Chi-Square Test and Contingency Analysis: Understanding Independence and Goodness-of-Fit, Slides of Statistics

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Contingency Analysis

& Goodness-of-Fit Tests

The Chi-square Distribution

• The chi-square distribution is a family of

distributions, depending on degrees of

freedom:

• d.f. = n - 1

2

0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28

d.f. = 1 d.f. = 5 d.f. = 15

χ^2 χ^2 χ^2

Finding the Critical Value

• The critical value, , is found from the

chi-square table

4

Do not reject H 0 Reject H 0

α

χ^2 α

χ^2

χ^2 α

Contingency Tables

Contingency Tables

• Situations involving multiple population

proportions

• Used to classify sample observations

according to two or more characteristics

• Also called a crosstabulation table.

Contingency Table Example

Sample results organized in a contingency table:

7

Gender

Hand Preference

Left Right

Female 12 108 120

Male 24 156 180

36 264 300

120 Females, 12 were left handed 180 Males, 24 were left handed

sample size = n = 300:

Logic of the Test

• If H 0 is true, then the proportion of left-handed

females should be the same as the proportion of left-

handed males

• The two proportions above should be the same as

the proportion of left-handed people overall

8

H 0 : Hand preference is independent of gender H (^) A: Hand preference is not independent of gender

Expected Cell Frequencies

• Expected cell frequencies:

10

Total sample size

(i Row total)(j Column total) e

th th

ij =

14. 4 300

( 120 )( 36 ) e 11 = =

Example:

Observed v. Expected Frequencies

Observed frequencies vs. expected frequencies:

11

Gender

Hand Preference Left Right

Female

Observed = 12 Expected = 14.

Observed = 108 Expected = 105.

120

Male

Observed = 24 Expected = 21.

Observed = 156 Expected = 158.

180

36 264 300

Observed v. Expected Frequencies

13

Gender

Hand Preference Left Right

Female

Observed = 12 Expected = 14.

Observed = 108 Expected = 105.

120

Male

Observed = 24 Expected = 21.

Observed = 156 Expected = 158.

180

36 264 300

1. 6848
2. 4

( 156 158. 4 )

21. 6

( 24 21. 6 )

105. 6

( 108 105. 6 )

14. 4

χ^2 = (^12 −^14.^4 )^2 + −^2 + −^2 + −^2 =

Contingency Analysis

14

χ^2 .05 = 3.841^ χ^2 Reject H 0

α = 0.

Decision Rule: If χ^2 > 3.841, reject H 0 , otherwise, do not reject H (^0)

χ^2 = 0. 6848 with d.f.=(r -1)(c -1) = (1)(1) = 1

Do not reject H 0

Here, χ^2 = 0. < 3.841, so we do not reject H (^0) and conclude that gender and hand preference are independent

• Are technical support calls equal across all days of the week? (i.e., do calls follow a uniform distribution?) - Sample data for 10 days per day of week: Sum of calls for this day: Monday 290 Tuesday 250 Wednesday 238 Thursday 257 Friday 265 Saturday 230 Sunday 192

16

Chi-Square Goodness-of-Fit Test

Σ = 1722

 If calls are uniformly distributed, the 1722 calls

would be expected to be equally divided across the

7 days:

 Chi-Square Goodness-of-Fit Test: test to see if the

sample results are consistent with the expected

results

17

Logic of Goodness-of-Fit Test

246 expectedcalls per day if uniform

Chi-Square Test Statistic

• The test statistic is

19

(where df k 1)

e

(o e )

i

2

χ 2 = i − i = −

where: k = number of categories oi = observed cell frequency for category i ei = expected cell frequency for category i

H 0 : The distribution of calls is uniform over days of the week H (^) A: The distribution of calls is not uniform

The Rejection Region

• Reject H 0 if

20

∑

− χ = i

2 (^2) i i e

(o e )

H 0 : The distribution of calls is uniform over days of the week H (^) A: The distribution of calls is not uniform

2 α

2 χ > χ

0

α

χ^2 α

Do not reject H Reject H 0 0

(with k – 1

degrees of

freedom)

χ^2