Chi-Square Tests3/14/12

• Testing the distribution of a single categorical variable : 2 goodness of fit

• Testing for an association between two categorical variables: 2 test for association

Section 7.1, 7.2 Professor Kari Lock MorganDuke University

For more information and to sign up, go to www.stat.duke.edu/datafest

• NOTE: This week only, my Friday office hours will be from 3:30 – 5:30 pm (NOT 1-3pm as usual)

Office Hours This Week

• Homework 6 (due Monday, 3/19)

• Anonymous Midterm Evaluation (due Monday, 3/19, 5pm)

• Project 1 (due Thursday, 3/22, 5pm)

To Do

Multiple Categories

•So far, we’ve learned how to do inference for categorical variables with only two categories

•Today, we’ll learn how to do hypothesis tests for categorical variables with multiple categories

Rock-Paper-Scissors (Roshambo)

Play a game!

Can we use statistics to help us win?

Rock-Paper-Scissors

Which did you throw?

a) Rockb) Paperc) Scissors

Rock-Paper-ScissorsROCK PAPER SCISSORS

34 24 40

How would we test whether all of these categories are equally likely?

Hypothesis Testing

1.State Hypotheses

2.Calculate a statistic, based on your sample data

3.Create a distribution of this statistic, as it would be observed if the null hypothesis were true

4.Measure how extreme your test statistic from (2) is, as compared to the distribution generated in (3)

test statistic

Hypotheses

Let pi denote the proportion in the ith category.

H0 : All pi s are the sameHa : At least one pi differs from the others

OR

H0 : Every pi = 1/3Ha : At least one pi ≠ 1/3

Test Statistic

Why can’t we use the familiar formula

to get the test statistic?

•More than one sample statistic•More than one null value

We need something a bit more complicated…

sample statistic null valueSE

Observed Counts•The observed counts are the actual counts observed in the study

ROCK PAPER SCISSORSObserved 34 24 40

Expected Counts•The expected counts are the expected counts if the null hypothesis were true

•For each cell, the expected count is the sample size (n) times the null proportion, pi

expected = inp

ROCK PAPER SCISSORSObserved 34 24 40Expected 33 33 33

Chi-Square Statistic

•A test statistic is one number, computed from the data, which we can use to assess the null hypothesis

•The chi-square statistic is a test statistic for categorical variables:

22 observed - expected

expected

Rock-Paper-ScissorsROCK PAPER SCISSORS

Observed 34 24 40Expected 33 33 33

2 2 22 34 33 24 33 40 33

3.9733 33 33

What Next?

We have a test statistic. What else do we need to perform the hypothesis test?

A distribution of the test statistic assuming H0 is true

How do we get this? Two options:1) Simulation2) Distributional Theory

Simulation

Let’s see what kind of statistics we would get if rock, paper, and scissors are all equally likely.

1) Take 3 scraps of paper and label them Rock, Paper, Scissors. Fold or crumple them so they are indistinguishable. Choose one at random.

2) What did you get?(a) Rock (b) Paper (c) Scissors

3) Calculate the 2 statistic.

4) Form a distribution.

5) How extreme is the actual test statistic?

RStudio

Chi-Square Distribution

Chi-Square Distribution•If each of the expected counts are at least 5, AND if the null hypothesis is true, then the 2 statistic follows a 2 –distribution, with degrees of freedom equal to

df = number of categories – 1

• Rock-Paper-Scissors:

df = 3 – 1 = 2

Chi-Square Distribution

Chi-Square p-values1.Applet: http://surfstat.anu.edu.au/surfstat-home/tables/chi.php

2.RStudio:pchisq(ts,df,lower.tail=FALSE)

3.TI-83:2nd DISTR 7:2cdf( lower bound, upper bound, df

Because the 2 is always positive, the p-value (the area as extreme or more extreme) is always the upper tail

t-distribution p-values and t*

1.Applet: http://surfstat.anu.edu.au/surfstat-home/tables/t.php

2.RStudio:p-value: pt(ts,df) (use lower.tail=TRUE or FALSE,

and double if two-tailed) t* for 95% CI: qt(0.975,df)

3.TI-83:p-value: 2nd DISTR 5: tcdf( lower bound, upper bound, dft*: Approximate with a normal (invnorm)

Goodness of Fit

• A 2 test for goodness of fit tests whether the distribution of a categorical variable is the same as some null hypothesized distribution

• The null hypothesized proportions for each category do not have to be the same

Chi-Square Test for Goodness of Fit1.State null hypothesized proportions for each category, pi.

Alternative is that at least one of the proportions is different than specified in the null.

2.Calculate the expected counts for each cell as npi . Make

sure they are all greater than 5 to proceed.

3.Calculate the 2 statistic:

4.Compute the p-value as the area in the tail above the 2 statistic, for a 2 distribution with df = (# of categories – 1)

5.Interpret the p-value in context.

22 observed - expected

expected

Mendel’s Pea Experiment

Source: Mendel, Gregor. (1866). Versuche über Pflanzen-Hybriden. Verh. Naturforsch. Ver. Brünn 4: 3–47 (in English in 1901, Experiments in Plant Hybridization, J. R. Hortic. Soc. 26: 1–32)

• In 1866, Gregor Mendel, the “father of genetics” published the results of his experiments on peas

• He found that his experimental distribution of peas closely matched the theoretical distribution predicted by his theory of genetics (involving alleles, and dominant and recessive genes)

Mendel’s Pea ExperimentMate SSYY with ssyy:Þ 1st Generation: all Ss Yy

Mate 1st Generation:Þ 2nd Generation

Second Generation

S, Y: Dominants, y: Recessive

Phenotype Theoretical Proportion

Round, Yellow 9/16

Round, Green 3/16

Wrinkled, Yellow 3/16

Wrinkled, Green 1/16

Mendel’s Pea ExperimentPhenotype Theoretical

ProportionObserved

Counts

Round, Yellow 9/16 315

Round, Green 3/16 101

Wrinkled, Yellow 3/16 108

Wrinkled, Green 1/16 32

Let’s test this data against the null hypothesis of each pi equal to the theoretical value, based on genetics

0 1 2 3 4

0

: 9 /16, 3 /16, 3 /16, 1/16: At is not as specified least one in a i

p p p pHH

Hp

Mendel’s Pea ExperimentPhenotype Theoretical

ProportionObserved

Counts

Round, Yellow 9/16 315

Round, Green 3/16 101

Wrinkled, Yellow 3/16 108

Wrinkled, Green 1/16 32

The 2 statistic is made up of 4 components (because there are 4 categories). What is the contribution of the first cell (315)?

(a) 0.016 (b) 1.05 (c) 5.21 (d) 107.2

Mendel’s Pea ExperimentPhenotype pi Observed

CountsExpected Counts

Round, Yellow 9/16 315 556 9/16 = 312.75 Round, Green 3/16 101 556 3/16 = 104.25

Wrinkled, Yellow 3/16 108 556 3/16 = 104.25Wrinkled, Green 1/16 32 556 1/16 = 34.75

556n expected = 556

22

2 2 2 2315 312.75 101 104.25 108 104.25 32 34.75312.75 104.25 104.25 34.7

observed expectedexpecte

50.470

d

Mendel’s Pea Experiment2 0.470

• Compare to a 2 distribution with 4 – 1 = 3 degrees of freedomhttp://surfstat.anu.edu.au/surfstat-home/tables/chi.php

p-value = 0.9254

Mendel’s Pea Experimentp-value = 0.9254

Does this prove Mendel’s theory of genetics? Or at least prove that his theoretical proportions for pea phenotypes were correct?

(a) Yes(b) No

Two Categorical Variables

• The statistics behind a 2 test easily extends to two categorical variables

• A 2 test for association (often called a 2 test for independence) tests for an association between two categorical variables

Rock-Paper-Scissors

• Did reading the example on Rock-Paper-Scissors in the textbook influence the way you play the game?

• Did you read the Rock-Paper-Scissors example in Section 6.3 of the textbook?

a) Yesb) No

Rock-Paper-ScissorsRock Paper Scissors Total

Read Example 2 10 5 17

Did Not Read Example 32 12 34 80

Total 34 24 39 97

H0 : Whether or not a person read the example in the textbook is not associated with how a person plays rock-paper-scissors

Ha : Whether or not a person read the example in the textbook is associated with how a person plays rock-paper-scissors

Expected Countscorow total ex lumn totapected =

sample el

siz

Maintains row and column totals, but redistributes the counts as if there were no association between the variables

Rock Paper Scissors Total

Read Example 2 10 5 17

Did Not Read Example 32 12 34 80

Total 34 24 39 97

Chi-Square

22 observed - expected

expected

Number of rows = Number of columns

df = ( 1) ( 1)

rc

r c

Chi-Square StatisticOBSERVED Rock Paper Scissors Total

Read Example 2 10 5 17

Did Not Read Example 32 12 34 80

Total 34 24 39 97

222222

225.96104.2156.843228.041419.793432.165.964.216.8428.0419.7932.16

15.4

EXPECTED Rock Paper Scissors Total

Read Example 4.21 6.84 17

Did Not Read Example 28.04 19.79 32.16 80

Total 34 24 39 97

17 5.9697

34

Rock-Paper-Scissors

df = (2 1) (3 1) 2

The smallest expected count is 5.96, which is greater than 5, so it is okay to use the chi-square distribution to find the p-value.

p-value = 0.0005

2 15.4

We have strong evidence that there is an association between reading the RPS example in the textbook and the first RPS throw. (This means you are paying attention and using what you learn in the textbook! )

Chi-Square Test for Association1.H0 : The two variables are not associated

Ha : The two variables are associated

2.Calculate the expected counts for each cell:

Make sure they are all greater than 5 to proceed.

3.Calculate the 2 statistic:

4.Compute the p-value as the area in the tail above the 2 statistic, for a 2 distribution with df = (r – 1) (c – 1)

5.Interpret the p-value in context.

22 observed - expected

expected

corow total ex lumn totapected = sample e

l siz

Metal Tags and PenguinsLet’s return to the question of whether metal tags are detrimental to penguins.

Source: Saraux, et. al. (2011). “Reliability of flipper-banded penguins as indicators of climate change,” Nature, 469, 203-206.

Survived Died Total

Metal Tag 33 134 167

Electronic Tag 68 121 189

Total 101 255 356

Is there an association between type of tag and survival?(a) Yes (b) No

Metal Tags and Penguins

Calculate expected counts (shown in parentheses). First cell:

Calculate the chi-square statistic:

p-value:

Survived Died Total

Metal Tag 33 134 167

Electronic Tag 68 121 189

Total 101 255 356

Survived Died Total

Metal Tag 33 (47.38) 134 (119.62) 167

Electronic Tag 68 (53.62) 121 (135.38) 189

Total 101 255 356

2 2 2 22 33 47.38 134 119.62 68 53.62 121 135.38

11.4847.38 119.62 53.62 135.38

df = (2 – 1) × (2 – 1) = 1 p-value = 0.0007

There is strong evidence that metal tags are detrimental to penguins.

row total 167 47.38sam

column total 1ple size 356

01

Metal Tags and Penguins

Difference in proportions test:z = -3.387 (we got something slightly different due to rounding)

p-value = 0.0007 (two-sided)

Chi-Square test:2 = 11.47p-value = 0.0007

Note: (-3.387)2 = 11.47

Two Categorical Variables with Two Categories Each

• When testing two categorical variables with two categories each, the 2 statistic is exactly the square of the z-statistic

• The 2 distribution with 1 df is exactly the square of the normal distribution

• Doing a two-sided test for difference in proportions or doing a 2 test will produce the exact same p-value.

Summary• The 2 test for goodness of fit tests whether one categorical variable differs from a hypothesized distribution

• The 2 test for association tests whether two categorical variables are associated

• For both, you calculate expected counts for each cell, compute the 2 statistic as

and find the p-value as the area above this statistic on a 2 distribution

22 observed - expected

expected