Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more. 
Home Multivariate Data Modeling Multiple Regression Estimation of New Observations  
See also: MLR, ANOVA  
MLR  Estimation of New ObservationsWhen calculating the regression parameters a_{i} of the general multiple linear regression equation one is not only interested in the actual parameters a_{i} but also in an estimate of the confidence interval of both the parameters a_{i} and the estimated target variable . While estimating the standard deviation of the parameters is quite simple, the estimation of the standard deviation of is rather complicated. The reason for this is that the distribution of depends on the particular set of a_{i}. In general, the multivariate distribution function of can be rather complex. However, there are two ways to estimate the standard deviation of , the first being rather easy to implement, the second one is more demanding: Rough Approximation: We can use the standard deviation s of the residuals to estimate the standard deviation of future values of y, i.e. . The interval of 2s can be interpreted as a rough approximation to the accuracy of the model (that is, the accuracy with which the model will predict future values of y for particular values of x_{i}). The calculation of s is easy and straightforward: with SSE being the sum of squared residuals, n being the number of observations, and k being the number of independent variables.Exact Solution: The exact way to calculate the confidence interval of can be seen as an extension of the WorkingHotelling confidence band of simple regression. In the case of multiple linear regression this band becomes a kdimensional volume. The estimated value falls within the (1α) confidence interval: with s being the standard deviation of the residuals, c being the augmented vector of the xvalues, (X^{T}X)^{1} being the inverse covariance matrix, and t_{α/2} being the quantile of the tdistribution at the probability α/2. 

Home Multivariate Data Modeling Multiple Regression Estimation of New Observations 