making change happen
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Making Change Happen
Kari Lock MorganSTAT 790: Teaching Statistics
10/24/12
Introducing Inference with Simulation Methods
Sleep versus Caffeine
Mednick, Cai, Kanady, and Drummond (2008). “Comparing the benefits of caffeine, naps and placebo on verbal, motor and perceptual memory,” Behavioral Brain Research, 193, 79-86.
• Students were given words to memorize, then randomly assigned to take either a 90 min nap, or a caffeine pill. 2 ½ hours later, they were tested on their recall ability.
• words
• Is sleep better than caffeine for memory?
Traditional Inference
1 22 21 2
1 2
s sn n
X X
2 23.31
15
3.551
.
2 12
25 12.25
2.14
1. Conditions Met?
3. Calculate numbers and plug into formula
4. Plug into calculator
5. Which theoretical distribution?6. df?7. find p-value
0.025 < p-value < 0.05
> pt(2.14, 11, lower.tail=FALSE)[1] 0.02780265
2. Which formula?
1 2 12n n
• Confidence intervals and hypothesis tests using the normal and t-distributions
• With a different formula for each situation, students often get mired in the details and fail to see the big picture
• Plugging numbers into formulas does little to help reinforce conceptual understanding
Traditional Inference
• Simulation methods (bootstrapping and randomization) are a computationally intensive alternative to the traditional approach
• Rather than relying on theoretical distributions for specific test statistics, we can directly simulate the distribution of any statistic
• Great for conceptual understanding!
Simulation Methods
To generate a distribution assuming H0 is true:
•Traditional Approach: Calculate a test statistic which should follow a known distribution if the null hypothesis is true (under some conditions)
• Randomization Approach: Decide on a statistic of interest. Simulate many randomizations assuming the null hypothesis is true, and calculate this statistic for each randomization
Hypothesis Testing
http://www.youtube.com/watch?v=3ESGpRUMj9E
Paul the Octopus
• Paul the Octopus predicted 8 World Cup games, and predicted them all correctly
• Is this evidence that Paul actually has psychic powers?
• How unusual would this be if he was just randomly guessing (with a 50% chance of guessing correctly)?
• How could we figure this out?
Paul the Octopus
• Students each flip a coin 8 times, and count the number of heads
• Count the number of students with all 8 heads by a show of hands (will probably be 0)
• If Paul was just guessing, it would be very unlikely for him to get all 8 correct!
• How unlikely? Simulate many times!!!
Simulate with Students
• In a randomized experiment on treating cocaine addiction, 48 people were randomly assigned to take either Desipramine (a new drug), or Lithium (an existing drug)
• The outcome variable is whether or not a patient relapsed
• Is Desipramine significantly better than Lithium at treating cocaine addiction?
Cocaine Addiction
R R R R R RR R R R R RR R R R R RR R R R R R
R R R R R RR R R R R RR R R R R RR R R R R R
R R R RR R R R R RR R R R R RR R R R R R
R R R RR R R R R RR R R R R RR R R R R R
New Drug Old Drug1. Randomly assign units to treatment groups
R R R RR R R R R RR R R R R RN N N N N N
RRR R R RR R R R N NN N N N N N
RR
N N N N N N
R = RelapseN = No Relapse
R R R RR R R R R RR R R R R RN N N N N N
RRR R R RR R R R RRR R N N N N
RR
N N N N N N
2. Conduct experiment3. Observe relapse counts in each group
Old DrugNew Drug
10 relapse, 14 no relapse 18 relapse, 6 no relapse
1. Randomly assign units to treatment groups
10 1824
ˆ ˆ
24.333
new oldp p
• Assume the null hypothesis is true
• Simulate new randomizations
• For each, calculate the statistic of interest
• Find the proportion of these simulated statistics that are as extreme as your observed statistic
Randomization Test
R R R RR R R R R RR R R R R RN N N N N N
RRR R R RR R R R N NN N N N N N
RR
N N N N N N
10 relapse, 14 no relapse 18 relapse, 6 no relapse
R R R R R RR R R R N NN N N N N NN N N N N N
R R R R R RR R R R R RR R R R R RN N N N N N
R N R NR R R R R RR N R R R NR N N N R R
N N N RN R R N N NN R N R R NR N R R R R
Simulate another randomization
New Drug Old Drug
16 relapse, 8 no relapse 12 relapse, 12 no relapse
ˆ ˆ16 1224 240.167
N Op p
R R R RR R R R R RR R R R R RN N N N N N
RRR R R RR N R R N NR R N R N R
RR
R N R N R R
Simulate another randomization
New Drug Old Drug
17 relapse, 7 no relapse 11 relapse, 13 no relapse
ˆ ˆ17 1124 240.250
N Op p
• Give students index cards labeled R (28 cards) and N (20 cards)
• Have them deal the cards into 2 groups
• Compute the difference in proportions
• Contribute to a class dotplot for the randomization distribution
Simulate with Students
• Why did you re-deal your cards?
• Why did you leave the outcomes (relapse or no relapse) unchanged on each card?
Cocaine AddictionYou want to know what would happen
• by random chance (the random allocation to treatment groups)
• if the null hypothesis is true (there is no difference between the drugs)
Simulate with StatKey
The probability of getting results as extreme or more extreme than those observed if the null hypothesis is true, is about .02. p-value
Proportion as extreme as observed statistic
observed statistic
www.lock5stat.com/statkey
Distribution of Statistic Assuming Null is True
Sleep versus Caffeine
Mednick, Cai, Kanady, and Drummond (2008). “Comparing the benefits of caffeine, naps and placebo on verbal, motor and perceptual memory,” Behavioral Brain Research, 193, 79-86.
• Students were given words to memorize, then randomly assigned to take either a 90 min nap, or a caffeine pill. 2 ½ hours later, they were tested on their recall ability.
• words
• Is sleep better than caffeine for memory?
Simulation Inference• How extreme would a sample difference of 3 be, if
there were no difference between sleep and caffeine for word recall?
• What kind of differences would we observe, just by random chance?
• Simulate many randomizations, assuming no difference
• Calculate the p-value as the proportion of simulated randomizations that yield differences as extreme as the observed 3
Randomization Test
p-valueProportion as extreme as observed statistic
observed statistic
Distribution of Statistic Assuming Null is True
StatKey at www.lock5stat.com
Traditional Inference
1 22 21 2
1 2
s sn n
X X
2 23.31
15
3.551
.
2 12
25 12.25
2.14
1. Conditions Met?
3. Calculate numbers and plug into formula
4. Plug into calculator
5. Which theoretical distribution?6. df?7. find p-value
0.025 < p-value < 0.05
> pt(2.14, 11, lower.tail=FALSE)[1] 0.02780265
2. Which formula?
1 2 12n n
Simulation vs Traditional• Simulation methods
• intrinsically connected to concepts• same procedure applies to all statistics• no conditions to check• minimal background knowledge needed
• Traditional methods (normal and t based)• familiarity expected after intro stats• needed for future statistics classes• only summary statistics are needed• insight from standard error
Simulation AND Traditional?
• We introduces inference with simulation methods, then cover the traditional methods
• Students have seen the normal distribution appear repeatedly via simulation; use this common shape to motivate traditional inference
• “Shortcut” formulas give the standard error, avoiding the need for thousands of simulations
• Students already know the concepts, so can go relatively fast through the mechanics
TopicsCh 1: Collecting DataCh 2: Describing DataCh 3: Confidence Intervals (Bootstrap)Ch 4: Hypothesis Tests (Randomization)Ch 5: Normal DistributionCh 6: Inference for Means and Proportions (formulas
and theory)Ch 7: Chi-Square TestsCh 8: ANOVACh 9: RegressionCh 10: Multiple Regression(Optional): Probability
• Normal and t-based inference after bootstrapping and randomization:
• Students have seen the normal distribution repeatedly – CLT easy!
• Same idea, just using formula for SE and comparing to theoretical distribution
• Can go very quickly through this!
Theoretical Approach
• Introduce new statistic - 2 or F
• Students know that these can be compared to either a randomization distribution or a theoretical distribution
• Students are comfortable using either method, and see the connection!
• If conditions are met, the randomization and theoretical distributions are the same!
Chi-Square and ANOVA
Chi-Square Statistic
Randomization Distribution
Chi-Square Distribution (3 df)
p-value = 0.357
2 statistic = 3.242
2 statistic = 3.242 p-value = 0.356
Student Preferences
Which way of doing inference gave you a better conceptual understanding of confidence intervals and hypothesis tests?
Bootstrapping and Randomization
Formulas and Theoretical Distributions
113 51
69% 31%
Student Preferences
Which way did you prefer to learn inference (confidence intervals and hypothesis tests)?
Bootstrapping and Randomization
Formulas and Theoretical Distributions
105 60
64% 36%Simulation Traditional
AP Stat 31 36
No AP Stat 74 24
Student Behavior• Students were given data on the second midterm and asked to compute a confidence interval for the mean
• How they created the interval:
Bootstrapping t.test in R Formula94 9 9
84% 8% 8%
A Student Comment
" I took AP Stat in high school and I got a 5. It was mainly all equations, and I had no idea of the theory behind any of what I was doing.
Statkey and bootstrapping really made me understand the concepts I was learning, as opposed to just being able to just spit them out on an exam.”
- one of my students
It is the way of the past…
"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
… and the way of the future“... 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
• How to assess the variability of sample statistics?
• Sample repeatedly from the population to create a sampling distribution to compute the standard error?
• Reality: We only have one sample!
• How do we determine how much statistics vary from sample to sample, if we only have one sample???
Confidence Intervals
Population
µ
BUT, in practice we don’t see the “tree” or all of the “seeds” – we only have ONE seed
Sampling Distribution
Bootstrap“Population”
What can we do with just one seed?
Grow a NEW tree!
𝑥
Estimate the distribution and variability (SE) of ’s from the bootstrap statistics
Bootstrap Distribution
Bootstrap statistics are to the original sample statistic
as the original sample statistic is to the population parameter
Bootstrap Distribution
•The variability of the bootstrap statistics is similar to the variability of the sample statistics
• The standard error of a statistic can be estimated using the standard deviation of the bootstrap distribution!
Standard Error
• Imagine the population is many, many copies of the original sample (what do you have to assume?)
• Sample repeatedly from this mock population
• This is done by sampling with replacement from the original sample
Bootstrap Distribution
Atlanta Commutes
Data: The American Housing Survey (AHS) collected data from Atlanta in 2004.
What’s the mean commute time for workers in metropolitan Atlanta?
Atlanta Commutes - Sample
Where might the “true” μ be?
Time20 40 60 80 100 120 140 160 180
CommuteAtlanta Dot Plot
Original Sample
“Population” many copies of sample
Sample from this “population”
Bootstrapping• A bootstrap sample is a random sample taken with replacement from the original sample, of the same size as the original sample
• A bootstrap statistic is the statistic computed on the bootstrap sample
• A bootstrap distribution is the distribution of many bootstrap statistics
Original Sample
BootstrapSample
BootstrapSample
BootstrapSample
.
.
.
Bootstrap Statistic
Sample Statistic
Bootstrap Statistic
Bootstrap Statistic
.
.
.
Bootstrap Distribution
Simulate with StatKeywww.lock5stat.com/statkey
Middle 95% of bootstrap statistics: (27.4, 30.9)
Other Changes• More Bayesian inference in undergraduate courses?
• More emphasis on analysis of big datasets?
• ?
Making Change Happen• You are the next generation of statistics professors… you are best suited to implement change!
• Think hard about the existing curriculum and pedagogy… is it best for the students?
• If not, how could it be made better?
• Talk to colleagues, attend conferences, …
Making Change Happen• The 2013 United States Conference on Teaching Statistics (USCOTS) will be in the Research Triangle Park!
• May 16-18, 2013
• Conference theme: “Making Change Happen”
• Plenary sessions, breakout sessions, posters
• You are all encouraged to attend!
Practice Teaching• 12 min, any statistics topic• “It’s not what you teach, it’s what they learn”• Motivate the topic• Use examples, and real data if possible• Stress conceptual understanding• Try visuals to clarify concepts• What are the key points?• Computer or board?• Engage the students!
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