If F SSR1SSEn 2 show that a Fni2R21 R2 and b R2 FF n 2S

If F = SSR/1/SSE/(n - 2), show that (a) F=(ni-2)R^2/1 - R^2 and (b) R^2 = F/F + n- 2.

Solution

Formulas:

SST = SSR + SSE

R² = Coefficient of Determination = SSR/SST or 1 – SSE/SST

a)

F = SSR(n-2)/SSE                       {Given}

= (SSR(n-2) / SST ) / (SSE/SST) {Divide numerator and denominator by SST and using formula 2}

= (R^2(n-2)) / (1 -R^2)   {Using formula 2}}

= ((n-2)R^2 )/ (1-R^2) Proved

b)

From part a,

F =((n-2)R^2 )/ (1-R^2)

=> F - FR^2 = nR^2 - 2R^2

=> R^2(-F+2-n) =-F

=> R^2 = -F / (-F+2-n)

=> R^2 = F /(F +n -2) Proved