Goodness of fit test - overview
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Goodness of fit test | Chi-squared test for the relationship between two categorical variables | $z$ test for a single proportion |
You cannot compare more than 3 methods |
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Independent variable | Independent /column variable | Independent variable | |
None | One categorical with $I$ independent groups ($I \geqslant 2$) | None | |
Dependent variable | Dependent /row variable | Dependent variable | |
One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with 2 independent groups | |
Null hypothesis | Null hypothesis | Null hypothesis | |
| H0: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
| 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. | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
| H1: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
| H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | |
Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | 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. | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells. | $z = \dfrac{p - \pi_0}{\sqrt{\dfrac{\pi_0(1 - \pi_0)}{N}}}$
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. | |
Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $z$ if H0 were true | |
Approximately the chi-squared distribution with $J - 1$ degrees of freedom | Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | Approximately the standard normal distribution | |
Significant? | Significant? | Significant? | |
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| Two sided:
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n.a. | n.a. | Approximate $C\%$ confidence interval for $\pi$ | |
- | - | Regular (large sample):
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n.a. | n.a. | Equivalent to | |
- | - |
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Example context | Example context | 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 $\pi_{high} = 0.2$? | Is there an association between economic class and gender? Is the distribution of economic class different between men and women? | 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. | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square...
| Analyze > Descriptive Statistics > Crosstabs...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
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Jamovi | Jamovi | Jamovi | |
Frequencies > N Outcomes - $\chi^2$ Goodness of fit
| Frequencies > Independent Samples - $\chi^2$ test of association
| Frequencies > 2 Outcomes - Binomial test
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Practice questions | Practice questions | Practice questions | |