SJFC MSTI 130 - Analyzing Data with Nonlinear Models

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Part IVAnalyzing Data with NonlinearModels307309In Unit One we began to see the world as data; in Unit Two we began to ask questionsof data in order to find out the story it has to tell about itself, and hence about the worldfrom which it was extracted. In Unit Three we began to make connections between setsof data, to see how the events in the situations from which the data were extracted mightbe related to each other. We began to analyze the relationships between sets of data bycapturing those relationships in regression models, simple linear ones at first involving adependent variable and a single explanatory variable, and then more complex linear oneswith a dependent variable and several explanatory variables. This unit investigates one ofthe four assumptions that underlie regression modeling and at the same time seeking todevelop the relationships between even more complex sets of data.One of the main assumptions about data when you construct a regression model is thatthe data is sampled from a linear relationship of some sort (either two-variable or more thantwo-variables). If this is not true, then your resulting regression model may seem to be okay,but it will exhibit problems of one of the following types:1. The model may be accurate for only a small slice of data. If we apply the model todata points outside this small slice, the resulting errors from the model may becomelarger and larger. This is related to having too small a sample of the data to noticethat it really does not exhibit linearity.2. The regression model consistently underestimates the data in certain regions and con-sistently overestimates it in other regions. This resulting pattern indicates that thereis a better model for the data than a linear model.In chapter 11 (page 311) we begin dealing with data that is not proportional, that is, datathat violates our first regression assumption that a linear model is an appropriate fit. We willstart by focusing on two-variable data and then learn how to extend this to multivariabledata. Even though most real data sets are multi-dimensional, there are solid reasons forbeginning our study with two-variable nonlinear data sets:• Not all data is multidimensional - sometimes two variables are enough.• Even in multidimensional data, we are often interested in the main effect first. Thatmeans looking at how the most significant variable relates to the dependent variable.• In many modeling applications, the data shows one dependent variable and two inde-pendent variables with a constraint (like total cost must be less than a fixed amount).In this case, the constraint relationship between the two independent variables can beused to reduce the number of independent variables to one, making the entire data settwo dimensional.• Finally, the models we are going to discuss are easy to picture in two dimensions; inmore dimensions, it is difficult to picture the models and develop an intuitive feel forwhat they can do. But the intuition we develop with two-variable data will help usinterpret the diagnostic graphs in StatPro’s regression output when we are dealing withmultidimensional models.310In much the same way that straight lines have parameters that can be chosen so asto match the line closely to the data, the basic nonlinear models we will introduce haveparameters that can serve the same purpose. By using these parameters to shift one of thebasic models horizontally and vertically and to stretch them and flip it, we can fit this basicfunction to a non-proportional data set.However, the regression routines in StatPro are only useful for producing linear models.In fact, we overcome this problem by transforming nonlinear data so that it becomes suitablylinear and then applying our regression model to this straightened out data. Thus, chapter12 (page 347) presents the key transformations that will convert many kinds of nonlineardata into linear data. This chapter also teaches us how to evaluate the quality of modelsbuilt from transformed data and then how to interpret these models. The unit closes withchapter 13 (page 379) on interpreting the relationships in nonlinear models with more thanone variable. We also discuss how to locate the maxima and minima of such


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SJFC MSTI 130 - Analyzing Data with Nonlinear Models

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