Introducing Concepts of Statistical Inference

Beth Chance, John Holcomb, Allan Rossman

Cal Poly – San Luis Obispo, Cleveland State University

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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)

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Is this feasible?

Experience at post-calculus level Developed spiral curriculum with logic of inference

(for 2×2 tables) in chapter 1 ISCAM: Investigating Statistical Concepts,

Applications, and Methods (Chance, Rossman) New project (funded by NSF/CCLI)

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

statistical process as a whole

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Workshop goals

Enable you to: Re-examine how you introduce concepts of

statistical inference to your students Help your students to understand fundamental

concepts of statistical inference Develop students’ understanding of the process of

statistical investigations Introduce normal-based methods of inference to

complement randomization-based ones

Workshop goals (cont.)

Enable you to: Implement activities based on real data from

genuine studies Assess student understanding of inference

concepts Make effective use of simulations, both tactile and

computer-based

6CAUSE Webinar April 2009 6

Agenda

Mon pm: Inference for proportion Overview, introductions Statistical significance via simulation Exact binomial inference CI for proportion Transition to normal-based inference for

proportion

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Agenda (cont.)

Tues am: Inference for 2×2 table Simulating randomization test Fisher’s exact test Observational studies, confounding Independent random samples

Tues pm: Comparing 2 groups with quant response Simulating randomization test Matched pairs designs

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Agenda (cont.)

Wed am: Assessment issues Strategies for assessing student

understanding/learning Preliminary findings

Wed pm: More inference scenarios Comparing several groups (ANOVA, chi-square) Correlation/regression Discussion of implementation issues

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Some notes

Agenda is always subject to change Already has changed some!

We’ll discuss some assessment, implementation issues throughout

Please offer questions, comments as they arise Be understanding when we don’t have all the

answers! We’ll also discuss some thorny issues that

we have not resolved among ourselves

Introductions

Who are you? Where/what do you teach? Why interested in this topic?

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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

Example 1: Helper/hinderer

Can do this on day 1 of course Logic of statistical inference/significance Null model, simulation, p-value, significance

Example 2: Kissing

Study: 8 of 12 kissing couples lean to right Does this provide evidence against 50/50

model? Does this provide evidence against 75/25

model? What models does this provide evidence

against?

Example 2: Kissing

Many new ideas here: Students describe rather than perform simulation Non-significant result (8/12) Null model other than 50/50 Looking at lower tail Sample size effect Big idea: Interval of plausible values (CI) Effect of confidence level Importance of random sampling

Transition to normal-based inference Two methods to find p-value for proportion:

Approximation by simulation Exact binomial calculation

Why should we present normal approx at all? Because it’s commonly used (not good reason) Because even minimally observant student will

notice similarities of these simulated distributions Because z-scores convey additional information

Distance from expected, measured in SDs

Example 1: Baseball Big Bang Some non-trivial aspects

Defining parameter Expressing hypotheses Sampling distribution

z = -5.75 conveys more information than p-value ≈ 0 95% CI:

Does this produce more/less understanding than forming CI by inverting test?

n

ppp

ˆ1ˆ96.1ˆ

Example 2: Which tire?

Which tire would you choose? Fun, simple in-class data collection

Almost always in conjectured direction May or may not be significant

Can use simulation or binomial or normal Investigate effect of sample size

Example 3: Cat Households

Sensible to use normal approx here H0: = 1/3, Ha: ≠ 1/3 z = -10.4, p-value ≈ .0000 99% CI: (.312, .320)

P-value and CI are complementary But provide different information

Statistical vs practical significance

Example 4: Female Senators

95% CI for : (.096, .244) Beware of biased sampling methods If you have access to entire population: no

inference to be drawn!

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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

Non-simulation approach

Exact randomization distribution Hypergeometric distribution Fisher’s Exact Test p-value =

= .0127 0.30

0.25

0.20

0.15

0.10

0.05

0.00

X

Pro

bability

10

0.0127

3

Distribution PlotHypergeometric, N=30, M=13, n=15

15

30

2

17

13

13

3

17

12

13

4

17

11

13

5

17

10

13

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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)

Example 2: Yawning

What’s different here? Group sizes not the same

So calculating success proportions more important

Experimental result not significant Lack of surprising-ness is harder for students to spot

than surprising-ness Well-stated conclusion is more challenging, subtle

Don’t want to “accept null model”

Example 3: Murderous Nurse?

Murder trial: U.S. vs. Kristin Gilbert Accused of giving patients fatal dose of heart stimulant Data presented for 18 months of 8-hour shifts

Relative risk: 6.34

Gilbert on shift Gilbert not on shift TotalDeath occurred 40 34 74

No death 217 1350 1567Total 257 1384 1641

Proportion 0.156 0.025

Example 3 (cont.)

Structurally the same as dolphin and yawning examples, but with one crucial difference No random assignment to groups

Observational study Allows many potential explanations other than “random

chance” Confounding variables Perhaps she worked intensive care unit or night shift

Is statistical significance still relevant? Yes, to see if “random chance” can plausibly be ruled

out as an explanation Some statisticians disagree

Example 4: Native Californians? What’s different here? Not random assignment to groups Independent random sampling from

populations So …

Scope of conclusions differs Generalize to larger populations, but no cause/effect

conclusions Use different kind of randomness in simulation

To model use of randomness in data collection

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Example 1: 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

68.2

93.5

92.15

1073.14

1117.12

90.382.1922

2

22

1

21

21

ns

ns

xxt

008.68.2Pr ? tvaluep

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Randomization approach

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

Start with tactile simulation using index cards Write each “score” on a card Shuffle the cards Randomly deal out 11 for deprived group, 10 for unrestricted

group Calculate difference in group means Repeat many times

Example 1 Sleep deprivation (cont.) Conclusion: Fairly strong evidence that sleep

deprivation produces lower improvements, on average, even three days later Justifcation: Experimental results as extreme as

those in the actual study would be quite unlikely to occur by chance alone, if there were no effect of the sleep deprivation

Exact randomization distribution

Exact p-value 2533/352716 = .0072

Example 2: Age discrimination? Employee ages:

25, 33, 35, 38, 48, 55, 55, 55, 56, 64 Fired employee ages in bold:

25, 33, 35, 38, 48, 55, 55, 55, 56, 64 Robert Martin: 55 years old Do the data provide evidence that the firing

process was not “random” How unlikely is it that a “random” firing process

would produce such a large average age?

Exact permutation distribution Exact p-value: 6 / 120 = .05

56524844403632

20

15

10

5

0

mean age (fired)

Frequency

Example 3: Memorizing letters You will be given a string of 30 letters Memorize as many as you can, in order, in 20

seconds

Confidence Intervals based on Randomization Tests (Quantitative) Invert randomization test

Subtract from all subjects in group B, re-randomize, add from all subjects in group B, compare to observed difference

Similar to binomial example (kissing study) Get standard error from randomization distribution

and use observed +- 2 SEs Get percentiles from randomization distribution and

use observed +- percentiles t-interval Bootstrapping

Series of Lab Assignments

Lab 1: Helper/Hinderer (Binomial test) Lab 2: Dolphin Therapy (2x2 table) Lab 3: Textbook prices (matched pairs from normal

population) or JFK/JFKC (randomization on quantitative variable)

Lab 4: Random Babies Lab 5: One-sample z-test for proportion (Reeses

Pieces) Lab 6: Sleepless nights (t-test, confidence interval) Lab 7: Sleep deprivation (randomization test) Lab 8: Study Hours and GPA (regression with

simulation and Minitab output)

Random Babies

Suppose that 4 mothers give birth to baby boys at the same hospital on the same night

Hospital staff returns babies to mothers at random!

How likely is it that … … nobody gets the right baby? … everyone gets the right baby? …

Random Babies

Last Names First Names Jones Jerry Miller Marvin Smith Sam Williams Willy

Random Babies

Last Names First Names

Jones Marvin

Miller

Smith

Williams

Random Babies

Last Names First Names

Jones Marvin

Miller Willy

Smith

Williams

Random Babies

Last Names First Names

Jones Marvin

Miller Willy

Smith Sam

Williams

Random Babies

Last Names First Names

Jones Marvin

Miller Willy

Smith Sam 1 match

Williams Jerry

Random Babies

1234 1243 1324 1342 1423 1432

2134 2143 2314 2341 2413 2431

3124 3142 3214 3241 3412 3421

4123 4132 4213 4231 4312 4321

Random Babies

1234 1243 1324 1342 1423 1432

4 2 2 1 1 2

2134 2143 2314 2341 2413 2431

2 0 1 0 0 1

3124 3142 3214 3241 3412 3421

1 0 2 1 0 0

4123 4132 4213 4231 4312 4321

0 1 1 2 0 0

Random Babies

Probability distribution 0 matches: 9/24=3/8 1 match: 8/24=1/3 2 matches: 6/24=1/4 3 matches: 0 4 matches: 1/24

Expected value 0(9/24)+1(8/24)+2(6/24)+3(0)+4(1/24)=1

Random Babies

First simulate, then do theoretical analysis Able to list sample space Short cuts when are actually equally likely Simple, fun applications of basic probability

Naming Presidents

List as many U.S. Presidents as you can in reverse chronological order (starting with the current President)

Score = # correct before first error

Naming Presidents

Obama Bush Clinton BushReagan Carter Ford NixonJohnson Kennedy Eisenhower TrumanRoosevelt Hoover Coolidge HardingWilson Taft Roosevelt McKinleyCleveland Harrison Cleveland ArthurGarfield Hayes Grant JohnsonLincoln Buchanan Pierce FillmoreTaylor Polk Tyler HarrisonVan Buren Jackson Adams MonroeMadison Jefferson Adams Washington

Naming Presidents

Use sample data to determine 90% t-interval What percentage of sample values are within

this interval? Is this close to 90%?

Naming Presidents

Lessons: Confidence interval is not a prediction interval Pay attention to what the parameter (“it”) is

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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

Emphasize purpose of simulation 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