ANCOVA - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
ANCOVA | $z$ test for a single proportion | Goodness of fit test |
You cannot compare more than 3 methods |
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Independent variables | Independent variable | Independent variable | |
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates) | None | None | |
Dependent variable | Dependent variable | Dependent variable | |
One quantitative of interval or ratio level | One categorical with 2 independent groups | One categorical with $J$ independent groups ($J \geqslant 2$) | |
THIS TABLE IS YET TO BE COMPLETED | Null hypothesis | Null 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. |
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n.a. | Alternative hypothesis | Alternative hypothesis | |
- | H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ |
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n.a. | Assumptions | Assumptions | |
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n.a. | Test statistic | Test statistic | |
- | $z = \dfrac{p - \pi_0}{\sqrt{\dfrac{\pi_0(1 - \pi_0)}{N}}}$
Here $p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size, and $\pi_0$ is the population proportion of successes according to the null hypothesis. | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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. | Sampling distribution of $z$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | |
- | Approximately the standard normal distribution | Approximately the chi-squared distribution with $J - 1$ degrees of freedom | |
n.a. | Significant? | Significant? | |
- | Two sided:
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n.a. | Approximate $C\%$ confidence interval for $\pi$ | n.a. | |
- | Regular (large sample):
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n.a. | Equivalent to | n.a. | |
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n.a. | Example context | Example context | |
- | Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? Use the normal approximation for the sampling distribution of the test statistic. | 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. | SPSS | SPSS | |
- | Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square...
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n.a. | Jamovi | Jamovi | |
- | Frequencies > 2 Outcomes - Binomial test
| Frequencies > N Outcomes - $\chi^2$ Goodness of fit
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Practice questions | Practice questions | Practice questions | |