Chapter 11 Chi Square Distribution and Its applications

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Chapter 11Chi Square Distributionand Its
applications
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What is a Chi Square?

• Chi Square is another probability distribution
  (like Normal and Student t)
• Symbol ?2
• Picture

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What is a Chi Square? (cont)

• Features Mode (i.e. Peak) at n 2.
• Degrees of Freedom n 2.

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Why ?2?

• The ?2 distribution allows you to test the
  independence of two factors.
• For instance, what about getting a tattoo and the
  frequency of Hepatitis C? What if you got the
  tattoo in a tattoo parlor? What if your cousin
  Dwayne gave you that great tattoo of Count
  Chocula?
• The Chi Squared test for independence allows you
  to decide whether the chance of having hepatitis
  is independent of the tattoo status.

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Requirements

• 1. Counted Data Condition
• Data must be in counts
• 2. Randomization Condition
• Individuals who have been counted should
  
• be a random sample from some
• population.
• 3. Expected Cell Frequency Condition
• The expected values (which you will
• calculate) should all be at least 5.

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What to do?

• Use a contingency table to find the differences
  between observed and expected values.

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What to do?

• Use a contingency table to find the differences
  between observed and expected values.

d.f. (R-1)(C-1) where R and C are the number of
rows/columns
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Lets Try one

• We have counts of 626 individuals categorized
  according to their tattoo status and their
  hepatitis status.

Hep C No Hep C Total
Tattoo, Parlor 17 35 52
Tattoo, Elsewhere 8 53 61
No Tattoo 22 491 513
Total 47 579 626
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• We want to test whether tattoo status and
  hepatitis status are independent.
• HO Tattoo status and hepatitis status are
  independent
• HA Tattoo status and hepatitis status are not
  independent
• Check the conditions
• 1. Counts of data categorized
• 2. While you dont have the information, you can
  assume that they were selected to avoid biases.
• 3. Well find out when we determine E for each
  cell

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Compute E
Hep C No Hep C Total
Tattoo, Parlor 17 35 52
Tattoo, Elsewhere 8 53 61
No Tattoo 22 491 513
Total 47 579 626
48.096
3.904
56.420
4.580
38.516
474.484
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Notice

• There are some values below 5 well continue
  from here for the sake of the data.
• What are the degrees of freedom?
• Find the ?2 value.
• Using a table, the value for P(?2 gt57.91)

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57.91
Less than .0001
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Conclusion

• The P-value is very small, so I reject the null
  hypothesis and conclude that hepatitis status is
  not independent of tattoo status.

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Resources

• www.microbiologybytes.com/maths/graphics/chi.gif