ANCOVA - overview

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ANCOVA
Multinomial logistic regression
Goodness of fit test
You cannot compare more than 3 methods
Independent variablesIndependent variablesIndependent variable
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variablesNone
Dependent variableDependent variableDependent variable
One quantitative of interval or ratio levelOne categorical with $J$ independent groups ($J \geqslant 2$)One categorical with $J$ independent groups ($J \geqslant 2$)
THIS TABLE IS YET TO BE COMPLETEDTHIS TABLE IS YET TO BE COMPLETEDNull hypothesis
--
  • H0: the population proportions in each of the $J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$
or equivalently
  • H0: the probability of drawing an observation from condition 1 is $\pi_1$, the probability of drawing an observation from condition 2 is $\pi_2$, $\ldots$, the probability of drawing an observation from condition $J$ is $\pi_J$
n.a.n.a.Alternative hypothesis
--
  • H1: the population proportions are not all as specified under the null hypothesis
or equivalently
  • H1: the probabilities of drawing an observation from each of the conditions are not all as specified under the null hypothesis
n.a.n.a.Assumptions
--
  • Sample size is large enough for $X^2$ to be approximately chi-squared distributed. Rule of thumb: all $J$ expected cell counts are 5 or more
  • Sample is a simple random sample from the population. That is, observations are independent of one another
n.a.n.a.Test statistic
--$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells.
n.a.n.a.Sampling distribution of $X^2$ if H0 were true
--Approximately the chi-squared distribution with $J - 1$ degrees of freedom
n.a.n.a.Significant?
--
  • Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or
  • Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
n.a.n.a.Example context
--Is the proportion of people with a low, moderate, and high social economic status in the population different from $\pi_{low} = 0.2,$ $\pi_{moderate} = 0.6,$ and $\pi_{high} = 0.2$?
n.a.n.a.SPSS
--Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square...
  • Put your categorical variable in the box below Test Variable List
  • Fill in the population proportions / probabilities according to $H_0$ in the box below Expected Values. If $H_0$ states that they are all equal, just pick 'All categories equal' (default)
n.a.n.a.Jamovi
--Frequencies > N Outcomes - $\chi^2$ Goodness of fit
  • Put your categorical variable in the box below Variable
  • Click on Expected Proportions and fill in the population proportions / probabilities according to $H_0$ in the boxes below Ratio. If $H_0$ states that they are all equal, you can leave the ratios equal to the default values (1)
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