BIOS14: Simple linear regression（一元线性回归） using R

NOTES

1 The relationship between variables

function relation: One can be described by $y=f(x)$.
correlation: There is no completely confirmed relaionship between two variables.

In the most cases, two variables can not describes by a function, but we can analysis their correlativity.

1. scatter plot: Visual inspection.

2. correlation coefficient: according to the data, calculating the degree of correlativity between variables.
   $$r=\frac{n\sum xy-\sum x\sum y}{\sqrt{n\sum x^2-(\sum x{)}^2}\cdot \sqrt{n\sum y^2-(\sum y{)}^2}}$$
   Here, $r$ is defined as linear correlation coefficient or Pearson’s correlation coefficient
   $|r|\ge 0.8$: high correlation
   $0.5\le |r|<0.8$: moderate correlation
   $0.3\le |r|<0.5$: low correlation
   $|r|<0.3$: no correlation

• When we use Pearson’s correlation coefficient, the data should be normality. If not, we can use Spearman correlation coefficient instaed. More details.

3. significance testing of $r$
   population correlationship coefficient: $\rho$
   sample correlationship coefficient: $r$
   hypothesis:
   $H_0:\rho =0$; $H_1:\rho \ne 0$
   statistics:
   $t=|r|\sqrt{\frac{n-2}{1-r^2}}\sim t(n-2)$
   decision
   If $|t|>t_{\alpha /2}$, reject $H_0$, there is a significantly linear ralationship between population variables.

2 Unary linear regression

2.1 Regression model

Population regression equation:
$$y_i={\beta }_0+{\beta }_1{x}_i+{\epsilon }_i$$
Estimated regression equation:
$${\hat{y}}_i={\hat{\beta }}_0+{\hat{\beta }}_1{x}_i$$
Assumption:
A1: dependent variable and independent variable is linear.
A2: The variable x is not random and must take at least two different values.
A3: $E(\epsilon )=0$
A4: The variation is constant around the regression line, independent of x. $var(y)={\sigma }^{}$