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Multinomial logistic regression is a particular solution to classification problems that use a linear combination of the observed features and some problem-specific parameters to estimate the probability of each particular value of the dependent variable.
In multilevel modeling, an overall change function (e.g. linear, quadratic, cubic etc.) is fitted to the whole sample and, just as in multilevel modeling for clustered data, the slope and intercept may be allowed to vary. For example, in a study looking at income growth with age, individuals might be assumed to show linear improvement over time.
This simple model is an example of binary logistic regression, and has one explanatory variable and a binary categorical variable which can assume one of two categorical values. Multinomial logistic regression is the generalization of binary logistic regression to include any number of explanatory variables and any number of categories.
The multinomial probit model is a statistical model that can be used to predict the likely outcome of an unobserved multi-way trial given the associated explanatory variables. In the process, the model attempts to explain the relative effect of differing explanatory variables on the different outcomes.
Example: On a 1-5 scale where 1 means disagree completely and 5 means agree completely, how much do you agree with the following statement. "The Federal government should do more to help people facing foreclosure on their homes." A multinomial discrete-choice model can examine the responses to these questions (model G, model H, model I ...
Administering the survey to a sample of respondents in any of a number of formats including paper and pen, but increasingly via web surveys; Analysing the data using appropriate models, often beginning with the Multinomial logistic regression model, given its attractive properties in terms of consistency with economic demand theory. [5]
However, it would also predict, for example, that a white person might have an average income $7,000 above a black person, and a 65-year-old might have an income $3,000 below a 45-year-old, in both cases regardless of location. A multilevel model, however, would allow for different regression coefficients for each predictor in each location.
P. Westfall, R. Tobias, R. Wolfinger (2011) Multiple comparisons and multiple testing using SAS, 2nd edn, SAS Institute; A gallery of examples of implausible correlations sourced by data dredging; An xkcd comic about the multiple comparisons problem, using jelly beans and acne as an example