Stat 10x

1
Stat 10x

• J. Chang
• Tuesday, 10/3/00

2
Today Producing DataSampling and Experimental
Design

• 3 Principles of Experimental Design
• Simple random samples
• Bias, variance
• Stratified sampling and blocking

Moore and McCabe Chapter 3.
3
Observation versus experiment

• Both attempt to study relationship between an
  explanatory variable and a response variable
• Experiment deliberately impose treatments on
  individuals to observe their responses.
• Observational study observe and measure what
  participants do naturally

4
An example experiment

• Wangensteen (1958) Gastric freezing. Experiment
  reported in JAMA treatment reduced ulcer pain.
  24 patients all said they felt better.
  Technique widely used for several years. OK?
• Several years later a different, larger study
  with a control group. Results
• 34 in treatment group improved.
• 38 in control group improved.
• Salk vaccine trial

5
Principle 1 Control or Comparison

• Comparison of different treatments.
• Want different treatment groups to be as similar
  as possible -- except for the treatments applied.
• Control effects of environmental or outside
  variables.
• Outside influences act the same on the different
  treatment groups. (E.g. placebo effect)

6
Bias

• How to assign experimental units to treatments?
• E.g.
• in comparing two medical treatments dont want to
  assign one treatment to sicker patients
• comparing seed varieties dont plant one in more
  fertile ground
• A study is biased if it systematically favors
  certain outcomes.
• How to avoid bias? Elaborate balancing?

7
Principle 2 Randomization

• Assign treatments randomly.
• Fair -- doesnt give an treatment a systematic
  advantage.
• But randomization balances out well only in the
  long run. So

8
Principle 3 Replication, or Sample size
Use sample sizes big enough so that we will be
able to distinguish a real effect from random
luck.
9
Its hard to be random
0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 0
1 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0
1 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 0 0 1
1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0
1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0
0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 0
1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 0 1
1 0 0 1 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1
0 1 0 1 1 0 0
10
Its hard to be random
not very creative
0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 0
1 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0
1 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 0 0 1
1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0
1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0
0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 0
1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 0 1
1 0 0 1 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1
0 1 0 1 1 0 0
11
Its hard to be random
not very creative
0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 0
1 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0
1 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 0 0 1
1 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0
1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0
0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 0
1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 0 1
1 0 0 1 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1
0 1 0 1 1 0 0
getting tired
12
Simple random samples

• Def A simple random sample of size n is a set of
  n individuals from a population chosen in such a
  way that each set of n individuals has an equal
  chance to be the sample actually selected.

Abbreviate simple random sample ? SRS
13
How to randomize
A natural way label individuals with 0, 1, 2,,
9.Take individual 1, then 9, then 2, then 3.
What if we had 25 individuals and wanted a SRS
of size 4?
19, 22, 05, 13
14
Blobs
What is the average area?
E.g. throwing darts leads to size-biased sampling.
15
Buses
Suppose average time between bus arrivals at a
stop is 20 minutes. You arrive at a random time.
What is your average waiting time until the next
bus?
10 minutes?
No -- in general its more.
Analogous to blobs
16
Sampling distributions
Say we want to estimate parameter p Probvote
for Bush
Here p 0.5. Pretend we dont know this.
17
Sampling distributions (cont.)
List possible SRSs and the corresponding
estimates.
Ind Vote 1 Bush2 Bush3 Gore4
Gore
18
Sampling distrib of p-hat from SRSs of size 2
19
Bias and variability of an estimator
E.g. recall true value was p 0.5. Sampling
distrib
Unbiased Mean of sampling distrib 0.5 true
value
Variability SD of sampling distrib ? 0.3
20
How about with SRSs of size 1?
Ind Vote 1 Bush2 Bush3 Gore4
Gore
21
Bias? Variability?
n 2
n 1
Neither is biased. Case n 2 has less
variability.
22
Bias and Variability

• Bias of an estimator (mean of sampling
  distrib) ? (true value of parameter)Statisti
  c is unbiased if bias 0.
• Variability of an estimator (SD of sampling
  distrib)Depends on sample size.

23
An example of a simulation

• Bias of estimators of variance -- use Minitab.

24
Stratified sampling

• E.g. estimate avg. salary of engineers at a
  company.Suppose 2 types of engineers junior
  and senior.Suppose company has 200 of each
  type. Want to est avg salary with a sample of
  size 10.
• Stratification idea combinea SRS of size 5 from
  junior engineers, anda SRS of size 5 from senior
  engineers.
• Is this a SRS of size 10?

25
Why stratify vs. take a SRS?

• Whats the advantage of stratifying?
• Bias?
• Variability?

26
Blocking in experimental design
3 types of seeds (treatments) A, B, C.And some
land to try them on
27
Blocking (cont.)
Suppose Worry about a fertility gradient ??
Believe field homogeneous ??
Partition experimental units into blocks.Assign
treatments randomly within each block.