The Analysis of Categorical Data and GoodnessofFit Tests

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Chapter 12

• The Analysis of Categorical Data and
  Goodness-of-Fit Tests

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12.1 Chi-Square Tests for Univariate Categorical
Data

• Examples of Univariate Categorical Data
• Each student in a sample of 100 is classified as
  full-time or part-time. (two categories)
• Each airline passenger in a sample of 50 is
  classified based on type of ticket-coach,
  business class, or first class. (three
  categories).
• Each voter in a sample of 100 is asked which of
  the five city council members he or she favors
  for mayor. (five categories).

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One-way Frequency Table for Univariate
Categorical Data

• Fees keep American taxpayers from using credit
  cards to make tax payments. 100 randomly selected
  taxpayers are asked if they will use a credit
  card to pay tax next year. The following are the
  outcome of the survey

The manager of a tax preparation company
might be interested in determining whether the
four possible responses occur equally often, that
is, the long-run proportion of responses in each
of the four categories is ¼.
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One-way Frequency Table for Univariate
Categorical Data

• Each item returned to a department store is
  classified according to how it was resolved cash
  refund, credit to charge account, merchandise
  exchange, or return refused. (four categories). A
  sample of 100 returns summarizes the observations
  in a one-way frequency table consisting of k 4
  cells

The customer relations manager for the
department store might be interested in
determining whether the four possible
dispositions for a return request occur equally
often, that is, the long-run proposition of
returns in each of the four categories is ¼.
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Notation

• k number of categories of a categorical
  variable,
• p1 true proportion for Category 1
• p2 true proportion for Category 2
• pk true proportion for Category k
• (Note p1 p2 pk 1.)
• The hypotheses to be tested have the form
• H0 p1 hypothesized proportion for Category 1.
  
• p2 hypothesized proportion for Category 2.
• pk hypothesized proportion for Category
  k.
• Ha H0 is not true. At least one of the true
  category proportion differs from the
  corresponding hypothesized value.

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Example 12.1 Births and the Lunar Cycle

• A common legend is that more babies than expected
  are born during 24 lunar cycle. The following
  data is from a sample of randomly selected births
  during 24 lunar cycles.

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Example Births and the Lunar Cycle

• If there is no relationship between number of
  births and the lunar cycle, then the number of
  births in each lunar cycle category should be
  proportional to the number of days included in
  that category.
• There are a total of 699 days in the 24 lunar
  cycles considered and 24 of those days are in the
  new moon category. If there is no relationship
  between number of births and lunar cycle
• Similarly, we can find the proportion of births
  during other lunar cycles.

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Example Births and the Lunar Cycle

• If there is no relationship between number of
  births and the lunar cycle, then
• H0 p1 0.0343, p2 0.2175, p3 0.0343, p4
  0.2132
• p5 0.0343, p6 0.2146, p7 0.0343, p8
  0.2175
• Ha H0 is not true.
• If H0 is true, the expected count for Category 1
  (new moon) is
• And the expected count for Category 2 is

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Example Births and the Lunar Cycle

• Expected counts for other categories are computed
  similarly.

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The Goodness-of-Fit Statistics ?2

• First we compute the quantity
• for each cell, where, for a sample of size n,
• The ?2 statistic is the sum of these quantities
  for all k cells

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Chi-Square Distribution

• The goodness-of-fit statistic, X2, is a
  quantitative measure of the extent to which the
  observed counts differ from those expected when
  H0 is true.
• Therefore, large values of X2 suggest rejection
  of H0.
• For a test procedure based on the X2 statistics,
  the associated P-value is the area under the
  appropriate chi-square curve and to the right of
  the computed X2 value. (Appendix Table 8)
• Reject H0 if P-value lt significance level a.

• Find the P-value if X2 is 4.93 and df 2.

From Appendix Table 8, P-value 0.085
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Goodness-of-Fit Test Procedure

• Hypotheses
• H0 p1 hypothesized proportion for Category 1
• pk hypothesized proportion for Category k
• Ha H0 is not true.
• Test statistic

P-value For a test procedure based on the
X2 statistics, the associated P-value is the area
under the appropriate chi-square curve and to the
right of the computed X2 value. (Appendix Table
8). Reject H0 if P-value lt significance
level a.
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Goodness-of-Fit Test Procedure

• When H0 is true and all expected counts are at
  least 5, ?2 has approximately a chi-square
  distribution with df k - 1.
• The P-value associated with the computed test
  statistic value is the area to the right of ?2
  under the df k - 1 chi-square curve.
• Upper-tail areas for chi-square distribution are
  found in Appendix Table 8.
• Assumptions
• Observed cell counts are based on a random
  sample.
• The sample size is large. The sample size is
  large enough for the chi-square test to be
  appropriate as long as every expected cell count
  is at least 5.

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Example Births and the Lunar Cycle Revisited

• Test the hypothesis that number of births is
  unrelated to lunar cycle using the data in
  Example 12.1. Choose a 0.05.
• df 8 - 1 7. The computed value of ?2 lt 12.01
  (the smallest entry in df 7 column), so P-value
  gt .10.
• Fail to reject H0, because P-value gt a.
• There is no enough evidence to conclude that
  number of births and lunar cycle are related.

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Example Hybrid Car Purchases

• The table on the right lists sales of hybrid cars
  in the top five states in 2004. Use ?2
  goodness-of-fit test and a significance level a
  .01 to test the hypothesis that hybrid sales for
  these 5 states are proportional to the 2004
  population (see table below) for these states.

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Solution to Example Hybrid Car Purchase

• If the hybrid sales for the 5 states are
  proportional to their 2004 population,
• then, the expected counts for hybrid sales in
  these states are
• Expected count for California 406(.495)
  200.970
• Expected count for Virginia 406(.103) 41.818
• Expected count for Washington 406(.085)
  34.510
• Expected count for Florida 406(.240) 97.440
• Expected count for Maryland 406(.077) 31.362

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Solution to Example Hybrid Car Purchase

• Let p1, p2, p3, p4, p5 denote the true proportion
  of hybrid car sales for California, Virginia,
  Washington, Florida and Maryland, respectively.
• Assumption The sample was a random sample. All
  expected counts are gt 5, so it is appropriate to
  use chi-square test.
• H0 p1 0.495, p2 0.103, p3 0.085, p4
  0.240, p5 0.077
• Ha H0 is not true.
• Test statistic

• P-value All expected counts exceed 5, so
  the P-value can be based on chi-square
  distribution with df 5 - 1 4. From Appendix
  Table 8, P-value lt 0.001 0 ( the test
  statistic 59.49 gt 13.81 and any value gt 13.81 has
  the right tail area lt 0.001).
• Conclusion Reject H0 since P-value lt a.
  There is evidence that hybrid car sales are not
  proportional to population size for at least one
  of the five states.

On next slides we use Excel to solve this problem.
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Click x, and an Insert Function dialog box
appears. Select Statistical in select a
category box. In the Select a function list
choose CHITEST. Then click OK.
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As soon as you input the Actual_range (observed
frequency) and Expected_range, you can see the
P-value in Formula result 3.70981E-12 (
3.70981 10-12 0 ).
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Exercise Color of Stolen Cars

• Does the color of a car influence the chance
  that it will be stolen? The AP reported that for
  a random sample of 830 stolen vehicles 140 were
  white, 100 were blue, 270 were red, 230 were
  black, and 90 were other colors. Use the X2
  goodness-of-fit test and a significance level of
  a.01 to test the hypothesis that proportions
  stolen are identical to population color
  proportions. It is known that 15 of all cars are
  white, 15 are blue, 35 are red, 30 are black
  and 5 are other colors.

Answer P-value lt .001. There is convincing
evidence that at least one of the color
proportions for stolen cars differs from the
corresponding proportions for all cars.