Goodness of fit test - overview

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Goodness of fit test
Binomial test for a single proportion$z$ test for a single proportion$z$ test for the difference between two proportionsGoodness of fit testChi-squared test for the relationship between two categorical variablesOne sample $z$ test for the meanOne sample $t$ test for the meanPaired sample $t$ testTwo sample $z$ testTwo sample $t$ test - equal variances not assumedTwo sample $t$ test - equal variances assumedOne way ANOVATwo way ANOVAPearson correlationRegression (OLS)Logistic regressionMann-Whitney-Wilcoxon testKruskal-Wallis testSign testMcNemar's testCochran's Q testMarginal Homogeneity test / Stuart-Maxwell testFriedman testWilcoxon signed-rank testOne sample Wilcoxon signed-rank testSpearman's rho
Independent variable
None
Dependent variable
One categorical with $J$ independent groups ($J \geqslant 2$)
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 $\pi_J$
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
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
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.
Sampling distribution of $X^2$ if H0 were true
Approximately the chi-squared distribution with $J - 1$ degrees of freedom
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 $\alpha$
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 $\pi_{high} = 0.2$?
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)
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)
Practice questions