Chi Square_Additional Lectures

8/3/2019 Chi Square_Additional Lectures

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Chi-Square

2

Introduction to Nonparametrics

Psychology 320

Gary S. Katz, Ph.D.

Definitions

Parametric Tests

Statistical tests that involve assumptions about

or estimations of population parameters.

(what weve been learning)

Nonparametric Tests

Also known as distribution-free tests

Statistical tests that do not rely on assumptions

of distributions or parameter estimates

(what were going to be learning)

More Definitions

The Chi-Square (X2) test is a nonparametric

test that is used to test hypotheses about

distributions of frequencies across

categories of data.

Different from what weve been learning

Then Now

Averages Frequencies

Scales Categories

Two Applications of the Chi-Square Test

The X2 goodness-of-fit test.

Used when we have distributions of frequencies

across two or more categories on one variable.

Test determines how well a hypothesized

distribution fits an obtained distribution.

The X2 test of independence.

Used when we compare the distribution of

frequencies across categories in two or more

independent samples.

Used in a single sample when we want to knowwhether two categorical variables are related.

The X2 Goodness-of-Fit Test

Flowers & GeneticsIn my backyard, I have a new hybrid rose bush.

I hypothesize that (according to Mendelian genetic

theory) that I should have 50% pink flowers,

25% white flowers, and 25% red flowers.

Pp Pp

PP Pp Pp pp


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FlowersI grow 120 of these plants from seed. The resulting

colors of flowers are as follows:

Pink White Red

75 20 25

Flowers - Reality & ExpectationsRecall, my expectations were 50% Pink, 25% White, 25% Red.

Pink White Red

75 20 25Observed

60 30 30Expected

So, if I planted 120 seeds, Id expectthis set of colored flowers.

Flowers - Reality & Expectations

Pink White Red

75 20 25Observed

60 30 30Expected

If my hypothesis is true (50%, 25%, 25%), how likely is it that I

could get this difference between my actual distribution

and my expected distribution of colored flowers?

The Chi-Square Test

Used to determine if the probability < , in

which case the hypothesis is rejected

or

if the probability > , in which case the

hypothesis is not rejected.

If my hypothesis is true (50%, 25%, 25%), how likely is it that I

could get this difference between my actual distribution

and my expected distribution of colored flowers?

The Chi-Square Test

Hypotheses

H0: P(pink, white, red) = .5, .25, .25

The population proportions of pink, white, and

red flowers are .5, .25, and .25, respectively.

H1: P(pink, white, red) .5, .25, .25The population proportions of pink, white, and

red flowers are something other than .5, .25,

and .25, respectively.

mutually-exclusive, exhaustive categories (P=1).

The Chi-Square Test

Notice that the hypotheses for the Chi-

Square Goodness-of-Fit Test are stated in

terms of proportions.

The Chi-Square TEST is conducted on

actualfrequencies not proportions.

Specifically, the X2 test operates on

differences between observed and expected

frequencies.

First - make sure everything is a frequency.


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The Chi-Square Test

Observed frequencies = O

Expected frequencies = E

E = N Expected Proportion

E = N P(cell)

Notice that E = O, always.

Pink White Red

75 20 25Observed

120(.5) = 60 120(.25) = 30 120(.25) = 30Expected

O = 120

E = 120

In the Chi Square Test

we calculate (O-E)2

/E in each cell, sum all of the (O-E)2/E values over all cells,

and compare this summed value to a critical

value.

( ) = EEOo2

2

The Chi-Square Distribution

Statisticians have found that if H0 is true

and we calculate the X2 statistic for all

possible samples of size N, the values for a

probability distribution called the X2

distribution.

Characteristics of the X2

distribution

A family of distributions varying in df (like

the tdistribution).

Positively skewed; the amount of skew

decreases as dfincreases.

Minimum value = 0 (X2 cant be negative)

Average (typical) value increases (the entire

distribution shifts to the right) as df

increases.

Characteristics of the X2 distribution

A family of distributions varying in df (like the tdistribution).

0

df=1df=3

df=5

df=10

Characteristics of the X2 distribution

As differences between Os and Es getbigger, X2 gets bigger.

Since we are only interested in rejecting H0if the differences between the obtained

frequencies and the expected frequencies is

greaterthan expected by chance, the

rejection region is in the upper tail.

( ) = EEOo2

2


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Chi-Square: Upper One-Tailed Test

X2c

Decision Rule: Reject H0 if X2

o > X2

c

Finding X2c

Table E.1 has the tabled values. df?

df = k - 1

Why?

If you have 3 categories, only the counts in 2 of

them are free to vary.

Choose , read down list ofdfto find X2c

Finding X2cdf 0.050 0.025 0.010 0.005

1 3.841 5.024 6.635 7.879

2 5.991 7.378 9.210 10.597

3 7.815 9.348 11.345 12.838

4 9.488 11.143 13.277 14.860

5 11.070 12.832 15.086 16.750

6 12.592 14.449 16.812 18.548

7 14.067 16.013 18.475 20.278

8 15.507 17.535 20.090 21.955

9 16.919 19.023 21.666 23.589

10 18.307 20.483 23.209 25.188

11 19.675 21.920 24.725 26.757

12 21.026 23.337 26.217 28.300

13 22.362 24.736 27.688 29.819

14 23.685 26.119 29.141 31.319

15 24.996 27.488 30.578 32.801

16 26.296 28.845 32.000 34.267

17 27.587 30.191 33.409 35.718

18 28.869 31.526 34.805 37.156

19 30.144 32.852 36.191 38.582

20 31.410 34.170 37.566 39.997

The Dreaded Six Steps

State H0 and H1.

Choose

Relevant probability distribution is X2 with

k- 1 df.

Find X2c & state decision rule: I will reject

H0 if X2o > X

2c

Calculate X2o

Apply decision rule.

Calculating X2o

Pink White Red

75 20 25Observed

60 30 30Expected

Finding X2oPink White Red

75 20 25Observed

60 30 30Expected

Pink White Red

15 -10 -5Observed - Expected

(O-E) = 0, always


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Finding X2o

Pink White Red

225 100 25(Observed - Expected)2

Pink White Red

15 -10 -5Observed - Expected

Finding X2o

3.75 3.33 .83(Observed - Expected)2 / Expected

Components of

X2

WhitePink Red

[(O-E)2/E] = X2o = 7.91

Since X2o > X2c, reject H0

Pink White Red

225 100 25(Observed - Expected)2

60 30 30Expected

Interpretation

Since we reject H0, the geneticists

hypothesis does not fit the data.

The population distribution across the three

categories is probably different than .50

pink, .25 white, .25 red.

Another Example: Breakfast!

The manufacturer of Posts Raisin Bran cereal, which lags

behind Kelloggs in sales, believe that, given the chance to

try both, most consumers will prefer Posts. They devise a

blind taste test. A sample of 100 people eat a small bowl

of each cereal, without knowing which is which, and they

are asked which cereal they like better. Fifty-seven people

say they like Posts better, while 43 choose Kelloggs.

Can the manufacturer advertise, More people prefer Posts?

H0: P(Posts) = P(Kelloggs) or P(Posts, Kelloggs) = .5, .5

H1: P(Posts) P(Kelloggs) or P(Posts, Kelloggs) .5, .5

Breakfast Answers

2) Use = .05

3) df = 1, X2 distribution with 1 df

4) X2c for = .05, df = 1, is 3.84; Decision rule: reject H0 if X2o > 3.84

5) Calculations

E(Posts) = E(Kelloggs) = 100 (.5) = 50

Cereal O E O-E (O-E) 2 (O-2) 2/E

Post's 57 50 7 49 0.98

Kellogg's 43 50 -7 49 0.98

100 100 0 1.96 = X2o

Since X2o < X2c (1.96 < 3.84), we retain H0.

The manufacturerscannot claim that more people prefer Posts.

Extra Credit Breakfast

In the Breakfast Example, we found that a 57 to 43

majority isnt enough to reject H0.

What is the smallest number of Posts preferences

that will lead to a significant finding (rejection of H0)

at = .05?

Correct and well-reasoned answers are worth 5pts

on top of your final (total) grade.


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Assumptions of the

Goodness-of-Fit Test

Observations in different categories are

independent.

Categories are mutually exclusive.

Categories are exhaustive.

No expected frequency < 2.

Few expected frequencies < 5.

The X2 distribution does not accurately

describe the probabilities of various sampling

outcomes if expected frequencies are small.

The X2 test of Independence

The X2 Test of Independence

Used when:

We want to compare the distribution of

frequencies across categories in two or more

independent samples.

We want to determine whether the paired

observations obtained in two or more

categorical variables are independent or

associated.

Same test.

Physical Contact in Neonates

A developmental psychologist hypothesizes that mothers who

have physical contact with their infants immediately after

birth are more likely to hold them on the left side, where the

sound of the mothers heartbeat is more pronounced, than

mothers who do not have such early contact with their infants.

She observes 125 early-contact mothers and 105 late-

contact mothers with the following results:

Late 55

Early 80

50

45

Left Right

105

125

230

Is there a

significant

difference?

Contingency Tables

This type of table is called a contingency

table.

We are trying to determine if the frequencies in

one variable are contingent upon the

frequencies of the other variable.

This is a 2 2 contingency table.

Late 55

Early 80

50

45

Left Right

Stating H0 & H1

When two or more groups are being

compared, H0 states that the population

distributions across all categories are thesame.

H1 states that the population distributions

differ.


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Stating H0 & H1

H0 : Early and late contact mothers do not

differ in how they hold their neonates.

H1 : Early and late contact mothers hold their

neonates differently.

OR

H0 : Group membership and distribution

across categories are unrelated.


across categories are related.

Stating H0 & H1


across categories are unrelated.


across categories are related.

OR

H0 : Time of first contact and how neonates

are held arenot related


are held are related.

Stating H0 & H1


are held arenot related


are held are related.

OR


are held are independent.


are held aredependent / correlated /related.

The X2 Test of Independence

Test statistic for the test of independence is

the same as in the goodness-of-fit test:

Two differences:

Calculation of expected frequencies

Calculation ofdf

( ) = EEOo2

2

Calculation of Expected Frequencies:

X2 test of Independence

For each cell,

( )( )N

Esumcolumnsumrow

=

Where N = total number of observations.

Calculation of Expected Frequencies:

X2 test of Independence

( )( )4.73

230

135125),( ==leftearlyE

Left Right Row Sums

Early 80 45 125Late 55 50 105

Column Sums 135 95 N = 230

( )( )6.51

230

95125),( ==rightearlyE

( )( )6.61

230

135105),( ==leftlateE

( )( )4.43

230

95105),( ==rightlateE


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Observed and Expected Frequencies

Left Right Row SumsEarly 80 45 125Late 55 50 105


Observed Frequencies

Left Right Row Sums

Early 73.4 51.6 125

Late 61.6 43.4 105


Expected Frequencies

Marginal Frequencies are Fixed in X2Analyses.

Why this Row and Column Sum?

N

rowPsumrow

)( =

N

columnPsumcolumn

)( =

NNP

sumcolumnsumrow)columnrow AND( =

Expected frequencies = N P

NNNE

sumcolumnsumrow)columnrow AND( =

NE

sumcolumnsumrow)columnrow AND(

=

dfin X2 Independence Tests

df = (# rows - 1) (# columns - 1)

Why?

Remember marginals are fixed in X2

independence tests.

100

100

50 60 90

How many cells are truly free to vary?

Mothers and Neonates


are held are independent.


are held aredependent / correlated /

related.

= .05

X2 distribution with 1 df

X2

c for

= .05, 1 df = 3.84; reject H0 if X

2o > 3.84

Mothers and Neonates

Left Right

Early 80 45Late 55 50

Observed FrequenciesLeft Right

Early 73.4 51.6Late 61.6 43.4

Expected Frequencies

Left Right

Early 6.6 -6.6

Late -6.6 6.6

Observed - Expected Left Right

Early 43.56 43.56

Late 43.56 43.56

(Observed - Expected) 2

Left Right

Early 0.59 0.84

Late 0.71 1.00

(O - E)^2 / E ( ) = EEOo2

2

X2o = 3.14

Decision: Retain H0, there is no relationship between side and time.

Overview of X2

X2 - a nonparametric test applied to

categorical, frequency data.

Relevant probability distribution is the X2

distribution.

A family of distributions varying in df

Positively skewed with minimum = 0

Skew decreases as dfincreases.

Center of distribution and critical values

increase as dfincreases.


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Overview of X2

Rejection region in the upper tail.

Decision rule: reject H0 if X2o > X2c

Two forms:

Goodness-of-fit

used to determine whether an obtained distribution

fits a hypothetical one.

Independence

used to test whether two categorical variables are

related

used to test whether two different samples are

related

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