y: dependent variable

X: independent variable

r: residual

$$y=X\beta+r$$

$$\beta=inv(X’*X)*X’*y$$

$$\sigma^2=r’*r/df$$

$$df=N-rank(X)$$

$$\sigma_\beta^2=\sigma^2inv(X’X)$$

$$T_\beta=\beta/\sigma_\beta$$

$$contrast variance = c’*\sigma_\beta^2*c$$

$$T_{contrast}=c’\beta/\sqrt{contrast variance}$$

$$r=\beta\frac{std(x)}{std(y)}$$

$$\beta=r\frac{std(y)}{std(x)}$$

$$var(y) = \beta^2var(x)+ResSS/(n)$$

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