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Multiple Linear Regression Analysis TESTS OF HYPOTHESES A PARTICULAR REGRESSION COEFFICIENT If the disturbances are mutually independent and normally distributed (with E(e) = 0 and var(e) = Isigma^2) and with a (preset) level of signifi- cance alpha, a significance test for a particular regression coefficient can be performed, or more specifically: the null hypothesis is: H0: ai = 0 (given that all other aj are in the model), which is tested against the alternative hypothesis: H1: ai is not equal to zero, by treating FRi = bi^2/var(bi) as a realization of a Fisher(1,n-p) variate. However, this test must be used with caution, because with the (preset) level of significance alpha, only one coefficient can be tested properly, while the computer output lists statistics for all coefficients. It seems very tempting to test the coefficients serially one at a time, but one must keep in mind that in doing so the level of significance of the whole test rises above the nominal value. ANALYSIS OF VARIANCE In the analysis of variance table the different contributions to the total uncorrected sum of squares Y'Y (which is the first part in the table) are given. The second part of the table assumes the presence of an (unknown) constant term in the model; if this term is absent, the 'mean'-line disappears and in the 'regression'-line p-1 changes into p and b'X'Y-nu^2 changes into b'X'Y. The third part of the table is only present when repeated observations for the dependent variable are available, in which case: k is the number of groups of replications, mi is the number of replications in group i, and W = (w1,...,wk)', with wi = Sum(j,1,mi,yij) / Sqrt(mi), for i = 1,...,k. The fourth part of the table is only present if a reduction is requested and possible. SSQ then stands for the residual sum of squares from a regression analysis with the first p-q out of the p independent variables (1 <= q <= p-1), while SSE stands for the residual sum of squares from a regression analysis with the original p independent variables. Analysis of variance source of right tail variation df sum of squares mean square F-ratio probability --------------------------------------------------------------------------- total n Y'Y --------------------------------------------------------------------------- mean 1 nu^2 MSM = nu^2 FRM P(FM>=FRM) regression p-1 b'X'Y - nu^2 MSR FRR P(FR>=FRR) residual n-p Y'Y - b'X'Y MSE = s^2 --------------------------------------------------------------------------- lack of fit k-p W'W - b'X'Y MSL FRL P(FL>=FRL) pure error n-k Y'Y - W'W MSP --------------------------------------------------------------------------- reduction q SSQ - SSE MSQ FRQ P(FQ>=FRQ) --------------------------------------------------------------------------- The column 'mean square' is obtained by division of the sums of squares by their corresponding degrees of freedom; The column 'F-ratio' is obtained by division of the mean squares by the residual mean square, except for the lack of fit F-ratio, which is obtained by division of the lack of fit mean square by the pure error mean square, thus: MSM = nu^2/1, MSR = (b'X'Y-nu^2)/(p-1), MSE = s^2 = (Y'Y-b'X'Y)/(n-p), MSL = (W'W-b'X'Y)/(k-p), MSP = (Y'Y-W'W)/(n-k), MSQ = (SSQ-SSE)/q, FRM = MSM/MSE, FRR = MSR/MSE, FRL = MSL/MSP and FRQ = MSQ/MSE. If the disturbances are mutually independent and normally distributed (with E(e) = 0 and var(e) = Isigma^2) and with a (preset) level of signifi- cance alpha, a significance tests can be performed for: 1. The mean of the observations for the dependent variable, or more speci- fically: the null hypothesis is: H0: E(u) = 0, which is tested against the alternative hypothesis: H1: E(u) is not equal to zero, by treating FRM as a realization of a Fisher(1,n-p) variate. 2. The regression equation, or more specifically: the null hypothesis is: H0: a1 = ... = ap = 0, except for the ai that denotes the constant term (if present), which is tested against the alternative hypothesis: H1: at least one of a1,...,ap is not equal to zero, by treating FRR as a realization of a Fisher(p-1,n-p) variate. 3. The adequacy (linearity) of the model, or more specifically: the null hypothesis is: H0: the linear model is adequate (that is, no model signifi- cantly improves the prediction of Y over the linear model), which is tested against the alternative hypothesis: H1: the linear model is not adequate, by treating FRL as a realization of a Fisher(k-p,n-k) variate. 4. A subset of regression coefficients, or more specifically: suppose without loss of generality that the subset consists of the last q coefficients, then the null hypothesis is: H0: ar = ... = ap = 0, with r = p-q+1, which is tested against the alternative hypothesis: H1: at least one of ar,...,ap is not equal to zero, by treating FRQ as a realization of a Fisher(q,n-p) variate. 5. A linear combination of the regression coefficients, or more specifi- cally: the null hypothesis is: H0: c'a = m, in which c is a vector of constants with order q+1, which is tested against the alternative hypotheses: H1: c'a is not equal to m, by substituting c'a = m in the original model, shifting the known terms to the left hand part, combining the corresponding terms in the right hand part, and testing the thus derived reduced model by treating FRQ as a realization of a Fisher(q,n-p) variate. In each case the right tail probability P(F >= FR) can be found in the last column of the analysis of variance table.