Stat 470-2
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Will Introduce two simple experiments, the appropriate
randomization and analysis procedures
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Know how to perform a t-test in your chosen computer package
Simple Comparative Experiments
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Overview
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–
Simple comparative experiments
Experiments that compare two treatments
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Two experimental designs:
1.
2.
completely randomized design
randomized paired comparison design
Example (Boys Shoes)
• Company wishes to run an experiment to determine if a new synthetic
material is better than the existing one used for making the soles of
boys' shoes
• Experiment was run to see if the new, cheaper sole wears at the same
rate at which the soles wear out
• Have enough resources to make 10 pairs of shoes
• How should one run the experiment?
Are These the Same?
Experiment 1:
Experiment 2
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10 boys were selected at random
•
10 boys were selected at random
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Each boy was given a pair of shoes
•
Each boy was given a pair of shoes
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5 boys received a pair of shoes with the
old sole (Sole A) and 5 boys received
shoes with the the new sole (sole B)
•
Each pair had 1 shoe with the old sole
and 1 shoe with the new sole
•
•
Each boy wears the shoes for 1 month
and the amount of wear is measured
For each pair of shoes, the sole type
was randomly assigned to the right or
left foot
•
Each boy wears the shoes for 1 month
and the amount of wear is measured
Analysis of Experiment #1
• How would you analyze the data from Experiment 1?
Completely Random Design
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Objective: Comparing two treatments - A and B …are they the same?
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Method:
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–
–
–
–
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N experimental units available for the experiment
Randomly assign treatment A to n1 exp. units and treatment B to n2 units
N = n1 + n 2
Conduct experiment
Results: A: yA1, yA2, …, yAn1; B: yB1, yB2, … yBn2
Analysis Objective:
– Compare the average responses, A vs. B
– Is there evidence that one treatment is better than other, on average? How much
better?
Model and Analysis
60
• What can you see from the
plot?
70 ExamScores 80
• Can compare distributions
using side-by-side box-plots
90
Useful Plot
Undergrads
Grads
Example (Tomato Fertilizer)
•
•
Experiment was conducted by a horticulturist
Has 2 types of fertilizer available for tomato production (A and B)
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Objective: Is one fertilizer better than the other – higher yield, on
average?
•
Has 11 tomato plots
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Experiment Procedure - specify fertilizer amounts each fertilizer; decide
upon number of pots to receive each fertilizer; randomly assign fertilizer to
pots
•
Response: yield – pounds of tomatoes
Example (Tomato Fertilizer)
• Data:
Pot
1
2
3
4
5
6
7
8
9
10
11
Treatment A
A
B
B
A
B
B
B
A
A
B
Yield
(lbs)
29.9 11.4 26.6 23.7 25.3 28.5
14.2 17.9 16.5 21.1 24.3
Example (Tomato Fertilizer)
40.0
30.0
YIELD (Pounds)
20.0
10.0
0.0
N =
TREATMENT
5
6
A
B
Example (Tomato Fertilizer)
Tomato Messages? It Depends
•
If we had high hopes, e.g., that fertilizer B would out-perform A, we’re disappointed.
Possible follow-on actions:
– whine, complain that experiment was flawed, propose repeat of experiment
– go back to the lab – develop another fertilizer
•
If we’re just trying to choose the better fertilizer to use in next year’s garden, results
mean that we can choose based on other factors:
– cost, environmental impact, …
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If we would like to broaden our results, we should run more experiments
– other soils, other fertilizer amounts, other watering schedules, …
Analysis of Experiment #2
• How would you analyze the data from Experiment 2?
• Can we use a 2-sample t-test?
• Would the 2-sample t-test detect a significant difference?
Paired Comparison Designs
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Objective - Compare two treatments
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Method
–
–
–
–
–
Select N experimental units
Each experimental unit receives both treatments
Conduct the experiment assigning the treatments in random order
Measure the responses
Results, N pairs: (yA1, yB1), (yA2, yB2), …, (yAN, yBN)
Model and Analysis
Benefits of Paired Experiment
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Paired experiment used to eliminate possible sources of variability (noise)
– If one receives sole A and another sole B, then the experimental error (variability
among experimental units that receive the same treatment) reflects variability
between boys and the variability within each boy
– If each boys receives both soles, then the comparison within boy eliminates the
variability among boys from the reference noise. The variability of repeated
measurements within each boy is the pertinent experimental error in this case
– This is an example of BLOCKING
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Can be cheaper
Data
A
13.20
8.20
10.90
14.30
10.70
6.60
9.50
10.80
8.80
13.30
B
14.00
8.80
11.20
14.20
11.80
6.40
9.80
11.30
9.30
13.60
Plot of Responses by Treatment
16
14
12
10
WEAR
8
6
SOLE
Comparison of Shoe Material Wear
Comparison of Shoe Wear, by Boy
16.0
15
15.0
14
14.0
13
13.0
11
Material A
Material B
10
9
Material B
Wear
12
12.0
11.0
10.0
9.0
8
8.0
7
7.0
6
0
1
2
3
4
5
6
Boy
7
8
9
10 11
6.0
6
8
10
12
Material A
14
16
Data
A
13.20
8.20
10.90
14.30
10.70
6.60
9.50
10.80
8.80
13.30
B
14.00
8.80
11.20
14.20
11.80
6.40
9.80
11.30
9.30
13.60
Diff=B-A
0.80
0.60
0.30
-0.10
1.10
-0.20
0.30
0.50
0.50
0.30
Analyzing the Data
Comments
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Experimental results must be interpreted and thought about in terms of the
subject-matter, not just the statistical results
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In a good experiment, the message should be reasonably clear in a good plot of
the data
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Formal statistical procedures quantify the impressions that good plots convey
You should know …
• how to design, conduct, and analyze:
– completely randomized design
– randomized paired comparison design
• how to recognize design from description of experiment
Blocking and Randomization
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Blocking
– eliminate sources of variability
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Randomization
– balance possible effects of uncontrolled sources of variability
– provide fair estimate of noise variability
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General Guidance:
“Block what you can and randomize what you cannot”
Something to Help You Get to Sleep
• Read the following news item and in groups of 2-4 discuss the
question below:
Headline: Xeriscaping May Use Up More Water
MESA, AZ – Desert landscaping (called xeriscaping), often planted by residents to conserve water,
may actually be using more water. ASU botanist Chris Martin and two students have been
measuring the amount of irrigation used in the yards of 18 homes in Tempe and Phoenix. Half
have desert plantings; the others have conventional plantings. In the 18 months of the study so
far, homeowners put an average of 2.24 gallons per square foot on the xeriscaped yards,
compared with 1.67 gallons per square foot on the other yards.
• What questions would you like to ask Prof. Martin to help you
interpret and evaluate these results?