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Dr. Kari Lock MorganDepartment of StatisticsPenn State University
USA
Making computing skills part of learning introductory stats
Royal Statistical Society10/13/16
ASA 2016 Recommendations for Intro StatGAISE: Guidelines for Assessment and Instruction in Statistics Education
1. Teach statistical thinking. ¡ Teach statistics as an investigative process of
problem-solving and decision making. ¡ Give students experience with multivariable thinking.
2. Focus on conceptual understanding. 3. Integrate real data with a context and purpose. 4. Foster active learning. 5. Use technology to explore concepts and analyze data. 6. Use assessments to improve and evaluate student learning.
1. Teach statistical thinking. ¡ Teach statistics as an investigative process of
problem-solving and decision making. ¡ Give students experience with multivariable thinking.
2. Focus on conceptual understanding. 3. Integrate real data with a context and purpose. 4. Foster active learning. 5. Use technology to explore concepts and analyze data. 6. Use assessments to improve and evaluate student learning.
Simulation-Based Inference
Question #1
Does drinking tea boost your immune system?
Tea and Immune Response
Antigensintea-BeveragePrimeHumanVγ2Vδ2TCellsinvitroandinvivoforMemoryandNon-memoryAntibacterialCytokineResponses,Kamath et.al.,ProceedingsoftheNationalAcademyofSciences,May13,2003.
• Participants were randomized to drink five or six cups of either tea (black) or coffee every day for two weeks (both drinks have caffeine but only tea has L-theanine)
• After two weeks, blood samples were exposed to an antigen, and immune system response was measured
• Does tea boost immunity (over coffee)?
Tea and Immune System
�̅�# − �̅�% = 34.82 − 17.70 = 17.12
1. Checkconditions
2. Computestatistic:chooseformula,plugandchug
3. Usetheoreticaldistribution(whichone?df?)
4. 0.025<p-value<0.05
Getting the p-value: Option 1
𝑡 =𝑥0 − 𝑥1
𝑠01𝑛0+ 𝑠11𝑛1
= 2.07
𝑛0 = 11𝑛1 = 10
So what’s a p-value???
• Plugging numbers into formulas does little to help reinforce conceptual understanding
• With a different formula for each test/interval, students often get mired in the details and fail to see the big picture
• We need a better way…
Traditional Inference
Actual ExperimentR R R R R
R R R R R
R R R R RR R R R R
Tea Coffee
R R RR R R R R
R R R
R R R R R
R
R
Actual Experiment
R R RR R R R R
R R R
R R R R R
R R R R13 18 205 1152 55 5647 48 R58
3 11 150 0
21 38 5216 21
Tea Coffee
Actual Experiment
R R RR R R R R
R R R
R R R R R
R R R R13 18 205 1152 55 5647 48 R58
3 11 150 0
21 38 5216 21
Tea Coffee
!!xT − xC =17.12
� Twoplausibleexplanations:¡Teaboostsimmunity
¡Randomchance Whatmighthappenjustbyrandomchance???
R R RR R R R R
R R R
R R R R R
R R R R13 18 205 1152 55 5647 48 R58
3 11 150 0
21 38 5216 21
Tea Coffee
R R RR R R R RR R 13 18 205 11
52 55 5647 48 R58
R R R
R R R R R
R R 3 11 150 0
21 38 5216 21
Simulation
Simulation
R R RR R R R R
R R R
R R R R R15 16 21
18 20 4721 13 R55
38 52 5
52 56 5811 48
Tea Coffee
R R RR R R R RR R 13 18 205 11
52 55 5647 48 R58
R
R R R R R
3 11 150 0
21 38 5216 21
Simulation
R R RR R R R R
R R R
R R R R R15 16 21
18 20 4721 13 R55
38 52 5
52 56 5811 48
Tea Coffee
0 3 R R0 11
!!xT − xC = −12.3
RepeatManyTimes!
We need technology!
StatKeywww.lock5stat.com/statkey
¡ Free¡ Easy to use¡ Online (or offline as chrome app)
p-valueProportionasextremeasobservedstatistic
observedstatistic
RandomizationTest
DistributionofstatisticifH0 true
If there were no difference between tea and coffee regarding immune system response, we would see results this extreme about 2.6% of the time
p-value: The chance of obtaining a statistic as extreme as that observed, just by random chance, if the null hypothesis is true
Simulation-Based Inference• Intrinsically connected to concepts
• Same procedure works for many statistics
• More generalizable (new statistics or designs)
• Minimal background knowledge needed
• Fewer conditions; conditions transparent
Question #2
What is the average mercury level of fish (Large Mouth
Bass) in Florida lakes?
Mercury Levels in Fish
!!n=53!!x =0.527!ppm
Lange, T.,Royals,H.andConnor,L.(2004). Mercuryaccumulation inlargemouthbass(Micropterus salmoides)inaFloridaLake.ArchivesofEnvironmentalContaminationandToxicology,27(4), 466-471.
!µ = ???
Getting a Margin of Error
Population Sample
Sample
Sample
SampleSampleSample
...
Calculatestatisticforeachsample
SamplingDistribution
StandardError(SE):standarddeviationofsamplingdistribution
MarginofError(ME)(95%CI:ME=2×SE)
statistic±ME
• Key idea: how much do statistics vary from sample to sample?• Problem?
• We can’t take lots of samples from the population!
Assessing Uncertainty
Getting a Margin of ErrorPopulation
(???)
statistic±ME
Sample
BestGuessatPopulation
Sample
Sample
Sample
SampleSampleSample
...
Distributionofthestatistic
Calculatestatisticforeachsample
StandardError(SE):standarddeviationofthestatistic
MarginofError(ME)(95%CI:ME=2×SE)
Bootstrapping� What is our best guess at the population,
given sample data?
¡The sample itself!
� Draw samples repeatedly from the sample data (of size n = 53)…
¡… with replacement! (bootstrapping)
� Calculate statistic for each bootstrap sample
� SE = standard deviation of these statistics
We Need Technology!� StatKey: lock5stat.com/statkey� Rossman/Chance: rossmanchance.com/applets� InZight: stat.auckland.ac.nz/~wild/iNZight� R: cran.r-project.org� RStudio: rstudio.com� Fathom: fathom.concord.org� Tinkerplots: tinkerplots.com� JMP: jmp.com� Minitab Express: minitab.com� StatCrunch: statcrunch.com
Red = Free
SE=0.047
statistic±2xSE0.527± 2x0.047(0.433,0.621)
MercuryLevelsinFish
Weare95%confidentthataveragemercurylevelinfishinFloridalakesisbetween0.433and0.621ppm.
95%ConfidenceInterval
Same process for every parameter!Estimatethemarginoferrorand/oraconfidenceintervalfor...• proportion(𝑝)• differenceinmeans(µ1 − µ2)• differenceinproportions(𝑝1 − 𝑝2)• standarddeviation(𝜎)• correlation(𝜌)• ... Samplewithreplacement
CalculatestatisticRepeat...
MercuryandpHinLakes
Lange,Royals,andConnor,TransactionsoftheAmericanFisheriesSociety(1993)
• ForFloridalakes,whatisthecorrelationbetweenaveragemercurylevel(ppm)infishtakenfromalakeandacidity(pH)ofthelake?
Givea95%CIforρ
r =-0.575
Simulation-Based Inference• Students leave the course with…
• Better conceptual understanding (Tintle et al, JSE, 2011; Maurer and Lock, TISE, 2016)
• Better retention of concepts (Tintle et al, SERJ, 2012)
• Broader ability to apply what they have learned
• Familiarity with modern computationally-intensive methods
Fall '14 Spring '15 Fall '15
2040
6080
100
Pos
t Tes
t Sco
re
Conceptual UnderstandingScores on a National Assessment
Averages:
p-value:0.00002
43% 60% 63% National:47%
National Assessment ResultsMost improvement on p-value questions!
T-test SBI Nat’l
Abletoreasonthatasmallerp-valueprovidesstrongerevidenceagainstthenullhypothesisthanalargerp-value.
29%
Abletoreasonaboutaconclusionbasedonastatisticallysignificantp-valueinthecontextofaresearchstudythatcomparestwogroups.
43%
45%
48%
68%
80%
Student Behavior• Students were given data on the second midterm (right after learning about t-intervals!) and asked to compute a confidence interval for the mean
• How they created the interval:Bootstrapping t.test inR Formula
84% 8% 8%
Use data from a sample survey to estimate a population mean or proportion; develop a margin of
error through the use of simulation methods for random sampling
Use data from a randomized experiment to compare two
treatments; use simulation to decide if differences between
parameters are significant
Common Core State Standards in Mathematics (High School)
Traditional Methods Still There� We use simulation-based inference to
introduce inference and build understanding
� We then cover traditional normal and t-based methods and SE formulas (goes quickly!)
� CLT easy to motivate after simulation
� Testing and intervals concepts already there
Sir R. A. Fisher
"Actually, the statistician does not carry out this very simple and very tedious process [the randomization test], but his conclusions have no justification beyond the fact that they agree with those which could have been arrived at by this elementary method."
-- Sir R. A. Fisher, 1936
George Cobb“... the consensus curriculum is still an unwitting prisoner of history. What we teach is largely the technical machinery of numerical approximations based on the normal distribution and its many subsidiary cogs. This machinery was once necessary, because the conceptually simpler alternative based on permutations was computationally beyond our reach. Before computers statisticians had no choice. These days we have no excuse. Randomization-based inference makes a direct connection between data production and the logic of inference that deserves to be at the core of every introductory course.”
-- Professor George Cobb, 2007
Want More?� Simulation-based inference blog:
causeweb.org/sbi
� Videos, presentations, and more: lock5stat.com
� Recordings from the 2016 Electronic Conference on Teaching Statistics (eCOTS): causeweb.org/cause/ecots/ecots16
� Email me: klm47@psu.edu
klm47@psu.edu
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