Anova Decomposition

source	SS	DF	MS
x(reg)	RSS	1	RSS
error	ESS	n-2	$s^{2}={{ESS} \over {n-2}}$$s^{2}={{ESS} \over {n-2}}$
total	TSS	n-1	-

• SS= Sum of squares
• DF= Degrees of freedom
• MS= Mean square

$TSS=RSS+ESS$$TSS=RSS+ESS$

$TSS=\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}$$TSS=\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}$

$RSS={b_{2}}^{2}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}$$RSS={b_{2}}^{2}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}$

$ESS=\sum _{i=1}^{n}(y_{i}-b_{1}-b_{2}x_{i})^{2}$$ESS=\sum _{i=1}^{n}(y_{i}-b_{1}-b_{2}x_{i})^{2}$

Goodness of fit:

Goodness of fit $R^{2}$$R^{2}$:

$R^{2}={{RSS} \over {TSS}}$$R^{2}={{RSS} \over {TSS}}$

Goodness of fit F-statistic:

$F={{RSS} \over {s^{2}}}$$F={{RSS} \over {s^{2}}}$

t-test:

${t_{obs}}={{b_{2}} \over {s_{B_{2}}}}$${t_{obs}}={{b_{2}} \over {s_{B_{2}}}}$

$pvalue=2P_{H_{0}}(T\geq {|t_{obs}|})$$pvalue=2P_{H_{0}}(T\geq {|t_{obs}|})$

Anova-test:

${f_{obs}}={{RSS} \over {s^{2}}}$${f_{obs}}={{RSS} \over {s^{2}}}$

$pvalue=P_{H_{0}}(F\geq {f_{obs}})$$pvalue=P_{H_{0}}(F\geq {f_{obs}})$

Confidence intervals:

$\beta _{1}{:}b_{1}\pm {t}_{{1+\gamma } \over {2}}(n-2)s_{B_{1}}$$\beta _{1}{:}b_{1}\pm {t}_{{1+\gamma } \over {2}}(n-2)s_{B_{1}}$

$\beta _{2}{:}b_{2}\pm {t}_{{1+\gamma } \over {2}}(n-2)s_{B_{2}}$$\beta _{2}{:}b_{2}\pm {t}_{{1+\gamma } \over {2}}(n-2)s_{B_{2}}$

Residuals:

$TSS=\sum _{i=1}^{N}(y_{i}-{\bar {y}})^{2}=\sum _{i=1}^{n}({\hat {y}}_{i}-{\bar {y}})^{2}+\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i})^{2}$$TSS=\sum _{i=1}^{N}(y_{i}-{\bar {y}})^{2}=\sum _{i=1}^{n}({\hat {y}}_{i}-{\bar {y}})^{2}+\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i})^{2}$

$RSS=\sum _{i=1}^{n}({\hat {y}}_{i}-{\bar {y}})^{2}$$RSS=\sum _{i=1}^{n}({\hat {y}}_{i}-{\bar {y}})^{2}$

$ESS=\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i})^{2}$$ESS=\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i})^{2}$

Forecast:

$x=x^{*}$$x=x^{*}$

${\hat {y}}=b_{1}+b_{2}x^{*}$${\hat {y}}=b_{1}+b_{2}x^{*}$

$var(B_{1}+B_{2}x^{*})=\sigma ^{2}[{{1} \over {n}}+{{(x^{*}-{\bar {y}})^{2}} \over {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}]$$var(B_{1}+B_{2}x^{*})=\sigma ^{2}[{{1} \over {n}}+{{(x^{*}-{\bar {y}})^{2}} \over {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}]$

$standarderror=s{\sqrt {{{1} \over {n}}+{{(x^{*}-{\bar {x}})^{2}} \over {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}}}$$standarderror=s{\sqrt {{{1} \over {n}}+{{(x^{*}-{\bar {x}})^{2}} \over {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}}}$