How to calculate dependence between type of hair and eyes?
Answers
lesson, we examine a data set of people (students in a large class)
who report their hair color and eye color. Our objective is to see if there is
some kind of association between the two variables and to try to characterize
that association. The original data are shown in Table 1. Also shown are the
expected counts under the independence assumption and the standardized
Pearson residuals (“z-values”). We have arranged the rows and columns from
what I somewhat subjectively have determined is darker to lighter, so they
are essentially ordinal variables.
Following typical practice, we test for independence. If we cannot reject
the null hypothesis, then there is probably not much point in going further
- we don’t have strong evidence for an association between hair color and
eye color. Any association we think we may see in the data could be due to
chance.
The observed value of the the χ
2
test statistic is 138.2925. The degrees
of freedom is ν = (4 − 1)(4 − 1) = 9. The p-value based on the asymptotic
χ
2
9 distribution is < 2.2 × 10−16, essentially 0. There is strong evidence for
some kind of association. All of the expected cell counts are quite large
(> 5) so there should be no difficulty with the χ
2 approximation of the null
distribution. Here is the copy-and-paste from the R-console:
> # have already entered the data into a 4 by 4 matrix:
> haireye
black brown red blond
brown 68 119 26 7
hazel 15 54 14 10
green 5 29 14 16
blue 20 84 17 94
> chisq.test(haireye)
Pearson’s Chi-squared test
data: haireye
X-squared = 138.29, df = 9, p-value < 2.2e-16
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Biology
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How to calculate dependence between type of hair and eyes?
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lesson, we examine a data set of people (students in a large class)
who report their hair color and eye color. Our objective is to see if there is
some kind of association between the two variables and to try to characterize
that association. The original data are shown in Table 1. Also shown are the
expected counts under the independence assumption and the standardized
Pearson residuals (“z-values”). We have arranged the rows and columns from
what I somewhat subjectively have determined is darker to lighter, so they
are essentially ordinal variables.
Following typical practice, we test for independence. If we cannot reject
the null hypothesis, then there is probably not much point in going further
- we don’t have strong evidence for an association between hair color and
eye color. Any association we think we may see in the data could be due to
chance.
The observed value of the the χ
2
test statistic is 138.2925. The degrees
of freedom is ν = (4 − 1)(4 − 1) = 9. The p-value based on the asymptotic
χ
2
9 distribution is < 2.2 × 10−16, essentially 0. There is strong evidence for
some kind of association. All of the expected cell counts are quite large
(> 5) so there should be no difficulty with the χ
2 approximation of the null
distribution. Here is the copy-and-paste from the R-console:
> # have already entered the data into a 4 by 4 matrix:
> haireye
black brown red blond
brown 68 119 26 7
hazel 15 54 14 10
green 5 29 14 16
blue 20 84 17 94
> chisq.test(haireye)
Pearson’s Chi-squared test
data: haireye
X-squared = 138.29, df = 9, p-value < 2.2e-16
1