サンプルサイズの問題
Sample size issue●Referenes, web sites●Reference in English"Chapter 14. Sample size issues." in Machin D, Campbell
MJ, Walters SJ (2007) Medical Statistics, 4th ed., Wiley,pp. 261-275.
– "Chapter 4. Comparing groups with p values: Reporting
hyothesis tests." in Lang TA, Secic M (2006) How to reportstatistics in medicine: Annotated guidelines for authors,editors, and reviewers. 2nd ed., American College ofPhysicians., pp.45-60.
– "Chapter 1. Research design" in Peacock JL, Peacock PJ(2011) Oxford handbook of medical statistics. Oxford Univ.
Press, pp.1-73 (especially 56-73).
–●
Minato Nakazawa, Ph.D.
<minato-nakazawa@people.kobe-u.ac.jp>●●25 April 2018●

※ この資料は英語なので，日本語がわかる方
は， pdf のテキストの第 4 章を読みながら話を聞いてください●●

Textbook in Japanese

●

–

●

–永田靖 (2003) サンプルサイズの決め方 . 朝倉書店
新谷歩 (2011) 今日から使える医療統計学講座【 Lesson 3 】
サンプルサイズとパワー計算 . 週刊医学界新聞， 2937 号
http://www.igaku-shoin.co.jp/paperDetail.do?id=PA02937_06

How should this student do?

New England Journal of Medicine

In the textbook of medical statistics...

This study was conducted as hypothesis testing.
It's well known that the large sample increases
statistical power, before conducting study,
determining the enough sample size is possible.
●
After getting the data, only possible way is writing
excuse.
●A study that is too small may be unethical, since it is
not powerful enough to demonstrate a worthwhile
correlation or difference.
●
Similarly, a study that is too large may also be
unethical since one may be giving people a treatmentthat could already have been proven to be inferior.
●Many journals now have checklists that include aquestion on whether the process of determining
sample size is included in the method section (and tobe reassured that it was carried out before the study
and not in retrospect).
●
The statistical guidelines for the British Medical Journalin Altman et al. (2000) state that: `Authors should●●– Too rare disease.
– Limitation of funding.
– As a habit in the specific research theme, small

sample size has been commonly accepted– etc.
●

Fictious example
Due to the limited number of patients within the study
period as graduate student, a student could obtain datafrom only 10 patients (with 10-30 healthy controls), to test
the null hypothesis X, that is a factor Y has no relation witha result Z (usually suffering from that disease).
●And, as the result of statistical test, the null hypothesis X
was not rejected at a significance level of 5%.
●
Such non-significant results should be submitted to thejournal, entitled as "Lack of association …", to avoid
publication bias.
●
However, the reviewer of the journal may criticize the nonsignificance due to too low statistical power by insufficient
sample size, then the paper will be rejected.
●
The graduate student may ask help for statistician, but it's
too late (at best, tell the way of excuse).
●
Such tragedy is frequently seen in the graduate students.
●

If the student is lucky and the study theme is quite
important, the paper may be accepted as researchnote or short report, instead of original article.

include information on … the number of subjects studied
and why that number of subjects was used.'

Excuses for no calculation of sample sizes

In commonly adopted guidelines...
●●●

STROBE Statement for Observational Study
https://strobe-statement.org/fileadmin/Strobe/uploads/checklists/
STROBE_checklist_v4_combined.pdf"10 Explain how the study size arrive at"
CONSORT Statement for Clinical Trials
http://www.consort-statement.org/"7a How sample size wasdetermined""7b When applicable, explanationof any interim analyses andstopping guidelines"
See, EQUATOR
http://www.equator-network.org/
which gives summary informationof many guidelines

A cynic once said that sample size calculations are aguess masquerading as mathematics. To perform such acalculation we often need information on factors such as
the standard deviation of the outcome which may not beavailable. Moreover the calculations are quite sensitive
to some of these assumptions.
●
Any study, whatever the size, contributes information, andtherefore could be worthwhile and several small studies,pooled together in a meta-analysis are moregeneralizable than one big study.
●
Often, the size of studies is determined by practicalities,
such as the number of available patients, resources, time
and the level of finance available.
●Studies, including clinical trials, often have severaloutcomes, such as benefit and adverse events, each ofwhich will require a different sample size.

Where the sample size calculation isunnecessary●●●

Qualitative studies / Case report
Small survey / pilot study– In descriptive study, usually previous information

about the measures is unavailable, so that thesample size calculation is impossible.
– Rules of thumb: at least 12 individuals in eachgroup●

List the main cross tabulations that will be needed toensure that total numbers will give adequate numbers in
the individual tables cells.

In exploratory studies ...
Two kinds of study●

Testing the null-hypothesis always requires the sample
size calculation before the study (already explained).
– Exploring the hidden hypothesis or describing estimateswith 95% confidence intervals may not always require thesample size calculation, but power analysis (to evaluate
sampling adequacy) after the study is possible.
–●

Values are obtained from previous studies in advance.
–

●
●
●

Necessary number of samples (n) is obtained by:
n = qnorm(1-alpha/2)^2 * 4 * SD^2 / d^2– (e.g.) Suppose we wish to estimate mean systolic bloodpressure in a patient group with a 10mmHg-wide (or5mmHg-wide) 95% confidence interval. Previous worksuggested using a standard deviation of 11.4.
n = 1.96^2 * 4 * 11.4^2 / 10^2 = 19.97... = 20
n = 1.96^2 * 4 * 11.4^2 / 5^2 = 79.88... = 80– Doubling the precision needs quadrupling the sample size.

Prevalence estimates from small samples will be imprecise
and may be misleading. For example, when we wish to get
the prevalence of a condition for which studies in othersettings have reported a prevalence of 10%. A small
sample of, say, 20 people, would be insufficient to produce
a reliable estimate since only 2 would be expected to havethe condition and ±1 would change the estimate by 5%.
– Sample size calculation determine the number of subjects
needed to give a sufficiently narrow confidence intervals.
–●

Example of sample size calculation inhypothesis testing

What kind of information is needed?
Method of statistical test (including null-hypothesis)
Type I error (alpha error: probability to reject the true nullhypothesis, in other words, false positive)– Type II error (beta error: probability to fail to reject the false nullhypothesis / false negative) = 1 – statistical power– Expected values from previous studies– Minimum differences of clinical importance●–4 degree difference has clinical importance.
Let alpha error 0.05 and statistical power 90%.
– SD of increase of elbow flexion is assumed as 5 degree.
–●Compare means by t-test:
n= 2*(zα-z1-β)2*SD2/d2 + zα2/4– Compare proportions by χ2 test:
n= (zα/2+z1-β)2*{p1(1-p1)+p2(1-p2)}/(p1-p2)2●●

http://biostat.mc.vanderbilt.edu/twiki/bin/view/Main/PowerSampleSize

Free Software
Survival (logrank test), ttest, Regression1,Regression2,Dichotomous (chisq-test),
Mantel-Haenszel areincluded.
●
An example of description
is given as text.
●
Ratio of two groups canbe specified as m●●n= 2*(zα-z1-β)2*SD2/d2 + zα2/4= 2*(-1.64-1.28)^2*5^2/4^2+(-1.64)^2/4= 27.3174 ~ 27

Based on this, 26 treated patients and 25 control patientswere measured. They showed increase in elbow flexsion by
16±4.5 and 6.5±3.4, respectively. The mean difference was9.5 (95%CI was 7.23 to 11.73), t-test resulted in t=8.43, df=49,
p<0.001.
●
Typical description of this design and statistical results should
be written as follows:●

Usually special softwares (nQuery, PASS, PS) or general
statistics softwares (SAS, SPSS, STATA, EZR, R, etc.) will beapplied.

PS: Power and Sample
Size Calculator

Calculation by the previous equation–Equations are quite different by statistical tests (and bytextbooks, because all of those are of approximation)

How to use “PS” software

Suppose we wish to compare the mean increased degree ofelbow flexion between stimulated and control patients.
––●

Expected population proportion (p)Desired width of confidence interval (d)– Confidence level (1-alpha)–Approximate equation to estimate the number of subjects
needed [ qnorm(1-alpha/2) means z1-α/2 ; ^2 = squared ]:n = qnorm(1-alpha/2)^2*4*p*(1-p) / d^2●(e.g.) Suppose we wish to estimate the prevalence ofasthma in an adult population with the width of the 95%
confidence interval 0.10, an accuaray of ±0.05. An
estimate of the population prevalence of asthma is 10%.
●––p = 0.10, d = 0.10, alpha = 0.05
n = qnorm(0.975)^2*4*0.1*0.9 / 0.1^2 = 1.96^2*36 = 138

Typical description
We designed the study to have 90% power to detect a 4degree difference between the groups in the increased
range of elbow flexion. Alpha was set at 0.05. (Methodssection)●Patients receiving electrical stimulation (n=26) increased
their range of elbow flexion by a mean of 16 degrees witha standard deviation of 4.5, whereas patients in thecontrol group (n=25) increased their range of flexion by amean of only 6.5 degrees with a standard deviation of
3.4. This 9.5-degree difference between means was
statistically significant (95%CI = 7.23 to 11.73 degrees;
two-tailed Student's t test, t=8.43; df=49; p<0.001).
(Results section)●–●

Required information from previous studies and study
purpose to estimate proportion–

Estimating proportions will be given in the next slide.

Principles of hypothesis testing●The standard deviation (SD) of the measure being estimated
The desired width of the confidence interval (d)The confidence level (usually 90, 95, or 99 %; 1-alpha)–

In the exploratory or descriptive studies●

When we would like to estimate a mean, the following 3
values are needed.

How to use “EZR” software●

EZR on Rcmdr is developed by Jichi Medical School●

http://www.jichi.ac.jp/saitama-sct/SaitamaHP.files/statmed.html––(Source: Lang and Secic, 2006, pp.47)
Here the expected standard deviation 5 nor applied
equation is not clearly written (both seems implicit).

Using R console to calculate sample size
●Free software; latest version is 1.35 on 1 March 2017.
●
Installation is very easy. Just click the downloaded executableinstaller. Messages are given in Japanese.
●
Usage is also easy. Introductory textbooks are available inJapanese.

> power.t.test(delta=4, sd=5, sig.level=0.05, power=0.9)●

Two-sample t test power calculationn = 33.82555
delta = 4
sd = 5sig.level = 0.05
power = 0.9
alternative = two.sided

1.0

NOTE: n is number in *each* group

0.8

0.6
Power

●(cont'd)Example of exploratory study

0.40.2●0.0102030N140

The result is 34 (similar to PS)