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 | Logistic regression | Two sample $z$ test |
You cannot compare more than 3 methods |
---|---|---|---|
Independent variable | Independent variables | Independent/grouping variable | |
None | One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables | One categorical with 2 independent groups | |
Dependent variable | Dependent variable | Dependent variable | |
One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with 2 independent groups | One quantitative of interval or ratio level | |
Null hypothesis | Null hypothesis | Null hypothesis | |
| Model chi-squared test for the complete regression model:
| H0: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2. | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
| Model chi-squared test for the complete regression model:
| H1 two sided: $\mu_1 \neq \mu_2$ H1 right sided: $\mu_1 > \mu_2$ H1 left sided: $\mu_1 < \mu_2$ | |
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. | Model chi-squared test for the complete regression model:
The wald statistic can be defined in two ways:
Likelihood ratio chi-squared test for individual $\beta_k$:
| $z = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 2, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1 - \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$. | |
Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $X^2$ and of the Wald statistic if H0 were true | Sampling distribution of $z$ if H0 were true | |
Approximately the chi-squared distribution with $J - 1$ degrees of freedom | Sampling distribution of $X^2$, as computed in the model chi-squared test for the complete model:
| Standard normal distribution | |
Significant? | Significant? | Significant? | |
| For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$:
| Two sided:
| |
n.a. | Wald-type approximate $C\%$ confidence interval for $\beta_k$ | $C\%$ confidence interval for $\mu_1 - \mu_2$ | |
- | $b_k \pm z^* \times SE_{b_k}$ where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $-z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval). | $(\bar{y}_1 - \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$
where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $-z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval). The confidence interval for $\mu_1 - \mu_2$ can also be used as significance test. | |
n.a. | Goodness of fit measure $R^2_L$ | n.a. | |
- | $R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ There are several other goodness of fit measures in logistic regression. In logistic regression, there is no single agreed upon measure of goodness of fit. | - | |
n.a. | n.a. | Visual representation | |
- | - | ||
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 $\pi_{high} = 0.2$? | Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes? | Is the average mental health score different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1 = 2$ amongst men and $\sigma_2 = 2.5$ amongst women. | |
SPSS | SPSS | n.a. | |
Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square...
| Analyze > Regression > Binary Logistic...
| - | |
Jamovi | Jamovi | n.a. | |
Frequencies > N Outcomes - $\chi^2$ Goodness of fit
| Regression > 2 Outcomes - Binomial
| - | |
Practice questions | Practice questions | Practice questions | |