CAUSE Webinar: Introducing Math Majors to Statistics Allan Rossman and Beth Chance Cal Poly – San Luis Obispo April 8, 2008

CAUSE Webinar 2 Outline Goals Guiding principles Content of an example course Assessment Examples (four)

April 8, 2008 CAUSE Webinar 3 Goals Redesign introductory statistics course for mathematically inclined students in order to:  Provide balanced introduction to the practice of statistics at appropriate mathematical level  Better alternative than “Stat 101” or “Math Stat” sequence for math majors’ first statistics course

April 8, 2008 CAUSE Webinar 4 Guiding principles (Overview) 1. Put students in role of active investigator 2. Motivate with real studies, genuine data 3. Repeatedly experience entire statistical process from data collection to conclusion 4. Emphasize connections among study design, inference technique, scope of conclusions 5. Use variety of computational tools 6. Investigate mathematical underpinnings 7. Introduce probability “just in time”

April 8, 2008 CAUSE Webinar 5 Principle 1: Active investigator Curricular materials consist of investigations that lead students to discover statistical concepts and methods  Students learn through constructing own knowledge, developing own understanding  Need direction, guidance to do that Students spend class time engaged with these materials, working collaboratively, with technology close at hand

April 8, 2008 CAUSE Webinar 6 Principle 2: Real studies, genuine data Almost all investigations focus on a recent scientific study, existing data set, or student collected data  Statistics as a science  Frequent discussions of data collection issues and cautions  Wide variety of contexts, research questions

April 8, 2008 CAUSE Webinar 7 Real studies, genuine data Popcorn and lung cancer Historical smoking studies Night lights and myopia Effect of observer with vested interest Kissing the right way Do pets resemble their owners Who uses shared armrest Halloween treats Heart transplant mortality Lasting effects of sleep deprivation Sleep deprivation and car crashes Fan cost index Drive for show, putt for dough Spock legal trial Hiring discrimination Comparison shopping Computational linguistics

April 8, 2008 CAUSE Webinar 8 Principle 3: Entire statistical process First two weeks:  Data collection Observation vs. experiment (Confounding, random assignment vs. random sampling, bias)  Descriptive analysis Segmented bar graph Conditional proportions, relative risk, odds ratio  Inference Simulating randomization test for p-value, significance Hypergeometric distribution, Fisher’s exact test Repeat, repeat, repeat, … Random assignment  dotplots/boxplots/means/medians  randomization test Sampling  bar graph  binomial  normal approximation

April 8, 2008 CAUSE Webinar 9 Principle 4: Emphasize connections Emphasize connections among study design, inference technique, scope of conclusions  Appropriate inference technique determined by randomness in data collection process Simulation of randomization test (e.g., hypergeometric) Repeated sampling from population (e.g., binomial)  Appropriate scope of conclusion also determined by randomness in data collection process Causation Generalizability

April 8, 2008 CAUSE Webinar 10 Principle 5: Variety of computational tools For analyzing data, exploring statistical concepts Assume that students have frequent access to computing  Not necessarily every class meeting in computer lab Choose right tool for task at hand  Analyzing data: statistics package (e.g., Minitab)  Exploring concepts: Applets (interactivity, visualization)  Immediate updating of calculations: spreadsheet (Excel)

April 8, 2008 CAUSE Webinar 11 Principle 6: Mathematical underpinnings Primary distinction from “Stat 101” course  Some use of calculus but not much  Assume some mathematical sophistication E.g., function, summation, logarithm, optimization, proof  Often occurs as follow-up homework exercises Examples  Counting rules for probability Hypergeometric, binomial distributions  Principle of least squares, derivatives to find minimum Univariate as well as bivariate setting  Margin-of-error as function of sample size, population parameters, confidence level

April 8, 2008 CAUSE Webinar 12 Principle 7: Probability “just in time” Whither probability?  Not the primary goal  Studied as needed to address statistical issues  Often introduced through simulation Tactile and then computer-based Addressing “how often would this happen by chance?”  Examples Hypergeometric distribution: Fisher’s exact test for 2×2 table Binomial distribution: Sampling from random process Continuous probability models as approximations

April 8, 2008 CAUSE Webinar 13 Chapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6 Data CollectionObservation vs. experiment, confounding, randomization Random sampling, bias, precision, nonsampling errors Paired dataIndependent random samples Bivariate Descriptive Statistics Conditional proportions, segmented bar graphs, odds ratio Quantitative summaries, transformations, z-scores, resistance Bar graphModels, Probability plots, trimmed mean Scatterplots, correlation, simple linear regression ProbabilityCounting, random variable, expected value empirical ruleBermoulli processes, rules for variances, expected value Normal, Central Limit Theorem Sampling/ Randomization Distribution Randomization distribution for Sampling distribution for X, Large sample sampling distributions for, Sampling distributions of, OR, Chi-square statistic, F statistic, regression coefficients ModelHypergeometricBinomialNormal, tNormal, t, log- normal Chi-square, F, t Statistical Inference p-value, significance, Fisher’s Exact Test p-value, significance, effect of variability Binomial tests and intervals, two-sided p- values, type I/II errors z-procedures for proportions t- procedures, robustness, bootstrapping Two-sample z- and t- procedures, bootstrap, CI for OR Chi-square for homogeneity, independence, ANOVA, regression Content of Example Course (ISCAM)

Assessments Investigations with summaries of conclusions Worked out examples Practice problems  Quick practice, opportunity for immediate feedback, adjustment to class discussion Homework exercises Technology explorations (labs)  e.g., comparison of sampling variability with stratified sampling vs. simple random sampling Student projects  Student-generated research questions, data collection plans, implementation, data analyses, report April 8, 2008 CAUSE Webinar 14

April 8, 2008 CAUSE Webinar 15 Example 1: Friendly Observers Psychology experiment  Butler and Baumeister (1998) studied the effect of observer with vested interest on skilled performance A: vested interest B: no vested interest Total Beat threshold 3811 Do not beat threshold 9413 Total12 24 How often would such an extreme experimental difference occur by chance, if there was no vested interest effect?

April 8, 2008 CAUSE Webinar 16 Example 1: Friendly Observers Students investigate this question through  Hands-on simulation (playing cards)  Computer simulation (Java applet)  Mathematical model counting techniques

April 8, 2008 CAUSE Webinar 17 Example 1: Friendly Observers Focus on statistical process  Data collection, descriptive statistics, inferential analysis Arising from genuine research study  Connection between the randomization in the design and the inference procedure used Scope of conclusions depends on study design  Cause/effect inference is valid Use of simulation motivates the derivation of the mathematical probability model  Investigate/answer real research questions in first two weeks

April 8, 2008 CAUSE Webinar 18 Example 2: Sleep Deprivation Physiology Experiment  Stickgold, James, and Hobson (2000) studied the long-term effects of sleep deprivation on a visual discrimination task sleep condition n Mean StDev Median IQR deprived unrestricted How often would such an extreme experimental difference occur by chance, if there was no sleep deprivation effect? (3 days later!)

April 8, 2008 CAUSE Webinar 19 Example 2: Sleep Deprivation Students investigate this question through  Hands-on simulation (index cards)  Computer simulation (Minitab)  Mathematical model p-value= p-value .002

April 8, 2008 CAUSE Webinar 20 Example 2: Sleep Deprivation Experience the entire statistical process again  Develop deeper understanding of key ideas (randomization, significance, p-value) Tools change, but reasoning remains same  Tools based on research study, question – not for their own sake Simulation as a problem solving tool  Empirical vs. exact p-values

Example 3: Infants’ Social Evaluation Sociology study  Hamlin, Wynn, Bloom (2007) investigated whether infants would prefer a toy showing “helpful” behavior to a toy showing “hindering” behavior  Infants were shown a video with these two kinds of toys, then asked to select one  14 of month-olds selected helper Is this result surprising enough (under null model of no preference) to indicate a genuine preference for the helper toy?

Example 3: Infants’ Social Evaluation Simulate with coin flipping Then simulate with applet

Example 3: Infants’ Social Evaluation Then learn binomial distribution, calculate exact p- value

Example 3: Infants’ Social Evaluation Learn probability distribution to answer inference question from research study Again the analysis is completed with  Tactile simulation  Technology simulation  Mathematical model Modeling process of statistical investigation  Examination of methodology, further questions in study Follow-ups  Different number of successes  Different sample size

April 8, 2008 CAUSE Webinar 25 Example 4: Sleepless Drivers Sociology case-control study  Connor et al (2002) investigated whether those in recent car accidents had been more sleep deprived than a control group of drivers No full night’s sleep in past week At least one full night’s sleep in past week Sample sizes “case” drivers (crash) “control” drivers (no crash)

April 8, 2008 CAUSE Webinar 26 Example 4: Sleepless Drivers Sample proportion that were in a car crash Sleep deprived:.581 Not sleep deprived:.484 Odds ratio: 1.48 How often would such an extreme observed odds ratio occur by chance, if there was no sleep deprivation effect?

April 8, 2008 CAUSE Webinar 27 Example 4: Sleepless Drivers Students investigate this question through  Computer simulation (Minitab) Empirical sampling distribution of odds-ratio Empirical p-value  Approximate mathematical model 1.48

April 8, 2008 CAUSE Webinar 28 Example 4: Sleepless Drivers SE(log-odds) = Confidence interval for population log odds:  sample log-odds + z* SE(log-odds)  Back-transformation 90% CI for odds ratio: 1.05 – 2.08

April 8, 2008 CAUSE Webinar 29 Example 4: Sleepless Drivers Students understand process through which they can investigate statistical ideas Students piece together powerful statistical tools learned throughout the course to derive new (to them) procedures  Concepts, applications, methods, theory

April 8, 2008 CAUSE Webinar 30 For more information Investigating Statistical Concepts, Applications, and Methods (ISCAM), Cengage Learning, Instructor resources:  Solutions to investigations, practice problems, homework exercises  Instructor’s guide  Sample syllabi  Sample exams