Linear regression examples and applications

Linear Regression

Predictions and Hypothesis Testing

What benefit does a variable provide when developing and examining models?

Statistics can be used to describe a phenomenon or conduct hypothesis testing, but statistics can also used to predict outcomes (Hanneman, Kposowa, & Riddle, 2012, p. 7). To enable predictions to be made, variables are required in the statistical model. Variables allow researchers to model the mathematical relationship between two or more phenomenon. For example, the relationship between tree circumference and the number of leaves could be described by the two variables girth and leaves. This relationship may or may not be causal, but it could be correlational. If a causal relationship is suspected, then maybe reducing the number of leaves inhibits expansion of tree girth. In this case, leaves would be the independent variable and girth the dependent variable. If the model is valid, then the tree girth could be predicted for any value of leaves. Variables therefore allow researchers to test hypotheses and make predictions.

2. Explain the purpose of simple linear regression and scatter diagrams. Please provide a simple linear regression model and define each variable used.

Linear regression is a statistical tool for quantifying the relationship between two variables (Journal of Tropical Pediatrics, n.d., p. 3). Linear regression is used to predict the mean response of the dependent variable (Y) given a change in the independent variable (X). Scatter diagrams are useful because the amount of error inherent to the prediction model can be visualized. If the best-fit line is included in the scatter plot, then the magnitudes of the error (residuals) for a given value X can be easily seen. Scatter plots are also useful for troubleshooting the prediction model.

If X = leaves and Y = tree girth, then Y = bX + a, where b = slope of the best-fit line and a = the y-intercept (Journal of Tropical Pediatrics, n.d., p. 5). This is the regression equation and a and b are the regression coefficients. If the regression coefficients were a = -2.5 and b = 0.05 for a give tree species, then the regression equation would be Y = 0.05*X + -2.5. Any value X could be entered into this formula and the predicted value of Y determined. For example, if the number of leaves on a tree were 250, then the predicted girth would be 0.05*250 + -2.5 = 10 cm.

3. Describe multiple regression analysis and discuss potential uses for this model.