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 | McNemar's test | Marginal Homogeneity test / Stuart-Maxwell test |
You cannot compare more than 3 methods |
---|---|---|---|
Independent variable | Independent variable | Independent variable | |
None | 2 paired groups | 2 paired groups | |
Dependent variable | Dependent variable | Dependent variable | |
One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with 2 independent groups | One categorical with $J$ independent groups ($J \geqslant 2$) | |
Null hypothesis | Null hypothesis | Null hypothesis | |
| Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options:
Other formulations of the null hypothesis are:
| H0: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$ | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
| The alternative hypothesis H1 is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are:
| H1: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group. | |
Assumptions | Assumptions | Assumptions | |
|
|
| |
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. | $X^2 = \dfrac{(b - c)^2}{b + c}$
Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0. | Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand. | |
Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | Sampling distribution of the test statistic if H0 were true | |
Approximately the chi-squared distribution with $J - 1$ degrees of freedom | If $b + c$ is large enough (say, > 20), approximately the chi-squared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$. | Approximately the chi-squared distribution with $J - 1$ degrees of freedom | |
Significant? | Significant? | Significant? | |
| For test statistic $X^2$:
| If we denote the test statistic as $X^2$:
| |
n.a. | Equivalent to | n.a. | |
- |
| - | |
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 $\pi_{high} = 0.2$? | Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders? | Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| |
Jamovi | Jamovi | n.a. | |
Frequencies > N Outcomes - $\chi^2$ Goodness of fit
| Frequencies > Paired Samples - McNemar test
| - | |
Practice questions | Practice questions | Practice questions | |