exploratory data analysis estimation...exploratory data analysis • before doing inference, always...

12/23/2012

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Statistics: Unlocking the Power of Data Lock5

STAT 101 Dr. Kari Lock Morgan

12/6/12

Synthesis

Big Picture Essential Synthesis

• Bayesian Inference (continued)

• Review

• Synthesis Activities


Data Collection • The way the data are/were collected determines the scope of inference

• For generalizing to the population: was it a random sample? Was there sampling bias?

• For assessing causality: was it a randomized experiment?

• Collecting good data is crucial to making good inferences based on the data


Exploratory Data Analysis • Before doing inference, always explore your data with descriptive statistics

• Always visualize your data! Visualize your variables and relationships between variables

• Calculate summary statistics for variables and relationships between variables – these will be key for later inference

• The type of visualization and summary statistics depends on whether the variable(s) are categorical or quantitative


Estimation

• For good estimation, provide not just a point estimate, but an interval estimate which takes into account the uncertainty of the statistic

• Confidence intervals are designed to capture the true parameter for a specified proportion of all samples

• A P% confidence interval can be created by

• bootstrapping (sampling with replacement from the sample) and using the middle P% of bootstrap statistics

• *statisti z Sc E


Hypothesis Testing • A p-value is the probability of getting a statistic as extreme as observed, if H0 is true

• The p-value measures the strength of the evidence the data provide against H0

• “If the p-value is low, the H0 must go”

• If the p-value is not low, then you can not reject H0 and have an inconclusive test


p-value • A p-value can be calculated by

• A randomization test: simulate statistics assuming H0 is true, and see what proportion of simulated statistics are as extreme as that observed

• Calculating a test statistic and comparing that to a theoretical reference distribution (normal, t, 2, F)

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Hypothesis Tests Variables Appropriate Test

One Quantitative Single mean (t)

One Categorical Single proportion (normal) Chi-square Goodness of Fit

Two Categorical Difference in proportions (normal) Chi-square Test for Association

One Quantitative, One Categorical

Difference in means (t) Matched pairs (t) ANOVA (F)

Two Quantitative Correlation (t) Slope in Simple Linear Regression (t)

More than two Multiple Regression (t, F)


Regression • Regression is a way to predict one response variable with multiple explanatory variables

• Regression fits the coefficients of the model

• The model can be used to

• Analyze relationships between the explanatory variables and the response

• Predict Y based on the explanatory variables

• Adjust for confounding variables

0 1 2 21 ... k k ix xxy


Probability

( or ) ( ) ( ) ( and )P A B P A P B P A B

( ) ( and ) + P(A and not )P A P A B B

( if ) (( and ) )P P A BA PB B

(not ) 1 ( )P A P A

( and ) ( if ) ( )

(( if )

) ( )

P A B P B A P A

PP A

BB

B P


Romance

• What variables help to predict romantic interest?

• Do these variables differ for males and females?

• All we need to figure this out is DATA!

(For all of you, being almost done with STAT 101, this is the case for many interesting questions!)


Speed Dating • We will use data from speed dating conducted at Columbia University, 2002-2004

• 276 males and 276 females from Columbia’s various graduate and professional schools

• Each person met with 10-20 people of the opposite sex for 4 minutes each

• After each encounter each person said either “yes” (they would like to be put in touch with that partner) or “no”


Speed Dating Data

What are the cases?

a) Students participating in speed dating b) Speed dates c) Ratings of each student

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Speed Dating

What is the population?

Ideal population?

More realistic population?


Speed Dating

It is randomly determined who the students will be paired with for the speed dates.

We find that people are significantly more likely to say “yes” to people they think are more intelligent.

Can we infer causality between perceived intelligence and wanting a second date?

a) Yes b) No


Successful Speed Date?

What is the probability that a speed date is successful (results in both people wanting a second date)?

To best answer this question, we should use

a) Descriptive statistics b) Confidence Interval c) Hypothesis Test d) Regression e) Bayes Rule


Successful Speed Date?

63 of the 276 speed dates were deemed successful (both male and female said yes).

A 95% confidence interval for the true proportion of successful speed dates is

a) (0.2, 0.3) b) (0.18, 0.28) c) (0.21, 0.25) d) (0.13, 0.33)

𝑝 ± 2𝑝 (1 − 𝑝 )

𝑛

𝑝 = 63/276 = 0.23

0.23 ±20.23 (1−0.23)

276

0.23 0.05 (0.18, 0.28)


Pickiness and Gender

Are males or females more picky when it comes to saying yes?

Guesses?

a) Males b) Females



Are males or females more picky when it comes to saying yes? How could you answer this?

a) Test for a single proportion b) Test for a difference in proportions c) Chi-square test for association d) ANOVA e) Either (b) or (c)

Yes No

Males 146 130

Females 127 149

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Do males and females differ in their pickiness? Using α = 0.05, how would you answer this?

a) Yes b) No c) Not enough information Statistics: Unlocking the Power of Data Lock5

Reciprocity

Are people more likely to say yes to someone who says yes back? How would you best answer this?

a) Descriptive statistics b) Confidence Interval c) Hypothesis Test d) Regression e) Bayes Rule

Male says Yes Male says No

Female says Yes 63 64

Female says No 83 66


Reciprocity

Are people more likely to say yes to someone who says yes back? How could you answer this?

a) Test for a single proportion b) Test for a difference in proportions c) Chi-square test for association d) ANOVA e) Either (b) or (c)

Male says Yes Male says No

Female says Yes 63 64

Female says No 83 66

p-value =0.3731


Reciprocity

Are people more likely to say yes to someone who says yes back?

p-value = 0.3731

Based on this data, we cannot determine whether people are more likely to say yes to someone who says yes back.


Race and Response: Females

Does the chance of females saying yes to males differ by race?

How could you answer this question?

a) Test for a single proportion b) Test for a difference in proportions c) Chi-square goodness of fit d) Chi-square test for association e) ANOVA

Asian Black Caucasian Latino Other

𝒑 0.50 0.57 0.42 0.48 0.53

p-value =0.69


Race and Response: Males

Each person rated their date on a scale of 1-10 based on how much they liked them overall.

Does how much males like females differ by race?

How would you test this?

a) Chi-square test b) t-test for a difference in means c) Matched pairs test d) ANOVA e) Either (b) or (d)

p-value =0.892

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Physical Attractiveness

Each person also rated their date from 1-10 on the physical attractiveness. Do males rate females higher, or do females rate males higher?

Which tool would you use to answer this question?

a) Two-sample difference in means b) Matched pair difference in means c) Chi-Square d) ANOVA e) Correlation


Physical Attractiveness

𝑥 𝑀 − 𝑥 𝐹 = 0.406

95% CI: (0.10, 0.71) p-value = 0.01


Other Ratings Each person also rated their date from 1-10 on

the following attributes:

Attractiveness Sincerity Intelligence How fun the person seems Ambition Shared interests

Which of these best predict how much someone will like their date?


Multiple Regression MALES RATING FEMALES:

FEMALES RATING MALES:


Ambition and Liking

How does the perceived ambition of a date relate to how much the date is liked?

How would you answer this question?

a) Inference for difference in means b) ANOVA c) Inference for correlation d) Inference for simple linear regression e) Either (b), (c) or (d)


Ambition and Liking

r = 0.44, SE = 0.05 Find a 95% CI for .

𝛽 1 =0.28, SE = 0.06 Test whether 1 differs from 0.

.44 2(.05) = (0.34, 0.54) t = 0.28/0.06 = 4.67 => significant

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Data!

If you have a question that needs answering…

ALL YOU NEED IS DATA!!!!


Final

Tuesday, December 11th, 2 – 5pm

No make-ups, no excuses

25% of your course grade

Cumulative from the entire course

Open only to a calculator and 3 double-sided pages of notes prepared only by you

StatKey will be available if needed for theoretical distributions, but a calculator will be sufficient


Office Hours Before Final

Sunday, 4 – 7pm, Tracy, Old Chem 211 A

Monday, 12 – 3pm, Prof Morgan, Old Chem 216

Monday, 4 – 6pm, Heather, Old Chem 211A

Monday, 6 – 9pm, Sam, Old Chem 211A

Tuesday, 12 – 1pm, Prof Morgan, Old Chem 216


To Do Project 2 individual grades on Sakai (due Monday,

12/10)

Do Homework 9 (all practice problems)

Study for final!

Do Big Picture Essential Synthesis problems (solutions)

Do Practice Final (solutions)

If you want more problems to do… any odd essential synthesis or review problems (solutions

under documents on course website) any problem in the book (solutions in my office – can

check during office hours on Monday)


Thank You!!!

http://stat.duke.edu/courses/Fall12/sta101.002/HW9.pdf

https://sakai.duke.edu/access/content/group/e941fd20-1580-4930-8a6a-225b4b69a70a/BigESsolutions.pdf

https://sakai.duke.edu/access/content/group/e941fd20-1580-4930-8a6a-225b4b69a70a/PracticeFinal.pdf

https://sakai.duke.edu/access/content/group/e941fd20-1580-4930-8a6a-225b4b69a70a/PracticeFinal_sol.pdf

exploratory data analysis estimation...exploratory data analysis • before doing inference, always...

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