Chi-Squared (?2) Analysis

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Chi-Squared (?2) Analysis
AP Biology Unit 4
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What is Chi-Squared?

• In genetics, you can predict genotypes based on
  probability (expected results)
• Chi-squared is a form of statistical analysis
  used to compare the actual results (observed)
  with the expected results
• NOTE ?2 is the name of the whole variable you
  will never take the square root of it or solve
  for ?

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Chi-squared

• If the expected and observed (actual) values are
  the same then the ?2 0
• If the ?2 value is 0 or is small then the data
  fits your hypothesis (the expected values) well.
• By calculating the ?2 value you determine if
  there is a statistically significant difference
  between the expected and actual values.

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Step 1 State a null hypothesis

• Your null hypothesis states that there is no
  difference between the observed and expected
  values.
• You will either accept or reject your null
  hypothesis based on the Chi squared value that
  you determine.

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Step 2 Calculating expected and determining
observed values

• First, determine what your expected and observed
  values are.
• Observed (Actual) values That should be
  something you get from data usually no
  calculations ?
• Expected values based on probability
• Suggestion make a table with the expected and
  actual values

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Step 1 Example

• Observed (actual) values Suppose you have 90
  tongue rollers and 10 nonrollers
• Expected Suppose the parent genotypes were both
  Rr ? using a punnett square, you would expect 75
  tongue rollers, 25 nonrollers
• This translates to 75 tongue rollers, 25
  nonrollers (since the population you are dealing
  with is 100 individuals)

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Step 1 Example

• Table should look like this

Expected Observed (Actual)
Tongue rollers 75 90
Nonrollers 25 10
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Step 2 Calculating ?2

• Use the formula to calculated ?2
• For each different category (genotype or
  phenotype calculate
• (observed expected)2 / expected
• Add up all of these values to determine ?2

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Step 2 Calculating ?2
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Step 2 Example

• Using the data from before
• Tongue rollers
• (90 75)2 / 75 3
• Nonrollers
• (10 25)2 / 25 9
• ?2 3 9 12

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Step 3 Determining Degrees of Freedom

• Degrees of freedom of categories 1
• Ex. For the example problem, there were two
  categories (tongue rollers and nonrollers) ?
  degrees of freedom 2 1
• Degrees of freedom 1

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Step 3 Determining Degrees of Freedom

• Degrees of freedom (df) refers to the number of
  values that are free to vary after restriction
  has been
• placed on the data. For instance, if you have
  four numbers with the restriction that their sum
  has to be 50,
• then three of these numbers can be anything, they
  are free to vary, but the fourth number
  definitely is
• restricted. For example, the first three numbers
  could be 15, 20, and 5, adding up to 40 then the
  fourth
• number has to be 10 in order that they sum to 50.
  The degrees of freedom for these values are then
  three.
• The degrees of freedom here is defined as N - 1,
  the number in the group minus one restriction (4
  - I ).
• Adapted by Anne F. Maben from "Statistics for the
  Social Sciences" by Vicki Sharp

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Step 4 Critical Value

• Using the degrees of freedom, determine the
  critical value using the provided table
• Df 1 ? Critical value 3.84

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Step 5 Conclusion

• If ?2 gt critical value
• there is a statistically significant difference
  between the actual and expected values.
• If ?2 lt critical value
• there is a NOT statistically significant
  difference between the actual and expected values.

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Step 5 Example

• ?2 12 gt 3.84
• There is a statistically significant difference
  between the observed and expected population

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Chi-squared and Hardy Weinberg

• Review If the observed (actual) and expected
  genotype frequencies are the same then a
  population is in Hardy Weinberg equilibrium
• But how close is close enough?
• Use Chi-squared to figure it out!
• If there isnt a statistically significant
  difference between the expected and actual
  frequencies, then it is in equilibrium

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Example

• Using the example from yesterday

Ferrets Expected Observed (Actual)
BB 0.45 x 164 74 78
Bb 0.44 x 164 72 65
bb 0.11 x 164 18 21
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Example

• ?2 Calculation
• BB (78 74)2 / 74 0.21
• Bb (72 65)2 / 72 0.68
• bb (18 21)2 / 18 0.5
• ?2 0.21 0.68 0.5 1.39
• Degrees of Freedom 3 1 2
• Critical value 5.99
• ?2 lt 5.99 ? there is not a statistically
  significant difference between expected and
  actual values ? population DOES SEEM TO BE in
  Hardy Weinberg Equilibrium (different answer from
  last lecture more accurate)