Estimates Reliability

The problem of determining the accuracy of estimates from the multiple regression is basically similar as for estimates from a simple regression equation. Since the correlation is rarely perfect, the estimates made from the regression equation will deviate from the correct value of the dependent variable. If an estimate is of maximum usefulness, it is very necessary to have some indication of its precision, as in the simple regression equation, the measure of reliability is an average of these deviations of the actual value of non-dependent variable from the estimate from the regression equation or, in another word, the standard error of estimate.

Coefficient of multiple determinations

The coefficient of multiple determinations is analogous to the coefficient of determination in the two-variable case. The fit of a straight line to the two-variable scatter was measured by the simple coefficient of determination r2 which was defined as the ratio of the stated sum of squares to the total sum of squares. In the same way we can define the coefficient of multiple determinations that is denoted by R2. Symbolically:

R₂ = SSR / SST = 1 - SSE / SST

Illustration in a trivariate distribution:

σ₁ = 2, σ₂ + σ₃ =3

r₁² = 0.7, r₂³ = r₃¹ = 0.5

Find (i) b_12.3 and (ii) b_13.2

Solution
   
b_12.3 = r_12.3 σ_1.3 / σ_2.3

r_12.3 = r₁₂ - r₁₃ r₂₃ / 1 - r₂₁₃ 1 - r₂₂₃ = 0.7 - (0.5) (0.5) / 1 - (0.5)2 1 - (0.5)²

=0.7 - 0.25 / 0.75 = 0.45 / 0.75 = 0.6

σ_1.3 = σ₁ (1 - r₂₂₃₎ = 2 1 - 0.25 = 1.732

σ_2.3 = σ₂ (1 - r₂₂₃) = 3 1 - 0.25 = 2.598

b_13.2 = 0.6 x 1.732 / 2.598 = 0.4

b_13.2 = r_13.2 σ_1.2 / σ_3.2

r_13.2 = r₁₃ - r₁₂ r₂₃ / 1 - r₂₁₂ 1 - r₂₂₃ = 0.5 - (0.7 x 0.5) / 1 - (0.7)2 1 - (0.5)2

= 0.15 / 0.51 0.75 = 0.243

σ_1.2 = σ 1 (1 - r213) = 2 1 - 0.49 = 1.428

σ_3.2 = σ3 (1 - r223 =3 1 - 0.25 = 2.598

b_13.2 = 0.243 x 1.458 / 2.598 = 0.134