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Multiple Linear Regression Analysis THE MODEL SPECIFICATION To let the program know between which variables the statistician expects a certain kind of relationship, he must provide a model specifi- cation, which consists of the keyword "Model" followed by a formula (the model statement), which resembles the notation of regression models in common statistical literature quite closely. For instance: "Model" y = alpha0 + alpha1 * x1 + alpha2 * x2; A model formula consists of an identifier to denote the dependent variable (the left hand part), followed by an '=' (equal), followed by the sum of a number of terms (the right hand part), while it is terminated with a ';' (semicolon). Each term must be the product of an identifier to denote the parameter (which is to be estimated) and an identifier to denote the independent variable. An exception is made for the optional constant term, which is given as a single identifier denoting that constant term, and which may be placed anywhere in the model. Each identifier must start with a letter and is allowed to contain any number of letters, digits and blanks. As most peripheral equipment of a computer is unable to process sub- or superscriptions or Greek letters, we write alpha0, alpha1 and alpha2. Identifiers have no inherent meaning, but serve for the identification of variables, parameters and functions. They may be chosen freely (except for the twentyone standard function names and the ten option names, cf. "Help"/Options). It is advised not to use the same identifier to denote two (or more) different quantities; for regression parameters, however, it will not lead to fatal errors, whereas for the dependent and independent variables distinguishable identifiers must be used indeed. Correct model formulae are for instance: "Model" y variable = constant + parameter * x variable; and "Model" depvar = const + beta1 * xvar1 + beta2 * xvar2; TRANSFORMATIONS Almost all transformations a user would like to perform on his input data fit quite naturally in the model formula: each transformation is expressed as a formula itself. If, for instance, the user wants to include in the model formula as an independent variable the natural logarithm of the sum of two other variables, he writes: (if those two other variables are called: xvar1 and xvar2) Ln (xvar1 + xvar2) . In model formulae the operators '+' (plus), '-' (minus), '*' (asterisk), and '/' (slash) are allowed, all with their conventional meaning of addition, subtraction, multiplication and division respectively. Of course the normal operator precedence rules are obeyed. Special operators are: ':' (colon), integer division and '^' (uparrow), exponentiation. The operation term : factor is defined only for operands both of type integer and will yield a result of type integer, with the same sign as would be obtained by normal division, while the magnitude is found by dividing the two quantities and taking the whole part; mathematically it can be defined as: a : b = Sign (a / b) * Entier (Abs (a / b)), for instance: 5 : 2 = 2 and -7 : 2 = -3. The operation factor ^ primary denotes exponentiation, where the factor is the base and the primary is the exponent, for instance: 5 ^ 2 = 25 and 2 ^ 3 ^ 2 = 64 but 2 ^ (3 ^ 2) = 512. Also the following twentyone standard functions are allowed: Abs (E), Sign (E), Sqrt (E), Sin (E), Cos (E), Tan (E), Ln (E), Log (E), Exp (E), Entier (E), Round (E), Mod (E1, E2), Min (E1, E2), Max (E1, E2), Arcsin (E), Arccos (E), Arctan (E), Sinh (E), Cosh (E), Tanh (E) and Indicator (E1, E2, E3) in which E, E1, E2 and E3 are expressions in terms of variables, operators and standard functions. Round (E) is defined as: Entier (E + 0.5) and Indicator (E1, E2, E3) is defined as: IF E1 <= E2 <= E3 THEN 1 ELSE 0. The dependent variable may be transformed in a similar way and as a consequence the model formula in its most general form looks like: "Model" G (y) = b0 + b1 * F1 (x1,...,xm) + ... + bp * Fp (x1,...,xm); Some examples of transformed model formulae are: "model" y = a0 + a1 * Sqrt (x1 + x2) + a2 * Sqrt (x3); and "MODEL" Arcsin (Sqrt (Y)) = A0 + A1 * X + A2 * X ^ 2; A user can specify model formulae in which terms with known regression coefficients appear, by subtracting those terms from the left hand part of the model formula, for instance: "Model" y - 5.4321 * x3 = a0 + a1 * x + a2 * x ^ 2; This applies especially to the constant term; if this term is known it must be shifted to the left hand part. If weights are present in the input data (or can be computed out of the input data), to indicate that the variances of the observations are not all equal (cf. "Help"/Theory), the left hand part of the model formula can be expanded with a so called weight part (which can be an expression), preceeded by a '&' (ampersand), for instance: "Model" Depvar & Max (Abs (Weight), 10) = Const + Param * Indepvar;