実験計画法(May 1, 2013)






実験計画の起源に関わる，ある伝説Fisherの３原則

実験的研究は，どんなものであれ，注意深くデザインされねばならない。

実験計画法は，R.A. Fisherがロザムステッドで行った農学研究に始まる。



保健医療分野では，この種のデザインは，毒性試験や臨床試験で用量反応関係を分析するために必須。

繰り返し（Replication）:各処理について最低２回以上の繰り返し測定が必要。

ケンブリッジのある晴れた日，多くの教授がアフタヌーンティーを楽しんでいた。と，ある婦人が，自分はミルクティーを飲めば，それがミルクが先か紅茶が先のどちらで淹れたものか判定できると主張した。

無作為化（Randomization）:実験の順序や区画（農業試験の場合）は無作為に割り付けねばならない。

教授たちの間で，彼女の主張を廻って大激論勃発。

局所コントロール（＝ブロック化：blocking）:大規模な実験の場合，サンプル全体の無作為化は不適切であり，代わりにいくつかのやり方で局所のブロックを作り，各ブロック内で無作為割り付けをすることで，ブロック間変動として，偏りを除去することができる。

もちろん，ヒトを対象にした研究は，実施前に倫理審査を通らねばならず，倫理審査に提出する書類には，適切なサンプルサイズの設定を含む，適切な研究デザインが記述されねばならない。

そこでR.A. Fisher曰く「実験したらどうだい？」

この能力は，ミルクを先に入れて作ったミルクティーと紅茶を先に淹れて作ったミルクティーを何杯か用意して，無作為な順番で飲んで貰えば検定可能。



能力の判定条件を考える必要あり。何回必要？

（参考）ミルクと紅茶の順番は本当に味に影響する？
George Orwell 「完璧な紅茶を淹れる11の法則」
10: ミルクを紅茶に加えるのだ。逆ではいけない。
英国の王立化学会がGeorge Orwellの生誕100年を記念するパーティを開いたとき，Dr. Andrew Stapley (2003)曰く冷えたミルクをカップの底に入れておいてから，熱い紅茶を注ぐのが良い。こうするとミルクが紅茶を冷ますことができる。逆だと熱い紅茶がミルクの温度を急に上げるのでミルクの風味が損なわれる。


情報源: BBC News
(http://news.bbc.co.uk/2/hi/uk/3016342.stm)

1. Use tea from India or Ceylon (Sri Lanka), not China



2. Use a teapot, preferably ceramic

3. Warm the pot over direct heat

4. Tea should be strong - six spoons of leaves per 1 litre

5. Let the leaves move around the pot - no bags or strainers

6. Take the pot to the boiling kettle

7. Stir or shake the pot

8. Drink out of a tall, mug-shaped tea cup9. Don't add creamy milk

10. Add milk to the tea, not vice versa



11. No sugar!

白黒付けるには何杯飲めばいい？


Correct judge of 1/1 may occur at 50%.

Correct judges of 2/2 may occur at 25%Correct judges of 3/3 may occur at 12.5%

Correct judges of 4/4 may occur at 6.25%

Correct judges of 5/5 may occur at 3.125％



Testing this hypothesis requires at least 5 cups.



本当はどちらが先だとより美味しいのか，試してみた日本人ブロガーがいた。130ccの紅茶と30ccのタカナシ低温殺菌牛乳を使用した（高温殺菌とかロングライフのミルクでは違いが分からないらしい）。
この人の主観的判断では，「ミルクが先」が美味。
http://blog.livedoor.jp/teatime312/archives/cat_123365.html

有名な実験計画デザイン


本当は判別能力が無いのに偶然５回連続で正解する，3.125% という値は，偶然で片付けるには稀すぎる。通常，
この判定基準は5%を切るかどうかにおく。これが有意水準（Fisher流）である。帰無仮説「彼女は判定能力をもっていない」が有意水準5%で棄却される。

完璧な一杯の紅茶の真実（続き）

George Orwell's 11 rulesOne group pretest-posttest design: paired t-test.

One group pretest-posttest designCompletely randomized design: t-test/ANOVA
for quantitive data, chi-square test for proportion
Randomized block design: similar to completelyrandomized design / considering block's effectCompare serum cortisol levels before and after thesurgery in rheumatoid arthritis patients
Compare the depression score before and after thesound-therapy in depression patients
Compare the simple calculation test score before andafter drinking coffee.

Latin-square: usually ANOVA
Crossover design: Matched (paired) analysis ofvariance (within-subject difference will be zero ornot, adjusted by the order of treatment)

The design enables a researcher to compute acontrast between means in which the pretestand posttest means are measured with thesame precision.

The statisical test is usually paired t-test.

Null hypothesis: the mean of difference is zero.

Example of paired t-test






Pararell group design(=Completely randomized design)

Exercise

We can use “survey” dataframe of MASS package in R(EZR), whereas it is the result of cross-sectional study.

In EZR, select [File] and [Read data included in package],then select [MASS] and [survey].

The “survey” contains the responses of 237 students at
the University of Adelaide to a number of questions(Venables and Ripley, 1999). Variables include the span
(distance from tip of thumb to tip of little finger of spreadhand) of writing hand as [Wr.Hnd] and that of non-writing
hand as [NW.Hnd].



Select [Statistical Analysis], [continuous variables], then[paired t-test]. Select [NW.Hnd] at left panel and [Wr.Hnd] atright panel and click [OK]. That's all.

Compare the results of simple
calculation test before and after
drinking coffee.
In EZR, at first, making data:
select [File], [New data],then enter the data as
right screen-capture.
Conducting paired t-test
can be done in similar manner as
NW.Hnd-Wr.Hnd









Dropouts sometimes occur. 50,000 residents who may potentially
The design is usuallyshown as diagram (right).
In this diagram, thequantitive data of thesubjects can be comparedby one-way ANOVA ,proportion by chi-square
test.
Unbalanced sample size
may reduce the statisticalpower.



If the sample size for each treatment is 15 and the kind oftreatment is 3, randomized block design randomly selectone of 6 possible blocks ({A, B, C}, {A, C, B}, {B, C, A}, {B,A, C}, {C, A, B}, {C, B, A}) 15 times. By doing so, if thestudy may suspend in the middle, the sample size
difference is at most 1 among groups. Description andanalysis can be similar to complete randomization, but theanalysis considering blocks' effect is also possible.

45 samples are supposed
to attend, but 42 in fact.

Treat B
13

Treat C
141113
15
completed completed completed
protocolprotocolprotocolLatin-square design（ラテン方格法）
Due to incompleted study, completely randomized design
may lead to unbalanced sample sizes among groups.

go to the target hospital

Treat A15

When the experiment have one treatment (A)with p≧2 levels, 2 nuisance variables (B, C)
each with p levels, this design is useful. The
name is originated from ancient puzzle.
Assume p is 3. Latin square isshown as right. Group 1 gets b1
the combination of treatmentb2
a1b1c1 for n1 individuals.
Following n2 individuals getb3
treatment a1b2c3 as group 2.

However, now we can use computer software. If we want toassign 45 subjects into 3 treatments, typematrix(sample(1:45, 45, replace=FALSE),3) in R.
> matrix(sample(1:45, 45, replace=FALSE),3) -> x
> x[1,][1] 34 13 37 17 3 11 4 23 18 39 42 40 36 8 25
> t(apply(x,1,sort))[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15][1,]
348 1113
17 18
2325
34
36
37
39
40
42[2,]
167 101415 161926
29
31
33
41
43
44[3,]
259 122021 222427283032353845[t = -2.862, df = 9, p-value =0.01872] mean significant
difference.

Factorial design

There are several randomization methods. Fleiss JL (1986)“The design and analysis of clinical experiments” recommends
to use “random permutation tables” instead of “random numbertables”.

Randomized block(s) design（乱塊法）

How to describe?
Very simple. The subjects who signed informed consent arecompletely randomly (not haphazardly) assigned to one of theseveral treatment (exposure).

c1c2c3a1a2a3

a2

a3

a1

a3

a1

a2

By doing so, the effects of B and C on theoutcome measure can be controlled in ANOVA.

Another method to keep size balance is “Minimizationdesign”. It minimize the sample size difference at eachtime of sampling.







Example of 2x2 factorial design
McMaster et al. (1985): a randomized trial toevaluate breast self-examination teaching materials
as leaflets or tape/slides.
The treatment groups were designed as fourpararell groups as
1. No leaflets nor tape/slides given (control)
2. Leaflets displayed
3. Tape/slides program
4. Both given
The effect of teaching can be evaluated using
ANOVA: two kinds of materials can be evaluated.

Crossover design（クロスオーバー法）


Subjects will be seriallytreated by 2 kinds ofintervention with properinterval (wash-out period toavoid carry-over) in different
order.
(Example) Hilman BC et al.
“Intracutaneous Immune
Serum Globulin Therapy inAllergic Children.”, JAMA.
1969; 207(5): 902-906.

Types of outcome measureSuperiority trials: The effect of new treatment issignificantly better than control or notEquivalence trials: The effect of new treatment
is similar to control or not
Non-inferiority trials: Special case ofequivalence trials.