Building Conceptual Understanding of Statistical Inference Patti Frazer Lock Cummings Professor of Mathematics St. Lawrence University Canton, New York.

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Building Conceptual Understanding of Statistical Inference Patti Frazer Lock Cummings Professor of Mathematics St. Lawrence University Canton, New York AMATYC November, 2013

The Lock 5 Team Dennis Iowa State Kari Harvard/Duke Eric UNC/Duke Robin & Patti St. Lawrence

New Simulation Methods “The Next Big Thing” United States Conference on Teaching Statistics, May 2011 Common Core State Standards in Mathematics Increasingly used in the disciplines

New Simulation Methods Increasingly important in DOING statistics Outstanding for use in TEACHING statistics Help students understand the key ideas of statistical inference

“New” Simulation Methods? "Actually, the statistician does not carry out this very simple and very tedious process, 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

Bootstrap Confidence Intervals and Randomization Hypothesis Tests

First: Bootstrap Confidence Intervals

Example 1: What is the average price of a used Mustang car? Select a random sample of n=25 Mustangs from a website (autotrader.com) and record the price (in $1,000’s) for each car.

Sample of Mustangs: Our best estimate for the average price of used Mustangs is $15,980, but how accurate is that estimate?

We would like some kind of margin of error or a confidence interval. Key concept: How much can we expect the sample means to vary just by random chance?

Traditional Inference 2. Which formula? 3. Calculate summary stats 6. Plug and chug 4. Find t * 5. df? OR t * = Interpret in context CI for a mean 1. Check conditions

“We are 95% confident that the mean price of all used Mustang cars is between $11,390 and $20,570.” Answer is good, but the process is not very helpful at building understanding. Our students are often great visual learners but get nervous about formulas and algebra. Can we find a way to use their visual intuition?

Bootstrapping Brad Efron Stanford University Key Idea: Assume the “population” is many, many copies of the original sample. “Let your data be your guide.”

Suppose we have a random sample of 6 people:

Original Sample A simulated “population” to sample from

Bootstrap Sample: Sample with replacement from the original sample, using the same sample size. Original SampleBootstrap Sample

Original Sample Bootstrap Sample

Original Sample Bootstrap Sample ●●●●●● Bootstrap Statistic Sample Statistic Bootstrap Statistic ●●●●●● Bootstrap Distribution

We need technology! StatKey (Free, easy-to-use, works on all platforms)

StatKey Standard Error

Using the Bootstrap Distribution to Get a Confidence Interval Keep 95% in middle Chop 2.5% in each tail We are 95% sure that the mean price for Mustangs is between $11,930 and $20,238

Example 2: What yes/no question do you want to ask the sample of people in this audience? MAYBE: Did you/are you going to dress up in any kind of costume this week? OR: Is this your first time at AMATYC? OR: Do you live in California?

Raise your hand if your answer to the question is YES. Example #2 : Find a 90% confidence interval for the proportion of people attending AMATYC interested in introductory statistics who would answer “yes” to this question.

Why does the bootstrap work?

Sampling Distribution Population µ BUT, in practice we don’t see the “tree” or all of the “seeds” – we only have ONE seed

Bootstrap Distribution Bootstrap “Population” What can we do with just one seed? Grow a NEW tree! µ

Example 3: Diet Cola and Calcium What is the difference in mean amount of calcium excreted between people who drink diet cola and people who drink water? Find a 95% confidence interval for the difference in means.

What About Hypothesis Tests?

P-value: The probability of seeing results as extreme as, or more extreme than, the sample results, if the null hypothesis is true. Say what????

Example 1: Beer and Mosquitoes Does consuming beer attract mosquitoes? Experiment: 25 volunteers drank a liter of beer, 18 volunteers drank a liter of water Randomly assigned! Mosquitoes were caught in traps as they approached the volunteers. 1 1 Lefvre, T., et. al., “Beer Consumption Increases Human Attractiveness to Malaria Mosquitoes, ” PLoS ONE, 2010; 5(3): e9546.

Beer and Mosquitoes Beer mean = 23.6 Water mean = Does drinking beer actually attract mosquitoes, or is the difference just due to random chance? Beer mean – Water mean = 4.38 Number of Mosquitoes BeerWater

Traditional Inference 2. Which formula? 3. Calculate numbers and plug into formula 4. Plug into calculator 5. Which theoretical distribution? 6. df? 7. find p-value < p-value < Check conditions

Simulation Approach Beer mean = 23.6 Water mean = Does drinking beer actually attract mosquitoes, or is the difference just due to random chance? Beer mean – Water mean = 4.38 Number of Mosquitoes BeerWater

Simulation Approach Number of Mosquitoes BeerWater Find out how extreme these results would be, if there were no difference between beer and water. What kinds of results would we see, just by random chance?

Simulation Approach Number of Mosquitoes BeerWater Find out how extreme these results would be, if there were no difference between beer and water. What kinds of results would we see, just by random chance? Number of Mosquitoes Beverage

Simulation Approach Beer Water Find out how extreme these results would be, if there were no difference between beer and water. What kinds of results would we see, just by random chance? Number of Mosquitoes Beverage

StatKey! P-value

Traditional Inference 1. Which formula? 2. Calculate numbers and plug into formula 3. Plug into calculator 4. Which theoretical distribution? 5. df? 6. find p- value < p-value < 0.001

Beer and Mosquitoes The Conclusion! The results seen in the experiment are very unlikely to happen just by random chance (just 1 out of 1000!) We have strong evidence that drinking beer does attract mosquitoes!

“Randomization” Samples Key idea: Generate samples that are (a)based on the original sample AND (a)consistent with some null hypothesis.

Example 2: Malevolent Uniforms Do sports teams with more “malevolent” uniforms get penalized more often?

Example 2: Malevolent Uniforms Sample Correlation = 0.43 Do teams with more malevolent uniforms commit more penalties, or is the relationship just due to random chance?

Simulation Approach Find out how extreme this correlation would be, if there is no relationship between uniform malevolence and penalties. What kinds of results would we see, just by random chance? Sample Correlation = 0.43

Randomization by Scrambling

StatKey P-value

Malevolent Uniforms The Conclusion! The results seen in the study are unlikely to happen just by random chance (just about 1 out of 100). We have some evidence that teams with more malevolent uniforms get more penalties.

P-value: The probability of seeing results as extreme as, or more extreme than, the sample results, if the null hypothesis is true. Yeah – that makes sense!

Example 3: Light at Night and Weight Gain Does leaving a light on at night affect weight gain? In particular, do mice with a light on at night gain more weight than mice with a normal light/dark cycle? Find the p-value and use it to make a conclusion.

Simulation Methods These randomization-based methods tie directly to the key ideas of statistical inference. They are ideal for building conceptual understanding of the key ideas. Not only are these methods great for teaching statistics, but they are increasingly being used for doing statistics.

How does everything fit together? We use these methods to build understanding of the key ideas. We then cover traditional normal and t- tests as “short-cut formulas”. Students continue to see all the standard methods but with a deeper understanding of the meaning.

It is the way of the past… "Actually, the statistician does not carry out this very simple and very tedious process, 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

Additional Resources Statkey Descriptive Statistics Sampling Distributions Normal and t-Distributions

Thanks for listening!