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Name-Brand vs. Off-Brand: A Twist on Taste Testing for aMathematical Statistics Course

Eric M. Reyes

Rose-Hulman Institute of Technology

JSM 2014

Outline

1 Background

2 Activity

3 Analysis

4 Results

Binary Choice Background

Dataset and Story

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The Mathematical Statistics Course

Students have a strong mathematics background including: Calculus,Probability, Programming, and Introductory Statistics, (possibly)Analysis.

Topics include probability, properties of random samples, estimation,and inference via maximum likelihood.

Exposes students to the theory underlying many methods encounteredin other courses.

It is really easy to divorce this course from the pedagogical tools weuse in an Introductory Statistics course.

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Background for Students

Name-Brand vs. Off-Brand: Can We Really Distinguish Between the Two?

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Binary Choice Activity

Activity and Assignment

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Data Collection

Question: What is the probability that a consumer can actually discernthe difference between Cheerios and an off-brand version?

Taste Test:

Participants were instructors at the university.

Double-blind.

Each participant was given m = 6 samples (3 name-brand and 3off-brand in randomized order).

Class Discussion:

What is the benefit of blinding and how might it be implemented?

How will randomization be utilized in this study?

What variables should be recorded in this study?

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Data Collection

Taste Test:

Double-blind.

Class Discussion:

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Data Collection

Taste Test:

Double-blind.

Class Discussion:

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Data Collection

Taste Test:

Double-blind.

Class Discussion:

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Data Collection

Taste Test:

Double-blind.

Class Discussion:

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The Data

0 1 2 3 4 5 6Number Correctly Categorized

Children No Children

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The Assignment

Describe the distribution of the probability of actually discerningbetween the name-brand and off-brand products.

Account for the fact that a correct response could be due to a truediscernment or a lucky guess.

Construct a poster-presentation summarizing your analysis and results.

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The Assignment

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The Assignment

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Binary Choice Analysis

Analysis

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Model for Data-Generating Process

Model Proposed by Morrison [Am. Stat. (1978)]

Pi is the probability the i-th subject can actually discern between thename-brand and off-brand.

Then, the probability of correctly categorizing the response is

Ci = Pi +1

2(1− Pi )

Assume PiIID∼ Beta(α, β).

Then, the number correct is Yi | Ci ∼ Bin (6,Ci ).

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Ci = Pi +1

2(1− Pi )

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Ci = Pi +1

2(1− Pi )

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Ci = Pi +1

2(1− Pi )

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Probabilistic Model

Letting f (y | α, β) represent the marginal density of Yi , then we havethat

`(α, β | y) =n∑

log [f (yi | α, β)]

Determining the marginal density is a good exercise in its own right(Casella and Berger [Statistical Inference (pg 197)]).

Students are expected to show that

f (y | α, β) =

)Γ(α + β)

2mΓ(α)Γ(β)

y∑k=0

)Γ(α + k)Γ(β + m − y)

Γ(α + β + m − y + k)

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Probabilistic Model

`(α, β | y) =n∑

f (y | α, β) =

)Γ(α + β)

2mΓ(α)Γ(β)

y∑k=0

)Γ(α + k)Γ(β + m − y)

Γ(α + β + m − y + k)

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Probabilistic Model

`(α, β | y) =n∑

f (y | α, β) =

)Γ(α + β)

2mΓ(α)Γ(β)

y∑k=0

)Γ(α + k)Γ(β + m − y)

Γ(α + β + m − y + k)

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Computation of MLE’s

No closed-form solution exists for α̂ and β̂.

Students constructed R programs to compute the estimates using thedata.

Due to numerical instability, care had to be given to the optimizationroutine chosen and the coding of the likelihood.

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Binary Choice Results

Results

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Results for Primary Question

0.0e+00

5.0e−07

1.0e−06

1.5e−06

2.0e−06

0.00 0.25 0.50 0.75 1.00Probability of Actually Discerning

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Results Comparing Those with and without Children

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Conclusions

Students gave positive feedback on the activity, particularly seeing aproblem through from design to results.

Activity ties together several topics used throughout the course, andis a nice capstone to maximum likelihood.

Lends itself well to a short poster presentation.

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Conclusions

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Conclusions

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Binary Choice References

References

I Casella G and Berger RL.Statistical Inference (2nd ed, pg 197).Pacific Grove, CA: Thomson Learning, 2002.

I Morrison DG.A Probability Model for Forced Binary Choices.The American Statistician, 32(1):23-25, 1978.

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Binary Choice References

Contact Information

Eric M. ReyesDepartment of MathematicsRose-Hulman Institute of Technology5500 Wabash Ave.Terre Haute, IN 47803

web: www.rose-hulman.edu/∼reyesememail: [email protected]

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