 5 -1 -1  1
# 6  1 -1  1
# 7 -1  1  1
# 8  1  1  1

A <- design$A
B <- design$B
C <- design$C

# Specify the other factors 4: up to 7 factors + intercept can be estimated
# from the 8 experiments.
D <- A*B
E <- A*C
F <- B*C
G <- A*B*C

y <- c(77.1, 68.9, 75.5, 72.5, 67.9, 68.5, 71.5, 63.7)
mod <- lm(y ~ A + B + C + D + E + F + G)
summary(mod)

library(lattice)
coeff.full <- coef(mod)[2:length(coef(mod))]
coeff <- sort(abs(coeff.full), index.return=TRUE)
barchart(coeff$x, 
         xlim=c(0, max(abs(coeff.full))+0.1),
         xlab=list("Magnitude of effect", cex=1.5), 
         ylab = list("Effect", cex=1.5),
         groups=(coeff.full>0)[coeff$ix], col=c("lightblue", "orange"),
         scales=list(cex=1.5)
)

# Factors B, D and F are not important. Remove and rebuild.
mod.update.1 <- lm(y ~ A + C + E + G)
confint(mod.update.1)

# The confidence interval for E spans zero. Might be of practical relevance,
# but it is not of statistical relevance.

mod.update.2 <- lm(y ~ A + C + G)
confint(mod.update.2)

Code for contour plots in response surface methods

R

MATLAB

# Initial set of 4 runs
T <- c(-1,    +1,   -1,   +1,    0)
S <- c(-1,    -1,   +1,   +1,    0)
y <- c(193,  310,  468,  571,  407)

# Visualize the surface
# ------------------------------------------------------
mod <- lm(y ~ T + S + T*S)
summary(mod)

N <- 50  # resolution of surface (higher values give smoother plots)

# The lower and upper bounds, in coded units, over which we want
# to visualize the surface
bound <- 3 
T_plot <- seq(-bound, bound ,length=N)
S_plot <- seq(-bound, bound, length=N)
grd <- expand.grid(T=T_plot, S=S_plot)

# Predict directly from least squares model
grd$y <- predict(mod, grd)

library(lattice)
contourplot(y ~ T * S, 
            data = grd,
            cuts = 10, 
            region = TRUE,
            col.regions = terrain.colors,
            xlab = "Temperature",
            ylab = "Substrate",
            main = "Predicted response to factors T and S, in coded units"
)

trellis.focus("panel", 1, 1, highlight=FALSE)
lpoints(T, S, pch="O", col="red", cex=3)
llines(c(-1, +1, +1, -1, -1), c(-1, -1, +1, +1, -1), col="red", lwd=3)
lpoints(1, 2.44, col="red", cex=5, pch="*")
ltext(0.4, 2.44, "Expt 5", cex=2, col="red")
trellis.unfocus()

X = [+1 +1 +1 +1 +1; ...
     -1 +1 -1 +1 0; ...
     -1 -1 +1 +1 0; ...
     +1 -1 -1 +1 0]';
 
% Profit values recorded from the factorial experiments
profit = [193 310 468 571 407]';
% Coefficient order = [Intercept, b_T, b_S, b_{TS}]
b = inv(X'*X)*X'*profit
 
% What does the model surface look like?
[T, S] = meshgrid(-2:0.1:2, -2:0.1:2);
Y = b(1) + b(2).*T + b(3).*S + b(4).*T.*S;
 
subplot(1, 2, 1)
surf(T, S, Y)
xlabel('T')
ylabel('S')
grid('on')
 
subplot(1, 2, 2)
contour(T, S, Y)
xlabel('T')
ylabel('S')
grid('on')
axis equal
colorbar

Extending the response surface around the next factorial point 6

# Factorial at point 6
T <- c(-1,    +1,   -1,   +1,    0)
S <- c(-1,    -1,   +1,   +1,    0)
y <- c(694, 725, 620, 642, 688)

# Build a least squares model for the local region
mod <- lm(y ~ T + S + T*S)
summary(mod)

next.12 <- data.frame(T=+1, S=-3.0)
pred.12 <- predict(mod, next.12)   # predicted $811, but actual was $716: far off!

# Visualize the surface
N <- 100  # resolution of surface (higher values give smoother plots)

# The lower and upper bounds, in coded units, over which we want
# to visualize the surface
bound <- 3.1
T_plot <- seq(-bound, bound ,length=N)
S_plot <- seq(-bound, bound, length=N)
grd <- expand.grid(T=T_plot, S=S_plot)

# Predict directly from least squares model
grd$y <- predict(mod, grd)

library(lattice)
contourplot(y ~ T * S, 
            data = grd,
            cuts = 10, 
            region = TRUE,
            col.regions = terrain.colors,
            xlab = "Temperature",
            ylab = "Substrate",
            main = "Predicted response to factors T and S, in coded units"
)

trellis.focus("panel", 1, 1, highlight=FALSE)
lpoints(T, S, pch="O", col="red", cex=3)
llines(c(-1, +1, +1, -1, -1), c(-1, -1, +1, +1, -1), col="red", lwd=3)
lpoints(1, -3, col="red", cex=5, pch="*")
ltext(1.2, -3, "12", cex=2, col="red")
trellis.unfocus()



# Central composite design: experiments 8 to 11, 6, and 13 to 16 [9 expts]

T <- c(-1,    +1,   -1,   +1,    0,        0, +sqrt(2),       0, -sqrt(2))
S <- c(-1,    -1,   +1,   +1,    0, -sqrt(2),        0, sqrt(2),        0)
y <- c(694,  725,  620,  642,  688,      720,      699,     610,      663)

# Add quadratic terms to the model
mod.quad <- lm(y ~ T + S + T*S + I(S^2) + I(T^2))
summary(mod.quad)
confint(mod.quad)

# Predict directly from least squares model
grd$y <- predict(mod.quad, grd)
library(lattice)
contourplot(y ~ T * S, 
            data = grd,
            cuts = 100,  # use more contour lines 
            region = TRUE,
            col.regions = terrain.colors,
            xlab = "Temperature",
            ylab = "Substrate",
            main = "Predicted response to factors T and S, in coded units"
)
trellis.focus("panel", 1, 1, highlight=FALSE)
lpoints(T, S, pch="O", col="red", cex=3)
llines(c(-1, +1, +1, -1, -1), c(-1, -1, +1, +1, -1), col="red", lwd=3)
llines(c(0, 0), c(-sqrt(2), sqrt(2)), col="red", lwd=3)
llines(c(-sqrt(2), sqrt(2)), c(0, 0),  col="red", lwd=3)
trellis.unfocus()

# Predicted profit for next experiment, number 17 at
next.17 <- data.frame(T=2.0, S=-1.9)
pred.17 <- predict(mod.quad, next.17)   # $736; the actual recorded profit was $738, so close!

@@ Line 55: / Line 55: @@
 | video_download_link12_MP4_size = 234 M
 | video_notes12 =}}
+<span style="color:#900000">{{Huge|This page is out of date.}}</span> {{Huge|Please see the [[Design_and_analysis_of_experiments |latest version of these notes]].}}
 == Course notes and slides ==
@@ Line 77: / Line 82: @@
 == Software source code ==
-Source code to estimate the DOE model
+Source code to estimate the DOE model (''scroll down if the code is not visible'')
 {| class="wikitable"
@@ Line 85: / Line 90: @@
 |-
 |
-<syntaxhighlight lang="sas">
+<syntaxhighlight lang="R">
 T <- c(-1, +1, -1, +1)  # centered and scaled temperature
 S <- c(-1, -1, +1, +1)  # centered and scaled substrate concentration
@@ Line 222: / Line 227: @@
 </rst>
-<syntaxhighlight lang="sas">
+<syntaxhighlight lang="R">
 # Create the design matrix in a quick way in R
 C <- T <- S <- c(-1, +1)
@@ Line 297: / Line 302: @@
 Fractional factorial code in R (see Saturated Design example in the slides)
-<syntaxhighlight lang="sas">
+<syntaxhighlight lang="R">
 A <- B <- C <- c(-1, +1)
 design <- expand.grid(A=A, B=B, C=C)
@@ Line 355: / Line 360: @@
 |-
 |
-<syntaxhighlight lang="sas">
+<syntaxhighlight lang="R">
 # Initial set of 4 runs
 T <- c(-1,    +1,   -1,   +1,    0)
@@ Line 430: / Line 435: @@
 Extending the response surface around the next factorial point 6
-<syntaxhighlight lang="sas">
+<syntaxhighlight lang="R">
 # Factorial at point 6
 T <- c(-1,    +1,   -1,   +1,    0)

Difference between revisions of "Design and analysis of experiments (2013)"

Latest revision as of 19:01, 27 January 2017

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