Shrikant I. Bangdiwala

Regression: simple linear

Regression is a statistical term used for describing models that estimate the relationships among variables. Linear regression models study the relationship between a single dependent variable Y an...

Regression: multiple linear

The simple linear model is described by the mathematical straight line Y 1⁄4 b0 þ b1X, where b0 is called the intercept and b1 is called the slope. The focus in such models is on the slope b1 since, if equal to zero, there is no relationship between X and Y. Geometrically, any given straight line is determined by two points (x1,y1) and (x2,y2) that lie on the two-dimensional X–Y plane, so that the simple linear regression slope coefficient can be written as b1 1⁄4 y2 y1 ð Þ x2 x1 ð Þ , providing us the interpretation of the slope as the change in Y relative to a change in X. The geometric extension of a straight line in the twodimensional X–Y plane to the (k + 1)-dimension determined by k independent variables X1, X2, ... , Xk and Y is a hyperplane. Since high dimensions are difficult to visualize, we only present the geometry for the three-dimensional case, that is, when we have k = 2 independent variables X1 and X2 and we are interested in their relationship with a single dependent continuous variable Y. See Figure 1(a) for a hypothetical data example of n = 32 injured drivers with X1 = age, X2 = speed and Y = injury severity score (ISS). In the case of a linear relationship, the mathematical model, also called the linear predictor, is given by

Regression: binary logistic

Simple and multiple linear regression models study the relationship between a single continuous dependent variable Y and one or multiple independent variables X, respectively (Bangdiwala, 2018a, 2018b). If the value of the dependent variable Y can be only one of two outcomes (i.e. a binary variable, such as dead/alive, injured/not injured, or crash/no crash), the linear predictor function Xb (which equals b0 þ b1X1 þ b2X2 when we have two independent variables X1 and X2) would need to map onto the two values. Typically we consider the dependent variable Y as an indicator variable and assign the value of 1 to the outcome one is trying to predict, and the value of 0 to the other outcome. Mapping a linear predictor to only two values is not possible, so we have it map to the range of values from 0 to 1. Since probabilities range from 0 to 1, we map the linear predictor to a probability. Two common methods are the logistic model and the probit model. In the logistic model, we assume that the probability of Y having the value of 1 is given by the inverse of the log-odds or logit function:

Interpretation of relative effects

Interpretation of relative effects Shrikant I. Bangdiwala Population Health Research Institute and Department of Health Research Methods, Evidence and Impact, McMaster University, Hamilton, ON, Canada; Institute for Social and Health Sciences, University of South Africa, Johannesburg, South Africa; Violence, Injury & Peace Research Unit, South Africa Medical Research Council, Tygerberg, South Africa