Pearson's product-moment correlation

# Data input

> Shigeru<-c(2,3,6,10,15,18,26)

> Hanano<-c(3,7,9,15,25,28,30)

# Data plot

> plot(Shigeru,Hanano)

# Pearson's correlation

> cor(Shigeru, Hanano, method="pearson")

[1] 0.9626748

# Pearson's product-moment correlation test

> cor.test(Shigeru, Hanano, alternative="two.sided",method="pearson")

Pearson's product-moment correlation

data: Shigeru and Hanano

t = 7.9532, df = 5, p-value = 0.0005067

alternative hypothesis: true correlation is not equal to 0

95 percent confidence interval:

0.7621040 0.9946566

sample estimates:

cor

0.9626748

# Decide parameters, "a" and "b" of Hanano =a * Shigeru +b

> lm(Hanano~Shigeru)

Call:

lm(formula = Hanano ~ Shigeru)

Coefficients:

(Intercept) Shigeru

2.970 1.203

Thus, Hanano = 0.08778 * Shigeru + 29.67667.

# add the formula "Hanano = 0.08778 * Shigeru + 29.67667" on Shigeru-Hanano plot.

> plot(Shigeru,Hanano)

> abline(lm(Hanano~Shigeru), lty=1)

# Summary

> result<-lm(Hanano~Shigeru)

> summary(result)

Call:

lm(formula = Hanano ~ Shigeru)

Residuals:

1 2 3 4 5 6 7

-2.375388 0.422001 -1.185830 0.003729 3.990677 3.382846 -4.238036

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 2.9702 2.1185 1.402 0.219839

Shigeru 1.2026 0.1512 7.953 0.000507 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.242 on 5 degrees of freedom

Multiple R-squared: 0.9267, Adjusted R-squared: 0.9121

F-statistic: 63.25 on 1 and 5 DF, p-value: 0.0005067

# Residuals

> residuals(result)

1 2 3 4 5 6 7

-2.375388440 0.422001243 -1.185829708 0.003729024 3.990677439 3.382846489 -4.238036047

# Histogram of residuals

> hist(residuals(result))

# plot of residuals

> plot(x, residuals(result))

# theoretical values of the "Hanano"

> predict(result)

1 2 3 4 5 6 7

5.375388 6.577999 10.185830 14.996271 21.009323 24.617154 34.238036

# make theoretical data with theoretical lines

> theoretical<-c(5.375388, 6.577999,10.185830, 14.996271, 21.009323, 24.617154, 34.238036)

> plot(Shigeru, theoretical)

> abline(lm(Hanano~Shigeru), lty=1)