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
Cochran's Q test
Marginal Homogeneity test / Stuart-Maxwell test
Two sample $t$ test - equal variances assumed
Independent variableIndependent/grouping variableIndependent variableIndependent/grouping variable
NoneOne within subject factor ($\geq 2$ related groups)2 paired groupsOne categorical with 2 independent groups
Dependent variableDependent variableDependent variableDependent variable
One categorical with $J$ independent groups ($J \geqslant 2$)One categorical with 2 independent groupsOne categorical with $J$ independent groups ($J \geqslant 2$)One quantitative of interval or ratio level
Null hypothesisNull hypothesisNull hypothesisNull 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: $\pi_1 = \pi_2 = \ldots = \pi_I$

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 $I.$
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 $j.$
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 hypothesisAlternative hypothesisAlternative hypothesisAlternative 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: not all population proportions are equalH1: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.H1 two sided: $\mu_1 \neq \mu_2$
H1 right sided: $\mu_1 > \mu_2$
H1 left sided: $\mu_1 < \mu_2$
AssumptionsAssumptionsAssumptionsAssumptions
  • 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 of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another
  • Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
  • Within each population, the scores on the dependent variable are normally distributed
  • The standard deviation of the scores on the dependent variable is the same in both populations: $\sigma_1 = \sigma_2$
  • 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 statisticTest statisticTest statisticTest 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.
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.
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.$t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ 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 $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1 - \bar{y}_2$ 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 $\mu_2 > \mu_1$.
n.a.n.a.n.a.Pooled standard deviation
---$s_p = \sqrt{\dfrac{(n_1 - 1) \times s^2_1 + (n_2 - 1) \times s^2_2}{n_1 + n_2 - 2}}$
Sampling distribution of $X^2$ if H0 were trueSampling distribution of $Q$ if H0 were trueSampling distribution of the test statistic if H0 were trueSampling distribution of $t$ if H0 were true
Approximately the chi-squared distribution with $J - 1$ degrees of freedomIf the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedomApproximately the chi-squared distribution with $J - 1$ degrees of freedom$t$ distribution with $n_1 + n_2 - 2$ degrees of freedom
Significant?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 $\alpha$
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 $\alpha$
If we denote the test statistic as $X^2$:
  • 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: Right sided: Left sided:
n.a.n.a.n.a.$C\%$ confidence interval for $\mu_1 - \mu_2$
---$(\bar{y}_1 - \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
where the critical value $t^*$ is the value under the $t_{n_1 + n_2 - 2}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20).

The confidence interval for $\mu_1 - \mu_2$ can also be used as significance test.
n.a.n.a.n.a.Effect size
---Cohen's $d$:
Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1 - \bar{y}_2}{s_p}$$ Cohen's $d$ indicates how many standard deviations $s_p$ the two sample means are removed from each other.
n.a.n.a.n.a.Visual representation
---
Two sample t test - equal variances assumed
n.a.Equivalent ton.a.Equivalent to
-Friedman test, with a categorical dependent variable consisting of two independent groups.-One way ANOVA with an independent variable with 2 levels ($I$ = 2):
  • two sided two sample $t$ test is equivalent to ANOVA $F$ test when $I$ = 2
  • two sample $t$ test is equivalent to $t$ test for contrast when $I$ = 2
  • two sample $t$ test is equivalent to $t$ test multiple comparisons when $I$ = 2
OLS regression with one categorical independent variable with 2 levels:
  • two sided two sample $t$ test is equivalent to $F$ test regression model
  • two sample $t$ test is equivalent to $t$ test for regression coefficient $\beta_1$
Example contextExample contextExample contextExample 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 $\pi_{high} = 0.2$?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?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?Is the average mental health score different between men and women? Assume that in the population, the standard deviation of mental health scores is equal amongst men and women.
SPSSSPSSSPSSSPSS
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 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
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
  • Put the two paired variables in the boxes below Variable 1 and Variable 2
  • Under Test Type, select the Marginal Homogeneity test
Analyze > Compare Means > Independent-Samples T Test...
  • Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable
  • Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
  • Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
  • Continue and click OK
JamoviJamovin.a.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)
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
-T-Tests > Independent Samples T-Test
  • Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
  • Under Tests, select Student's (selected by default)
  • Under Hypothesis, select your alternative hypothesis
Practice questionsPractice questionsPractice questionsPractice questions