Download STAB22 Statistics I Lecture 13 1

STAB22 Statistics I
Lecture 13
1
Principles of Experimental
Design
1.
2.
Control: control sources of variation,
besides factors, by making conditions as
similar as possible for all treatment groups
Randomize: helps equalize effects of
unknown/uncontrollable sources of variation

Note: Randomization does not eliminate the
effects of these sources, but tries to spread
them out across the treatment levels so that we
can see past them
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Principles of Experimental
Design
3.
4.
Replicate: get several measurements of
response for each treatment
Blocking: for variables we can identify but
cannot control and which affect response,
divide subjects into groups of same variable
values (a.k.a. blocks) and randomize within
each block

Removes much of the variability due to the
difference among the blocks.
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Experimental Designs

Completely Randomized Design (CRD):
All experimental units are allocated at
random among all treatments

Randomized Block Design (RBD):
Random assignment of units to treatments is
carried out separately within each block.
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Blocking

Assume 36 adult & 6 child participants in
previous insomnia experiment
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If you just randomize, you could get all children in
one treatment
How would you block age?
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Experiments

Placebo: “fake” treatment designed to look
like a real one, used when just knowledge of
receiving any treatment can affect response


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E.g. Used to test effectiveness of pain medication
For comparing results, often use current
standard treatment as baseline
Subjects getting placebo/standard treatment
called control group
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Blinding

Knowledge of assigned treatment can often
influence the assessment of the response

Two classes of individuals can affect experiment
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Those who influence results (e.g. subjects, nurses)
Those who evaluate results (e.g. physicians, judges)
Blinding avoids bias from knowing treatment
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Single-blind: every individual in either one of the
two classes doesn’t know treatments
Double-blind: every individual in both of the two
classes doesn’t know treatments
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Experiments

Golden standard for experiments:
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randomized
double-blind
● comparative
● placebo-controlled.
Even so, could have confounding problems
Confounding variable: variable associated
with both factor & response

Cannot tell whether effect on response is caused
by factor or confounding variable

E.g. Subject’s weight might be is associated with both
insomnia & diet (veg/non-veg)
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Probability

Many things in life are random (uncertain): roll of
a die, tomorrow’s weather, etc
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Don’t know for certain what will happen, but
Could know how likely is something to happen
Probability describes how likely is “something”
(event) to happen.

Probabilities take values between 0 and 1:
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A value of 1 means event is sure to happen
A value of 0 means event is not going to happen
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Probability Terminology

Random phenomenon: process whose result is not
known beforehand
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Trial: particular realization of random phenomenon
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E.g. Particular roll of a standard die
Outcome: basic result of a trial that cannot be
broken down to simpler results

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E.g. Rolling a standard (6-sided) die
E.g. “Rolling a 1” is an outcome
Sample Space: collection of all possible outcomes
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Sample space usually denoted by S
E.g. for rolling standard die, S={1, 2, 3, 4, 5, 6}
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Events
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Event: collection of one or more outcomes
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Usually denoted by capital letters (A, B, etc)
E.g. A = {2, 4, 6} = “Rolling an even number”
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Events can contain just single outcome, e.g. A = {1}
Probabilities are assigned to events
Example: Student’s course grade is random
phenomenon w/ sample space S={A, B, C, D, F}
Are the following events of outcomes?
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“Getting a passing grade”
“Getting a failing grade”
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Empirical Probability
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In short term, unpredictable
In long run, predictable
Limiting rel. frequency value
called empirical probability
of event A, denoted by P(A)
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0.8
0.6
0.4
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Rel. Freq. {Heads}
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0.2
Relative Frequency of
# times A occurs
event A:
total # of trials
1.0
Observe sequence of coin tosses (trials) &
count # of times of event A={Heads}
0.0

0
200
400
600
800
# of trials
1000
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Law of Large Numbers (LLN)

LLN guarantees relative frequency will
eventually settle on specific value
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LLN holds if trials are independent (i.e. outcome
of one trial does not influence that of another)
LLN does not apply in the short-run
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E.g. Assume you flip fair coin, i.e. P(Heads)=1/2,
and first 10 outcomes are “Tails”.
Is next flip more likely to be “Heads”? NO
LLN only applies in the (infinitely) long run
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Other Ways to Assign
Probabilities

Theoretical Probability: Sometimes P(A)
can be deduced from mathematical model

For equally likely outcomes (e.g. rolling fair die)
# of outcomes in A
probability is: P  A  
total # outcomes
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Personal / Subjective Probability: Assign
P(A) based on personal beliefs

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Typically expert’s views (e.g. sports analyst’s
chances of Maple Leafs winning Stanley Cup)
Most general case (can always be used)
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Example

Consider rolling 2 fair dice: find (theoretical)
probability of their sum being 4
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