The data for this homework is drawn from a classic
analysis which I'll describe in more detail later.
The data come from an experiment in worsted yarn,
where pieces of yarn were subjected to repeated
loads (pulling) and the response was the number
of cycles until failure (the yarn broke).
The initial variables were:
a1: length of the test specimen in mm
a2: amplitude of the load cycle in mm
a3: the load in g.
These variables were coded by subtracting off a
mean value and dividing by one-half the range of
the data (units cancel), so that in this case
x1 = (a1 - 300)/50,
x2 = (a2 - 9),
x3 = (a3 - 45)/5.
As can be seen from the layout below, this was
a designed experiment - the levels of the x_i's
were chosen specifically to investigate the relationship
between the x_i's and the response. The design
used is known as a 3^3 factorial design - three
factors were measured at all combinations of three
levels each, low, medium and high. This deisgn
allows for the estimation of both first and second
order models.
Some parts of the homework: Fit the first and second
order models (including interactions) to the data.
Test for the significance of the higher order effects
as a group (partial F test). Plot the studentized
residuals versus the fitted response. Comment.
Examine the possibility of transforming the data.
Produce a plot of the scaled Residual Sums of Squares
using the Box-Cox transforms for both first and second
order models, superimposed on a single plot (like the
one I handed out in class). Using the transformation
that you deem most appropriate, refit the least
squares models and check for significance. Comment.
Produce an ANOVA table for the final model you choose,
and plot the sorted studentized residuals against
normal quantiles.
x1 x2 x3 y
-1 -1 -1 674
0 -1 -1 1414
1 -1 -1 3636
-1 0 -1 338
0 0 -1 1022
1 0 -1 1568
-1 1 -1 170
0 1 -1 442
1 1 -1 1140
-1 -1 0 370
0 -1 0 1198
1 -1 0 3184
-1 0 0 266
0 0 0 620
1 0 0 1070
-1 1 0 118
0 1 0 332
1 1 0 884
-1 -1 1 292
0 -1 1 634
1 -1 1 2000
-1 0 1 210
0 0 1 438
1 0 1 566
-1 1 1 90
0 1 1 220
1 1 1 360