Introducing Concepts of Statistical Inference Via

Introducing Concepts of Statistical Inference Via Randomization Tests John Holcomb - Cleveland State University Beth Chance, Allan Rossman, Emily Tietjen - Cal Poly State University George Cobb - Mount Holyoke College http://statweb.calpoly.edu/csi/

Introduction  2005 USCOTS Cobb proposed the idea of a new

introductory curriculum based on randomization methods.  Cobb (2007) “Our curriculum is needlessly complicated because we put the normal distribution, as an approximate sampling distribution for the mean, instead of putting the core logic of inference at the center.”  Why? “Tyranny of the computable.”  How????

Curriculum Goal  Use technology simulations to lead students

to develop an understanding of the concepts of statistical significance and p-values.  Focus on statistical process.  Repeated exposure throughout course.

Learning Process  Research study and data.  Tactile simulation and class discussion of

results.  Simulation using a tailored applet.  Empirical p-value and discussion.  Conclusion in context.

Classroom Activities (Example 1): Naughty or Nice?  Hamlin, Wynn, and Bloom (Nature, 2007)  Videos available at www.yale.edu/infantlab/socialevaluation/Helper-Hinderer.html

 Do the experimental results (14 out of 16)

provide convincing evidence that infants have a genuine preference for the helper toy rather than the result occurring by chance alone?  Inference for a binomial proportion.

Randomization approach • How likely such an extreme result would be

under the null model of no preference? • Begin with each student flipping a coin 16 times. • Combine results from class with a dotplot. • Move to applet to simulate tossing 16 coins for 1000 repetitions. • Determine the proportion of repetitions14 or more flips were heads.

Example 2: Sleep Deprivation?  Stickgold, James, & Hobson, (Nature Neuroscience,

2000)  21 subjects randomly assigned to one of two groups: sleep deprived group and unrestricted sleep.  Both groups then allowed as much sleep as wanted on the following two nights.  All subjects re-tested on the third day.

Example 2: Sleep Deprivation?  Randomized experiment.  Compared mean improvement in

response time to a visual stimulus on a computer screen.  Unrestricted sleep group: 19.82 ms.  Sleep deprived group: 3.90 ms.

 Inference for the difference of two means

from independent samples.

Randomization approach  How likely such an extreme result

(Diff=15.92 ms) would be under the null model of no treatment effect?  21 improvement scores written on index cards.  Randomly “deal” cards to two groups and record difference of group means.  Combine results of the class with a dotplot.

Randomization approach  Use an applet to simulate randomization

process 1000 times and explore the frequency of a difference in means of 15.92 or more extreme by chance alone.  http://www.rossmanchance.com/applets

Research Goal  Make evidence-based curriculum decisions.  Implement and collect data from small-scale classroom experiments.  Identify issues that create difficulties for student understanding.  Formulate questions about the most pedagogically effective way to implement this approach.

Curriculum Design Issue #1: First Activity  Should the first example that students

encounter be one where the result is statistically significant or one where the result is not significant at all?

Advantages  Students may find it easier to judge when

an observed result is surprising under a null model.  Starting with an insignificant result may reinforce students’ natural inclinations to regard a p-value as the probability the null model is true.

Classroom Experiment  Four sections of introductory statistics at Cal

Poly.  Half the students told 9 of the 16 infants chose the helper toy (non-significant result group).  Half the students told 14 of the 16 infants chose the helper toy (significant result group).  Students given the activity and told to work in pairs.

Classroom Experiment • Two instructors: one randomized across

sections and the other randomized by individuals. • Follow-up Quiz questions the next class period.

Question 1

9 of 16

14 of 16

When I conducted the simulation using 1,000,000 repetitions, I obtained a proportion (empirical p-value) of .402 (.002). Based on this result, which assumes the null model of no genuine preference, the actual result obtained by the researchers,9 of 16 (14 of 16), choosing the helper is a) impossible c) somewhat surprising

b) very surprising d) not at all surprising

Results  60.6% (n = 71) students in the “9 of 16”

group answered correctly.  77.5% (n = 71) students in the “14 of 16” group answered correctly.  Two-sided p-value is approximately 0.030.  Interpretation: students find it easier to interpret a surprising outcome than a nonsurprising one.

Question 2 Fill in the blanks in the following sentence to interpret this proportion from part (l). This proportion says that in about (1) % of ___(2)___, the researchers would get __(3)___ who choose the helper toy, assuming that ________(4)______.

Results This proportion says that in about (1) % of ___(2)___, the researchers would get __(3)___ who choose the helper toy, assuming that ______(4)______. 9 of 16

14 of 16

(1) 40.2% vs. 0.2%: Non-significant group performed better, but may be an artifact of the difficulty of expressing 0.002 as a percentage.

Results This proportion says that in about (1) % of ___(2)___, the researchers would get __(3)___ who choose the helper toy, assuming that ______(4)______.

(2) Both groups about 50% correctly answering 1,000,000 repetitions.

Results This proportion says that in about (1) % of ___(2)___, the researchers would get __(3)___ who choose the helper toy, assuming that ______(4)______.

(3) “14 of 16” group performed better (54.2% vs. 38.9%, p-value = .066) in citing the observed result or more extreme.

Results This proportion says that in about (1) % of ___(2)___, the researchers would get __(3)___ who choose the helper toy, assuming that ______(4)______.

(4) Both groups about 76% correctly answering “assuming no preference.”

Interpretation  Although students seem to equally

understand the null model, they did differ slightly in realizing what the simulation told them.

Question 3 Based on your answer to (1) and (2), which of the following would you consider the most appropriate conclusion from this study? (choose one) (a) These 16 infants have no genuine preference and therefore there’s no reason to doubt that the researchers’ result is different from .5 just by random chance. (b) The researchers’ results would be very surprising if there was no genuine preference for the helper and therefore I believe there is a preference. (c) There is a large (small) chance that there is a genuine preference for the helper.

Results  Approximately 77% in each group answered

the correct answer: (a) for the “9 of 16” group and (b) for the “14 of 16.”

Curriculum Design Issue #2: Tactile Simulations  We have suggested beginning each

simulation with a tactile version before turning to technology, but does the tactile aspect really add value to the students’ learning experience?

Should we do tactile first?  Potential advantages:  Students are in a better position to

understand what the technology simulation is doing if they have first performed a tactile simulation themselves.  We use applets that mirror the hands-on activity as closely as possible so the technology is not a “black box.”  Potential disadvantage: tactile simulations take valuable class time.

Classroom Experiment  Randomly assigned 43 students to two

treatment groups.  Class topic was investigating the sampling distribution of a single proportion.

Classroom Experiment  Tactile group (20 students) students

each determined the sample proportion of orange candies among 25 actual Reese’s Pieces. Students created a class dotplot of sample proportions. Then turned to applet for simulation of many samples of size 25.

Classroom Experiment  Non-tactile group (23) immediately moved

to simulation with applet without working with Reese’s Pieces and creating pooled dotplot.

Results  Students in both groups given a quiz the

next day.  Five questions that involved a single sample proportion.  Independent and blinded statistics instructor graded the quizzes.  No statistically significant differences were found.

Results  Interesting outcome was that both groups

seemed to finish the activity in about the same amount of time.  May suggest the tactile aspect does not take more time and does not hinder learning.

Results  In a follow-up questionnaire to a different

class, approximately 50% of the students indicated the tactile component of the activity was helpful.

Curriculum Design Issue #3  Should the first activity that students

encounter focus on inference for a single proportion, as in the “Naughty or Nice” example, or on a comparison of two groups, as in the “Sleep Deprivation” example?

Curriculum Design Issue #4  In the case of a simulation involving a single

proportion, as with the “Naughty or Nice” example, how should the tactile simulation be conducted?  16 students each tossing one coin  Each student tosses 1 coin 16 times

 Each student tosses 16 coins

Curriculum Design Issue #5  In the case of a randomization test for a 2×2

table, what statistic should the students calculate in their simulations?  Difference in conditional proportions of

success.  Ratio of conditional proportions (Relative risk).  Number of successes in group A.

Curriculum Design Issue #6  How much of the work should the

technology do automatically (calculating empirical p-values)?  Push a button (e.g., based on two-way

table).  Specify observed result to count beyond.

Curriculum Design Issue #7  Should the type of randomness used in the

simulation always reflect the role of randomness used in the actual data collection process?  Randomizing (assignment) when data arises from experiments. Bootstrapping and/or sampling from finite populations when data arises from samples.  Always randomizing group membership.

Summary  Feasibility of random assignment in

classroom experiments.  Focused and relevant research questions.  Direct link between research and classroom

practice.  Assessment instruments described in Holcomb, Chance, Rossman, and Cobb (2010, Proceedings).

References  Cobb, George W. (2007). The Introductory Statistics Course: A

Ptolemaic Curriculum? Technology Innovations in Statistics Education, 1(1),www.escholarship.org/uc/item/6hb3k0nz  Holcomb, J., Chance, B., Rossman, A., & Cobb, G., (2010). Assessing student learning about statistical inference, Proceedings of the 8th International Conference on Teaching Statistics.  Hamlin, J. K., Wynn, K., & Bloom, P. (2007). Social evaluation by preverbal infants. Nature, 450, 557-559  Stickgold, R., James, L., & Hobson, J.A. (2000). Visual discrimination learning requires post-training sleep. Nature Neuroscience, 2, 1237-1238.  http://statweb.calpoly.edu/csi Thanks to National Science Foundation DUE/CCLI #0633349