STATISTICAL MODELLING Appendix C Analysis of designed experiments in R
Generally the analysis begins with an initial graphical exploration of the data — boxplots for experiments with a single treatment factor and interaction plots for factorial experiments. Then the aov function is used to analyse the data from the experiment. The analysis is specified using a model formula of the form:
Response variable ~ explanatory variables (and operators)
There is a subtlety in the type of the explanatory variables that arises in connection with the analysis of designed experiments. If the explanatory variable is a numeric, such as a numeric vector, then R fits just one coefficient for it. So for a single explanatory variable, a straight-line relationship between the response and explanatory variables is fitted. On the other hand, if the explanatory variable is categorical, such as a factor, a coefficient is fit for each level of the variable. Being stored in a factor object signals to R that it should use indicator variables, or their equivalent, instead of values for the variable itself in estimating the parameters. Most often explanatory variables will be factors. Also different forms of the analysis are obtained depending on whether the Error function is used as part of the explanatory variable specification. Generally, the form with the Error function is preferred. When you use the Error function you must also include outside the Error function those fixed terms from inside the Error function. The aov function produces an aovlist object when the Error function is used — it is a list of aov objects, one for each Error term in the analysis. Without the Error function a single aov object is produced. Next diagnostic checking based on the residuals is performed — residual-versus-fitted-values and residual-versus-factors plots, normal probability plot of the residuals and, for all but the CRD, Tukey’s one-degree-of-freedom-for-nonadditivity. The functions resid.errors and fitted.errors must be used to extract the residuals and fitted values when the Error function is used in a call to aov. Also the function tukey.1df performs the nonadditivity test — you must specify the error.term when using tukey.1df following an aov function in which and Error function is employed. These are nonstandard functions in the package dae available
from the web site atically in the pools where R is available. For instructions on installation see the help(function) in the R console. The final stage of the analysis will be the further, detailed examination of treatment differences — either multiple comparisons analysis, the fitting of polynomial trends or, in some special cases, investigating specific contrasts. To fit polynomial trends the each quantitative factor must be converted to ordered and polynomial contrasts associated with the ordered. Then the split argument is used in the summary function. For example, if there were 7 treatments then the following function will fit a cubic and compute the Deviations line: > summary(Experiment.aov, split = list(Treatment = list(L = 1, Q = 2, C = 3, Dev = 4:6))) Because there are 7 treatments the maximum degree polynomial that can be fitted is of degree 6 and so those above cubic, degrees 4 to 6, are combined into the Deviations line. In some examples, more than one of these methods for examining treatment differences is given for illustrative purposes only. In each case, the most appropriate one should be selected for reporting the results of the analysis. Also, plots will usually be performed to illustrate the conclusions about treatment differences. Usually, bar charts (Bar Y Min Base) will be used for qualitative factors and scatter plots with fitted polynomials (Poly Fit) for quantitative factors. Note that must call aov without using the Error function for the multiple comparisons function, multicomp, to work.
C.1. Completely randomized design
Example II.2 Caffeine effects on students
To illustrate the procedures, I am going to use an experiment in which the effect of orally ingested caffeine on a physical task was investigated (Draper and Smith, 1981, sec.9.1). Thirty healthy male college students were selected and trained in finger tapping. Ten men were randomly assigned to receive one of three doses of caffeine (0, 100 or 200 mg). The number of finger taps after ingesting the caffeine was recorded for each student and the data were as follows: The R expressions required to produce the complete analysis of this example are as follows: R expressions to produce analysis
#set up data.frame with factors Students and Dose and response variable Taps CRDCaff.dat <- data.frame(Students = factor(1:30), Dose = factor(rep(c(0,100,200), times=10))) CRDCaff.dat$Taps <- c(242,248,246,245,246,248,244,245,250,248,247,252,247,248,248, 248,250,250,242,247,246,244,246,248,246,243,245,242,244,250)
The following output shows how CRDCaff.dat is set up: > CRDCaff.dat Students Dose Taps 1 1 0 242 2 2 100 248 3 3 200 246 4 4 0 245 5 5 100 246 6 6 200 248 7 7 0 244 8 8 100 245 9 9 200 250 10 10 0 248 11 11 100 247 12 12 200 252 13 13 0 247 14 14 100 248 15 15 200 248 16 16 0 248 17 17 100 250 18 18 200 250 19 19 0 242 20 20 100 247 21 21 200 246 22 22 0 244 23 23 100 246 24 24 200 248 25 25 0 246 26 26 100 243 27 27 200 245 28 28 0 242 29 29 100 244 30 30 200 250
Further expressions:
The example below is for a standard randomized complete block design. However, the same expressions would be used for a generalized randomized complete block design. Example IV.1 Penicillin yield
The R expressions required to produce the complete analysis of this example are given below. They assume that the data frame RDBDPen.dat has been appropriately set up prior to using them. One way to set it up is as follows: > attach(RDBDPen.dat) > RDBDPen.dat Blend Flask Treat Yield 1 1 1 A 89 2 1 2 B 88 3 1 3 C 97 4 1 4 D 94 5 2 1 A 84 6 2 2 B 77 7 2 3 C 92 8 2 4 D 79 9 3 1 A 81 10 3 2 B 87 11 3 3 C 87 12 3 4 D 85 13 4 1 A 87 14 4 2 B 92
15 4 3 C 89 16 4 4 D 84 17 5 1 A 79 18 5 2 B 81 19 5 3 C 80 20 5 4 D 88
The form of the aov function depends on whether Blend is fixed or random as Blend must be included outside the Error function if Blend is fixed, but not if it is random. Thus for Blend fixed the form of the aov function is
RCBDPen.aov <- aov(Yield ~ Blend + Treat + Error(Blend/Flask), RDBDPen.dat)
RCBDPen.aov <- aov(Yield ~ Treat + Error(Blend/Flask), RDBDPen.dat)
R expressions to produce analysis
attach(RDBDPen.dat) boxplot(split(Yield, Blend), xlab="Blend", ylab="Yield") boxplot(split(Yield, Treat), xlab="Treatment", ylab="Yield") RCBDPen.aov <- aov(Yield ~ Blend + Treat + Error(Blend/Flask), RDBDPen.dat)
or RCBDPen.aov <- aov(Yield ~ Treat + Error(Blend/Flask), RDBDPen.dat) summary(RCBDPen.aov) #Compute Blend F and p Blend.F <- 66/18.833 Blend.p <- 1-pf(Blend.F, 4, 12) data.frame(Blend.F,Blend.p) # # Diagnostic checking # res <- resid.errors(RCBDPen.aov) fit <- fitted.errors(RCBDPen.aov) data.frame(Blend,Flask,Treat,Yield,res,fit) plot(fit, res, pch=16) qqnorm(res, pch=16) qqline(res) tukey.1df(RCBDPen.aov, RCBDPen.dat, error.term="Blend:Flask") tukey.1df(RCBDPen.BlendRandom.aov, RCBDPen.dat, error.term="Blend:Flask") # # Plotting Treat means # RCBDPen.tab <- model.tables(RCBDPen.aov, type="means") RCBDPen.Treat.Mean <- data.frame(Treat.lev = levels(Treat), Treat.Mean = as.vector(RCBDPen.tab$tables$Treat)) barchart(Treat.Mean ~ Treat.lev, ylim=c(0,90), xlab="Treatment", ylab="Yield (%)", main="Fitted values for Yield", data=RCBDPen.Treat.Mean)
C.3. Latin square design
Example V.2 Pollution effects of petrol additives
The R expressions required to produce the complete analysis of this example are given below. They assume that the data frame LSPolut.dat has been appropriately set up prior to using them. One way to set it up is as follows: > load("LSPolut.dat.rda") > attach(LSPolut.dat) > LSPolut.dat Standard.Order Random.Order Drivers Cars Additives Reduct.NO 15 15 1 1 1 A 21 14 14 2 1 2 D 20 16 16 3 1 3 B 26 13 13 4 1 4 C 25 11 11 5 2 1 D 23 10 10 6 2 2 A 20 12 12 7 2 3 C 26 9 9 8 2 4 B 27 7 7 9 3 1 B 15 6 6 10 3 2 C 16 8 8 11 3 3 A 16 5 5 12 3 4 D 13 3 3 13 4 1 C 17 2 2 14 4 2 B 20 4 4 15 4 3 D 20 1 1 16 4 4 A 15
Again the form of the aov function depends on whether Drivers and Cars are fixed or random. Delete the random ones from outside the Error function. For example if both are random the call to aov would become
LSPolut.aov <- aov(Reduct.NO ~ Additives + Error(Drivers*Cars), LSPolut.dat)
R expressions to produce analysis
boxplot(split(Reduct.NO, Drivers), xlab="Drivers", ylab="Reduction in NO") boxplot(split(Reduct.NO, Cars), xlab="Cars", ylab="Reduction in NO") boxplot(split(Reduct.NO, Additives), xlab="Additives", ylab="Reduction in NO") LSPolut.aov <- aov(Reduct.NO ~ Drivers + Cars + Additives + Error(Drivers*Cars), LSPolut.dat) summary(LSPolut.aov) #Compute Drivers and Cars Fs and p-values Drivers.F <- 72/2.667 Drivers.p <- 1-pf(Drivers.F, 3, 6) Cars.F <- 8/2.667 Cars.p <- 1-pf(Cars.F, 3, 6) data.frame(Drivers.F,Drivers.p,Cars.F,Cars.p) # # Diagnostic checking # res <- resid.errors(LSPolut.aov) fit <- fitted.errors(LSPolut.aov) data.frame(Drivers,Cars,Additives,Reduct.NO,res,fit) plot(fit, res, pch=16) qqnorm(res, pch=16) qqline(res) tukey.1df(LSPolut.aov, LSPolut.dat, error.term = "Drivers:Cars") # # multiple comparisons # model.tables(LSPolut.aov, type="means") q <- qtukey(0.95, 4, 6) q
# # Plotting Treat means # LSPolut.tab <- model.tables(LSPolut.aov, type="means") LSPolut.Adds.Mean <- data.frame(Adds.lev = levels(Additives), Adds.Mean = as.vector(LSPolut.tab$tables$Additives)) LSPolut.Adds.Mean <- LSPolut.Adds.Mean[order(LSPolut.Adds.Mean$Adds.Mean),] #use factor to order bars LSPolut.Adds.Mean$Adds.lev <- factor(LSPolut.Adds.Mean$Adds.lev, levels=LSPolut.Adds.Mean$Adds.lev) barchart(Adds.Mean ~ Adds.lev, xlab="Additives", ylim=c(0,25), ylab="NO Reduction", main="Fitted values for Nitrous Oxide Reduction", data=LSPolut.Adds.Mean)
C.4. A set of Latin squares design
In general, the analysis for a set of the Latin squares is obtained by using a model formula in which the explanatory variables are specified to be randomized structure formula + Error(unrandomized structure formula) In addition, fixed terms in the unrandomized structure formula need to be also included with the terms outside the Error function, as mentioned in the introduction to this Appendix. Example Case 2 — same Cars different Drivers
As an example of an experiment involving a set of Latin squares, we illustrate the analysis for the case in which petrol additives are assigned to the combinations of four drivers and four cars using a Latin square and this is done for two different occasions with same 4 cars on both occasions, but with the drivers on one occasion unconnected with those on the other. As a result the rows of the square, but not the columns, are rerandomized on the second occasion. The R expressions required to produce the complete analysis of this example are given below. They assume that the data window LSRepeat2.dat has been appropriately set up prior to using them. One way to set it up follows — note that Data is randomly generated data to enable an analysis to be performed. > LSRepeat2.dat <- data.frame(LSRepeat2.lay, Data = rnorm(n)) > attach(LSRepeat2.dat) > LSRepeat2.dat Units Permutation Occasion Drivers Cars Additives Data 1 1 21 1 1 1 C 1.33299051 2 2 24 1 1 2 A 1.52009255 3 3 22 1 1 3 B 0.18680922 4 4 23 1 1 4 D 0.26281501 5 5 17 1 2 1 A -0.07712774 6 6 20 1 2 2 C -1.31362565 7 7 18 1 2 3 D -0.71681803 8 8 19 1 2 4 B 0.16637414 9 9 29 1 3 1 B -0.59372326 10 10 32 1 3 2 D -0.76442376 11 11 30 1 3 3 C 0.77602428 12 12 31 1 3 4 A 0.66442845 13 13 25 1 4 1 D 0.90195336 14 14 28 1 4 2 B 0.12874893 15 15 26 1 4 3 A -0.68993861 16 16 27 1 4 4 C -0.76842248 17 17 5 2 1 1 D 0.41973900
18 18 8 2 1 2 B -0.39102629 19 19 6 2 1 3 A -0.41841197 20 20 7 2 1 4 C -0.24032621 21 21 13 2 2 1 A -0.28418206 22 22 16 2 2 2 C -0.25538454 23 23 14 2 2 3 D -0.71548353 24 24 15 2 2 4 B 1.01486123 25 25 9 2 3 1 C -0.41233805 26 26 12 2 3 2 A -1.23423748 27 27 10 2 3 3 B 0.26930868 28 28 11 2 3 4 D 0.99496028 29 29 1 2 4 1 B 1.90147149 30 30 4 2 4 2 D 0.49720016 31 31 2 2 4 3 C 0.08308735 32 32 3 2 4 4 A 2.00740238
Again the form of the aov function depends on whether Occasion, Drivers and Cars are fixed or random — the expressions below contain the aov function if all terms except Drivers:Cars[Occasion] are fixed. If some of Occasion, Drivers and Cars are random, delete the random factors from outside the Error function, except that Occasion can only be deleted if Drivers is also random. For example, if all three factors are random, the call to aov would become
LSRepeat2.aov <- aov(Data ~ Additives + Error((Occasion/Drivers) * Cars),
R expressions to produce analysis
boxplot(split(Data, Occasion), xlab="Occasion", ylab="Data") boxplot(split(Data, Drivers), xlab="Drivers", ylab="Data") boxplot(split(Data, Cars), xlab="Cars", ylab="Data") boxplot(split(Data, Additives), xlab="Additives", ylab="Data") LSRepeat2.aov <- aov(Data ~ Occasion/Drivers + Occasion*Cars + Additives + Error((Occasion/Drivers)*Cars), LSRepeat2.dat) summary(LSRepeat2.aov) # # Diagnostic checking # res <- resid.errors(LSRepeat2.aov) fit <- fitted.errors(LSRepeat2.aov) plot(fit, res, pch=16) qqnorm(res, pch=16) qqline(res) tukey.1df(LSRepeat2.aov, LSRepeat2.dat, error.term = "Occasion:Drivers:Cars") # # multiple comparisons # model.tables(LSRepeat2.aov, type="means") q <- qtukey(0.95, 4, 15) q # # Plotting Treat means # LSRepeat2.tab <- model.tables(LSRepeat2.aov, type="means") LSRepeat2.Adds.Mean <- data.frame(Adds.lev = levels(Additives), Adds.Mean = as.vector(LSRepeat2.tab$tables$Additives)) LSRepeat2.Adds.Mean <- LSRepeat2.Adds.Mean[order(LSRepeat2.Adds.Mean$Adds.Mean),] #use factor to order bars LSRepeat2.Adds.Mean$Adds.lev <- factor(LSRepeat2.Adds.Mean$Adds.lev, levels=LSRepeat2.Adds.Mean$Adds.lev) barchart(Adds.Mean ~ Adds.lev, xlab="Additives", ylab="NO Reduction", main="Fitted values for Nitrous Oxide Reduction", data=LSRepeat2.Adds.Mean)
C.5. Factorial experiments
Layouts for factorial experiments can be obtained in R using the expressions for the chosen design when only a single-factor is involved. The difference with factorial experiments is that the several treatment factors are entered in standard order. Initial graphical exploration for factorial experiments utilises interaction plots, produced with the function interaction.plot for two factors or the nonstandard function interaction.ABC.plot, available from the dae library, for more than two factors. We do not produce boxplots in this case, in part because they look at just the overall effects of one factor and are only relevant if the factors are independent. Note that for factorial experiments the model formula used in the aov function should involve the terms arising from the design employed (Blocks for the RCBD and Rows + Columns for the Latin Square) as well as the treatment factors (for example A * B * C). As usual, multiple comparisons procedures can be employed and R used to obtain the tables of means and studentized ranges needed in the calculations. Fitting polynomial submodels for factorial experiments in R is an extension of the procedure for a single factor described in the introduction to this appendix. You have to a) specify the polynomial contrasts for each quantitative factor, b) use the crossed (*) operator between factors in the model formula (for example A * B), and c) specify all the terms involving the quantitative factors in the list argument of the summary function. The general form of the summary function for one factor, B say, quantitative is: summary(Experiment.aov, split = list(B = list(L = 1, Q = 2, Dev = 3:(b-1))), "A:B" = list(L = 1, Q = 2, Dev = 3:(b-1)))) and for two factors, A and B say, quantitative is summary(Experiment.aov, split = list(A = list(L = 1, Q = 2, Dev = 3:(a-1)), B = list(L = 1, Q = 2, Dev = 3:(b-1)))) Note the use of Dev to collect together the remaining degrees of freedom. Of course, if a or b is less than 3, the Dev term(s) should be omitted; there is no point in using the split function for a factor with 2 levels. Also, quotation marks are needed for the terms involving two or more factors. As for the single-factor case, the fitted equation is obtained using the lm function where the independent variables are the values of the quantitative factors and appropriate powers and products, saved in numeric vectors prior to invoking lm. Thus if R.Fac and S.Fac are two quantitative factors, then expressions for obtaining terms for a model involving all possible terms for polynomials up to order 2 and the associated lm function are:
R <- as.numeric(as.vector(R.Fac)) S <- as.numeric(as.vector(S.Fac)) R2 <- R*R; S2 < S*S R1S1 <- R*S; R1S2 <- R*S2; R2S1 <- R2*S; R2S2 <- R2*S2 lm(y ~ R + R2 + S + S2 + R1S1 + R1S2 + R2S1 +R2S2, data.frame) You will need to delete terms deemed unnecessary by the hypothesis testing. To plot the fitted surface use commands of the following form: Experiment.lm <- lm(y ~ poly(R, u) + poly(S, v) + poly(R, w) *
poly(S, x), singular.ok=T)
Experiment.grid <- list(R = seq(min(R), max(R), length = 40), S = seq(min(S), max(S), length = 40)) Experiment.surf <- expand.grid(Experiment.grid) Experiment.surf$y <- as.vector(predict(Experiment.lm,
wireframe(y ~ R*S, data= Experiment.surf, drape=TRUE) You will need to substitute new names for Experiment, y, R, S, u, v, w, x. Note that u and v are the orders of the main-effect terms and w and x are the maximum orders of each factor in cross-product terms. See the practical for this chapter for examples.
Example VII.5 Animal survival experiment The R expressions required to produce the complete analysis of this example are as follows. They assume that the Data window Fac2Pois.dat has been appropriately set up prior to using them. One way to set it up is as follows: > attach(Fac2Pois.dat) > Fac2Pois.dat Animals Poison Treat Surv.Time 1 1 1 1 0.31 2 2 1 2 0.82 3 3 1 3 0.43 4 4 1 4 0.45 5 5 1 1 0.45 6 6 1 2 1.10 7 7 1 3 0.45 8 8 1 4 0.71 9 9 1 1 0.46 10 10 1 2 0.88 11 11 1 3 0.63 12 12 1 4 0.66 13 13 1 1 0.43 14 14 1 2 0.72 15 15 1 3 0.76 16 16 1 4 0.62 17 17 2 1 0.36 18 18 2 2 0.92 19 19 2 3 0.44 20 20 2 4 0.56 21 21 2 1 0.29 22 22 2 2 0.61 23 23 2 3 0.35 24 24 2 4 1.02 25 25 2 1 0.40 26 26 2 2 0.49 27 27 2 3 0.31 28 28 2 4 0.71 29 29 2 1 0.23 30 30 2 2 1.24 31 31 2 3 0.40 32 32 2 4 0.38 33 33 3 1 0.22 34 34 3 2 0.30 35 35 3 3 0.23 36 36 3 4 0.30 37 37 3 1 0.21 38 38 3 2 0.37 39 39 3 3 0.25 40 40 3 4 0.36 41 41 3 1 0.18 42 42 3 2 0.38 43 43 3 3 0.24 44 44 3 4 0.31 45 45 3 1 0.23 46 46 3 2 0.29 47 47 3 3 0.22 48 48 3 4 0.33 R expressions to produce analysis
attach(Fac2Pois.dat) Fac2Pois.dat interaction.plot(Poison, Treat, Surv.Time, lwd=4) boxplot(split(Surv.Time, Poison), xlab="Poison", ylab="Survival time (10 hours)") boxplot(split(Surv.Time, Treat), xlab="Treatment", ylab="Survival time (10 hours)") Fac2Pois.aov <- aov(Surv.Time ~ Poison * Treat + Error(Animals), Fac2Pois.dat) summary(Fac2Pois.aov) # # Diagnostic checking # res <- resid.errors(Fac2Pois.aov) fit <- fitted.errors(Fac2Pois.aov) data.frame(Animals,Poison,Treat,Surv.Time,res,fit) plot(fit, res, pch=16) plot(as.numeric(Poison), res, pch=16) plot(as.numeric(Treat), res, pch=16) qqnorm(res, pch=16) qqline(res) Fac2Pois.NoError.aov <- aov(Surv.Time ~ Poison * Treat, Fac2Pois.dat) library(MASS) boxcox(Fac2Pois.NoError.aov, lambda=seq(from = -2.5, to = 2.5, len=20), plotit=T) # # re-analysis # detach(Fac2Pois.dat) Fac2Pois.dat$Death.Rate <- 1/Fac2Pois.dat$Surv.Time attach(Fac2Pois.dat) interaction.plot(Poison, Treat, Death.Rate, lwd=4) Fac2Pois.DR.aov <- aov(Death.Rate ~ Poison * Treat + Error(Animals), Fac2Pois.dat) summary(Fac2Pois.DR.aov) res <- resid.errors(Fac2Pois.DR.aov) fit <- fitted.errors(Fac2Pois.DR.aov) plot(fit, res, pch=16) plot(as.numeric(Poison), res, pch=16) plot(as.numeric(Treat), res, pch=16) qqnorm(res, pch=16) qqline(res) # # multiple comparisons # model.tables(Fac2Pois.DR.aov, type="means") q.PT <- qtukey(0.95, 12, 36) q.PT q.P <- qtukey(0.95, 3, 36) q.P q.T <- qtukey(0.95, 4, 36) q.T # # Plotting means # Fac2Pois.DR.tab <- model.tables(Fac2Pois.DR.aov, type="means") Fac2Pois.DR.Poison.Means <- data.frame(Poison = levels(Poison), Death.Rate = as.vector(Fac2Pois.DR.tab$tables$Poison)) barchart(Death.Rate ~ Poison, main="Fitted values for Death rate", ylim=c(0,4), data=Fac2Pois.DR.Poison.Means) Fac2Pois.DR.Treat.Means <- data.frame(Treatment = levels(Treat), Death.Rate = as.vector(Fac2Pois.DR.tab$tables$Treat)) barchart(Death.Rate ~ Treat, main="Fitted values for Death rate", ylim=c(0,4), data=Fac2Pois.DR.Treat.Means)
Example C.1 Grafting experiment
The R expressions required to produce the complete analysis of this example are as follows. They assume that the Data window Fac2Take.dat has been appropriately set up prior to using them. One way to set it up is as follows: > attach(Fac2Take.dat) > Fac2Take.dat Blocks Plots A B Take Cell.1.1 Treats 1 1 1 1 1 64 1 1 2 1 2 2 1 23 2 3 3 1 3 1 2 30 2 2 4 1 4 2 2 15 2 4 5 2 1 1 1 75 1 1 6 2 2 2 1 14 2 3 7 2 3 1 2 50 2 2 8 2 4 2 2 33 2 4 9 3 1 1 1 76 1 1 10 3 2 2 1 12 2 3 11 3 3 1 2 41 2 2 12 3 4 2 2 17 2 4 13 4 1 1 1 73 1 1 14 4 2 2 1 33 2 3 15 4 3 1 2 25 2 2 16 4 4 2 2 10 2 4
Note that if Blocks are random the call to aov would become
Fac2Take.aov <- aov(Take ~ A * B + Error(Blocks/Plots), Fac2Take.dat)
R expressions to produce analysis
attach(Fac2Take.dat) Fac2Take.dat interaction.plot(A, B, Take, lwd=4) Fac2Take.aov <- aov(Take ~ Blocks + A * B + Error(Blocks/Plots), Fac2Take.dat) summary(Fac2Take.aov) # # recompute for missing value # MSq <- c(73.729, 4795.6, 1387.6, 1139.1, 2.8797) Res <- c(rep(819.6/8, 4), 816.6828/7) df.num <- c(3,rep(1,4)) df.den <- c(rep(8, 4),7) Fvalue <- MSq/Res pvalue <- 1-pf(Fvalue, df.num, df.den) data.frame(MSq,Res,df.num,df.den,Fvalue,pvalue) # # Diagnostic checking # res <- resid.errors(Fac2Take.aov) fit <- fitted.errors(Fac2Take.aov) data.frame(Blocks,Plots,A,B,Take,res,fit) plot(fit, res, pch=16) plot(as.numeric(A), res, pch=16) plot(as.numeric(B), res, pch=16) qqnorm(res, pch=16) qqline(res) tukey.1df(Fac2Take.aov, Fac2Take.dat, error.term = "Blocks:Plots") # # multiple comparisons # Fac2Take.tab <- model.tables(Fac2Take.aov, type="means") Fac2Take.tab$tables$"A:B" q <- qtukey(0.95, 4, 8) q
# # reanalysis for one-cell interaction model # Fac2Take.dat$Cell.1.1 <- factor(1 + as.numeric(A != "1" | B != "1")) Fac2Take.dat$Treats <- fac.combine(list(A, B)) detach(Fac2Take.dat) attach(Fac2Take.dat) Fac2Take.dat Fac2Take.aov <- aov(Take ~ Blocks + Cell.1.1/Treats + Error(Blocks/Plots), Fac2Take.dat) summary(Fac2Take.aov) # # recompute for missing value # MSq <- c(73.729,6556.7,382.8) Res <- rep(819.6/8, 3) df.num <- c(3, 1, 2) Fvalue <- MSq/Res pvalue <- 1-pf(Fvalue, df.num, 8) data.frame(MSq,Res,df.num,Fvalue,pvalue)
C.6. Two-level factorial experiments
a) Replicated two-level factorial experiments
Example VIII.1 23 pilot plant experiment (continued)
The R expressions required to produce the complete analysis of this example are as follows. # #obtain randomized layout # n <- 16 mp <- c("-", "+") Fac3Pilot.ran <- fac.gen(generate = list(Te = mp, C = mp, K = mp), each = 2, order="yates") Fac3Pilot.unit <- list(Tests = n) Fac3Pilot.lay <- fac.layout(unrandomized = Fac3Pilot.unit, randomized = Fac3Pilot.ran, seed = 897) #sort treats into Yates order Fac3Pilot.lay <- Fac3Pilot.lay[Fac3Pilot.lay$Permutation,] #add Yield Fac3Pilot.dat <- data.frame(Fac3Pilot.lay, Yield = c(59, 61, 74, 70, 50, 58, 69, 67, 50, 54, 81, 85, 46, 44, 79, 81)) #re-sort into randomized order Fac3Pilot.dat <- Fac3Pilot.dat[Fac3Pilot.dat$Units,] attach(Fac3Pilot.dat) interaction.ABC.plot(Yield, Te, C, K, data=Fac3Pilot.dat, title="Effect of Temperature(Te), Concentration(C) and
Fac3Pilot.aov <- aov(Yield ~ Te * C * K + Error(Tests), Fac3Pilot.dat) summary(Fac3Pilot.aov) round(yates.effects(Fac3Pilot.aov, error.term = "Tests", data=Fac3Pilot.dat), 2) # # Diagnostic checking # res <- resid.errors(Fac3Pilot.aov) fit <- fitted.errors(Fac3Pilot.aov) plot(fit, res, pch=16) plot(as.numeric(Te), res, pch=16) plot(as.numeric(C), res, pch=16) plot(as.numeric(K), res, pch=16) qqnorm(res, pch=16) qqline(res) # # treatment differences # Fac3Pilot.means <- model.tables(Fac3Pilot.aov, type="means") Fac3Pilot.means$tables$"Grand mean" Fac3Pilot.means$tables$"Te:K" Fac3Pilot.means$tables$"C" q <- qtukey(0.95, 4, 8) q
Note the use of the round function with the yates.effects function to obtain nicer output by rounding the effects to 2 decimal places.
Unreplicated two-level factorial experiments
The analysis of single replicate of a 2k factorial experiments is based the normal probability plot of Yates effects, followed by an ANOVA for fitted model so that tables of means and residuals can be obtained and diagnostic checking performed. Example VIII.2 A 24 process development study (continued) The R expressions required to produce the complete analysis of this example are as follows. # # set up data frame and obtain initial analysis # mp <- c("-", "+") fnames <- list(Catal = mp, Temp = mp, Press = mp, Conc = mp) Fac4Proc.Treats <- fac.gen(generate = fnames, order="yates") Fac4Proc.dat <- data.frame(Runs = factor(1:16), Fac4Proc.Treats) remove("Fac4Proc.Treats") Fac4Proc.dat$Conv <- c(71,61,90,82,68,61,87,80,61,50,89,83,59,51,85,78) attach(Fac4Proc.dat) Fac4Proc.dat Fac4Proc.aov <- aov(Conv ~ Catal * Temp * Press * Conc + Error(Runs), Fac4Proc.dat) summary(Fac4Proc.aov) round(yates.effects(Fac4Proc.aov, error.term="Runs", data=Fac4Proc.dat), 2) # Perform analysis assuming 3- & 4-factor interactions negligible Fac4Proc.TwoFac.aov <- aov(Conv ~ (Catal + Temp + Press + Conc)^2 + Error(Runs), Fac4Proc.dat) summary(Fac4Proc.TwoFac.aov) # #Yates effects probability plot # qqnorm.yeffects(Fac4Proc.aov, error.term="Runs", data=Fac4Proc.dat) # # Diagnostic checking # Fac4Proc.Fit.aov <- aov(Conv ~ Temp * Conc + Catal + Press + Error(Runs), Fac4Proc.dat) summary(Fac4Proc.Fit.aov) tukey.1df(Fac4Proc.Fit.aov, Fac4Proc.dat, error.term="Runs") res <- resid.errors(Fac4Proc.Fit.aov) fit <- fitted.errors(Fac4Proc.Fit.aov) plot(fit, res, pch=16) qqnorm(res, pch=16) qqline(res) plot(as.numeric(Temp), res, pch=16) plot(as.numeric(Conc), res, pch=16) plot(as.numeric(Catal), res, pch=16) plot(as.numeric(Press), res, pch=16) # # treatment differences # Fac4Proc.means <- model.tables(Fac4Proc.aov, type="means") Fac4Proc.means$tables$"Grand mean" Fac4Proc.means$tables$"Temp:Conc" Fac4Proc.means$tables$"Catal" Fac4Proc.means$tables$"Press" interaction.plot(Temp, Conc, Conv) q <- qtukey(0.95, 4, 10) q Confounded two-level factorial experiments
In general, the analysis of confounded designs depends on whether only a single replicate of the treatments has been observed in the experiment or several replicates of the complete set of treatments have been observed. If a single replicate of the treatments has been observed, then the analysis must be based on the normal probability plot of the Yates effects. On the other hand, if two or more complete sets of treatments have been observed, the analysis can be based on an analysis of variance table. To get the correct analysis, you must use an Error function as part of the model formula. Example VIII.7 Partial confounding in a repeated four block experiment (continued) As four complete sets of treatments have been observed, the analysis will use an analysis of variance table. Having set up the factors and data in an R data.frame, the instructions to produce this analysis are as follows: attach(Fac3Conf.4Blocks.Partial.dat) Fac3Conf.4Blocks.Partial.dat Fac3Conf.4Blocks.Partial.aov <- aov(Yield ~ A * B * C + Error(Blends/Runs),
summary(Fac3Conf.4Blocks.Partial.aov) # # Diagnostic checking # tukey.1df(Fac3Conf.4Blocks.Partial.aov, Fac3Conf.4Blocks.Partial.dat,
res <- resid.errors(Fac3Conf.4Blocks.Partial.aov) fit <- fitted.errors(Fac3Conf.4Blocks.Partial.aov) plot(fit, res, pch=16) qqnorm(res, pch=16) qqline(res) plot(as.numeric(A), res, pch=16) plot(as.numeric(B), res, pch=16) plot(as.numeric(C), res, pch=16)
Note the inclusion of Error(Blends/Runs) in the model formula. d) Fractional two-level factorial experiments
Generally the analysis of fractional 2k designs is based the normal probability plot of Yates effects, followed by an ANOVA for fitted model so that tables of means and residuals can be obtained and diagnostic checking performed. Example VIII.10 A bike experiment The following expressions are used to analyse the first fraction: # # set up data.frame # mp <- c("-", "+") fnames <- list(Seat = mp, Dynamo = mp, Handbars = mp) Frf7Bike.Treats <- fac.gen(generate = fnames, order="yates") attach(Frf7Bike.Treats) Frf7Bike.Treats$Gear <- factor(mpone(Seat)*mpone(Dynamo), labels = mp)
Frf7Bike.Treats$Raincoat <- factor(mpone(Seat)*mpone(Handbars), labels = mp) Frf7Bike.Treats$Brekkie <- factor(mpone(Dynamo)*mpone(Handbars), labels = mp) Frf7Bike.Treats$Tyres <- factor(mpone(Seat)*mpone(Dynamo)*mpone(Handbars), labels = mp) detach(Frf7Bike.Treats) Frf7Bike.dat <- data.frame(Runs = factor(1:8), Frf7Bike.Treats) Frf7Bike.dat$Time <- as.vector(c(69, 52, 60, 83, 71, 50, 59, 88)) Frf7Bike.dat # # analyse # Frf7Bike.aov <- aov(Time ~ (Seat + Dynamo + Handbars + Gear + Raincoat + Brekkie + Tyres)^2 + Error(Runs), Frf7Bike.dat) summary(Frf7Bike.aov) qqnorm.yeffects(Frf7Bike.aov, error.term = "Runs", data=Frf7Bike.dat) round(yates.effects(Frf7Bike.aov, error.term="Runs", data=Frf7Bike.dat), 2)
Note the use of ‘+’ and ^2 in the model formula for aov so that the analysis only takes into account main effects and two-factor interactions. Diagnostic checking is not performed in this case because of the very low residual degrees of freedom. The following expressions are used to combine the two fractions and analyse the combined data. # # combine fractions # Frf7Bike.Both.dat <- rbind(Frf7Bike.dat,Frf7Bike2.dat) Frf7Bike.Both.dat <- data.frame(Block = factor(rep(1:2, each=8)), Frf7Bike.Both.dat) Frf7Bike.Both.dat # # analyse # Frf7Bike.Both.aov <- aov(Time ~ Block + (Seat + Dynamo + Handbars + Gear + Raincoat + Brekkie + Tyres)^2 + Error(Block/Runs), Frf7Bike.Both.dat) summary(Frf7Bike.Both.aov) qqnorm.yeffects(Frf7Bike.Both.aov, error.term = "Block:Runs", data=Frf7Bike.Both.dat) round(yates.effects(Frf7Bike.Both.aov, error.term="Block:Runs", data=Frf7Bike.Both.dat), 2) # # re-do analysis for just fitted model followed by diagnostic checking # Frf7Bike.Both.Fit.aov <- aov(Time ~ Block + Dynamo + Gear + Error(Block/Runs), Frf7Bike.Both.dat) summary(Frf7Bike.Both.Fit.aov) tukey.1df(Frf7Bike.Both.Fit.aov, Frf7Bike.Both.dat, error.term="Block:Runs") res <- resid.errors(Frf7Bike.Both.Fit.aov) fit <- fitted.errors(Frf7Bike.Both.Fit.aov) plot(fit, res, pch=16) qqnorm(res, pch=16) qqline(res) plot(as.numeric(Frf7Bike.Both.dat$Seat), res, pch=16) plot(as.numeric(Frf7Bike.Both.dat$Dynamo), res, pch=16) plot(as.numeric(Frf7Bike.Both.dat$Handbars), res, pch=16) plot(as.numeric(Frf7Bike.Both.dat$Gear), res, pch=16) plot(as.numeric(Frf7Bike.Both.dat$Raincoat), res, pch=16) plot(as.numeric(Frf7Bike.Both.dat$Brekkie), res, pch=16) plot(as.numeric(Frf7Bike.Both.dat$Tyres), res, pch=16)
Again ‘+’ and ^2 are used in the model formula for aov to restrict the analysis to main effects and two-factor interactions.
C.7. Split-plot experiment
Example IV.1 Production rate experiment
attach(SPLProd.dat) interaction.plot(Methods, Sources, Prodn, lwd = 4) SPLProd.aov <- aov(Prodn ~ Factories + Methods * Sources + Error(Factories/Areas/Parts), SPLProd.dat) summary(SPLProd.aov) #Compute Factories and Areas[Factories] Fs and p-values Factories.F <- 424.07/315.7 Factories.p <- 1-pf(Factories.F, 3, 6) Factories.Areas.F <- 315.7/136.94 Factories.Areas.p <- 1-pf(Factories.Areas.F, 6, 18) data.frame(Factories.F,Factories.p,Factories.Areas.F,Factories.Areas.p) # # Diagnostic checking # tukey.1df(SPLProd.aov, SPLProd.dat, "Factories:Areas:Parts") res <- resid.errors(SPLProd.aov) fit <- fitted.errors(SPLProd.aov) plot(fit, res, pch=16) qqnorm(res, pch=16) qqline(res) plot(as.numeric(Methods), res, pch=16) plot(as.numeric(Sources), res, pch=16) # # tables of means # SPLProd.means <- model.tables(SPLProd.aov, type="means") SPLProd.means qtukey(0.95, 3, 6) qtukey(0.95, 3, 18)
Note that Factories occurs both inside and outside the Error function in the model formula used in the aov function for analysing this experiment. This is because Factories has been designated as a fixed, unrandomized factor. Also, the F-ratios for Factories and Areas[Factories] have to be computed explicitly because they are not given in the output from the summary function.
Pregnancy + Natural Progesterone = Superior Baby By: Sterling Morgan What if there was a natural substance that when used by a pregnant woman could: * Protect the fetus from miscarriage * Increase the feeling of well being of the mother * Increase the potential IQ of the child, and * Produce calmer, less colicky babies! That substance exists, and it is natural progesterone. We are not speaking
1 Assenmacher, Christian 2 Baitz, Christoph 3 Baitz, Michael 4 Bandrowski, Rudi 5 Berg, Richard 6 Blum, Katharina 7 Boonen, Noel 8 Braun, Michael 9 Breitbach, Christoph 10 Breitbach, Thomas 11 Bumke, Heike 12 Bumke, Margit 13 Carstesen, Uwe 14 Cipriani, Ciovanni 15 Corleis, Reiner 16 Dammers, Wolfgang 17 Dombois, Alexander von 18 Eichberger, Mir