1. Chapter 4: Randomized Blocks, Latin Squares, and Related Designs
Design & Analysis of Experiments 8E 2012 Montgomery
STT 511-STT411: Design of Experiments and Analysis of Variance Dr. Cuixian Chen
Chapter 4
Chapter 4: Randomized Blocks, Latin Squares, and Related Designs

2. Example 1-- Review: Design Of Experiments (Chap2)
Completely randomized designs (CRD)
Random Allocation
Compare Response
Repeat
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3. Example2 -- Review: Design Of Experiments (Chap2)
Completely randomized designs (CRD)
Randomized complete block design (RCBD)
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4. Review: Design Of Experiments (Matched pairs designs)
Matched pairs: Choose pairs of subjects that are closely matched—e.g., same sex, height, weight, age, and race.
Within each pair, randomly assign who will receive which treatment.
It is also possible to just use a single person, and give the two treatments to this person over time in random order. In this case, the “matched pair” is just the same person at different points in time.
The most closely
matched pair
studies use
identical twins.

5. Randomized complete block design (RCBD)

6. Design of Engineering Experiments – The Blocking Principle
Text Reference, Chapter 4
Blocking and nuisance factors
The randomized complete block design or the RCBD
Extension of the ANOVA to the RCBD
Other blocking scenarios…Latin square designs
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7. The Blocking Principle
Blocking is a technique for dealing with nuisance factors
A nuisance factor is a factor that probably has some effect on the response, but it’s of no interest to the experimenter… however, the variability it transmits to the response needs to be minimized
Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units, or locations.
Many industrial experiments involve blocking (or should)
Failure to block is a common flaw in designing an experiment (consequences?)
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8. The Hardness Testing Example
Text reference, pg 139, 140
We wish to determine whether 4 different tips produce different (mean) hardness reading on a Rockwell hardness tester
Gauge & measurement systems capability studies are frequent areas for applying DOE
Assignment of the tips to an experimental unit; that is, a test coupon
Structure of a completely randomized experiment
The test coupons are a source of nuisance variability
Alternatively, the experimenter may want to test the tips across coupons of various hardness levels
The need for blocking
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9. The Hardness Testing Example
Test coupon
Tips
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10. The Hardness Testing Example
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11. The Hardness Testing Example
Determine whether 4 different tips produce different (mean) hardness reading on a Rockwell hardness tester.
Test coupons are a source of nuisance variability.
Alternatively, experimenter may want to test tips across coupons of various hardness levels.
The need for blocking.
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12. The Hardness Testing Example
Determine whether 4 different tips produce different (mean) hardness reading on a Rockwell hardness tester.
Test coupons are a source of nuisance variability.
Alternatively, experimenter may want to test tips across coupons of various hardness levels.
The need for blocking.
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13. The Hardness Testing Example
To conduct this experiment as a RCBD, assign all 4 tips to each coupon
Each coupon is called a “block”; that is, it’s a more homogenous experimental unit on which to test the tips
Variability between blocks can be large, variability within a block should be relatively small
In general, a block is a specific level of the nuisance factor
A complete replicate of the basic experiment is conducted in each block
A block represents a restriction on randomization
All runs within a block are randomized
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14. The Hardness Testing Example
Suppose that we use b = 4 blocks:
Notice the two-way structure of the experiment
Once again, we are interested in testing the equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks)
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15. Coded Data=(#-9.5)*10
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16. Extension of the ANOVA to the RCBD
Suppose single factor, a treatments (factor levels) and b blocks
A statistical model (effects model) for the RCBD is
The relevant (fixed effects) hypotheses are
Model Assumption:
Constraints:
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17. Extension of the ANOVA to the RCBD
ANOVA partitioning of total variability:
Let’s add and subtract three terms:
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18. Extension of the ANOVA to the RCBD
Let’s add and subtract three terms here…
ANOVA partitioning of total variability:
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19. Extension of the ANOVA to the RCBD
The degrees of freedom for the sums of squares in
are as follows:
Therefore, ratios of sums of squares to their degrees of freedom result in mean squares and the ratio of the mean square for treatments to the error mean square is an F statistic that can be used to test the hypothesis of equal treatment means
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20. ANOVA Display for the RCBD
Manual computing (ugh!)…see Equations (4-9) – (4-12), page 144
Use software to analyze the RCBD (Design-Expert, JMP)
ANOVA Display for the RCBD
Manual computing (ugh!)…see Equations (4-9) – (4-12), page 144
The reference distribution for F0 is the F(a-1, (a-1)(b-1)) distribution
Reject the null hypothesis (equal treatment means) if
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21. ANOVA Display for the RCBD
Manual computing:
Chapter 4
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22. Example 4.1: CRBD Analysis with vascular grafts
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23. Vascular Graft Example (pg. 145)
To conduct this experiment as a RCBD, assign all 4 pressures to each of the 6 batches of resin
Each batch of resin is called a “block”; that is, it’s a more homogenous experimental unit on which to test the extrusion pressures
Q: SSTotal = , SStreat = , SSblock = , first find SSerror; then find MStreat, MSerror, F0 and p-value.
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24. Example 4.1: CRBD Analysis with vascular grafts
we conclude that extrusion pressure affects the mean yield. The P- value for the test is also quite small.
Also, the resin batches (blocks) seem to differ significantly, because the mean square for blocks is large relative to error.
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25. Vascular Graft Example (pg. 145)
p-value=1-pf(8.1067,3,15) =
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26. Example 4.1: Incorrect approach to vascular grafts
The incorrect analysis of these data as a completely randomized single-factor design is shown in Table 4.5.
MSerror has more than doubled, from 7.33 in RCBD to in CRD.
All variability due to blocks is now in error term. Call RCBD a noise-reducing design technique;
It effectively increases signal-to-noise ratio, or it improves precision with which treatment means are compared.
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27. Example 4.1: CRBD Analysis with vascular grafts
Note that the square for error has more than doubled, increasing from 7.33 in the RCBD to
All of the variability due to blocks is now in the error term. This makes it easy to see why we sometimes call the RCBD a noise- reducing design technique; it effectively increases the signal-to- noise ratio in the data, or it improves the precision with which treatment means are compared.
This example also illustrates an important point. If an experimenter fails to block when he or she should have, the effect may be to inflate the experimental error, and it would be possible to inflate the error so much that important differences among the treatment means could not be identified.
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28. Vascular Graft Example: Input data and ANOVA for CRBD
################ The Vascular Graft example ####################
batch=rep(1:6, 4);
pressure=c(rep(8500, 6), rep(8700, 6), rep(8900, 6), rep(9100, 6));
y=c(90.3,89.2,98.2,93.9,87.4,97.9,92.5,89.5,90.6,94.7,87.0,95.8,85.5,90.8,89.6,86.2,88.0,93.4,82.5,89.5,85.6,87.4,78.9,90.7)
anova(lm(y~as.factor(pressure)+as.factor(batch)));
### or
tab4.3<-read.table("\\\\bearsrv\\classrooms\\Math\\wangy\\stt4511\\Vascular-Graft.TXT",header=TRUE); anova(lm(tab4.3$Yield~as.factor(tab4.3$Pressure)+as.factor(tab4.3$Batch)));
anova(lm(tab4.3[,3]~as.factor(tab4.3[,2])+as.factor(tab4.3[,1])));
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29. Vascular Graft Example Design-Expert Output
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30. Vascular Graft Example JMP output
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31. Residual Analysis for the Vascular Graft Example
Basic residual plots to check whether normality, constant/equal variance assumptions are satisfied
Check randomization
Check: ideally No patterns in residuals vs. block
Can also plot residuals versus pressure (i.e. residuals vs. factor)
These plots provide more information about the constant variance assumption, possible outliers
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32. Residual Analysis for the Vascular Graft Example
Model fits well and model assumptions are
Satisfied.
A normal probability plot of residuals
No severe indication of non-normality, nor is there any evidence pointing to possible outliers.
If model is correct and the assumptions are satisfied, this plot should be structureless.
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33. Residual Analysis for the Vascular Graft Example
Plots of the residuals by treatment (extrusion pressure) and by batch of resin or block.
If there is more scatter in residuals for a particular treatment, it indicates this treatment produces more erratic response readings than others. More scatter in residuals for a particular block could indicate that block is not homogeneous. However, Figure 4.6 gives no indication of inequality of variance by treatment but there is an indication that there is less variability in yield for batch 6. However, since all other residual plots are satisfactory, we will ignore this.
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34. ANOVA for CRBD for the Vascular Graft Example: Residual Analysis in R
batch=rep(1:6, 4);
pressure=c(rep(8500, 6), rep(8700, 6), rep(8900, 6), rep(9100, 6));
y=c(90.3,89.2,98.2,93.9,87.4,97.9,92.5,89.5,90.6,94.7,87.0,95.8,85.5,90.8,89.6,86.2,88.0,93.4,82.5,89.5,85.6,87.4,78.9,90.7);
anova(lm(y~as.factor(pressure)+as.factor(batch)));
### analysis after CRBD:
#### Residual analysis
model.4.3<-lm(y~as.factor(pressure)+as.factor(batch));
res.4.3<-resid(model.4.3);
pred.4.3<-model.4.3$fitted;
par(mfrow=c(3,2))
qqnorm(res.4.3); qqline(res.4.3); ##for residuals##
plot(c(1:24),res.4.3);
plot(pred.4.3,res.4.3);
plot(pressure, res.4.3);
plot(batch,res.4.3);
dev.off();
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35. Residual Analysis for the Vascular Graft Example
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36. Post-ANOVA Comparison of Means for RCBD
Determining which specific means differ, following an ANOVA is called the multiple pairwise comparisons b/w all a populations means.
Suppose we are interesting in comparing all pairs of a treatment means and that the null hypothesis that we wish to test
H0: µi=µj v.s. Ha: µi ≠ µj for all i≠j.
We will use pairwise t-tests on means…sometimes called Fisher’s Least Significant Difference (or Fisher’s LSD) Method
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37. Multiple Comparisons: Fisher’s Least Significant Difference for RCBD with exact p-value
Test H0: µi=µj v.s. Ha: µi ≠ µj, for all i≠j.
~t( df=(a-1)(b-1) )
Then we can find exact p-value from this approach. b
If ANOVA for RCBD indicates a significant difference in treatment means, we are usually interested in multiple comparisons to discover which treatment means differ.
Any of multiple comparison procedures discussed in Section 3.5 may be used, by simply replacing # of replicates (n) in single-factor CRD by the number of blocks (b) in RCBD.
Also, remember to use # of error df for RCBD with [(a-1)(b-1)] instead of those for CRB [a(n-1)].
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38. Fisher’s LSD for Vascular Graft Example
(yi.)-bar’s are given:
Use Fisher’ LSD method to make comparison among all 4 pressure treatments to determine specially which designs differ in the mean response rate.
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39. Multiple Comparisons: Fisher’s Least Significant Difference
Test H0: µi=µj v.s. Ha: µi ≠ µj, for all i≠j. df
df=(a-1)(b-1) b b
df=(a-1)(b-1) df b
If ANOVA for RCBD indicates a significant difference in treatment means, we are usually interested in multiple comparisons to discover which treatment means differ.
Any of multiple comparison procedures discussed in Section 3.5 may be used, by simply replacing # of replicates (n) in single-factor CRD by the number of blocks (b) in RCBD.
Also, remember to use # of error df for RCBD with [(a-1)(b-1)] instead of those for CRB [a(n-1)].
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40. ANOVA for CRBD for the Vascular Graft Example: Fisher's ‘Least Significant Difference’ method
batch=rep(1:6, 4);
pressure=c(rep(8500, 6), rep(8700, 6), rep(8900, 6), rep(9100, 6));
y=c(90.3,89.2,98.2,93.9,87.4,97.9,92.5,89.5,90.6,94.7,87.0,95.8,85.5,90.8,89.6,86.2,88.0,93.4,82.5,89.5,85.6,87.4,78.9,90.7);
anova(lm(y~as.factor(pressure)+as.factor(batch)));
#### post RBCD analysis
tapply(y, pressure, mean)
pairwise.t.test(y, as.factor(pressure), p.adjust.method ="none")
By Fisher’s LSD
µ1≠ µ4
µ2≠ µ4
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41. ANOVA for CRBD for the Vascular Graft Example: STT511: Tukey's ‘Honest Significant Difference’ method
################ The Vascular Graft example ####################
batch=rep(1:6, 4);
pressure=c(rep(8500, 6), rep(8700, 6), rep(8900, 6), rep(9100, 6));
y=c(90.3,89.2,98.2,93.9,87.4,97.9,92.5,89.5,90.6,94.7,87.0,95.8,85.5,90.8,89.6,86.2,88.0,93.4,82.5,89.5,85.6,87.4,78.9,90.7);
anova(lm(y~as.factor(pressure)+as.factor(batch)));
#### post RBCD analysis
### analysis after RBD: #### Residual analysis
model.4.3<-lm(y~as.factor(pressure)+as.factor(batch)); tapply(y, pressure, mean)
TukeyHSD(aov(model.4.3), "as.factor(pressure)")
## Create a set of confidence intervals on differences between means of levels of a factor with specified
## family-wise probability of coverage.
## The intervals are based on the Studentized range statistic, Tukey's ‘Honest Significant Difference’ method.
## Each component is a matrix with columns diff giving the difference in the observed means,
## lwr giving the lower end point of the interval,
## upr giving the upper end point and
## p adj giving the p-value after adjustment for the multiple comparisons.
## It will list all possible pairwise comparison within different groups ##
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42. Also see Figure 4.2, Design-Expert output
Multiple Comparisons for the Vascular Graft Example – Which Pressure is Different with Fisher’s LSD?
Also see Figure 4.2, Design-Expert output
By Design Expert package
µ1≠ µ3
µ1≠ µ4
µ2≠ µ4
By Tukey’s HSD:
µ1≠ µ4
µ2≠ µ4
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43. ANOVA for RCBD: Example 2

44. Example 2: RCBD analysis in R
######## another example
dat4.4<-read.table("\\\\bearsrv\\classrooms\\Math\\wangy\\stt4511\\BHH2-Data\\tab0404.dat",skip=1)[,2:4];
names(dat4.4)<-c("blend","treatment","value");
##### a$blend<-factor(a$blend); #When R read the data file, it automatically take
##### character as factor, but it is not the case for numbers. We are suppose to
### transform it "by hand"#####
summary(aov(value~blend+treatment,data=dat4.4)); summary(aov(value~factor(blend)+treatment,data=dat4.4)); anova(lm(value~factor(blend)+treatment,data=dat4.4));
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45. Example 2: RCBD analysis in SAS
data tab4p4;
infile '\\bearsrv\classrooms\Math\wangy\stt4511\BHH2-Data\tab0404.dat' firstobs=2;
input run blend$ treat$ value;
run;
proc print data=tab4p4;
proc anova data=tab4p4 ; /* instead of "anova", people also use "glm" */
class blend treat;
model value = blend treat;
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46. ANOVA for RCBD: Example 3 (for students’ practice)

47. Example 3: Review: The Hardness Testing Example
Suppose that we use b = 4 blocks:
Notice the two-way structure of the experiment
Once again, we are interested in testing the equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks)
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48. Example 3: ANOVA for the RCBD in R
############## How to code the data??? #########
Tip=c(rep(1,4), rep(2,4), rep(3,4), rep(4,4)); print(Tip)
Coupon=rep(1:4, 4); print(Coupon)
Hardness=c(9.3,9.4,9.6,10.0,9.4,9.3,9.8,9.9,9.2,9.4,9.5,9.7,9.7,9.6,10.0,10.2); print(Hardness);
cbind(Tip, Coupon, Hardness)
h.data = data.frame(cbind(Tip, Coupon, Hardness)) ## for more complicated operation
## OR read in the table:
h.data<-read.table("\\\\bearsrv\\classrooms\\Math\\wangy\\stt4511\\Hardness-Testing1.TXT", header=TRUE);
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49. Example 3: ANOVA for the RCBD in R
############## Randomized block design in R #########
h.data<-read.table("\\\\bearsrv\\classrooms\\Math\\wangy\\stt4511\\Hardness-Testing1.TXT", header=TRUE);
anova(lm(h.data[,3]~as.factor(h.data[,2])+as.factor(h.data[,1])));
## or ##
anova(lm(h.data$Hardness~as.factor(h.data$Coupon)+as.factor(h.data$Tip)));
### the connection with one way anova when the block effect is ignored.... anova(lm(h.data$Hardness~as.factor(h.data$Tip)));
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50. Example 3: ANOVA for the RCBD in R
How are the numbers calculated?
Which p-value is the more important one that we should consider?
Are the degrees of freedom correct?
May we ignore the block effect?
What would be the ANOVA table if we want to ignore the block effect no matter what?
Q:What will be the conclusion then, if we ignore blocking?
Chapter 4
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51. Example 3: ANOVA for the RCBD in R – Residual Analysis for CRBD
h.data<-read.table("\\\\bearsrv\\classrooms\\Math\\wangy\\stt4511\\Hardness-Testing1.TXT",header=TRUE);
model.h<-lm(h.data$Hardness~as.factor(h.data$Coupon)+as.factor(h.data$Tip));
res.h<-resid(model.h);
pred.h<-model.h$fitted;
par(mfrow=c(3,2))
qqnorm(res.h);qqline(res.h);
plot(c(1:16),res.h);
plot(pred.h,res.h);
plot(h.data$Coupon,res.h);
plot(h.data$Tip,res.h);
dev.off();
## Or you can do the following ##
par(mfrow=c(2,2))
plot(model.h)
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52. Example 3: Residual analysis
Equal variance?
Chapter 4
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53. Example 3: Post RCBD analysis Verify:
Chapter 4
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54. Example 3: Post RBCD analysis
h.data<-read.table("\\\\bearsrv\\classrooms\\Math\\wangy\\stt4511\\Hardness-Testing1.TXT",header=TRUE);
model.h<-lm(h.data$Hardness~as.factor(h.data$Coupon)+as.factor(h.data$Tip));
#### post RBCD analysis
tapply(h.data$Hardness,h.data$Tip, mean)
TukeyHSD(aov(model.h),"as.factor(h.data$Tip)")
#library(agricolae);
#out=LSD.test(h.data$Hardness, as.factor(h.data$Tip), 9, ) #### verify
#print(out);
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55. Example 4 of blocking: more than two treatment levels
Note: 88 is the response, while (2) is the random order of RCBD.
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56. Connection b/w one-sample paired t-test with RCBD

57. Matched pairs t procedures for dependent sample
Subjects are matched in “pairs” and
outcomes are compared within each unit
Example: Pre-test and post-test studies look at data collected on the same sample elements before and after some experiment is performed.
Example: Twin studies often try to sort out the influence of genetic factors by comparing a variable between sets of twins.
We perform hypothesis testing on the difference in each unit

58. Matched pairs t test
The variable studied becomes Xdifference = (X1 − X2). The null hypothesis of NO difference between the two paired groups. H0: µdifference= 0 ; Ha: µdifference>0 (or <0, or ≠0)
When stating the alternative, be careful how you are calculating the difference (after – before or before – after).
Conceptually, this is not different from tests on one population.

59. Matched Pairs t-test (one-sample t-test after finding differences)
If we take diff=After – Before, and we want to show that the “After group” has increased over the “Before group”
Ha: m > 0
“After group” has decreased
Ha: m < 0
The two groups are different
Ha: m ≠0
Note: we have n-pairs of observations!

60. Blocking: with matched pairs two treatment levels
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61. Example: The paired sample t-test in R
paired.data<-read.table("\\\\bearsrv\\classrooms\\Math\\wangy\\stt4511\\BHH2-Data\\tab0305.dat",header=TRUE);
t.test(paired.data$diff, mu=0);
## Or do Matched pairs t test manually ##
d<-paired.data$diff;
t.val<-mean(d)/(sd(d)/sqrt(10))
p.val<-2*(1-pt(abs(t.val),9))
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62. Connection b/w one-sample paired t-test with RCBD
Q: what is the connection between two-sample paired t-test with RCBD
Q: what would be the data transformation?
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63. Connection b/w one-sample paired t-test with RCBD in R
####### link with two sample paired t-test
paired.data<-read.table("\\\\bearsrv\\classrooms\\Math\\wangy\\stt4511\\BHH2-Data\\tab0305.dat",header=TRUE);
boy<-c(c(1:10),c(1:10));
mat<-c(rep('A',10),rep('B',10))
wear<-c(paired.data[,2], paired.data[,4])
wear.dat<-data.frame(wear,boy,mat)
anova(lm(wear.dat$wear~as.factor(wear.dat$boy)+as.factor(wear.dat$mat)));
## For one sample matched pair t-test ##
t.test(paired.data$diff)
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64. Connection b/w one-sample paired t-test with RCBD in R
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65. Latin Square Design: other type of block design

66. Review: The Hardness Testing
Tips
Test coupon
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67. §4.2: Latin Square Design: other type of block design
Text reference, Section 4.2, pg. 158
These designs are used to simultaneously control (or eliminate) two sources of nuisance variability/blocks
(eg: batches of raw materials and operators)
Use Latin Square Design to eliminate two nuisance sources of variability: Use blocking in rows and columns.
A significant assumption is that the three factors (treatments, TWO nuisance factors) do not interact
If this assumption is violated, the Latin square design will not produce valid results
Latin squares are not used as much as the RCBD in industrial experimentation
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68. The Rocket Propellant Problem – A Latin Square Design
Suppose that an experimenter is studying effects of five different formulations of a rocket propellant used in aircrew escape systems. Each formulation is mixed from a batch of raw material that is only large enough for five formulations to be tested.
Furthermore, the formulations are prepared by several operators, and there may be substantial differences in the skills and experience of the operators. Thus, it would seem that there are two nuisance factors: batches of raw material and operators.
The appropriate design for this problem consists of testing each formulation exactly once in each batch of raw material and for each formulation to be prepared exactly once by each of five operators. The resulting design is called a Latin square design.
Notice that the design is a square arrangement and that five formulations (or treatments) are denoted by the Latin letters A, B, C, D, and E; hence the name Latin square.
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69. The Rocket Propellant Problem – A Latin Square Design
Both batches of raw material (rows) and operators (columns) are orthogonal to
treatments
Chapter 4
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70. The Rocket Propellant Problem – A Latin Square Design
Row
factor
Column factor
Both batches of raw material (rows) and operators (columns) are orthogonal to
treatments
Treatment
factor
This is a
Page 159 shows some other Latin squares
Table 4-13 (page 162) contains properties of Latin squares
Statistical analysis?
Chapter 4
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71. The Latin Square Design
One treatment and two block factors!
How do we denote the r.v.?
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72. The Rocket Propellant Problem – A Latin Square Design
Chapter 4
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73. Statistical Analysis of the Latin Square Design
The statistical (effects) model is
yijk is the observation in ith row and kth column for jth treatment, µ is the overall mean, αi is the ith row effect, Tj is the jth treatment effect, βk is the kth column effect, and Eijk is the random error.
statistical analysis (ANOVA) is much like analysis for RCBD.
See the ANOVA table, page 160 (Table 4.10)
The analysis for the rocket propellant example follows
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74. Statistical Analysis of the Latin Square Design~
Chapter 4
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75. Latin Square Design for the Rocket Propellant Problem
Q: ANOVA for Latin Square: We have
SSTotal = 676, SSBatches = 68, SSOperators = 150, SStreat = 330,
first find SSerror;
then find MStreat, MSerror, F0 and p-value.
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76. Statistical Analysis of the Latin Square Design
P-value=1-pf(7.734,4,12)=
Q: what would be the output if Batches is not considered as a block?
Chapter 4
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77. How to input data for Rocket Propellant Problem
Operator=rep(1:5, 5); print(Operator);
Batches=c(rep(1, 5), rep(2, 5), rep(3, 5), rep(4, 5), rep(5, 5)); print(Batches);
Treat=c("A","B","C","D","E","B","C","D","E","A","C","D","E","A","B","D","E","A","B","C","E","A","B","C","D"); print(Treat);
y=c(24, 20, 19, 24, 24, 17, 24, 30, 27, 36, 18, 38, 26, 27, 21, 26, 31, 26, 23, 22, 22, 30, 20, 29, 31);
anova(lm(y ~ as.factor(Treat)+as.factor(Batches)+factor(Operator)));
Chapter 4
Design & Analysis of Experiments 8E 2012 Montgomery

78. Statistical Analysis of the Latin Square Design
## Latin Square in R ##
rocket<-read.table("\\\\bearsrv\\classrooms\\Math\\wangy\\stt4511\\Rocket-Propellant.TXT",header = TRUE);
anova(lm(rocket$BRate~rocket$Formulation+as.factor(rocket$Batches)+factor(rocket$Operator)));
Design & Analysis of Experiments 8E 2012 Montgomery
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79. Eg2: Automobile Emissions
Use ANOVA for Latin Square Method to analyze this dataset
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 4

80. Example 2: Statistical Analysis of Latin Square Design
## Latin Square in R ##
dat4.8<-read.table("\\\\bearsrv\\classrooms\\Math\\wangy\\stt4511\\BHH2-Data\\tab0408.dat",header = TRUE);
y=dat4.8$y;
additive=dat4.8$additive;
cars=dat4.8$cars;
driver=dat4.8$driver;
anova(lm(y ~ as.factor(additive) + as.factor(cars)+as.factor(driver)))
## Or ##
summary(aov(y~(additive)+factor(cars)+factor(driver),data=dat4.8)); #
anova(lm(y~(additive)+factor(cars)+factor(driver),data=dat4.8));
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 3

81. Statistical Analysis of the Latin Square Design
data tab4p8;
infile '\\bearsrv\classrooms\Math\wangy\stt4511\BHH2-Data\tab0408.dat' firstobs=2;
input driver cars additive $ y;
run;
proc print data=tab4p8;
proc anova data=tab4p8 ; /* instead of "anova", people also use "glm" */
class driver cars additive;
model y = driver cars additive;
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 3

82. Statistical Analysis of the Latin Square Design
The post ANOVA analysis is the same with ANOVA for CRB and ANOVA for RCBD!
Chapter 4
Design & Analysis of Experiments 8E 2012 Montgomery

83. Other Topics Missing values in blocked designs
RCBD
Latin square
Replication of Latin Squares
Crossover designs
Graeco-Latin Squares
Incomplete block designs
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 4