# Goodness of fit test - 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

Goodness of fit test | Goodness of fit test | Goodness of fit test |
You cannot compare more than 3 methods |
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Independent variable | Independent variable | Independent variable | |

None | None | None | |

Dependent variable | Dependent variable | Dependent variable | |

One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with $J$ independent groups ($J \geqslant 2$) | |

Null hypothesis | Null hypothesis | Null hypothesis | |

- H
_{0}: the population proportions in each of the $J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$
- H
_{0}: 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 $\pi_J$
| - H
_{0}: the population proportions in each of the $J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$
- H
_{0}: 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 $\pi_J$
| - H
_{0}: the population proportions in each of the $J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$
- H
_{0}: 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 $\pi_J$
| |

Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |

- H
_{1}: the population proportions are not all as specified under the null hypothesis
- H
_{1}: the probabilities of drawing an observation from each of the conditions are not all as specified under the null hypothesis
| - H
_{1}: the population proportions are not all as specified under the null hypothesis
- H
_{1}: the probabilities of drawing an observation from each of the conditions are not all as specified under the null hypothesis
| - H
_{1}: the population proportions are not all as specified under the null hypothesis
- H
_{1}: the probabilities of drawing an observation from each of the conditions are not all as specified under the null hypothesis
| |

Assumptions | Assumptions | 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
| - 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
| - 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
| |

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 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 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. | |

Sampling distribution of $X^2$ if H_{0} were true | Sampling distribution of $X^2$ if H_{0} were true | Sampling distribution of $X^2$ if H_{0} were true | |

Approximately the chi-squared distribution with $J - 1$ degrees of freedom | Approximately the chi-squared distribution with $J - 1$ degrees of freedom | Approximately the chi-squared distribution with $J - 1$ degrees of freedom | |

Significant? | Significant? | 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 $\alpha$
| - 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 $\alpha$
| - 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 $\alpha$
| |

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 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 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 $\pi_{high} = 0.2$? | |

SPSS | SPSS | 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)
| 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)
| 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)
| |

Jamovi | Jamovi | 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)
| 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)
| 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)
| |

Practice questions | Practice questions | Practice questions | |