Contingency Tables

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• Contingency Tables
• For
• Tests of Independence

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Multinomials Over Various Categories

• Thus far the situation where there are multiple
  outcomes for the qualitative variable without
  regard to anything else has been discussed.
• Now we discuss whether or not two qualitative
  variables are related, i.e are they independent?

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EXAMPLES

• (1) Can it be concluded that cola preference and
  gender are dependent?
• (2) Can it be concluded that cola preference and
  age are dependent?

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RULE OF 5

• ?2 (Chi-squared) is actually only an approximate
  distribution for the test statistic.
• To be a valid approximation
• ALL eis should be ? 5
• If the rule of 5 is violated, combine some
  categories so that the condition is met.

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COLA PREFERENCE VS. GENDER

• The 1000 cola drinkers were further classified as
  to whether they were male or female.
• COLA MALE FEMALE ROW TOTAL
• Coke 240 170 r1
  410
• Pepsi 200 150 r2 350
• RC 50 30 r3 80
• Shasta 35 15 r4
  50
• Jolt 75 35 r5 110
• COLUMN
• TOTAL c1 600 c2 400 n
  1000

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HYPOTHESIS TESTCan we Conclude Cola Preference
and Gender Are Dependent?

• H0 (NO) Cola preference and gender are
  independent
• HA (YES) Cola preference and gender are
  dependent
• ? .05
• Reject H0 if ?2 gt ?2.05,DF
• The correct DF (r-1)(c-1) (5-1)(2-1) (4)(1)
  4
• where r rows and c columns
• Reject H0 if ?2 gt ?2.05,4 9.48773

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HOW DO WE GET THE eijs?

• Let P(A) Probability a respondent favors Coke
• Let P(B) Probability a respondent is a male
• If H0 is true The classifications are
  independent
• Thus P(A and B) P(A)P(B)
• Best guess for P(A) ? 410/1000 .41
• Best guess for P(B) ? 600/1000 .6
• Thus P(A and B) ? (.41)(.6) .246
• Expected number (Coke and male) e11
  1000(.246) 246
• This can be gotten by r1c1/n (410)(600)/1000
  246

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CONTIGENCY TABLES

• Contingency tables are a convenient way of
  expressing the results when there are two
  classifications
• It is the equivalent of a multinomial table for
  two classifications
• We put the eijs in parentheses under (or next
  to) the fijs in the table then we calculate

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eijs for Cola vs. Gender

• Coke/Male e11 (410)(600)/1000 246
• Coke/Female e12 (410)(400)/1000 164
• Pepsi/Male e21 (350)(600)/1000 210
• Pepsi/Female e22 (350)(400)/1000 140
• RC/Male e31 ( 80)(600)/1000 48
• RC/Female e32 ( 80)(400)/1000 32
• Shasta/Male e41 ( 50)(600)/1000 30
• Shasta/Female e42 ( 50)(400)/1000 20
• Jolt/Male e51 (110)(600)/1000 66
• Jolt/Female e52 (110)(400)/1000 44

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Notes on Calculating es

• The column totals may be set in advance or may be
  random based on the survey.
• These eijs were all whole numbers -- if they are
  not DO NOT ROUND TO WHOLE NUMBERS.
• All these es ? 5 but suppose e52 were actually
  3
• We might combine the results from Shasta and Jolt
  colas.
• This would reduce the number of rows and hence
  the degrees of freedom.
• e52 is not less than 5 here, so we do not have to
  do this.

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CONTINGENCY TABLE FORCOLA vs. GENDER

• Men Women Total
• Coke 240 170 410
• (246) (164)
• Pepsi 200 150 350
• (210) (140)
• RC 50 30 80
• ( 48) ( 32)
• Shasta 35 15 50
• ( 30) ( 20)
• Jolt 75 35 110
• ( 66) ( 44)
• Total 600 400 1000

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?2 for Cola vs. Gender

• ?2 (240-246)2/246 (170-164)2/164
  (200-210)2/210 (150-140)2/140
  ( 50 - 48)2/ 48 ( 30- 32)2/ 32 (
  35 - 30)2/ 30 ( 15- 20)2/ 20 (
  75- 66)2/ 66 ( 35- 44)2/ 44
  6.92
• ?2 6.92 lt ?2.05,4 9.48773
• There is not enough evidence to conclude gender
  and cola preference are dependent.

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COLA PREFERENCE vs. AGE

• Survey results
• lt20 20-40 40-60 gt60 TOTAL
• Coke 155 140 75 40 410
• Pepsi 155 95 75 25 350
• RC 30 20 15 15 80
• Shasta 20 15 10 5 50
• Jolt 40 30 25 15 110
• TOTAL 400 300 200 100 1000

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HYPOTHESIS TEST

• There are r 5 rows and c 4 columns
• H0 (NO) Cola preference and age are independent
• H1 (YES) Cola preference and age are dependent
• ? .05
• Reject H0 if ?2 gt ?2.05,DF
• DF (r-1)(c-1) (5-1)(4-1) (4)(3) 12
• Reject H0 if ?2 gt ?2.05,12 21.0261

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Sample eijs

• e34 (Row 3 Total)(Column 4 Total)/(Grand Total)
  
• (80) (100) / 1000 8
• e41 (Row 4 Total)(Column 1 Total)/(Grand Total)
  
• (50) (400) / 1000
  20

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CONTINGENCY TABLE FORCOLA vs. AGE

• lt20 20-40 40-60 gt60 Total
• Coke 155 140 75 40 410
• (164) (123) (82) (41)
• Pepsi 155 95 75 25 350
• (140) (105) (70) (35)
• RC 30 20 15 15 80
• ( 32) ( 24) (16) ( 8)
• Shasta 20 15 10 5 50
• ( 20) ( 15) (10) ( 5)
• Jolt 40 30 25 15 110
• ( 44) ( 33) (22) (11)
• Total 400 300 200 100 1000

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?2 for Cola vs. Age

• ?2 (155-164)2/164 (140-123)2/123
  (75-82)2/82 (40-41)2/41 (
  40 - 44)2/ 44 ( 30- 33)2/ 33 (
  25- 22)2/ 22 ( 15- 11)2/ 11
  18.72
• ?2 18.72 lt ?2.05,12 21.0261
• There is not enough evidence to conclude cola
  preference and age are dependent.

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Excel

• CHITEST gives the p-value for the test
• CHITEST(Observed Values, Expected Values)
• Must first calculate the expected values, eijs
• See next slide for easy way to calculate these
  values.

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Review

• Contingency tables allow for comparisons to
  determine if two different categories are
  independent
• Excel -- CHITEST is used to generate the p-values
  for the chi-squared test
• Expected Values
• (Row Total)(Column Total)/n
• By hand -- total degrees of freedom (r-1)(c-1)
• and the ?2 statistic is calculated by