Exam II Marks

Chapter 20.1 Correlation

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.

No causal ordering

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

−1≤ 𝜌≤1

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

Estimate

�̂�=𝑟= 1𝑛−1

∙∑ (𝑌 1−𝑌 1 ) ∙ (𝑌 2−𝑌 2 )

𝑠1 ∙𝑠2

Compute t

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

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

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.