Multiple Simple Linear Regression Functions

COMPARISON OF MULTIPLE SIMPLE LINEAR REGRESSION FUNCTIONS 

Furthermore, if any confidence intervals of y and overlap, the y values on that specific x value are considered equivalent at a. Note that the 

FIGURE 2.38 Other possible comparisons between regression lines. 

At the beginning of this chapter, we learned to evaluate hi to assure that the slope was not 0. Now we expand this process slightly to compare two slopes, b a and hi. The test hypothesis for a two-tail test will be 

However, the test can be adapted to perform one-tail tests, too. 

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This simple linear regression can be readily adapted to multiple regression, i.e., to develop the best fit to an equation involving more than one variable, as in multicomponent spectrophotometric analysis (see Chapter 13), or in most complex curve fitting where the results are related to the independent variable but not in a linear fashion. The well-known Taylor, McLauren, or related theorems about infinite series state that most mathematical functions, f(x), can be expressed as the sum of series of terms in x". Many of these result in converging series, i.e., only several terms are necessary to represent the function with reasonable precision. 

If the regression assumes a linear relationship then it is referred to as simple linear regression if there is one independent variable or, multiple linear regression if there is more than one independent variable. If the regression model has a non-linear function then the regression is referred to as non-linear regression. 

Figure 11. Total sulfur (dry weight basis) as a function of chloride salinity and organic matter. Arrows indicate points taken from Frazier and Osanik (30). Salinity of the most saline site (line CB) was estimated using data from Brupbacher, Sedberry, and Willis (31). The plane A-B-C-D represents the simple multiple linear regression of total sulfur on chloride salinity and organic matter and the 4 lines in the plane the regression values for total sulfur at the 4 sites.

Excel provides several ways to find the coefficients that provide the best fit of a function to a set of data points — a process sometimes referred to as curve fitting. The "best fit" of the curve is considered to be foimd when the sum of the squares of the deviations of the data points from the calculated curve is a minimum. In the field of statistics, finding the least-squares best-fit parameters that describe a data set is known as regression analysis. In this chapter you ll learn how to perform simple and multiple linear regression... 

Partial and total order ranking strategies, which from a mathematical point of view are based on elementary methods of Discrete Mathematics, appear as an attractive and simple tool to perform data analysis. Moreover order ranking strategies seem to be a very useful tool not only to perform data exploration but also to develop order-ranking models, being a possible alternative to conventional QSAR methods. In fact, when data material is characterised by uncertainties, order methods can be used as alternative to statistical methods such as multiple linear regression (MLR), since they do not require specific functional relationship between the independent variables and the dependent variables (responses). 

If y is a function of more than two variables (e.g., quaternary alloy), then geometric intuition is lost even for the simple case of multiple linear regression since more than three dimensions are required to plot the data. 

Differences in calibration graph results were found in amount and amount interval estimations in the use of three common data sets of the chemical pesticide fenvalerate by the individual methods of three researchers. Differences in the methods included constant variance treatments by weighting or transforming response values. Linear single and multiple curve functions and cubic spline functions were used to fit the data. Amount differences were found between three hand plotted methods and between the hand plotted and three different statistical regression line methods. Significant differences in the calculated amount interval estimates were found with the cubic spline function due to its limited scope of inference. Smaller differences were produced by the use of local versus global variance estimators and a simple Bonferroni adjustment. 

The last example shows how to fit a polynomial to data. The same thing can be done when the functions are not simple powers, but are more complicated functions. However, to keep the problem linear, the unknown coefficients must be coefficients of those functions that is, the functions are completely specified. Multiple regression simply determines how much of each one is needed. Thus, the form of the equation is... 

Linear interpolation is simple but not accurate enough to be recommended. Market analysts use multiple regression or spline-based methods instead. One technique is to assume that the discount factors represent a functional form—that is, a higher-order function that takes... 

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