16: Binomial Dependent Variables
- Page ID
- 57785
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In the previous chapter, we examined what we can do if the dependent variable is dichotomous, has only two possible outcomes. The response variable followed a Bernoulli distribution. In this chapter, we extend this idea to where the outcome variable follows a Binomial distribution --- conditional on the values of the independent variable(s).
In this chapter, we first define a Binomial random variable, we then show that the Binomial distribution is a member of the Exponential Class of distributions (EC). With that, we can look at the assumptions and a couple extended examples.
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In the previous chapter, we explored a situation where the classical linear model utterly failed. That failure has been known since the early parts of the 20th century. As a result, statisticians created several specialized fitting techniques for binary dependent variables. The importance of the generalized linear model is not that it can fit models with a binary dependent variable, but that it is a technique that can fit those models and many others. This allows the researcher to pay closer attention to the dependent variable and use more information.
In this chapter, we cover the case where the dependent variable represents a count of successes out of a known number of attempts, \(n\).
While least squares can be used in such cases, even without clearly violating its assumptions, it is inefficient. You will be throwing out important information, namely the distribution of the dependent variable. Statisticians hate to throw out information.
This chapter starts with exploring the most common type of variable representing a count of successes out of a known number of attempts. After working with the Binomial, it looks into what happens when the data show the wrong level of variability... and what to do about it.