Goodness of fit test overview

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Goodness of fit test	Chi-squared test for the relationship between two categorical variables	Two sample $z$ test
Independent variable	Independent /column variable	Independent/grouping variable
None	One categorical with $I$ independent groups ($I \geqslant 2$)	One categorical with 2 independent groups
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 quantitative of interval or ratio level
Null hypothesis	Null hypothesis	Null hypothesis
H₀: the population proportions in each of the $J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$ or equivalently H₀: 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 $\pi_J$	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: the distribution of the dependent variable is the same in each of the $I$ populations If there is one random sample of size $N$ from the total population: H₀: the row and column variables are independent	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
H₁: the population proportions are not all as specified under the null hypothesis or equivalently H₁: the probabilities of drawing an observation from each of the conditions are not all as specified under the null 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: the distribution of the dependent variable is not the same in all of the $I$ populations If there is one random sample of size $N$ from the total population: H₁: the row and column variables are dependent	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
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 under the null hypothesis. Rule of thumb: 2 $\times$ 2 table: all four expected cell counts are 5 or more Larger than 2 $\times$ 2 tables: average of the expected cell counts is 5 or more, smallest expected cell count is 1 or more There are $I$ independent simple random samples from each of $I$ populations defined by the independent variable, or there is one simple random sample from the total population	Within each population, the scores on the dependent variable are normally distributed Population standard deviations $\sigma_1$ and $\sigma_2$ are known Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, 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 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.	$z = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$ 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 $X^2$ 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	Standard normal distribution
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 $\alpha$	Two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
n.a.	n.a.	$C\%$ confidence interval for $\mu_1 - \mu_2$
-	-	$(\bar{y}_1 - \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$ 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.	n.a.	Visual representation
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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 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... 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 > Descriptive Statistics > Crosstabs... Put one of your two categorical variables in the box below Row(s), and the other categorical variable in the box below Column(s) Click the Statistics... button, and click on the square in front of Chi-square Continue and click OK	-
Jamovi	Jamovi	n.a.
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 > Independent Samples - $\chi^2$ test of association Put one of your two categorical variables in the box below Rows, and the other categorical variable in the box below Columns	-
Practice questions	Practice questions	Practice questions

Goodness of fit test - overview