## Stat.usu.edu

**Expected Completion of Content**
**Assignment Due (by 11:59 P.M.):**
**Directions: **In completing these Exercises, use the SAS tools presented in class. You may

ﬁnd the SAS Help useful. Where appropriate, you may perform calculations by hand. Note

that neatness and format (including SAS code in an appendix) will contribute 10 points to

the total score. This assignment will be graded out of 75 points (60 points for 5500 students).

For purposes of attribution, Exercises 1 and 2 are based on data referred to in Sidik and

Jonkman 2005, JRSS Applied Statistics 54(2):367-384; the article will not help with this

assignment.

1. Spirometry involves measuring lung function, and values can be reported in terms of
peak ﬂow rate (liters per minute). Researchers in the Williams lab wanted to seeif aminophylline treatment aﬀects spirometric [peak ﬂow rate] values in severe acuteasthmatic patients any diﬀerently than did salbutamol, another potential asthma treat-ment. Nine patients were randomly assigned to receive aminophylline, and eleven ran-domly assigned to receive salbutamol. Each patient’s spirometric value was recorded,then they received their designated treatment, and their spirometric value was re-measured one hour later. The aminophylline group saw a mean spirometric increaseof 44, with a standard deviation of 20. The salbutamol group saw a mean increase of86, with a standard deviation of 15.

(a) (5 points) Conduct an appropriate test of signiﬁcance for the Williams lab, using
signiﬁcance threshold

*α *=

*.*05.

Be sure to specify (i) the test, (ii) null and
alternative hypotheses, (iii) test statistic, (iv) p-value, and (v) conclusion (incontext). You may assume the spirometric increase values for both groups ofpatients are approximately normally distributed.

(b) (5 points) Clearly explain what your p-value from 1a means. Do not just repeat
what your conclusion is – what does the number itself mean?
(c) (5 points) Assuming a common standard deviation as estimated from the data,
what would be the power of this test if the true eﬀect of aminophylline were anincrease of 20 liters per minute more than salbutamol (i.e., the mean diﬀerence is20) ?
(d) (5 points) Clearly explain what power means in this context.

(e) (5 points) Construct a clear visual representation of how the power of this test
varies with the underlying true mean diﬀerence.

2. Twelve other labs independently assessed the same eﬀect of aminophylline treatment
on spirometric values in severe acute asthmatic patients, as compared to another po-tential treatment. The p-values from these twelve other tests of signiﬁcance are in thefollowing table:
Add your p-value from Exercise 1a to this list of p-values.

(a) (5 points) In considering this family of 13 tests, explain clearly why it might be
worthwhile to adjust for multiple hypothesis tests.

(b) (5 points) Report a table of raw and adjusted p-values, using the following ad-
justments: Bonferroni, Sidak, Holm (step-down Bonferroni), Hochberg (step-upBonferroni), and False Discovery Rate (Benjamini-Hochberg).

(c) (5 points) Which of these adjustments would you recommend, and why?
(d) (5 points) After making this adjustment, what change, if any, would you make to
your conclusion regarding the aminophylline eﬀect in the Williams lab?
(e) (5 points) Using Fisher’s composite testing method, evaluate the composite null
of these 13 tests. Report the test statistic, p-value, and conclusion (in context).

3. (6500 students only; 15 points) First, read the paper by Raimund Alt provided on
the course website, paying particular attention to his “closed LSD-procedure”. Thenconsider a multiple regression model with many predictors:

*Yi *=

*β*0 +

*β*1

*Xi,*1 +

*β*2

*Xi,*2 +

*. . . *+

*βkXi,k *+

*ϵi.*
When we ﬁt such a model and look for signiﬁcant predictors, we test a family of nullhypotheses

*H*0

*,j *:

*βj *= 0 for

*j ∈ {*1

*. . . k}*. However, the typical approach is to notadjust the resulting set of

*k *p-values for multiple hypothesis testing. Is this lack ofadjustment justiﬁed by Alt’s “closed LSD-procedure” approach? Write a convincingparagraph explaining why or why not. As part of your discussion, report the conclu-sions of the two approaches when applied to the two sample data sets on the coursewebsite (both use

*k *= 3). Do the two approaches yield equivalent conclusions in bothdata sets? You may ﬁnd the following SAS code to be a helpful template.

Source: http://www.stat.usu.edu/jrstevens/biostat/Homework1.pdf

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