ExptDesign 复旦大学生物统计学课件

Restraints on scale of experiments?
Why experimental scale matters?
• balance between costs of a large scale and gains in experimental efficiency
• large scale may cause un-control of experimental errors, for example
xy (md)
t=
0 whenever m=d
sxy
sxy
Thus, with randomization, background of experimental treatments can be homogenized and thus systematic errors can be effectively removed!
(ii). influences of replication in an experimental design (explained by the cat food example)
• increasing replicate number increases accuracy of mean estimate because
s
2 x
s x2
/
n
• increasing replication increases d.f. of t test statistic (2n-2) and in turn decreases the threshold to disprove Ho, for example
df P 0.05
❖ The design II:
• Patients with gastric ulcers are divided into two groups. • Patients in group I are given balloons containing refrigerated
liquid and those in group II are given balloons containing nothing (placebo). None of the patients are told which balloon are given!
❖ need of randomization to homogenize background of experimental treatment(s) and to remove systematic errors (R)
❖ need of replication to increase experimental power (R)
❖ Data:
M group: x1, x2, ….,xn D group: y1, y2, ….,yn
x nxi andsx 2 nxix2/(n1)
i 1
i 1
n
yyi
andsy 2 nyiy2/(n1)
i 1
i 1
However, if the cats are randomly grouped
• How many replicates are needed to detect a genuine difference in preference between the two cat foods with a given confidence of 1-a?
You should know (a). x = E(x), y = E(y) and (b). s2 = Var(ei)
源自文库
Blocks (B)
4
VxB
4
Replicates 10
Total
19
SS 174.05 429.50
70.0 171.0 844.5
MS 174.05 107.38
17.50 17.10
FP 10.2 <0.01 6.28 <0.01 1.02 ns
Effect of blocking
❖ The replicate error within varieties is very much reduced and so the ‘Variety’ effects may be detected with larger chance (experimental power)
Solution: divide experiment into several sub-experiments “blocks” and dissect variation of block component from total variation
Low
Gradient in fertility of soil High
s
2 x
s
2 y
4
a t2 n 2 sx x/ y n 2 7 4 .5 / n 2 50 .6 n 2 5 td f 2 n 2()
The above inequality equation can be solved for any given a. For example, when a =0.05, n 16
❖ Increases Power of test
❖ Called a Randomised Complete Block Expt. (RCB).
Types of ‘block’
❖ Shelves in incubator ❖ Different occasions or sites ❖ Different operators
• Data based on this experiment showed 34% improved with cold treatment but 34% improvement was also observed with placebo!
An appropriate control is needed to make a sensible comparison!
Randomized blocks
Block at right angles to gradient if known
Low
Gradient in fertility of soil
High
Yes
No
Design of randomised blocks
❖ Each block would contain the same number of subjects (plants).
Example I. Does gastric freezing relieve gastric ulcers?
❖ The design I: • Patients with gastric ulcers swallow deflated balloon into which a refrigerated liquid was introduced • Data published in Journal of American Medical Association showed that patients so treated had less stomach acid and pain than untreated patients
x i m i f i s e i a n d y i d i f i s e i
Where i = 1 if cat i is fat and 0 otherwise; i = 1 if cat i is slim and 0 otherwise and
Pr{i = 1} = Pr{i = 0} = Pr{i = 1} = Pr{i = 0} =1/2, i.e. fat or slim cats are equally likely given “Mousey” or “Doggy”, then the expected value of the t statistic is now
One Way ANOVA
Item df
SS
MS
F
Prob.
Between 1 varieties Within 18 varieties
Total 19
174.05 174.05 4.67 0.05-0.01 670.50 37.25 844.55
ANOVA table
Item
df
Varieties (V) 1
A field trial to test yield performance of a set of varieties
Low
Gradient in fertility of soil
High
Small area has a homogeneous environment so small s2
Large area has a heterogeneous environment so s2 large
❖ Each block would contain all the treatments. ❖ The plants and treatments in each block would
be separately randomised. ❖ So, each block is a complete but small version
Example II. Do cats prefer Mousey (M) to Doggy (D)?
❖ The design:
• two cat foods: Mousey (M) & Doggy (D) • to measure amount of food eaten in grams when offered to a cat. • to give M and D to n cats each. So we need 2n cats (n 2). • we have to group the 2n cats randomly into the two feeding groups
The “1C + 3Rs” principles of experimental designs
❖ need of “positive” and/or “negative” control to contrast genuine effects of experimental treatments (C)
1
12.706
2
4.303
4
2.776
6
2.447
8
2.306
10
2.228
20
2.086
30
2.042
1.960
P 0.01 63.657 9.925 4.604 3.707 3.355 3.169 2.845 2.750 2.576
A modest number of replicates lowers the “hurdle” for the test statistic to jump over to reach significance
They can be obtained from published data or estimated from a trial.
For example, one plans to test 10% difference between the two cat food
with y
= 25 and
Experimental Design
Objective: Planning experiments to allow collection of valid data & powerful statistical analysis of the data
Professor Zewei Luo, Office: S224, Tel: 45404 E-mail: z.luo@bham.ac.uk
of the experiment. ❖ The environmental effect of the gradient is
therefore removed and s2 minimised.
The example
Gradient
Completely randomised
5 randomised blocks