STAT 516 hw 7

Author

Karl Gregory

Students in several classes were asked to measure on their left hands the distances $B$ and $A$ depicted in the diagram below:

Do our fingers grow according to the golden ratio?

It is of interest to see whether our (humans’) fingers grow according to the golden ratio such that the mean of the ratio $B / A$ in the adult human population is equal to $(1 + \sqrt{5}) / 2 \approx 1.618$ . Due to possible inconsistencies in the collection of these numbers in the several classes, assume the measurements have arisen according to the one-way random effects model $Y_{i j} = μ + A_{i} + ε_{i j}$ for $i = 1, \dots, a$ , $j = 1, \dots, n_{i}$ , where the $A_{i}$ are independent $N (0, σ_{A}^{2})$ random variables, the $ε_{i j}$ are independent $N (0, σ_{ε}^{2})$ random variables, $a$ is the number of classes and the $n_{i}$ are the numbers of students in the classes.

A comma-separated-values file containing the data can be downloaded from here. It can be read into R with the following code:

gr <- read.csv(pathtofile, # replace pathtofile 
               colClasses = c("numeric","numeric","factor"))
gr$r <- gr$B / gr$A # compute ratio B / A

There were different numbers of students in each class, so the design is unbalanced. In the first part of this homework we will make a balanced version of the data set; in the second part we will work with the full data set.

Part I

Construct a balanced data set by running the code below, which draws (without replacement) a subsample of size $n = 15$ from each data set.

# fix the random seed so the same subsets 
# are drawn every time the code runs
set.seed(1) 

classes <- levels(gr$class)
a <- length(classes)
n <- 15
gr_bal <- data.frame()
for(i in 1:a){
  
  ind <- which(gr$class == classes[i])
  gr_bal <- rbind(gr_bal,gr[sample(ind,n),])  
  
}

head(gr_bal)
table(gr_bal$class)

1.

Using the balanced data, report the mean of the ratio $B / A$ for the subsample drawn from each class.

2.

Produce side-by-side boxplots of the observed responses in the different classes.

3.

Obtain method of moments estimators for $σ_{A}^{2}$ and $σ_{ε}^{2}$ .

4.

Check whether these are the same as the restricted maximum likelihood estimates obtained with lmer().

5.

Report the test statistic and $p$ value for testing whether class-to-class variability accounts for any significant portion of the variability in the measurements. What do you conclude?

6.

Obtain for each class a prediction (a guess) of the realized value of the class random effect $A_{i}$ .

7.

Construct a $95 %$ confidence interval for the overall mean of the ratio $B / A$ . State whether you think it is plausible that our fingers grow according to the golden ratio such that the population mean of $B / A$ is equal to $1.618$ .

8.

Remove the set.seed(1) command (so that new subsamples of data are drawn in the construction of the balanced data set) and run your analysis again: Do you get the same results? How stable is the analysis? Do you have any concerns?

Part II

Now use the entire data set.

1.

Produce side-by-side boxplots of the responses in the treatment groups.

2.

Report the number of observations in each class.

3.

Check whether the assumptions of the one-way random effects model hold for these data.

4.

Obtain a $95 %$ confidence interval for the overall mean and interpret the interval.

5.

Consider the interval computed by the code below. Explain the strategy behind the construction of this interval and carefully explain whether you think it is appropriate or inappropriate (trustworthy or untrustworthy).

y <- gr$r
y.. <- mean(y)
sn <- sd(y)
N <- length(y)
alpha <- 0.05
tval <- qt(1-alpha/2,N-1)
lo <- y.. - tval * sn / sqrt(N)
up <- y.. + tval * sn / sqrt(N)
c(lo,up)

[1] 1.568001 1.635472