Concepts of Statistical Inference: A Randomization-Based Curriculum

Allan Rossman, Beth Chance, John Holcomb

Cal Poly – San Luis Obispo, Cleveland State University

2CAUSE Webinar April 2009 2

Outline

Overview, motivation Three examples Merits, advantages Five questions Assessment issues Conclusions, lessons learned Q&A

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Ptolemaic Curriculum?

“Ptolemy’s cosmology was needlessly complicated, because he put the earth at the center of his system, instead of putting the sun at the center. Our curriculum is needlessly complicated because we put the normal distribution, as an approximate sampling distribution for the mean, at the center of our curriculum, instead of putting the core logic of inference at the center.”

– George Cobb (TISE, 2007)

4

Is randomization-based approach feasible? Experience at post-calculus level

Developed spiral curriculum with logic of inference (Fisher’s Exact Test) in chapter 1

ISCAM: Investigating Statistical Concepts, Applications, and Methods

New project Rethinking for lower mathematical level More complete shift, including focus on entire

statistical process as a whole

4

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Example 1: Helper/hinderer?

Sixteen infants were shown two videotapes with a toy trying to climb a hill One where a “helper” toy pushes the original toy up One where a “hinderer” toy pushes the toy back down

Infants were then presented with the two toys as wooden blocks Researchers noted which toy infants chose

http://www.yale.edu/infantlab/socialevaluation/Helper-Hinderer.html

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Example 1: Helper/hinderer?

Data: 14 of the 16 infants chose the “helper” toy Core question of inference:

Is such an extreme result unlikely to occur by chance (random selection) alone …

… if there were no genuine preference (null model)?

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Analysis options

Could use a binomial probability calculation We prefer a simulation approach

To emphasize issue of “how often would this happen in long run?”

Starting with tactile simulation

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Strategy

Students flip a fair coin 16 times Count number of heads, representing choices of

“helper” toy Fair coin represent null model of no genuine

preference Repeat several times, combine results

See how surprising to get 14 or more heads even with “such a small sample size”

Approximate (empirical) P-value Turn to applet for large number of repetitions:

http://statweb.calpoly.edu/bchance/applets/BinomDist3/BinomDist.html

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Results

Pretty unlikely to obtain 14 or more heads in 16 tosses of a fair coin, so …

Pretty strong evidence that infants do have genuine preference for helper toy and were not just picking at random

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Example 2: Dolphin therapy?

Subjects who suffer from mild to moderate depression were flown to Honduras, randomly assigned to a treatment

Is dolphin therapy more effective than control? Core question of inference:

Is such an extreme difference unlikely to occur by chance (random assignment) alone (if there were no treatment effect)?

Dolphin therapy Control group TotalSubject improved 10 3 13Subject did not 5 12 17

Total 15 15 30Proportion 0.667 0.200

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Some approaches

Could calculate test statistic, P-value from approximate sampling distribution (z, chi-square) But it’s approximate But conditions might not hold But how does this relate to what “significance” means?

Could conduct Fisher’s Exact Test But there’s a lot of mathematical start-up required But that’s still not closely tied to what “significance” means

Even though this is a randomization test

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Alternative approach

Simulate random assignment process many times, see how often such an extreme result occurs Assume no treatment effect (null model) Re-randomize 30 subjects to two groups (using cards)

Assuming 13 improvers, 17 non-improvers regardless Determine number of improvers in dolphin group

Or, equivalently, difference in improvement proportions Repeat large number of times (turn to computer) Ask whether observed result is in tail of distribution

Indicating saw a surprising result under null model Providing evidence that dolphin therapy is more effective

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Analysis

http://www.rossmanchance.com/applets/Dolphins/Dolphins.html

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Conclusion

Experimental result is statistically significant And what is the logic behind that?

Observed result very unlikely to occur by chance (random assignment) alone (if dolphin therapy was not effective)

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Example 3: Lingering sleep deprivation? Does sleep deprivation have harmful effects

on cognitive functioning three days later? 21 subjects; random assignment

Core question of inference: Is such an extreme difference unlikely to occur by

chance (random assignment) alone (if there were no treatment effect)?

improvement

sleep c

onditio

n

4032241680-8-16

deprived

unrestricted

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One approach

Calculate test statistic, p-value from approximate sampling distribution

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93.5

92.15

1073.14

1117.12

90.382.1922

2

22

1

21

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ns

ns

xxt

008.68.2Pr ? tvaluep

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Another approach

Simulate randomization process many times under null model, see how often such an extreme result (difference in group means) occurs

difference in group means by random assignment

num

ber

of ra

ndom

izations

181260-6-12-18

120

100

80

60

40

20

0

= 13 / 1000approx p-value

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Advantages

You can do this at beginning of course Then repeat for new scenarios with more richness Spiraling could lead to deeper conceptual understanding

Emphasizes scope of conclusions to be drawn from randomized experiments vs. observational studies

Makes clear that “inference” goes beyond data in hand Very powerful, easily generalized

Flexibility in choice of test statistic (e.g. medians, odds ratio) Generalize to more than two groups

Takes advantage of modern computing power

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Question #1

Should we match type of randomness in simulation to role of randomness in data collection? Major goal: Recognize distinction between random

assignment and random sampling, and the conclusions that each permit

Or should we stick to “one crank” (always re-randomize) in the analysis, for simplicity’s sake?

For example, with 2×2 table, always fix both margins, or only fix one margin (random samples from two independent groups), or fix neither margin (random sampling from one group, then cross-classifying)

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Question #2

What about interval estimation? Estimating effect size at least as important as assessing

significance How to introduce this?

Invert test Test “all” possible values of parameter, see which do not put

observed result in tail Easy enough with binomial, but not as obvious how to

introduce this (or if it’s possible) with 2×2 tables Alternative: Estimate +/- margin-of-error

Could estimate margin-of-error with empirical randomization distribution or bootstrap distribution

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Question #3

How much bootstrapping to introduce, and at what level of complexity? Use to approximate SE only? Use percentile intervals? Use bias-correction?

Too difficult for Stat 101 students? Provide any helpful insights?

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Question #4

What computing tools can help students to focus on understanding ideas? While providing powerful, generalizable tool?

Some possibilities Java applets, Flash

Very visual, contextual, conceptual; less generalizable Minitab

Provide students with macros? Or ask them to edit? Or ask them to write their own?

R Need simpler interface?

Other packages? StatCrunch, JMP have been adding resampling capabilities

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Question #5

What about normal-based methods? Do not ignore them!

Introduce after students have gained experience with randomization-based methods

Students will see t-tests in other courses, research literature

Process of standardization has inherent value A common shape often arises for empirical

randomization/sampling distributions Duh!

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Assessment: Developing instruments that assess … Conceptual understanding of core logic of inference

Jargon-free multiple choice questions on interpretation, effect size, etc.

“Interpret this p-value in context”: probability of observed data, or more extreme, under randomness, if null model is true

Ability to apply to new studies, scenarios Define null model, design simulation, draw conclusion More complicated scenarios (e.g., compare 3 groups)

Understanding of components of activity/simulation Designed for use after an in-class activity using

simulation. Example Questions

What did the cards represent? What did shuffling and dealing the cards represent? What implicit assumption about the two groups did the

shuffling of cards represent? What observational units were represented by the dots on

the dotplot? Why did we count the number of repetitions with 10 or

more “successes” (that is, why 10)?

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Conducting small classroom experiments Research Questions:

Start with study that has with significant result or non? Start with binomial setting or 2×2 table? Do tactile simulations add value beyond computer

ones? Do demonstrations of simulations provide less value

than student-conducted simulations?

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Conclusions/Lessons Learned

Put core logic of inference at center Normal-based methods obscure this logic Develop students’ understanding with

randomization-based inference Emphasize connections among

Randomness in design of study Inference procedure Scope of conclusions

But more difficult than initially anticipated “Devil is in the details”

Conclusions/Lessons Learned

Don’t overlook null model in the simulation Simulation vs. Real study Plausible vs. Possible

How much worry about being a tail probability How much worry about p-value = probability

that null hypothesis is true

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Thanks very much!

Thanks to NSF (DUE-CCLI #0633349) Thanks to George Cobb, advisory group More information: http://statweb.calpoly.edu/csi

Draft modules, assessment instruments Questions/comments:

arossman@calpoly.edu bchance@calpoly.edu j.p.holcomb@csuohio.edu