I have a data series for a dependent and two independent variables. How to identify the best fit relationship.
A scatter plot is often used to present bivariate quantitative data (pairs of linked numerical observations). Each variable is represented on an axis and the axes are labeled accordingly.
A scatter plot displays data as points on a grid using the associated numbers as coordinates. The way the points are arranged by themselves in a scatter plot may or may not suggest a relationship between the two variables. . Bivariate data may have an underlying relationship that can be modeled by a mathematical function. You will consider linear models.
A line of best fit (trend or regression line) is a straight line that best represents the data on a scatter plot. This line may pass through some of the points, none of the points, or all of the points. When a linear model is indicated there are several ways to find a function that approximates the y-value for any given x-value. A method called regression is the best way to find a line of best fit, but it requires extensive computations and is generally done on a computer or graphing calculator. A good approximation can be found by estimating the line that best fits the data and writing its equation using two points on the line.
Dear Sir, what if the relationship is not linear? In my case the linear relationship does not show the best fit relationship. Rather, square root of the variable explains the dependent variable properly. How to identify the proper relationship?
The fundamental difference between linear and nonlinear regression, and the basis for the analyses' names, are the acceptable functional forms of the model. Specifically, linear regression requires linear parameters while nonlinear does not. Use nonlinear regression instead of linear regression when you cannot adequately model the relationship with linear parameters.
A nonlinear equation can take many different forms. In fact, because there are an infinite number of possibilities, you must specify the expectation function Minitab uses to perform nonlinear regression. These examples illustrate the variability (θ 's represent the parameters):y = θ1X1 (Convex 2, 1 parameter, 1 predictor)
y = θ1 * X1 / ( θ2 + X1 ) (Michaelis-Menten equation, 2 parameters, 1 predictor)
y = θ1 - θ2 * ( ln ( X1 + θ3 ) - ln ( X2 )) (Nernst equation, 3 parameters, 2 predictors)
Your choice for the expectation function often depends on previous knowledge about the response curve's shape or the behavior of physical and chemical properties in the system. Potential nonlinear shapes include concave, convex, exponential growth or decay, sigmoidal (S), and asymptotic curves. You must specify the function that satisfies both the requirements of your previous knowledge and the nonlinear regression assumptions.
While the flexibility to specify many different expectation functions is very powerful, it can also require great effort to determine the function that provides the optimal fit for your data. This often requires additional research, subject area knowledge, and trial and error analyses. In addition, for nonlinear equations, determining the effect each predictor has on the response can be less intuitive than it is for linear equations.
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