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How should we refer to a logistic model where the dependent variable has more than two categories and you have more than one independent variables? Could it multivariable multinomial logistic regression?

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Yes. "Multivariable" is almost unnecessary as most statisticians would consider multinomial (aka polytomous) logistic regression to already signal that multiple predictors are being modeled. Note that multinomial regression is limited by the lowest frequency of Y, unlike the case where you are able to consider Y to be ordinal and hence use an ordinal regression model such as the proportional odds ordinal logistic regression model.

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  • $\begingroup$ How about if one fits the model to assess the effect of each variable on the desired outcome- the bivariable case $\endgroup$
    – Moses
    Feb 21, 2022 at 14:03
  • $\begingroup$ I thought it would be relevant to say bivariable multinomial logistic regression.; similar to what you would say logistic regression (two categories) $\endgroup$
    – Moses
    Feb 21, 2022 at 14:05
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    $\begingroup$ @Moses “Multinomial” refers to having three or more categories in your “y” variable, such as dog/cat/alligator instead of dog/cat. The number of predictor variables has no bearing on the name. // If you want to extend something like bivariate probit to considering two dependent but separate categorical variables, that is a different story (e.g., $y_1\in\{dog,cat,alligator\}$, $y_2\in\{house, yard, zoo\}$, where cats are most likely to be in the house, dogs in the yard, and alligators in the zoo, but those rules do not always hold). $\endgroup$
    – Dave
    Feb 21, 2022 at 14:38

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