Researchers taught 18 monkeys to distinguish each of 100 pairs of objects, 20 pairs each at 16, 12, 8, 4, and 2 weeks prior to a treatment. After this training, they blocked access to the hippocampal formation in 11 of the monkeys. All monkeys were then tested on their ability to distinguish the objects. The five-dimensional response for each monkey is the number of correct objects distinguished among those taught at 16, 12, 8, 4, and 2 weeks prior to treatment.

1 |

A data frame with 18 observations on the following 7 variables.

- Monkey
Monkey name

- Treatment
a treatment factor with levels

`"Control"`

and`"Treated"`

- Week2
percentage of 20 objects taught 2 weeks prior to treatment that were correctly distinguished in the test

- Week4
percentage of 20 objects taught 4 weeks prior to treatment that were correctly distinguished in the test

- Week8
percentage of 20 objects taught 8 weeks prior to treatment that were correctly distinguished in the test

- Week12
percentage of 20 objects taught 12 weeks prior to treatment that were correctly distinguished in the test

- Week16
percentage of 20 objects taught 16 weeks prior to treatment that were correctly distinguished in the test

Ramsey, F.L. and Schafer, D.W. (2013). *The Statistical Sleuth: A
Course in Methods of Data Analysis (3rd ed)*, Cengage Learning.

Sola-Morgan, S. M. and Squire, L. R. (1990). The Primate Hippocampal
Formation: Evidence for a Time-limited Role in Memory Storage, *Science*
**250**: 288–290.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | ```
str(case1601)
attach(case1601)
## EXPLORATION
short <- (Week2 + Week4)/2
long <- (Week8 + Week12 + Week16)/3
myPointCode <- ifelse(Treatment=="Control",15,16)
myPointColor <- ifelse(Treatment=="Control","orange","green")
plot(long ~ short, pch=myPointCode, col=myPointColor, cex=2)
abline(h=mean(long),lty=2)
abline(v=mean(short),lty=2)
identify(short,long,labels=Monkey) # Identify outliers; press Esc when done
## INFERENCE USING HOTELLING's T-SQUARED TEST
myLm1 <- lm(cbind(short,long) ~ Treatment) # Full model
myLm2 <- lm(cbind(short,long) ~ 1) # Reduced model, with only intercept
anova(myLm2, myLm1, test="Hotelling") # p-value for Treatment effect
# confidence intervals
n1 <- sum(Treatment=="Control") # 7 control monkeys
n2 <- sum(Treatment=="Treated") # 11 treated monkeys
multiplier <- sqrt(2*((n1+n2-2)/(n1+n2-3))*qf(.95,2,n1+n2-3)) # Sleuth p. 492
summary(myLm1)
shortEffect <- myLm1$coef[2,1] # Difference in sample averages; Short
seShortEffect <- 3.352 # Read this from summary(myLm1)
halfWidth <- multiplier*seShortEffect # Half width of 95% confidence interval
shortEffect + c(-1,1)*halfWidth #95% CI for effect of treatment on Short
longEffect <- myLm1$coef[2,2] # Difference in sample averages; Long
seLongEffect <- 3.2215 # Read this from summary(myLm1)
halfWidth <- multiplier*seLongEffect # Half width of 95% confidence interval
longEffect + c(-1,1)*halfWidth #95% CI for effect of treatment on Long
## GRAPHICAL DISPLAY FOR PRESENTATION
myPointCode <- ifelse(Treatment=="Control",21,22)
myPointColor <- ifelse(Treatment=="Control","green","orange")
plot(long ~ jitter(short),
xlab="Short-Term Memory Score (Percent Correct)",
ylab="Long-Term Memory Score (Percent Correct)",
main="Memory Scores for 11 Hippocampus-Blocked and 7 Control Monkeys",
pch=myPointCode, bg=myPointColor, cex=2.5, lwd=3)
identify(short,long,labels=Monkey) # Label the outliers; press Esc when done
legend(52,54,legend=c("Control","Hippocampus Blocked"), pch=c(21,22),
pt.bg=c("green","orange"), pt.cex=c(2.5,2.5), pt.lwd=c(3,3), cex=1.5)
## ADVANCED: RANDOMIZATION TEST FOR EQUALITY OF BIVARIATE RESPONSES
myAnova <- anova(myLm2, myLm1, test="Hotelling") #Hotelling Test for Treatment
myAnova$approx[2] #[1] 12.32109: F-statistic
numRep <- 50 # Number of random regroupings (change to 50,000)
FStats <- rep(0,numRep) # Initialize a variable for storing the F-statistics
myLmReduced <- lm(cbind(short,long) ~ 1)# Fit the reduced model once
for (rep in 1:numRep) { # Do the following commands in parenthese num.rep times
randomGroup <- rep("Group1",18) # Set randomGroup initially to all "Group1"
randomGroup[sample(1:18,7)] <- "Group2" # Change 7 at random to "Group2"
randomGroup <- factor(randomGroup) # Make the character variable a factor
myLmFull <- lm(cbind(short,long) ~ randomGroup) # Fit full model
myAnova2 <- anova(myLmReduced, myLmFull, test="Hotelling") # Hotelling's test
FStats[rep] <- myAnova2$approx[2] # Store the F-statistic
} # If numRep = 50,000, go get a cup of coffee while you wait for this.
hist(FStats, main="Approx. Randomizatin Dist of F-stat if No Treatment Effect")
abline(v=12.32109) # Show actually observed Hotelling F-statistic
pValue <- sum(FStats >= 12.32109)/numRep
pValue # Approximate randomization test p-value (no distributional assumptions)
detach(case1601)
``` |

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