Goodness of fit test overview

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Goodness of fit test	Paired sample $t$ test	Kruskal-Wallis test	McNemar's test
Independent variable	Independent variable	Independent/grouping variable	Independent variable
None	2 paired groups	One categorical with $I$ independent groups ($I \geqslant 2$)	2 paired groups
Dependent variable	Dependent variable	Dependent variable	Dependent variable
One categorical with $J$ independent groups ($J \geqslant 2$)	One quantitative of interval or ratio level	One of ordinal level	One categorical with 2 independent groups
Null hypothesis	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: $\mu = \mu_0$ Here $\mu$ is the population mean of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.	If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations: H₀: the population medians for the $I$ groups are equal Else: Formulation 1: H₀: the population scores in any of the $I$ groups are not systematically higher or lower than the population scores in any of the other groups Formulation 2: H₀: P(an observation from population $g$ exceeds an observation from population $h$) = P(an observation from population $h$ exceeds an observation from population $g$), for each pair of groups. Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.	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: First score of pair is 0, second score of pair is 0 First score of pair is 0, second score of pair is 1 (switched) First score of pair is 1, second score of pair is 0 (switched) First score of pair is 1, second score of pair is 1 The null hypothesis H₀ is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) = 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 the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the null hypothesis are: H₀: $\pi_1 = \pi_2$, where $\pi_1$ is the population proportion of ones for the first paired group and $\pi_2$ is the population proportion of ones for the second paired group H₀: for each pair of scores, P(first score of pair is 1) = P(second score of pair is 1)
Alternative hypothesis	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 two sided: $\mu \neq \mu_0$ H₁ right sided: $\mu > \mu_0$ H₁ left sided: $\mu < \mu_0$	If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations: H₁: not all of the population medians for the $I$ groups are equal Else: Formulation 1: H₁: the poplation scores in some groups are systematically higher or lower than the population scores in other groups Formulation 2: H₁: for at least one pair of groups: P(an observation from population $g$ exceeds an observation from population $h$) $\neq$ P(an observation from population $h$ exceeds an observation from population $g$)	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: $\pi_1 \neq \pi_2$ H₁: for each pair of scores, P(first score of pair is 1) $\neq$ P(second score of pair is 1)
Assumptions	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	Difference scores are normally distributed in the population Sample of difference scores is a simple random sample from the population of difference scores. That is, difference scores are independent of one another	Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, observations 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
Test statistic	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.	$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$ Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores). The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.	$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ Here $N$ is the total sample size, $R_i$ is the sum of ranks in group $i$, and $n_i$ is the sample size of group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N (N + 1)} \times \sum \frac{R^2_i}{n_i}$ and then subtract $3(N + 1)$. Note: if ties are present in the data, the formula for $H$ is more complicated.	$X^2 = \dfrac{(b - c)^2}{b + c}$ 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.
Sampling distribution of $X^2$ if H₀ were true	Sampling distribution of $t$ if H₀ were true	Sampling distribution of $H$ if H₀ were true	Sampling distribution of $X^2$ if H₀ were true
Approximately the chi-squared distribution with $J - 1$ degrees of freedom	$t$ distribution with $N - 1$ degrees of freedom	For large samples, approximately the chi-squared distribution with $I - 1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used.	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 $b$.
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 $\alpha$	Two sided: Check if $t$ observed in sample is at least as extreme as critical value $t^$ or Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $t$ observed in sample is equal to or larger than critical value $t^$ or Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$	For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$: 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 $\alpha$	For test statistic $X^2$: 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 $\alpha$ If $b + c$ is small, the table for the binomial distribution should be used, with as test statistic $b$: Check if $b$ observed in sample is in the rejection region or Find two sided $p$ value corresponding to observed $b$ and check if it is equal to or smaller than $\alpha$
n.a.	$C\%$ confidence interval for $\mu$	n.a.	n.a.
-	$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$ where the critical value $t^$ is the value under the $t_{N-1}$ 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 $\mu$ can also be used as significance test.	-	-
n.a.	Effect size	n.a.	n.a.
-	Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0.$	-	-
n.a.	Visual representation	n.a.	n.a.
-		-	-
n.a.	Equivalent to	n.a.	Equivalent to
-	One sample $t$ test on the difference scores. Repeated measures ANOVA with one dichotomous within subjects factor.	-	Stuart-Maxwell test, with a categorical dependent variable consisting of two independent groups Cochran's Q test, with two related groups Two sided sign test: $b = W$, and $X^2 = z^2$
Example context	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 the average difference between the mental health scores before and after an intervention different from $\mu_0 = 0$?	Do people from different religions tend to score differently on social economic status?	Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?
SPSS	SPSS	SPSS	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)	Analyze > Compare Means > Paired-Samples T Test... Put the two paired variables in the boxes below Variable 1 and Variable 2	Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples... Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable Click on the Define Range... button. If you can't click on it, first click on the grouping variable so its background turns yellow Fill in the smallest value you have used to indicate your groups in the box next to Minimum, and the largest value you have used to indicate your groups in the box next to Maximum Continue and click OK	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 McNemar test
Jamovi	Jamovi	Jamovi	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)	T-Tests > Paired Samples T-Test Put the two paired variables in the box below Paired Variables, one on the left side of the vertical line and one on the right side of the vertical line Under Hypothesis, select your alternative hypothesis	ANOVA > One Way ANOVA - Kruskal-Wallis Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable	Frequencies > Paired Samples - McNemar test Put one of the two paired variables in the box below Rows and the other paired variable in the box below Columns
Practice questions	Practice questions	Practice questions	Practice questions

Goodness of fit test - overview