Multinomial logistic regression - overview

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Multinomial logistic regression
Binomial test for a single proportion
Independent variablesIndependent variable
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variablesNone
Dependent variableDependent variable
One categorical with $J$ independent groups ($J \geqslant 2$)One categorical with 2 independent groups
THIS TABLE IS YET TO BE COMPLETEDNull hypothesis
-H0: $\pi = \pi_0$

Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis.
n.a.Alternative hypothesis
-H1 two sided: $\pi \neq \pi_0$
H1 right sided: $\pi > \pi_0$
H1 left sided: $\pi < \pi_0$
n.a.Assumptions
-
  • Sample is a simple random sample from the population. That is, observations are independent of one another
n.a.Test statistic
-$X$ = number of successes in the sample
n.a.Sampling distribution of $X$ if H0 were true
-Binomial($n$, $P$) distribution.

Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis).
n.a.Significant?
-Two sided:
  • Check if $X$ observed in sample is in the rejection region or
  • Find two sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
Right sided:
  • Check if $X$ observed in sample is in the rejection region or
  • Find right sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
Left sided:
  • Check if $X$ observed in sample is in the rejection region or
  • Find left sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
n.a.Example context
-Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$?
n.a.SPSS
-Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
  • Put your dichotomous variable in the box below Test Variable List
  • Fill in the value for $\pi_0$ in the box next to Test Proportion
n.a.Jamovi
-Frequencies > 2 Outcomes - Binomial test
  • Put your dichotomous variable in the white box at the right
  • Fill in the value for $\pi_0$ in the box next to Test value
  • Under Hypothesis, select your alternative hypothesis
Practice questionsPractice questions