1. How to use the PS sample size software for advanced applications
****************
William D. Dupont
Department of Biostatistics
****************
GCRC Research-Skills Workshop
October 10, 2003

2. Additional follow-up F
Survival Analysis
Follow-up time
Fate at exit
Statistic: Log-rank test
Power: Schoenfeld & Richter, Biometrics 1982
Accrual time A
Additional follow-up F
Sample Size
Time

3. Median survival for experimental subjects Median survival for controls
Relative Risk (Hazard Ratio) for controls relative to experimental subjects
Median survival for experimental subjects
Median survival for controls =
= 2 6 3
Median Survival
Experimental treatment
Control treatment

4. For t tests power calculations for increased or decreased response relative to control response are symmetric.
i.e. The power to detect is the same as the power to detect
This is not true for Survival Analysis.
The power to detect a two-fold increase in hazard does not equal the power to detect a 50% decrease in hazard.

5. Power to detect treatment 1 vs. control greater than treatment 1 vs
Power to detect treatment 1 vs. control greater than treatment 1 vs.control
Treatment 1
Control
Relative Risk
Treatment 1 vs Control 0.5
Treatment 2 vs. Control 2
Control vs. Treatment 1 2
Treatment 2

6. It is important to read the parameter definitions carefully.
Multiple power curves can be plotted on a single graph.
Black and white plots can be produced.

7. Power Calculations for Linear Regression

8. = standard deviation of x variable
= standard deviation of y variable s x =
4.75 20 25 30 35 40 45 5 10 15
Estriol (mg/24 hr)
Birthweight (g/100)
Rosner Table 11.1
Am J Obs Gyn 1963;85:1-9

9. Estimated by s = root mean squared error (MSE)
= standard deviation of the regression errors
Estimated by s = root mean squared error (MSE)

10. r = 1 r = -1 r = 0 0 < r < 1 c) Correlation coefficient
is estimated by
{1.2}
r = 0
r = -1
0 < r < 1
r = 1

11. Measures of association between continuous variables
Correlation vs. linear regression 5 10 15 20 25 30 35 2 4 6 8 12
Independent Variable x
Response Variable y 1 3 7 9 11
Treatment r A
0.9 B
0.6
Both treatments have
identical values of
mx = 6, my = 20
sx = 2, sy = 5

12. Treatment l r s A
2.25
0.9
2.18 B
1.5
0.6
4.0 35 30 25
Response Variable y 20
Both treatments have
identical values of
mx = 6, my = 20
sx = 2, sy = 5 15 10 5 1 2 3 4 5 6 7 8 9 10 11 12
Independent Variable x

13. • y j j th Regression Error l g + l x j 1 Unit Patient Response of
Dosage g
Regression
Line
g + l x x j
Treatment Dose Level

14. c) Slope parameter estimate
l (a.k.a. ) is estimated by b = r sy /sx

15. l is estimated by b = r sy /sx

16. Studies can be either experimental or observational

17. Normally distributed unless investigator chooses level
Chosen by investigator
Treatment level

18. Estimating s directly is often difficult
If we can estimate or s = s =
Warning:
If the anticipated value of in your experiment is different from that found in the literature then your value of r will also be different.

19. 35 30 25
Response Variable y 20 15 10 5 1 2 3 4 5 6 7 8 9 10 11 12
Independent Variable x

20. Relationship between BMI and exercise time
n = 100 women willing to follow a diet exercise program for six months
Interquartile range (IQR) of exercise time = 15 minutes (pilot data)
= (IQR / 2) / z0.25 = 7.5 / = 11.11
= 4.0 kg/m2 = standard deviation of BMI for women obtained by Kuskowska-Wolk et al. Int J Obes 1992;16:1-9
Would like to detect a true drop in BMI of = kg/m2 per minute of exercise
(1/2 hour of exercise per day induces a drop of 2 kg/m2 over 6 months)

21. When s = 1 2 -2 Interquartile Range In general 0.25 0.25 -0.675 0.675
= (IQR / 2) / z0.25
When s = 1
0.25
0.25 2 -2
-0.675
0.675

22. Impaired Antibody Response to Pneumococcal Vaccine
after treatment for Hodgkin’s Disease
Siber et al. N Engl J Med 1978;299:
n = 17 patients treated with subtotal radiation. vaccinated 8 to 51 months later
A linear regression of log antibody response against time from radiation to vaccination gave
Suppose we wanted to determine the sample size for a new study with
patients randomized to vaccination at 10, 30, or 50 weeks

23. Composing Slopes of Two Linear Regressin Lines
Armitage and Berry (1994) gave age and pulmonary vital capacity for
28 cadmium industry workers with > 10 years of exposure
44 workers with no exposure

24. Need 167 exposed workers and
(unexposed)
(exposed)
pooled error variance
How many workers do we need to detect with
ratio of unexposed workers m = 44/28 = 1.57?
Need 167 exposed workers and
167 x 1.57 = 262 unexposed workers