exam ii marks. chapter 20.1 correlation correlation is used when we wish to know whether two...
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Exam II Marks
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Chapter 20.1 Correlation
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Correlation
• Correlation is used when we wish to know whether two randomly distributed variables are associated with each other
• Example– Total length Y1 of aphid stem
mothers and mean thorax length Y2 of their parthenogenetic offspring.
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No causal ordering
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Contrast to regression
𝜌=√𝛽1,2∙ 𝛽2,1
𝜌=cos (𝜃)
𝑖𝑓 𝜃=0𝑜 , h𝑡 𝑒𝑛 𝜌=1𝑖𝑓 𝜃=90𝑜 , h𝑡 𝑒𝑛 𝜌=0𝑖𝑓 𝜃=180𝑜 , h𝑡 𝑒𝑛𝜌=−1
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−1≤ 𝜌≤1
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Formal model
Regression• randomly distributed
response variable ~ fixed explanatory variable
Correlation• two random response
variables• No causal ordering, thus no
explanatory variable
[𝑌 1 ,𝑌 2 ]=𝑃𝐶 ∙ λ+𝜀
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Estimate
�̂�=𝑟= 1𝑛−1
∙∑ (𝑌 1−𝑌 1 ) ∙ (𝑌 2−𝑌 2 )
𝑠1 ∙𝑠2
Compute t
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State HA/Ho pair
HA:
Ho:
Crunch the numbers
�̂�=𝑟= 115−1
∙∑ ([8.7 ,8.5 ,… ]−9 ) ∙ ([5.95 ,5.65 ,… ]−5.73 )
1.88 ∙0.59
𝑟=10.0715.49
=0.650
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More number crunching> cor.test(dat$th.length,dat$tot.length)
Pearson's product-moment correlation
data: dat$th.length and dat$tot.length t = 3.0867, df = 13, p-value = 0.008666alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.2070464 0.8720726 sample estimates: cor 0.6503335
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Conclusions
• r = 0.650, n = 15, p = 0.0086• Thorax length of offspring is positively related
to stem mother total length. • The relation of offspring thorax length to size
of aphid stem mothers is monotonic but not necessarily linear.
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