A study was carried out to compare four different weight-reduction programs (1-4

ID: 3052631 • Letter: A

Question

A study was carried out to compare four different weight-reduction programs (1-4). Twenty-four
overweight people were chosen for the study and each was then randomly assigned to take one of the
four programs for one year (ie, having six people in each program). Data obtained is given below (negative
and positive values indicate weight losses and gains in lb respectively). It is also stored in the file,
weightloss.MTW, available on the unit iLearn.

Program
1 -59 22 -29 10 -23 72
2 -7 25 7 -8 40 64
3 -21 -54 -1 -44 -22 79
4 -27 -2 -77 -46 -11 93

a) Explain what design is used here.

b) Examine the data graphically and comment.

c) Carry out an analysis of variance using the data, and explain how the degrees of freedom of the
three sum of squares (Treatment, Error and Total) in the resulting ANOVA table are computed.

d) Check and comment on model assumptions.

e) Based on the results in the ANOVA table obtained, test if there is any difference in mean weight
loss/gain across the four programs at the 5% significance level. Interpret the results.

f) Comment on whether it is any need to carry out Tukey’s simultaneous tests or compute Tukey’s
simultaneous confidence intervals to compare all pair-wise program means here. Also comment
on whether a Tukey procedure is appropriate for this dataset. (Note: you do not have to carry out
the tests/compute the CIs here.)

g) Suppose that this study was initially planned to compare weight-reduction Program 3 with
Program 4, as well as Program 2 with the other three programs. Write down the two
corresponding contrasts. Show if the two contrasts are orthogonal to each other, and suggest an
appropriate method (test) to test simultaneously whether each of the two contrasts differs from
zero.

h) Based on the graphical display in part b), suppose we would like to test another contrast of
C 3 = (?1 ? ?2) . Show whether this contrast together with the two contrasts above in part g) form
a full orthogonal contrast set for this experiment/study.

i) Which test/method will you consider to investigate the three contrasts simultaneously? Why?

Explanation / Answer

SOLUTION

The design is Completely Randomised Design with 4 treatments and equal number of observations (6) per treatment. ANSWER

ANOVA

Source

Fcrit

p-value

Programmes

3911.5

1303.819

0.610081

3.098391

0.61627

Error

42743

2137.125

Total

46654

2028.433

Note on Degrees of Freedom

Total: Total number of observations – 1 = 24 – 1 = 23.

Treatments (Programmes): Total number of treatments – 1 = 4 – 1 = 3.

Error: Total – Treatment = 23 – 3 = 20.

Conclusion from ANOVA

Since F value < Fcrit, null hypothesis that the mean gain in weight all four programmes are the same is accepted.

Tukey’s test

There is no need for this since ANOVA has accepted the null hypothesis.

DONE