Sample Size

1
Sample Size

• Brian Yuen
• Public Health Sciences Medical Statistics

2
Learning outcomes

• By the end of this session you should
• understand why determination of sample size is
  important
• appreciate some statistical concepts
• be aware of considerations needed to perform
  sample size calculations
• be able to perform a sample size calculation to a
  precision with continuous or binary outcome
• be able to perform a sample size calculation
  based on two independent groups with continuous
  or binary outcome
• be appreciate with the useful resources relating
  to this topic
• be able to use PS to assist calculating sample
  size

3
Contents

• Introduction
• Reasons and objectives
• Sample size calculations
• for a precision with continuous outcome
• for a precision with binary outcome
• practical session
• Considerations for two independent groups
• 7 ingredients for sample size calculations
• Sample size calculations
• for two independent groups with continuous
  outcome
• for two independent groups with binary outcome
• practical session
• Fixing sample size and other methods
• Useful resources
• Sample size calculations for two independent
  groups in PS
• practical session
• Solutions to exercises

4
Why sample size calculation?

• Required by ethical committees
• need approval when doing a confirmatory trial or
  study
• to document your estimation of the required
  sample size, and they will not grant approval for
  research projects with too few or too many
  subjects.
• Required for grant application for a study
• Required by lots of journals, one of the
  checklist for writing up paper
• A thoroughly designed study should consider
  sample size calculation
• Need to consider this carefully as recruiting too
  few or too many subjects could cause ethical
  problems
• too few can't be specific enough about the size
  of the effect in the population, hence the study
  would become meaningless, unethical, and waste of
  resources
• too many comparatively more patients would be
  allocated to the inferior treatment, hence
  unethical, also waste of resources

5
Sample size Objectives

• Aim to find out a reasonably large enough sample
  size in which would give us high probability to
  detect a clinically worthwhile treatment effect
  if it exists
• Some information is required in order to perform
  the calculation
• Generally relates to a single primary outcome
• occasionally more than one is considered

6
Using sample estimates

• Uncertainty is often introduced when using a
  sample to make inferences about the population,
  because the information we collect is only an
  estimate
• In order to get the best estimate, we need a
  representative, unbiased and a reasonably sized
  sample
• We can quantify uncertainty through
• standard error
• confidence interval
• and use these to calculate sample size

7
Sample Size Calculations for a Precision
8
with continuous outcome
Objective Estimate a population mean to a
required (or pre-defined) precision

• SE of mean
• 95 CI

If we know a sensible value for SD (?) and the
desired confidence interval (CI) width, then we
can obtain n, the number of observations required.
9
with continuous outcome

• Example Peak Expiratory Flow Rate in young men
• Standard deviation 48 litres/min (Gregg et al,
  BMJ, 1973)
• Desired 95 confidence interval width ?20
  litres/min
• CI width
• 20
• Now, solve for n gives
• n
• Hence, a sample of 23 would enable us to estimate
  the population PEFR mean to within 20 litres/min
  (with 95 probability)

10
with binary outcome
Objective Estimate a population proportion to a
required precision

• SE of proportion
• 95 CI

This is useful when estimating a prevalence from
a survey, however, the standard error of a
proportion depends on the proportion itself, that
is the quantity we are trying to estimate.
Hence, we need an initial estimate of p.
11
with binary outcome

• Example Prevalence of pancreatic cancer
• Suppose we are trying to estimate the prevalence
  of pancreatic cancer, which we suspect to be
  about 3, and we want the 95 confidence interval
  width to be ?0.5
• CI width
• 0.5
• Now, solve for n gives
• n
• Hence, 4472 subjects are required to estimate the
  proportion of prevalence cases to within 0.5
  (with 95 probability).

12
Considerations Needed for Two Independent Groups
13
7 ingredients for sample size calculations

• Research question to be answered
• Outcome measure
• Effect size
• Variability success proportions
• For continuous outcome
• For binary outcome
• Type I error
• Type II error
• Other factors

14
Further explanations of ingredient 1

• Research question to be answered
• Translate the question into a clear hypothesis!
• For example,
• H0 there is no difference between treatment and
  control
• H1 there are differences between treatment and
  control
• Hypothesis ? Statistical results ? Conclusion
• statistically significant result (that is,
  plt0.05)
• ? enough evidence to reject H0 ? accept H1
• statistically non-significant result (that is,
  pgt0.05)
• ? no evidence to reject H0

15
Further explanations of ingredient 2

• Outcome measures
• Should only have one primary outcome measure per
  study!
• Could have a secondary outcome measure, but we
  can only sample sizing/powering for the primary
  outcome
• May not have enough power for any results
  relating to the secondary outcome
• Recall the two types of variables
• Continuous
• Categorical
• If the variable has 2 categories ? Binary

16
Further explanations of ingredient 3

• Effect Size (d) from the word difference
• The magnitude of difference that we are looking
  for
• Clinically important difference
• For 2 treatment arms
• difference in means if continuous outcome
• difference in success proportions if binary
  outcome
• Minimum value worth detecting
• Decide what the minimum better means by looking
  at the endpoint and by considering background
  noise
• Headache? or Moderate severe headache? or
  Migraine?
• Values could be found in previous literatures if
  they were doing similar study or can be estimated
  base on clinical experience but make sure it is
  reasonable (Remember GIGO!)

17
Further explanations of ingredient 3

• Effect Size (d)
• Example In previous study, morbidity of a
  certain illness under conventional care is known
  to be 73
• Interested in reducing morbidity to 50
  (clinically important)
• Therefore the effect size is 23
• A difference between these morbidities
• Example Summarising all the studies with similar
  setting and characteristics regarding to a
  specific outcome measure, e.g. pain relief
• The overall response rate on Placebo is 32
• The overall response rate on Active is 50
• The overall estimate of the difference between
  Active and Placebo is 18
• Of all the differences that are found in these
  studies, the smallest difference observed is 12
• Could be the minimum value worth detecting

18
Further explanations of ingredient 4.1

• Variability (s) pronounce as Sigma
• For continuous outcome only!
• Standard deviation (s) or variance (s2)
  represents the spread of the distribution of a
  continuous variable
• Values can usually be found in previous
  literatures or can be estimated base on clinical
  experience but make sure it is reasonable (GIGO!)
• Choose the largest or pool the values if the
  standard deviation of the outcome were reported
  in each group or treatment arm in a literature
• For further details on pooling standard
  deviations, read the next slide
• Consider carefully if different standard
  deviations of the outcome were found in different
  studies
• ask yourself if the design or study population of
  those studies were similar to yours
• if so, be conservative and choose the reasonably
  largest to estimate
• If none can be found from previous literature,
  but range of the outcome is available, can then
  divide the range by 6 (remember mean?3SD ? 99.7
  of observations) to get an estimate of the
  standard deviation

19
Pooled standard deviation

• If there are several studies with variance
  estimates available it is recommended that an
  overall estimate of the population variance or
  the pooled variance estimates, sp2, is obtained
  from the following formula
• where k is the number of studies, si2 is the
  variance estimate from the ith study and dfi is
  the degrees of freedom about this variance (which
  is the corresponding number of observations in
  the group minus 1, i.e. (ni - 1)).

20
Pooled standard deviation

• Example The following descriptive statistics
  (number of subjects, mean standard deviation)
  of an outcome measure for each treatment arm were
  reported,
• Treatment A nA 83, meanA sA 40.98 22.52
• Treatment B nB 87, meanB sB 37.89 19.74
• Using the formula above, the pooled variance
  (sp2) and the pooled SD (sp) is

21
Further explanations of ingredient 4.2

• Success proportions (p)
• For binary outcome only!
• Normally concerning Cured/Not Cured,
  Alive/Deadetc
• Require to know the success proportion of the
  binary outcome for each group or treatment arm
  first, can be found in previous literatures or
  estimate with clinical experience (GIGO!)
• In the above table, suppose we are interested in
  the proportion of Alive, then the success
  proportions in each treatment are pA and pB for
  treatment A and B respectively
• Denote is the average success proportion,
  i.e. (pA pB)/2
• We can use these information to find out the
  effect size and the standard deviation
• The effect size is the difference of the two
  success proportions, i.e. pA - pB
• The estimated standard deviation is
  , where is between 0 and 100

22
Further explanations of ingredients 5 6

• Type I error (a) Type II error (ß)
• You should have heard these mentioned in the
  Hypothesis Testing session, hence this is just a
  reminder
• Due to the fact that we are sampling from a
  population
• Uncertainty is introduced
• Quality of the sample will have an impact on our
  conclusion
• Error does exist
• There are two types of error
• Type I error (a) observed something in our
  sample but not exist in the population (the
  truth)
• e.g. drinking water leads to cancer
• Type II error (ß) observed nothing in our sample
  but something exist in the population (the truth)
• e.g. smoking doesnt lead to cancer

23
Further explanations of ingredients 5 6

• Type I error (a) Type II error (ß)
• Type I error (a) usually allow for 5
• Significant level a ? cut-off point for
  p-value, i.e. 0.05

24
Further explanations of ingredients 5 6

• Type I error (a) Type II error (ß)
• Type II error (ß) usually allow for 10 or 20,
  more than Type I error (since Type I error is
  referred as society risk and hence more crucial
  to pharmaceutical company financially)
• Power of the study 1-Type II error 1-ß,
  usually use 80 or 90, the probability of
  detecting a difference in our study if there is
  one in the whole population

25
Further explanations of ingredient 7

• Other factors
• Calculated sample size meaning the number of
  subjects required during the analysis, not the
  number to start with for recruiting subjects, if
  you want to detect a certain effect size with a
  specific significance and power
• Study design
• Response rate data gathering affect the response
  rate, e.g. about 50 response rate by postal
  questionnaire
• Drop-out rate due to following subjects for a
  long period of time, e.g. cohort study, usually
  20 - 25
• Can increase the sample size by a suitable
  percentage to allow for these problems
• Beware for example, increase calculated sample
  size (n) by 25

26
Sample Size Calculations forTwo Independent
Groups
27
Formula for 2 independent groups

• From the 7 ingredients, there are 4 crucial
  factors involve in the actual sample size
  calculation
• Effect size (d) the size of the difference we
  want to be able to detect
• Variability (s) or ( ) the
  standard deviation of the continuous outcome or
  the estimation for the binary outcome
• Level of significance (a) the risk of a Type I
  error we will accept
• Power (1-ß) the risk of a Type II error we will
  accept

28
Formula for 2 independent groups

• We use these 4 factors to generalise a formula to
  calculate sample size for 2 groups with
  continuous or binary outcome
• The formula is
• where ? is the standardised effect size
• i.e. effect size / variability
• ? d/? for continuous outcome
• ? for binary outcome

29
What is z-score?
z-score

• Z-score is the number of standard deviations
  above/below the mean. z (x ?)/?

30
What is z(1-?/2) and z(1-?)?

• z(1-?/2) is a value from the Normal distribution
  relating to significance level
• If the level of significance is set to 5, then ?
  0.05
• For 2-sided test, z(1-?/2) z0.975 1.9600
• If the level of significance is set to 1, then ?
  0.01
• For 2-sided test, z(1-?/2) z0.995 2.5758
• z(1-?) is a value from the Normal distribution
  relating to power
• If ? is set to 10, then the power is 90, so 1-
  ? 0.90
• For 1-sided test, z(1-?) z0.90 1.2816
• If ? is set to 20, then the power is 80, so 1-
  ? 0.80
• For 1-sided test, z(1-?) z0.80 0.8416

31
Table of z-scores
z-score
32
The quick formula

• We can pre-calculate z(1-?/2) z(1-?)2, and
  call this k, using the relevant z-scores provided
  in the table from the previous slide for
  different combination of level of significance ?
  and power 1-?, the formula then becomes
• n (per group) 2k/?2
• Remember to multiply the calculated sample size
  (n) by 2 to allow for 2 groups!
• Always round up your final sample size

where ? is effect size / variability ? d/? for
continuous outcome ? for binary outcome
33
Even simpler!

• For 5 significance level and power of 80
• n 2 ? (2 ? 7.85)/?2
• ? 32/?2 (Total for 2 groups)
• For 1 significance level and power of 90
• n 2 ? (2 ? 11.68)/?2
• ? 60/?2 (Total for 2 groups)
• A sample size of n within two groups will have
  80 (and 90 respectively) power to detect the
  standardised effect size ?, and that the test
  will be performed at the 5 (and 1 respectively)
  significance level (two-sided). Note that
  ??/?, hence the required sample size increases
  as ? increases, or as ? decreases.

34
The 4 factors sample size

• Referring to the quick formula, we can predict
  the effect on the sample size if we
  increase/decrease the value of each of the 4
  factors
• If the level of significance (?) decrease, e.g.
  from 5 to 1
• sample size increase
• If Type II error rate (?) decrease, power (1- ?)
  increase, e.g. from 80 to 90
• sample size increase
• If the effect size (d) decrease, e.g. detecting a
  smaller difference between the 2 groups
• sample size increase
• If the variability (?) decrease, e.g. assuming
  the outcome measure has a smaller spread or less
  vary
• sample size decrease

35
Confident intervals sample size

• Recall when calculating confidence intervals,
  standard error (SE) is used
• SE , which involves n
• 95 CI point estimate
• If you want more confident on your point estimate
  
• i.e. narrower confidence intervals (e.g. 95 to
  99)
• ? increase sample size

36
with continuous outcome

• Example Differences between means
• In a trial to compare the effects of two oral
  contraceptives on blood pressure (over one year),
  it is anticipated that one drug will increase
  diastolic blood pressure by 3mmHg, and the other
  will not change it. The standard deviation (of
  the changes in blood pressure) in both groups is
  expected to be 10mmHg. How many patients are
  required for this difference to be significant at
  the 5 level (with 80 power)?
• women per group
• and a total of 350 women need to be recruited.

37
with binary outcome

• Example Difference between proportions
• In a randomised clinical trial, the placebo
  response is anticipated to be 25, and the active
  treatment response 65. How many patients are
  needed if a two-sided test at the 1 level is
  planned, and a power of 90 is required?
• so n47 per group and a total of 94 patients are
  needed for this study.

38
Fixing sample size

• The maximum sample size is often fixed by
  practical constraints.
• The research question could then become
• What power will I have to detect a (specified)
  clinically important difference?
• What is the smallest difference I will be able
  to detect?

39
Other methods

• Other sample sizing methods are available for
• equivalence studies, where the aim is to show
  that two groups do not differ by more than a
  specified amount, ?.
• matched case-control studies with a binary
  outcome (exposed or unexposed), which requires
  specification of the anticipated odds ratio and
  the proportion of pairs with differing outcomes.
• crossover studies with a binary outcome (success
  or failure) (essentially the same as above).

40
Useful resources

• Machin D, Campbell M, Fayers P, Pinol A. Sample
  size tables for clinical studies. 2nd Ed. 1997.
  Blackwell Science.
• Sampsize included with book.
• St. Georges, Uni. of London (Statistics Guide
  for Research Grant Applicants)
• http//www.sgul.ac.uk/depts/chs/chs_research/stat_
  guide/size.cfm
• PS Power and Sample Size
• http//biostat.mc.vanderbilt.edu/twiki/bin/view/Ma
  in/PowerSampleSize
• Java applets for power and sample size
• http//www.stat.uiowa.edu/rlenth/Power/
• web browser on your PC is required to be able to
  run Java applets (version 1.1 or higher) which
  can be downloaded from java.sun.com

41
Useful commercial resources

• nQuery Advisor (Statistical Solutions)
• http//www.statsol.ie/nquery/nquery.htm
• PASS (NCSS Inc. 7 days free trial
• http//www.ncss.com/pass.html
• SamplePower (SPSS Inc. 10 days free trial)
• http//www.spss.com/spower/
• Stata help sampsi
• http//www.stata.com/
• http//www.stata.com/help.cgi?sampsi

42
Free software PS

• http//biostat.mc.vanderbilt.edu/twiki/bin/view/Ma
  in/PowerSampleSize

43
Two groups with continuous outcomes using PS
44
Two groups with continuous outcomes using PS

• Example Differences between means
• In a trial to compare the effects of two oral
  contraceptives on blood pressure (over one year),
  it is anticipated that one drug will increase
  diastolic blood pressure by 3mmHg, and the other
  will not change it. The standard deviation (of
  the changes in blood pressure) in both groups is
  expected to be 10mmHg. How many patients are
  required for this difference to be significant at
  the 5 level (with 80 power)?
• Recall from our calculation with the quick
  formula, we would need 175 women per group and
  with a total of 350 women to be recruited.

45
Two groups with continuous outcomes using PS
46
Two groups with binary outcomes using PS
47
Two groups with binary outcomes using PS

• Example Difference between proportions
• In a randomised clinical trial, the placebo
  response is anticipated to be 25, and the active
  treatment response 65. How many patients are
  needed if a two-sided test at the 1 level is
  planned, and a power of 90 is required?
• Recall from our calculation with the quick
  formula, we would need 47 patients per group and
  with a total of 94 patients to be recruited.

48
Two groups with binary outcomes using PS
49
Summary

• You should now be able to
• understand why determination of sample size is
  important
• appreciate some statistical concepts
• be aware of considerations needed to perform
  sample size calculations
• be able to perform a sample size calculation to a
  precision with continuous or binary outcome
• be able to perform a sample size calculation
  based on two independent groups with continuous
  or binary outcome
• be appreciate with the useful resources relating
  to this topic
• be able to use PS to assist calculating sample
  size

50
References

• Kirkwood B.R. Sterne J. A.C. Essential Medical
  Statistics, 2nd Edition. Oxford Blackwell 2004
  (Chapter 4 8, 35)
• Bland M. An Introduction to Medical Statistics,
  3rd Edition. Oxford Oxford University Press
  2000. (Chapters 8, 9 18)
• Altman D.G. Practical Statistics for Medical
  Research. London Chapman Hall 1999.
  (Chapters 8 15)
• Machin D., Campbell M., Fayers P. Pinol A.
  Sample Size Tables for Clinical Studies, 2nd
  Editio. Oxford Blackwell Science 1997.

51
Exercises
Brian Yuen Public Health Sciences Medical
Statistics
52
Exercise 1

• Suppose we are trying to estimate the prevalence
  of a disease in a country, which we suspect the
  prevalence to be 5, and would like to estimate
  it to within 1 of the true value (with 95
  confidence). How many patients are required?

53
Exercise 2

• A psychologist wishes to test the IQ of a certain
  population. His null hypothesis is that the mean
  IQ is 100, and he wishes to be able to detect a
  fairly small difference, with standard deviation
  of 0.2 and 95 confidence interval width of
  0.028, so that if he gets a non-significant
  result from his analysis, he can be sure the mean
  IQ from this population lies very close to 100.
  How many subjects should he recruit?

54
Exercise 3

• An investigator compares change in blood pressure
  due to placebo with that due to a drug. If
  investigator is looking for difference between
  groups of 5 mmHg, and between-subject SD (?) is
  10 mmHg. Assuming a two-sided test at the 5
  level (?0.05), and a power of 90 (1-?0.9), how
  many patients should be recruited?

55
Exercise 4

• IgA nephropathy (IgAN) is a world-wide disease
  and the cause of end-stage renal failure (ESRF)
  in 15 to 20 of patients within 10 years. No
  specific treatment has yet been established but
  many approaches have been investigated.
• A new two-arm parallel group RCT is going to
  randomise patients into either the
  immunosuppressive agent (steroid) group or the
  placebo group. Assume the placebo group will have
  15 of patents end up having ESRF within 10
  years, how many patients are required so that we
  can anticipate a reduction of ESRF by 5 (15 to
  10) in the steroid group with 5 significance
  level and 90 power?
• The Cochrane Database of Systematic Reviews
  2005, Issue 2. Samuels JA, Strippoli GFM, Craig
  JC, Schena FP, Molony DA. Immunosuppressive
  agents for treating IgA nephropathy.
• The Cochrane Database of Systematic Reviews 2003,
  Issue 4. Art. No. CD003965. DOI
  10.1002/14651858.CD003965.

56
Exercise 5

• Use PS to calculate the sample size for the
  following exercise.
• In a randomised clinical trial, we are interested
  in detecting a difference of a least 15 between
  two proportions. We are expecting the two success
  proportions to be equal to 60 and 75. How many
  patients are needed if we are going to perform a
  two-sided test at the 5 level with a power of
  80?

57
Exercise 6

• Use PS to calculate the sample size for the
  following exercise.
• We are developing a study to investigate the
  relationship of constant exposure to stress
  (within a month) and blood pressure level for men
  aged 25-34. From a pilot study, it was
  determined that the mean blood pressure of the
  group which were constantly under stress were
  132.86, while that of the group which were not
  constantly under stress were found to be 127.44.
  The common standard deviation is 16.79. How many
  men are required for this larger scale study,
  given the level of significance is set to 5 with
  90 power?

58
Homework
Brian Yuen Public Health Sciences Medical
Statistics
59
Exercise 7

• A survey is being planned to estimate the
  prevalence of secondary infertility amongst
  couples aged 20-45. The investigators expect the
  prevalence to be 10, and would like to estimate
  it to within 5 of the true value (with 90
  confidence). How many couples are required?

60
Exercise 8

• Suppose we want to sample a stable process that
  deposits a 500 Angstrom film on a semiconductor
  wafer in order to determine the process mean so
  that we can set up a control chart on the
  process. We want to estimate the mean within 10
  Angstroms of the true mean with 95 confidence.
  Our initial guess regarding the variation in the
  process is that one standard deviation is about
  20 Angstroms. How many sample do we need to take?

61
Exercise 9

• A previous small study was investigating the
  total cost of 10 specific items of fruits and
  vegetables which were sold in 3 supermarkets
  compare to that were sold in 3 local corner
  shops.
• On average, the total cost of 10 specific items
  of fruits and vegetables which were sold in
  supermarkets was 5.50. It was suggested that
  the total cost of those fruits and vegetables
  which were sold in the supermarkets was 1.6 times
  cheaper than those were sold in the local corner
  shops, with a common standard deviation of 1.65.
  
• Use the suggested formula and PS to find out how
  many supermarkets and local corner shops are
  required (assume equal group), if we are going to
  conduct a similar study to investigate the
  difference in cost of this size, with 90 power
  and using a 1 significance level?

62
Exercise 10

• In a pilot study, it was shown that adult
  Cypriots who had followed a traditional Cypriot
  Mediterranean type of diet had a 27 risk of
  hypertension, 7 lower than those who had
  followed a Western diet.
• Use the suggested formula and PS to find out the
  number of subjects required (assume equal group),
  if we conduct a similar but larger scale study,
  to evaluate the difference in the risk of
  hypertension of this magnitude with 80 power and
  using a 5 significance level?
• A postal questionnaire will be sent to the
  target population, assuming a high response rate
  of 70, how many questionnaires should be sent
  out?

63
Solutions
Brian Yuen Public Health Sciences Medical
Statistics
64
Exercise 1

• Solution
• Problem type Precision with binary outcome
• Thus, require 1825 patients in the study.

65
Exercise 2

• Solution
• Problem type Precision with continuous outcome
• Thus, he should recruit 196 subjects.

66
Exercise 3

• Solution
• Problem type Two independent groups with
  continuous outcome
• Thus, require 85 patients per treatment, i.e. a
  total of 170 patients need to be recruited.

67
Exercise 4

• Solution
• Problem type Two independent groups with binary
  outcome
• Therefore, 920 patients per group are needed and
  1840 patients in total for the new trial.

68
Exercise 5

• Solution
• Problem type Two independent groups with binary
  outcome
• ? 0.05
• power 0.80
• p0 0.6
• p1 0.75
• m 1
• n (per group) 152

69
Exercise 6

• Solution
• Problem type Two independent groups with
  continuous outcome
• ? 0.05
• power 0.90
• ? 5.42
• ? 16.79
• m 1
• n (per group) 203

70
Exercise 7

• Solution
• Problem type Precision with binary outcome
• Thus, require 97 patients in the study.

71
Exercise 8

• Solution
• Problem type Precision with continuous outcome
• Therefore, we need to take at least 16 samples
  from this process.

72
Exercise 9

• Solution
• Problem type Two independent groups with
  continuous outcome
• The average total cost of fruits and vegetables
• in supermarkets is 5.50
• in local corner shops is 5.50?1.6 8.8
• the difference in total cost 3.3
• common standard deviation 1.65
• To detect such difference with 90 power and
  using a 1 significance level
• n (per group) (2?14.88) / (3.3/1.65)2
• n (per group) 7.44 ? 8
• Therefore we will require 8 supermarkets and 8
  local corner shops.

73
Exercise 9

• Solution
• Problem type Two independent groups with
  continuous outcome
• ? 0.01
• power 0.90
• ? 3.3
• ? 1.65
• m 1
• n (per group) 9

74
Exercise 10

• Solution
• Problem type Two independent groups with binary
  outcome
• Proportion who follow
• a traditional Cypriot Mediterranean diet is 27,
  i.e. pA 27
• a Western diet is 34, i.e. pB 277 34
• the difference in proportion 7
• (2734)/2 30.5
• estimation of standard deviation
  ?30.5(100-30.5) 46.04
• To detect such difference with 80 power and
  using a 5 significance level
• n (per group) (2?7.85) / (7/46.04)2
• n (per group) 679.16 ? 680
• Therefore, in total, we will require 1360
  subjects for both groups.
• With 70 response rate, we should send out
  1360/0.7 1943 questionnaires.

75
Exercise 10

• Solution
• Problem type Two independent groups with binary
  outcome
• ? 0.05
• power 0.80
• p0 0.27
• p1 0.34
• m 1
• n (per group) 678