Synthesis and Review 2/20/12

• Hypothesis Tests: the big picture• Randomization distributions• Connecting intervals and tests• Review of major topics• Open Q+A

Section 4.4, 4.5, ES 2 Professor Kari Lock MorganDuke University

• Make one double-sided page of notes for in-class exam

• WORK PRACTICE PROBLEMS! (old exams and solutions to review questions

under Documents on course website)

• Read sections corresponding to anything you are still confused about

• Practice using technology to summarize, visualize, and perform inference on data

To Do

• Today, 3 – 4 pm (Prof Morgan) • Today, 4 – 6 pm (Christine)• Tuesday, 3 – 6 pm (Prof Morgan)• Tuesday, 6 – 8 pm (Yue)• Tuesday, 8 – 9 pm (Michael)

(My office hours this week have been moved to Monday and Tuesday to answer questions before the exam)

Office Hours this Week

Hypothesis Testing1. Define the parameter(s) of interest2. State your hypotheses3. Set significance level, (usually 0.05 if unspecified)4. (Collect your data)5. Plot your data6. Calculate the observed sample statistic7. Create a randomization distribution8. Calculate the p-value9. Assess the strength of evidence against H0

10. Make a formal decision based on the significance level11. Interpret the conclusion in context

Exercise and GenderAmong college students, does one gender exercise more than the other?

Mean number of exercise hours per week for femalesMean number of exercise hours per week for males

F

M

0 : 0: 0

F M

a F M

HH

Exercise and Gender

3.00M FX X

Exercise and Gender

www.lock5stat.com/statkey

p-value = 0.218 Little evidence against H0

Do not reject H0

This study does not provide evidence that there is any association between gender and exercise times among college students

Conclusion:

Results this extreme would happen about 22% of the time just by random chance if H0 were true, so this study does not provide adequate evidence against H0

Think:

• A randomization distribution is the distribution of statistics that would be observed, just by random chance, if the null hypothesis were true

1. Simulate randomizations assuming the null hypothesis is true

2. Calculate the statistic for each simulated randomization

Randomization Distribution

• In a randomized experiment the “randomness” is the random allocation of cases to treatment groups

• If the null hypothesis is true, it doesn’t make any difference which treatment group you get placed in

• Simulate randomizations assuming H0 is true by reallocate units to treatment groups, and keeping the response values the same

Randomized Experiments

• In observational studies, there is no random allocation to treatment groups

• In observational studies, what does “by random chance” even mean? What is random???

• How could we generate a randomization distribution for observational studies?

Observational Studies

• When data is collected by random sampling, without random allocation between groups, we can bootstrap to see what would happen by random chance

• Bootstrapping (resampling with replacement) simulates the distribution of the sample statistic that we would observe when taking many random samples of the population

Bootstrapping

• For a randomization distribution however, we need to know the distribution of the sample statistic, when the null hypothesis is true

• How could we bootstrap assuming the null hypothesis is true?

• Add/subtract values to each unit first to make the null hypothesis true (“shift the distribution”)

Bootstrapping

Reallocating versus Resampling

What is random? How do we simulate “random chance”?

Randomized Experiments

Random assignment to treatment groups

Reallocate (rerandomize)

Observational Studies

Random sampling from the population

Resample (bootstrap)

• In both cases, we need to make the null hypothesis true for a randomization distribution

Was the exercise by gender data collected via a randomized experiment?

(a)Yes(b) No(c) There is no way to tell

Exercise by Gender

• The randomness is not who is which gender (as with randomized experiments), but who is selected to be a part of the study

• Male sample mean: 12.4 hours• Female sample mean: 9.4 hours

• Add 3 hours to all the females, and then resample using bootstrapping

• www.lock5stat.com/statkey

Exercise by Gender

• Reallocating and resampling usually give similar answers in terms of a p-value

• For this class, it is fine to just use reallocating for tests, even if it is not actually a randomized experiment

• The point is to understand the reason for generating a randomization distribution

Method of Randomization

• Let’s return to the body temperature data

• Using bootstrapping, we found a 95% confidence interval for the mean body temperature to be (98.05, 98.47 )

• Let’s do a hypothesis test to see how much evidence this data provides against = 98.6

H0 : = 98.6Ha : ≠ 98.6

Body Temperatures

• How would we create a randomization distribution?

• The sample mean is 98.26. Add 0.34 to each unit so we can sample with replacement mimicking sampling from a population with mean 98.6

• Take many bootstrap samples to create a randomization distribution

Body Temperatures

Randomization Distribution

p-value = 0.002

Two Distributions

• If a (1 – α)% confidence interval does not contain the value of the null hypothesis, then a two-sided hypothesis test will reject the null hypothesis using significance level α• Intervals provide a range of plausible values for the population parameter, tests are designed to assess evidence against a null hypothesis

Intervals and Tests

• Using bootstrapping, we found a 95% confidence interval for the mean body temperature to be (98.05, 98.47 )

H0 : = 98.6Ha : ≠ 98.6

• At α = 0.05, we would reject H0

Body Temperatures

REVIEW

Population

Sample

Sampling

Statistical Inference

The Big Picture

Exploratory Data Analysis

Was the sample randomly selected?

Possible to generalize to

the population

Yes

Should not generalize to

the population

No

Was the explanatory variable randomly

assigned?

Possible to make

conclusions about causality

Yes

Can not make conclusions

about causality

No

Data Collection

Variable(s) Visualization Summary StatisticsCategorical bar chart,

pie chartfrequency table,

relative frequency table, proportion

Quantitative dotplot, histogram,

boxplot

mean, median, max, min, standard deviation,

z-score, range, IQR,five number summary

Categorical vs Categorical

side-by-side bar chart, segmented bar chart,

mosaic plot

two-way table, proportions

Quantitative vs Categorical

side-by-side boxplots statistics by group

Quantitative vs Quantitative

scatterplot correlation

Descriptive StatisticsThink of a topic or question you would like to use data to help you answer.

– What would the cases be? – What would the variables be? (Limit to one or two variables)

Descriptive StatisticsHow would you visualize and summarize the variable or relationship between variables?

a) bar chart/pie chart, proportions, frequency table/relative frequency table

b) dotplot/histogram/boxplot, mean/median, sd/range/IQR, five number summary

c) side-by-side or segmented bar plots/mosaic plots, difference in proportions, two-way table

d) side-by-side boxplot, stats by groupe) scatterplot, correlation

Statistic vs Parameter

• A sample statistic is a number computed from sample data.

• A population parameter is a number that describes some aspect of a population

Sampling Distribution

• A sampling distribution is the distribution of statistics computed for different samples of the same size taken from the same population

• The spread of the sampling distribution helps us to assess the uncertainty in the sample statistic

• In real life, we rarely get to see the sampling distribution – we usually only have one sample

• 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

Bootstrap

Original Sample

BootstrapSample

BootstrapSample

BootstrapSample

.

.

.

Bootstrap Statistic

Sample Statistic

Bootstrap Statistic

Bootstrap Statistic

.

.

.

Bootstrap Distribution

Confidence Interval

• A confidence interval for a parameter is an interval computed from sample data by a method that will capture the parameter for a specified proportion of all samples

• A 95% confidence interval will contain the true parameter for 95% of all samples

Standard Error

• The standard error (SE) is the standard deviation of the sample statistic

• The SE can be estimated by the standard deviation of the bootstrap distribution

• For symmetric, bell-shaped distributions, a 95% confidence interval is

statistic 2 SE

Percentile Method

• If the bootstrap distribution is approximately symmetric, a P% confidence interval can be gotten by taking the middle P% of a bootstrap distribution

Bootstrap DistributionBest Guess at Sampling Distribution

Statistic

2 3 4 5 6 7 8

Best Guess at Sampling Distribution

Statistic

2 3 4 5 6 7 8

Observed Statistic

Best Guess at Sampling Distribution

Statistic

2 3 4 5 6 7 8

Observed Statistic

P%

Best Guess at Sampling Distribution

Statistic

2 3 4 5 6 7 8

Observed Statistic

P%P%P%

Upper BoundUpper Bound

Lower Bound

• How unusual would it be to get results as extreme (or more extreme) than those observed, if the null hypothesis is true?

• If it would be very unusual, then the null hypothesis is probably not true!

• If it would not be very unusual, then there is not evidence against the null hypothesis

Hypothesis Testing

• The p-value is the probability of getting a statistic as extreme (or more extreme) as that observed, just by random chance, if the null hypothesis is true

• The p-value measures evidence against the null hypothesis

p-value

Hypothesis TestingDistribution of Statistic Assuming Null

Statistic

-3 -2 -1 0 1 2 3

Observed Statistic

Distribution of Statistic Assuming Null

Statistic

-3 -2 -1 0 1 2 3

Distribution of Statistic Assuming Null

Statistic

-3 -2 -1 0 1 2 3

Observed Statistic

p-value

• A randomization distribution is the distribution of sample statistics we would observe, just by random chance, if the null hypothesis were true

• The p-value is calculated by finding the proportion of statistics in the randomization distribution that fall beyond the observed statistic

Randomization Distribution

Statistical ConclusionsStrength of evidence against H0:

Formal decision of hypothesis test, based on = 0.05 :

statistically significant

not statistically significant

Formal Decisions

For a given significance level, ,

p-value < Reject Ho

p-value > Do not Reject Ho

Errors

Reject H0 Do not reject H0

H0 true

H0 false

TYPE I ERROR

TYPE II ERRORTrut

h

Decision

0If true, probability = H

0If true,

probability = 1H

If true, probability =

ap

Hower

If true, probability = 1

ap r

Howe

QUESTIONS???