Multilevel multinomial logistic regression - overview

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Multilevel multinomial logistic regression
One sample $t$ test for the mean
Independent variablesIndependent variable
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables, plus at least one random factorNone
Dependent variableDependent variable
One categorical with $J$ independent groups ($J \geqslant 2$)One quantitative of interval or ratio level
THIS TABLE IS YET TO BE COMPLETEDNull hypothesis
-H0: $\mu = \mu_0$

Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.
n.a.Alternative hypothesis
-H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$
n.a.Assumptions
-
  • Scores are normally distributed in the population
  • Sample is a simple random sample from the population. That is, observations are independent of one another
n.a.Test statistic
-$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $s$ is the sample standard deviation, and $N$ is the sample size.

The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.
n.a.Sampling distribution of $t$ if H0 were true
-$t$ distribution with $N - 1$ degrees of freedom
n.a.Significant?
-Two sided: Right sided: Left sided:
n.a.$C\%$ confidence interval for $\mu$
-$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N-1}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20).

The confidence interval for $\mu$ can also be used as significance test.
n.a.Effect size
-Cohen's $d$:
Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0.$
n.a.Visual representation
-
One sample t test
n.a.Example context
-Is the average mental health score of office workers different from $\mu_0 = 50$?
n.a.SPSS
-Analyze > Compare Means > One-Sample T Test...
  • Put your variable in the box below Test Variable(s)
  • Fill in the value for $\mu_0$ in the box next to Test Value
n.a.Jamovi
-T-Tests > One Sample T-Test
  • Put your variable in the box below Dependent Variables
  • Under Hypothesis, fill in the value for $\mu_0$ in the box next to Test Value, and select your alternative hypothesis
Practice questionsPractice questions