# 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 | Kruskal-Wallis test | Cochran's Q test |
You cannot compare more than 3 methods |
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Independent variable | Independent/grouping variable | Independent/grouping variable | |

None | One categorical with $I$ independent groups ($I \geqslant 2$) | One within subject factor ($\geq 2$ related groups) | |

Dependent variable | Dependent variable | Dependent variable | |

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

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 $\pi_J$
| If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
- H
_{0}: the population medians for the $I$ groups are equal
Formulation 1: - H
_{0}: the population scores in any of the $I$ groups are not systematically higher or lower than the population scores in any of the other groups
- H
_{0}: P(an observation from population $g$ exceeds an observation from population $h$) = P(an observation from population $h$ exceeds an observation from population $g$), for each pair of groups.
| H_{0}: $\pi_1 = \pi_2 = \ldots = \pi_I$
Here $\pi_1$ is the population proportion of 'successes' for group 1, $\pi_2$ is the population proportion of 'successes' for group 2, and $\pi_I$ is the population proportion of 'successes' for group $I.$ | |

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
| If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
- H
_{1}: not all of the population medians for the $I$ groups are equal
Formulation 1: - H
_{1}: the poplation scores in some groups are systematically higher or lower than the population scores in other groups
- H
_{1}: for at least one pair of groups: P(an observation from population $g$ exceeds an observation from population $h$) $\neq$ P(an observation from population $h$ exceeds an observation from population $g$)
| H_{1}: not all population proportions are equal | |

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
| - Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, observations are independent of one another
| - Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks 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. | $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | If a failure is scored as 0 and a success is scored as 1:
$Q = k(k - 1) \dfrac{\sum_{groups} \Big (\mbox{group total} - \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k - \mbox{block total})}$ Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores. Before computing $Q$, first exclude blocks with equal scores in all $k$ groups. | |

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

Approximately the chi-squared distribution with $J - 1$ degrees of freedom | For large samples, approximately the chi-squared distribution with $I - 1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used. | If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ 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 $\alpha$
| For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
- 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 $\alpha$
| If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
- 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. | Equivalent to | |

- | - | Friedman test, with a categorical dependent variable consisting of two independent groups. | |

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 $\pi_{high} = 0.2$? | Do people from different religions tend to score differently on social economic status? | Subjects perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks? | |

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 > K Independent Samples...
- Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
- Click on the Define Range... button. If you can't click on it, first click on the grouping variable so its background turns yellow
- Fill in the smallest value you have used to indicate your groups in the box next to Minimum, and the largest value you have used to indicate your groups in the box next to Maximum
- Continue and click OK
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
- Put the $k$ variables containing the scores for the $k$ related groups in the white box below Test Variables
- Under Test Type, select Cochran's Q test
| |

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)
| ANOVA > One Way ANOVA - Kruskal-Wallis
- Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
| Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to:
ANOVA > Repeated Measures ANOVA - Friedman - Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures
| |

Practice questions | Practice questions | Practice questions | |