• Transmission of the HIV-1 virus via sex workers contributes to the rapid spread of AIDS in Africa
• The spermicide nonoxynol-9 had shown in vitro activity against HIV-1, which motivated a clinical trial by van Damme et al. (2002).
• They tested whether a vaginal gel containing the chemical would reduce the risk of acquiring the disease by female sex workers.
• Data were gathered on a volunteer sample of 765 HIV-free sex-workers in six clinics in Asia and Africa.
• Two gel treatments were assigned randomly to women at each clinic.
• One gel contained nonoxynol-9 and the other contained a placebo.
• Neither the subjects nor the researchers making observations at the clinics knew who received the treatment and who got the placebo.

The goal of experimental design is to eliminate bias and to reduce sampling error when estimating and testing effects of one variable on another.

• To reduce bias, the experiment included:
  • Simultaneous control group: study included both the treatment of interest and a control group (the women receiving the placebo).
  • Randomization: treatments were randomly assigned to women at each clinic.
  • Blinding: neither the subjects nor the clinicians knew which women were assigned which treatment.
• To reduce the effects of sampling error, the experiment included:
  • Replication: study was carried out on multiple independent subjects.
  • Balance: number of women was nearly equal in the two groups at every clinic.
  • Blocking: subjects were grouped according to the clinic they attended, yielding multiple repetitions of the same experiment in different settings (“blocks”).

• In clinical trials either a placebo or the currently accepted treatment should be provided.
• In experiments requiring intrusive methods to administer treatment, such as
  • injections
  • surgery
  • restraint
  • confinement
• the control subjects should be perturbed in the same way as the other subjects, except for the treatment itself, as far as ethical considerations permit.
• The “sham operation”, in which surgery is carried out without the experimental treatment itself, is an example.
• In field experiments, applying a treatment of interest may physically disturb the plots receiving it and the surrounding areas, perhaps by trampling the ground by the researchers.
• Ideally, the same disturbance should be applied to the control plots.

• In a single-blind experiment, the subjects are unaware of the treatment that they have been assigned.
• Treatments must be indistinguishable to subjects, which prevents them from responding differently according to knowledge of treatment.
• Blinding can also be a concern in non-human studies where animals respond to stimuli
• In a double-blind experiment the researchers administering the treatments and measuring the response are also unaware of which subjects are receiving which treatments
  • Researchers sometimes have pet hypotheses, and they might treat experimental subjects in different ways depending on their hopes for the outcome
  • Many response variables are difficult to measure and require some subjective interpretation, which makes the results prone to a bias
  • Researchers are naturally more interested in the treated subjects than the control subjects, and this increased attention can itself result in improved response

• Replication is the assignment of each treatment to multiple, independent experimental units.
• Without replication, we would not know whether response differences were due to the treatments or just chance differences between the treatments caused by other factors.
• Studies that use more units (i.e. that have larger sample sizes) will have smaller standard errors and a higher probability of getting the correct answer from a hypothesis test.
• Larger samples mean more information, and more information means better estimates and more powerful tests.
• Replication is not about the number of plants or animals used, but the number of independent units in the experiment. An “experimental unit” is the independent unit to which treatments are assigned.
• The figure shows three experimental designs used to compare plant growth under two temperature treatments (indicated by the shading of the pots). The first two designs are un-replicated.

• A study design is balanced if all treatments have the same sample size.
• Conversely, a design is unbalanced if there are unequal sample sizes between treatments.
• Balance is a second way to reduce the influence of sampling error on estimation and hypothesis testing.
• To appreciate this, look again at the equation for the standard error of the difference between two treatment means.

• For a fixed total number of experimental units, n1 + n2, the standard error is smallest when n1 and n2 are equal.
• Balance has other benefits. For example, ANOVA is more robust to departures from the assumption of equal variances when designs are balanced or nearly so.

• Nested ANOVA or nested design
  • factors might be hierarchical - in other words nested - within one another
  • The sources of variance are therefore hierarchical too
• The factorial ANOVA design is the most common experimental design used to investigate more than one treatment variable
  • In a factorial design every combination of treatments from two (or more) treatment variables is investigated.
  • The main purpose of a factorial design is to evaluate possible interactions between variables.
  • An interaction between two explanatory variables means that the effect of one variable on the response depends on the state of a second variable.

• Nested designs are hierarchical
  • often contain sub-replicates that are random, uncontrolled, nuisance effects
  • but the nested factors can be of interest too
• Factorial designs are
  • all pairwise combinations,
  • and often involve all combinations of factor levels
  • when each factor is fixed interactions can be assessed
• Completely nested designs therefore have no interaction terms, whereas factorial designs do
• Mixed models can have a combination of fixed and random factors that are more complicated

• Example 1: Study of “repeatability” (simple nested design)
• The walking stick, Timema cristinae, is a wingless herbivorous insect on plants in chaparral habitats of California.
• Nosil and Crespi (2006) measured individuals using digital photographs.
• To evaluate measurement repeatability they took two separate photographs of each specimen.
• After measuring traits on one set of photographs, they repeated the measurements on the second set.

andrew_data <- read.table('andrew.tsv', header=T, sep=‘\t')
head(andrew_data)

• There are four variables: ‘TREAT’, ‘PATCH’, ‘QUAD’ and ‘ALGAE’
• The main effect factor is TREAT
• Make a simplified factor called TREAT2, in which 0% and 33% are a level called “low” and 66% and 100% are “high”

andrew_data$TREAT2 <- factor(c(rep(“low”,40),rep(“high”,40))

• The nested factor is PATCH - also need to turn this into a factor

andrew_data$PATCH <- factor(andrew_data$PATCH)

• In this case, our response variable is ALGAE
• Look at the distribution of ALGAE for the two levels of TREAT2 using boxplots based on the patch means, which are the replicates in this case.

andrew.agg <- with(andrew_data, aggregate(data.frame(ALGAE), 
                  by = list(TREAT2=TREAT2, PATCH=PATCH), mean)

library(nlme)
andrew.agg <- gsummary(andrew_data, groups=andrew_data$PATCH)

boxplot(ALGAE ~ TREAT2, andrew.agg)

• Evaluate assumptions based on the boxplots
• Is the design balanced (equal numbers of sub-replicates per PATCH)?

• Run the nested ANOVA:

nested.aov <- aov(ALGAE ~ TREAT2 + Error(PATCH), data=andrew_data)
summary(nested.aov)

• Do we detect an effect of TREAT2 (high vs low sea urchin density)?
  
• Estimate variance components to assess relative contributions of the random factors

library(nlme)
VarCorr(lme(ALGAE ~ 1, random = ~1 | TREAT2/PATCH, andrew_data))

• Calculate the % of variation due to between-treatment differences vs. due to among patches within treatment differences.
• See pg. 302 in Logan if you need help.
• What do these variance component estimates tell us???

• For example, Relyae (2003) looked at how a moderate dose (1.6mg/L) of a commonly used pesticide, carbaryl (Sevin), affected bullfrog tadpole survival.
• In particular, the experiment asked how the effect of carbaryl depended on whether a native predator, the red-spotted newt, was also present.
• The newt was caged and could cause no direct harm, but it emitted visual and chemical cues to other tadpoles
• The experiment was carried out in 10-L tubs (experimental units), each containing 10 tadpoles.
• The four combinations of pesticide treatment (carbaryl vs. water only) and predator treatment (present or absent) were randomly assigned to tubs.
• The results showed that survival was high except when pesticide was applied together with the predator.
• Thus, the two treatments, predation and pesticide, seem to have interacted.

continuous response and two main effect variables

rnadata <- read.table('RNAseq.tsv', header=T, sep='')
gene <- rnadata$Gene80
microbiota <- rnadata$Microbiota
genotype <- rnadata$Genotype

make new “pseudo factor,” combining genotype and microbiota

gxm <- interaction(genotype,microbiota)
levels(gxm)
boxplot(gene ~ gxm)

specify the following 2 contrasts

contrasts(gxm) <- cbind(c(2, -1, 0, -1), c(-1, -1, 3, -1))

Fit the factorial linear model

rna_aov <- aov(gene ~ gxm)

Examine the ANOVA table, using supplied contrasts. Figure out the appropriate titles to give them.

summary(rna_aov, split = list(gxm = list('xxx'=1,'xxx'=2)))

What does the contrast summary tell you about the nature of the interaction?

rnadata <- read.table('RNAseq.tsv', header=T, sep='')
head(rnadata)

variables excluding first 5 and last 5 observations

gene <- rnadata$Gene80[6:75] 
microbiota <- rnadata$Microbiota[6:75]
genotype <- rnadata$Genotype[6:75]
boxplot(gene ~ microbiota)
boxplot(gene ~ genotype)
boxplot(gene ~ microbiota*genotype)

Estimate the variance components using Restricted Maximum Likelihood (REML)

library(lme4)
lmer(gene ~ 1 + (1 | microbiota) + (1 | genotype) + (1 | microbiota:genotype))

Based on the REML sd estimates, what are the relative contributions of the factors to total variance in gene expression?

longevity_data <- read.table('longevity.csv', header=T, sep=',')
head(longevity_data)

Variables

long <- longevity_data$LONGEVITY
treat <- longevity_data$TREATMENT
thorax <- longevity_data$THORAX

• check to see if the covariate should be included

boxplot(long ~ treat)
plot(long ~ thorax)

• assess assumptions of normality and homogeneity of variance

plot(aov(long ~ thorax + treat ), which = 1)

• †ry it again with a transformed response variable

plot(aov(log10(long) ~ thorax + treat ), which = 1)

• visually assess linearity, homogenetiy of slopes and covariate range equality

library(lattice)
print(xyplot(log10(long) ~ thorax | treat, type = c("r", "p")))

• formally test homogenetiy of slopes by testing the interaction term

anova(aov(log10(long) ~ thorax*treat))

• formally test covariate range disparity by modeling the effect of the treatments on the covariate

anova(aov(thorax ~ treat))

• FINALLY, set up contrasts, fit the additive model and visualize the results (pg. 459 and 460 of your Logan book)
• Summarize the trends in a nice plot (pg. 461 of your Logan book)