STAT 516 hw 6

Thirty sheets of paper were printed, each with one of the following instructions printed on top, such that each instruction appeared on five sheets of paper:

1. “Draw a triangle”
2. “Draw any triangle”
3. “Draw literally any triangle”
4. “Draw a three-sided shape”
5. “Draw any three-sided shape”
6. “Draw literally any three-sided shape”

The sheets of paper were stacked such that the instructions were inserted in the above order, cyclically, five times in the stack, and the sheets were handed out one-by-one to students in a class, such that the desk at which a student sat determined which instruction he or she was given. After following the instructions, the students were asked to measure the lengths of the sides ($a$, $b$, and $c$) of their shapes and record them. Twenty-two students attended the class, which resulted in an unbalanced design; to make the design balanced, two statistics graduate students were accosted in the hallway and asked to participate.

The data are in the file triangle_threesided_shape_draw.csv, which can be downloaded from this folder. The R code below will prepare the data for analysis: It will compute on each set of side lengths the value $d$, where $d^2$ is defined as $$d^2=\frac{(a-b{)}^2+(b-c{)}^2+(a-c{)}^2}{3(a^2+b^2+c^2)}.$$ Thus $d$ will be equal to zero if a triangle has equal sides ($a=b=c$), so triangles similar to equilateral triangles will have small values of $d$. The values of $d$ will be used as the response in a two-way ANOVA model.

tri0 <- read.csv(pathtofile) # replace pathtofile 

a <- tri0$a
b <- tri0$b
c <- tri0$c
d <- sqrt(((a - b)^2 + (b - c)^2 + (a - c)^2 ) / (3*(a^2 + b^2 + c^2)))
tri <- data.frame(d,
                  F1 = as.factor(ifelse(tri0$F1=="threesided","thr","tri")),
                  F2 = as.factor(tri0$F2))
tri

Consider fitting the two-way treatment effects model which assumes the responses arise as $$Y_{ijk}=\mu +{\tau }_i+{\gamma }_j+(\tau \gamma {)}_{ij}+{\epsilon }_{ijk}$$ for $k=1,\dots ,n_{ij}$, $i=1,\dots ,a$, $j=1,\dots ,b$, where ${\epsilon }_{ijk}$ are independent error terms having the $\mathcal{N}(0,{\sigma }^2)$ distribution.