Section 15.5 Summary
Subsection 15.5.1 Glossary
- Mosaic Plot
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A plot that describes the relation between a response factor and an explanatory variable. Vertical rectangles represent the distribution of the explanatory variable. Horizontal rectangles within the vertical ones represent the distribution of the response.
- Logistic Regression
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A type of regression that relates between an explanatory variable and a response of the form of an indicator of an event.
Subsection 15.5.2 Discuss in the Forum
In the description of the statistical models that relate one variable to the other we used terms that suggest a causality relation. One variable was called the “explanatory variable” and the other was called the “response”. One may get the impression that the explanatory variable is the cause for the statistical behavior of the response. In negation to this interpretation, some say that all that statistics does is to examine the joint distribution of the variables, but causality cannot be inferred from the fact that two variables are statistically related. What do you think? Can statistical reasoning be used in the determination of causality?
As part of your answer in may be useful to consider a specific situation where the determination of causality is required. Can any of the tools that were discussed in the book be used in a meaningful way to aid in the process of such determination?
Notice that the last 3 chapters dealt with statistical models that related an explanatory variable to a response. We considered tools that can be used when both variable are factors and when both are numeric. Other tools may be used when one of the variables is a factor and the other is numeric. An analysis that involves one variable as the response and the other as explanatory variable can be reversed, possibly using a different statistical tool, with the roles of the variables exchanged. Usually, a significant statistical finding will be still significant when the roles of a response and an explanatory variable are reversed.
Subsection 15.5.3 Formulas
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Logistic Regression, (Probability): \(p_i = \frac{e^{a + b \cdot x_i}}{1 + e^{a + b\cdot x_i}}\text{.}\)
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Logistic Regression, (Predictor): \(\log(p_i/[1-p_i]) = a + b\cdot x_i\text{.}\)
