Stat 470-2 Will Introduce two simple experiments, the appropriate randomization and analysis procedures Know how to perform a t-test in your chosen computer.

Stat 470-2

• Will Introduce two simple experiments, the appropriate randomization and analysis procedures

• Know how to perform a t-test in your chosen computer package

Simple Comparative Experiments

• Overview – Simple comparative experiments

– Experiments that compare two treatments

– Two experimental designs:1. completely randomized design

2. 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:

• 10 boys were selected at random

• Each boy was given a pair of shoes

• 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 boy wears the shoes for 1 month and the amount of wear is measured

Experiment 2

• 10 boys were selected at random

• Each boy was given a pair of shoes

• Each pair had 1 shoe with the old sole and 1 shoe with the new sole

• 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

• Objective: Comparing two treatments - A and B …are they the same?

• Method:– N experimental units available for the experiment– Randomly assign treatment A to n1 exp. units and treatment B to n2 units – N = n1 + n2

– 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

Useful Plot

• Can compare distributions using side-by-side box-plots

• What can you see from the plot?

6070

8090

Undergrads Grads

Exam

Score

s

Example (Tomato Fertilizer)

• Experiment was conducted by a horticulturist• Has 2 types of fertilizer available for tomato production (A and B)

• Objective: Is one fertilizer better than the other – higher yield, on average?

• Has 11 tomato plots

• 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)

65N =

TREATMENT

BA

YIE

LD

(P

ou

nd

s)40.0

30.0

20.0

10.0

0.0

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, …

• 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

• Objective - Compare two treatments

• 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

• 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

• Can be cheaper

Data

A B13.20 14.008.20 8.8010.90 11.2014.30 14.2010.70 11.806.60 6.409.50 9.8010.80 11.308.80 9.3013.30 13.60

Plot of Responses by Treatment

SOLE

WE

AR

16

14

12

10

8

6

Comparison of Shoe Material Wear

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

15.0

16.0

6 8 10 12 14 16

Material A

Ma

teri

al

B

Comparison of Shoe Wear, by Boy

6

7

8

9

10

11

12

13

14

15

0 1 2 3 4 5 6 7 8 9 10 11

Boy

Wea

r Material A

Material B

Data

A B Diff=B-A13.20 14.00 0.808.20 8.80 0.6010.90 11.20 0.3014.30 14.20 -0.1010.70 11.80 1.106.60 6.40 -0.209.50 9.80 0.3010.80 11.30 0.508.80 9.30 0.5013.30 13.60 0.30

Analyzing the Data

Comments

• Experimental results must be interpreted and thought about in terms of the subject-matter, not just the statistical results

• In a good experiment, the message should be reasonably clear in a good plot of the data

• 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

• Blocking– eliminate sources of variability

• Randomization– balance possible effects of uncontrolled sources of variability

– provide fair estimate of noise variability

• 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?