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Chi-Squared Test

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Chi-Squared Test Chi-Square Test Compare observed data with data you would expect to obtain according to a specific hypothesis How much deviation can occur before you ... – PowerPoint PPT presentation

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Title: Chi-Squared Test


1
Chi-Squared Test
2
Chi-Square Test
  • Compare observed data with data you would expect
    to obtain according to a specific hypothesis
  • How much deviation can occur before you conclude
    something other than chance at work?
  • ?2 S(o-e)2 / e
  • Tests the null hypothesis
  • States that there is no significant difference
    between the expected and observed result

3
Lizard Example
  • Male phenologies in a population of side-blotched
    lizards
  • Three male phenotypes
  • Very aggressive orange-throat males
  • Moderately aggressive blue-throat males
  • Sneaker yellow-throat males
  • The three phenotypes vary in their frequencies
    over time

4
Scientific Hypothesis
  • Scientific hypothesis
  • The three male phenotypes are present in equal
    frequencies in the population
  • Research question
  • Are there differences in the frequencies of male
    phenotypes in the population of lizards
  • Chi-square (goodness of fit)
  • How well does our observed frequency distribution
    match the expected frequency distribution

5
Step 1
  • Determine number of observed and expected in each
    category
  • Obtained a sample of 51 males from the population
  • 12 orange-throat males
  • 30 blue-throat males
  • 9 yellow-throat males
  • Expected frequency is 111
  • 51/3 17
  • We expect 17 of each individuals

6
Step 2
  • Calculate X2

Orange Blue Yellow
Observed (o) 12 30 9
Expected (e) 17 17 17
Deviation (o-e) -5 13 -8
Deviation2 (d2) 25 169 64
d2/e 1.47 9.94 3.76
X2 Sd2/e 15.17 X2 Sd2/e 15.17
7
Step 3
  • Determine degrees of freedom (df)
  • Number of values we can pick freely without being
    constrained by other values
  • Number of categories in the problem minus 1
  • Three categories (orange, blue, yellow)
  • 2 df

8
Step 4
  • Determine a relative standard to serve as the
    basis for rejecting the hypothesis
  • Commonly used in biological research plt0.05
  • P value probability of rejecting the null
    hypothesis (no difference between observed and
    expected) when it is actually true
  • Probability of making an error of falsely stating
    that there is a significant difference between e
    and o when there is not a significant difference.
  • p lt 0.05
  • Stating that there is a less than 5 chance of
    error of stating there is a difference when there
    actually isnt a significant difference

9
Step 5
  • Refer to chi-square distribution table
  • Locate appropriate df (2)
  • Locate the value corresponding to the p value of
    0.05

10
Conclusions
  • If your X2 gt distribution table
  • Conclude observed numbers significantly different
    from the expected
  • X2 (15.17) is greater than 5.99 (df2, p0.05)
  • Reject the hypothesis that the three male
    phenotypes are present in equal frequencies
  • Significant difference in frequencies among the
    three phenotypes

Degrees of Freedom (df) Probability (p) Probability (p) Probability (p) Probability (p) Probability (p) Probability (p) Probability (p) Probability (p) Probability (p) Probability (p) Probability (p)
Degrees of Freedom (df) 0.95 0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.01 0.001
1 0.004 0.02 0.06 0.15 0.46 1.07 1.64 2.71 3.84 6.64 10.83
2 0.10 0.21 0.45 0.71 1.39 2.41 3.22 4.60 5.99 9.21 13.82
3 0.35 0.58 1.01 1.42 2.37 3.66 4.64 6.25 7.82 11.34 16.27
4 0.71 1.06 1.65 2.20 3.36 4.88 5.99 7.78 9.49 13.28 18.47
5 1.14 1.61 2.34 3.00 4.35 6.06 7.29 9.24 11.07 15.09 20.52
Nonsignificant Nonsignificant Nonsignificant Nonsignificant Nonsignificant Nonsignificant Nonsignificant Nonsignificant Significant Significant Significant
11
Plant and Environment Interaction
  • Galapagos Tree Fern
  • Present in 4 life zones on the islands
  • Miconia Zone
  • Brown Zone
  • Scalesia Zone
  • Transition Zone

12
  • Hypothesis
  • The cactus tree fern is present in equal
    frequencies across the life zones
  • Observed data
  • Miconia Zone - 13
  • Brown Zone - 15
  • Scalesia Zone - 22
  • Transition Zone - 18
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