Anova Decomposition source SS DF MS x(reg) RSS 1 RSS error ESS n-2 s 2 = E S S n − 2 {\displaystyle s^{2}={{ESS} \over {n-2}}} total TSS n-1 - SS= Sum of squares DF= Degrees of freedom MS= Mean square T S S = R S S + E S S {\displaystyle TSS=RSS+ESS} T S S = ∑ i = 1 n ( y i − y ¯ ) 2 {\displaystyle TSS=\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}} R S S = b 2 2 ∑ i = 1 n ( x i − x ¯ ) 2 {\displaystyle RSS={b_{2}}^{2}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}} E S S = ∑ i = 1 n ( y i − b 1 − b 2 x i ) 2 {\displaystyle ESS=\sum _{i=1}^{n}(y_{i}-b_{1}-b_{2}x_{i})^{2}} Goodness of fit:[edit | edit source] Goodness of fit R 2 {\displaystyle R^{2}} : R 2 = R S S T S S {\displaystyle R^{2}={{RSS} \over {TSS}}} Goodness of fit F-statistic: F = R S S s 2 {\displaystyle F={{RSS} \over {s^{2}}}} t-test:[edit | edit source] t o b s = b 2 s B 2 {\displaystyle {t_{obs}}={{b_{2}} \over {s_{B_{2}}}}} p v a l u e = 2 P H 0 ( T ≥ | t o b s | ) {\displaystyle pvalue=2P_{H_{0}}(T\geq {|t_{obs}|})} Anova-test:[edit | edit source] f o b s = R S S s 2 {\displaystyle {f_{obs}}={{RSS} \over {s^{2}}}} p v a l u e = P H 0 ( F ≥ f o b s ) {\displaystyle pvalue=P_{H_{0}}(F\geq {f_{obs}})} Confidence intervals:[edit | edit source] β 1 : b 1 ± t 1 + γ 2 ( n − 2 ) s B 1 {\displaystyle \beta _{1}{:}b_{1}\pm {t}_{{1+\gamma } \over {2}}(n-2)s_{B_{1}}} β 2 : b 2 ± t 1 + γ 2 ( n − 2 ) s B 2 {\displaystyle \beta _{2}{:}b_{2}\pm {t}_{{1+\gamma } \over {2}}(n-2)s_{B_{2}}} Residuals:[edit | edit source] T S S = ∑ i = 1 N ( y i − y ¯ ) 2 = ∑ i = 1 n ( y ^ i − y ¯ ) 2 + ∑ i = 1 n ( y i − y ^ i ) 2 {\displaystyle 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}} R S S = ∑ i = 1 n ( y ^ i − y ¯ ) 2 {\displaystyle RSS=\sum _{i=1}^{n}({\hat {y}}_{i}-{\bar {y}})^{2}} E S S = ∑ i = 1 n ( y i − y ^ i ) 2 {\displaystyle ESS=\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i})^{2}} Forecast:[edit | edit source] x = x ∗ {\displaystyle x=x^{*}} y ^ = b 1 + b 2 x ∗ {\displaystyle {\hat {y}}=b_{1}+b_{2}x^{*}} v a r ( B 1 + B 2 x ∗ ) = σ 2 [ 1 n + ( x ∗ − y ¯ ) 2 ∑ i = 1 n ( x i − x ¯ ) 2 ] {\displaystyle 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}}}]} s t a n d a r d e r r o r = s 1 n + ( x ∗ − x ¯ ) 2 ∑ i = 1 n ( x i − x ¯ ) 2 {\displaystyle standarderror=s{\sqrt {{{1} \over {n}}+{{(x^{*}-{\bar {x}})^{2}} \over {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}}}}