17.2 Common Non-Linear Models and Their Applications

Non-linear models capture complex relationships between variables that don't follow straight lines. These models, like exponential, logarithmic, and polynomial, are crucial for understanding real-world phenomena where change isn't constant.

Applying non-linear models involves choosing the right type, estimating parameters, and interpreting results in context. Logistic regression, a special case for binary outcomes, is widely used in fields like medicine and finance to predict probabilities.

Non-linear models: Types vs Applications

Types of non-linear models

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• Non-linear models describe relationships between variables that do not follow a straight line pattern
  • Used to model complex, real-world phenomena where the rate of change between the independent and dependent variables is not constant
• Exponential models used when the rate of change of the dependent variable is proportional to its current value
  • Characterized by the equation $y = ab^x$, where $a$ is the initial value, $b$ is the growth or decay factor, and $x$ is the independent variable
  • Exponential growth models describe situations where the rate of change increases over time (population growth, compound interest)
  • Exponential decay models describe situations where the rate of change decreases over time (radioactive decay, drug elimination from the body)
• Logarithmic models are the inverse of exponential models
  • Used when the rate of change of the dependent variable decreases as the independent variable increases
  • Characterized by the equation $y = a + b \ln(x)$, where $a$ is the y-intercept, $b$ is the slope, and $x$ is the independent variable
  • Often used to describe situations where the rate of change slows down over time (relationship between body mass and metabolic rate in animals)
• Polynomial models used when the relationship between the dependent and independent variables is curvilinear
  • Described by a polynomial equation of degree $n$, such as $y = a + bx + cx^2 + ... + nx^n$
  • Quadratic models (second-degree polynomials) describe relationships with a single turning point (trajectory of a thrown object, profit of a company as a function of production)
  • Higher-degree polynomial models can describe more complex curvilinear relationships but may be prone to overfitting and difficult to interpret

Applying non-linear models to real-world data

• Applying non-linear models involves selecting an appropriate model based on the observed pattern of the data and the underlying theoretical assumptions
• Parameters of the non-linear model can be estimated using various methods
  • Least squares regression or maximum likelihood estimation minimize the difference between the observed and predicted values
• Interpreting the results requires understanding the meaning of the estimated parameters in the context of the real-world problem
  • In exponential models, the growth or decay factor ($b$) represents the rate at which the dependent variable changes with respect to the independent variable (population growth model with a growth factor of 1.05 indicates a 5% increase per unit of time)
  • In logarithmic models, the slope ($b$) represents the change in the dependent variable associated with a one-unit increase in the natural logarithm of the independent variable (interpretation depends on the specific context)
  • In polynomial models, the coefficients of the polynomial terms represent the effect of the independent variable on the dependent variable at different orders (in a quadratic model, the coefficient of the squared term determines the direction and steepness of the curvature)
• The fitted non-linear model can be used to make predictions for new values of the independent variable
  • Accuracy of the predictions depends on the quality of the model fit and the range of the data used to estimate the parameters

Logistic regression for binary outcomes

Properties of logistic regression

• Logistic regression is a type of non-linear model used to predict the probability of a binary outcome based on one or more predictor variables (success or failure, presence or absence)
• Based on the logistic function, which maps the linear combination of the predictor variables to a probability value between 0 and 1
  • Logistic function defined as $p(x) = 1 / (1 + e^{-(b_0 + b_1x_1 + ... + b_nx_n)})$, where $p(x)$ is the probability of the outcome, $b_0$ is the intercept, $b_1$ to $b_n$ are the coefficients of the predictor variables $x_1$ to $x_n$, and $e$ is the base of the natural logarithm
• Coefficients in a logistic regression model are estimated using maximum likelihood estimation, which finds the values that maximize the likelihood of observing the data given the model
• Interpretation of the coefficients is based on the odds ratio, which represents the change in the odds of the outcome for a one-unit increase in the predictor variable, holding all other variables constant
  • An odds ratio greater than 1 indicates an increase in the odds of the outcome
  • An odds ratio less than 1 indicates a decrease in the odds

Applications of logistic regression

• Logistic regression can model the relationship between a binary outcome and categorical or continuous predictor variables, making it a versatile tool for various applications
  • Medical diagnosis: Predicting the presence or absence of a disease based on patient characteristics and test results
  • Marketing: Predicting the likelihood of a customer purchasing a product based on demographic and behavioral data
  • Credit risk assessment: Predicting the probability of default on a loan based on the applicant's financial and personal information
• Logistic regression models can be extended to handle multi-category outcomes or ordinal outcomes by modifying the link function and the interpretation of the coefficients
  • Multinomial logistic regression for multi-category outcomes
  • Ordinal logistic regression for ordinal outcomes

Goodness-of-fit and predictive power of non-linear models

Evaluating goodness-of-fit

• Evaluating the goodness-of-fit involves assessing how well the model captures the underlying pattern of the data and how much of the variability in the dependent variable is explained by the model
• Coefficient of determination ($R^2$) is a commonly used measure of goodness-of-fit for non-linear models
  • Represents the proportion of the variance in the dependent variable that is explained by the model
  • Should be used with caution for non-linear models, as it may not have the same interpretation as in linear regression
• Residual analysis is another approach to assessing the goodness-of-fit
  • Residuals are the differences between the observed and predicted values of the dependent variable
  • A well-fitting model should have residuals that are randomly distributed around zero, with no systematic patterns or trends
  • Plotting residuals against the predicted values or the independent variable can help identify non-random patterns (heteroscedasticity, non-linearity) indicating a poor model fit
  • Residual plots can also detect outliers or influential observations that may have a disproportionate impact on the model fit

Assessing predictive power

• Predictive power of a non-linear model can be evaluated using cross-validation techniques (k-fold cross-validation, leave-one-out cross-validation)
  • Involve splitting the data into training and testing sets, fitting the model on the training set, and evaluating its performance on the testing set
• Metrics such as mean squared error (MSE), root mean squared error (RMSE), or mean absolute error (MAE) can quantify the predictive accuracy of the model on the testing set
• For logistic regression models, the area under the receiver operating characteristic curve (AUC-ROC) is a commonly used measure of predictive power
  • Represents the model's ability to discriminate between the two outcome classes
• Comparing the goodness-of-fit and predictive power of different non-linear models can help select the most appropriate model for a given problem
  • Choice of the model should also consider the interpretability, parsimony, and theoretical justification of the model in the context of the research question or application