This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the righthand column. To practice with a specific method click the button at the bottom row of the table
One categorical with $I$ independent groups ($I \geqslant 2$)
One within subject factor ($\geq 2$ related groups)
One categorical with $I$ independent groups ($I \geqslant 2$)
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables
Dependent variable
Dependent variable
Dependent variable
Dependent variable
Dependent variable
One categorical with $J$ independent groups ($J \geqslant 2$)
One of ordinal level
One categorical with 2 independent groups
One quantitative of interval or ratio level
One categorical with 2 independent groups
Null hypothesis
Null hypothesis
Null hypothesis
Null hypothesis
Null hypothesis
H_{0}: the population proportions in each of the $J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$
or equivalently
H_{0}: 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 $\pi_J$
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:
H_{0}: the population medians for the $I$ groups are equal
Else:
Formulation 1:
H_{0}: 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:
H_{0}:
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.
H_{0}: $\pi_1 = \pi_2 = \ldots = \pi_I$
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 $I.$
ANOVA $F$ test:
H_{0}: $\mu_1 = \mu_2 = \ldots = \mu_I$
$\mu_1$ is the population mean for group 1; $\mu_2$ is the population mean for group 2; $\mu_I$ is the population mean for group $I$
$t$ Test for contrast:
H_{0}: $\Psi = 0$
$\Psi$ is the population contrast, defined as $\Psi = \sum a_i\mu_i$. Here $\mu_i$ is the population mean for group $i$ and $a_i$ is the coefficient for $\mu_i$. The coefficients $a_i$ sum to 0.
$t$ Test multiple comparisons:
H_{0}: $\mu_g = \mu_h$
$\mu_g$ is the population mean for group $g$; $\mu_h$ is the population mean for group $h$
Model chisquared test for the complete regression model:
H_{0}: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
Wald test for individual regression coefficient $\beta_k$:
H_{0}: $\beta_k = 0$
or in terms of odds ratio:
H_{0}: $e^{\beta_k} = 1$
Likelihood ratio chisquared test for individual regression coefficient $\beta_k$:
H_{0}: $\beta_k = 0$
or in terms of odds ratio:
H_{0}: $e^{\beta_k} = 1$
in the regression equation
$
\ln \big(\frac{\pi_{y = 1}}{1  \pi_{y = 1}} \big) = \beta_0 + \beta_1 \times x_1 + \beta_2 \times x_2 + \ldots + \beta_K \times x_K
$. Here $ x_i$ represents independent variable $ i$, $\beta_i$ is the regression weight for independent variable $ x_i$, and $\pi_{y = 1}$ represents the true probability that the dependent variable $ y = 1$ (or equivalently, the proportion of $ y = 1$ in the population) given the scores on the independent variables.
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
H_{1}: the population proportions are not all as specified under the null hypothesis
or equivalently
H_{1}: the probabilities of drawing an observation from each of the conditions are not all as specified under the null hypothesis
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:
H_{1}: not all of the population medians for the $I$ groups are equal
Else:
Formulation 1:
H_{1}:
the poplation scores in some groups are systematically higher or lower than the population scores in other groups
Formulation 2:
H_{1}:
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 $g$)
H_{1}: not all population proportions are equal
ANOVA $F$ test:
H_{1}: not all population means are equal
$t$ Test for contrast:
H_{1} two sided: $\Psi \neq 0$
H_{1} right sided: $\Psi > 0$
H_{1} left sided: $\Psi < 0$
$t$ Test multiple comparisons:
H_{1}  usually two sided: $\mu_g \neq \mu_h$
Model chisquared test for the complete regression model:
H_{1}: not all population regression coefficients are 0
Wald test for individual regression coefficient $\beta_k$:
H_{1}: $\beta_k \neq 0$
or in terms of odds ratio:
H_{1}: $e^{\beta_k} \neq 1$
If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested:
H_{1} right sided: $\beta_k > 0$
H_{1} left sided: $\beta_k < 0$
Likelihood ratio chisquared test for individual regression coefficient $\beta_k$:
H_{1}: $\beta_k \neq 0$
or in terms of odds ratio:
H_{1}: $e^{\beta_k} \neq 1$
Assumptions
Assumptions
Assumptions
Assumptions
Assumptions
Sample size is large enough for $X^2$ to be approximately chisquared 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
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 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks 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 each of the populations: $\sigma_1 = \sigma_2 = \ldots = \sigma_I$
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
In the population, the relationship between the independent variables and the log odds $\ln (\frac{\pi_{y=1}}{1  \pi_{y=1}})$ is linear
The residuals are independent of one another
Often ignored additional assumption:
Variables are measured without error
Also pay attention to:
Multicollinearity
Outliers
Test statistic
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.
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.
If a failure is scored as 0 and a success is scored as 1:
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.
ANOVA $F$ test:
$\begin{aligned}[t]
F &= \dfrac{\sum\nolimits_{subjects} (\mbox{subject's group mean}  \mbox{overall mean})^2 / (I  1)}{\sum\nolimits_{subjects} (\mbox{subject's score}  \mbox{its group mean})^2 / (N  I)}\\
&= \dfrac{\mbox{sum of squares between} / \mbox{degrees of freedom between}}{\mbox{sum of squares error} / \mbox{degrees of freedom error}}\\
&= \dfrac{\mbox{mean square between}}{\mbox{mean square error}}
\end{aligned}
$
where $N$ is the total sample size, and $I$ is the number of groups.
Note: mean square between is also known as mean square model, and mean square error is also known as mean square residual or mean square within.
$t$ Test for contrast:
$t = \dfrac{c}{s_p\sqrt{\sum \dfrac{a^2_i}{n_i}}}$
Here $c$ is the sample estimate of the population contrast $\Psi$: $c = \sum a_i\bar{y}_i$, with $\bar{y}_i$ the sample mean in group $i$. $s_p$ is the pooled standard deviation based on all the $I$ groups in the ANOVA, $a_i$ is the contrast coefficient for group $i$, and $n_i$ is the sample size of group $i$.
Note that if the contrast compares only two group means with each other, this $t$ statistic is very similar to the two sample $t$ statistic (assuming equal population standard deviations). In that case the only difference is that we now base the pooled standard deviation on all the $I$ groups, which affects the $t$ value if $I \geqslant 3$. It also affects the corresponding degrees of freedom.
$t$ Test multiple comparisons:
$t = \dfrac{\bar{y}_g  \bar{y}_h}{s_p\sqrt{\dfrac{1}{n_g} + \dfrac{1}{n_h}}}$
$\bar{y}_g$ is the sample mean in group $g$, $\bar{y}_h$ is the sample mean in group $h$,
$s_p$ is the pooled standard deviation based on all the $I$ groups in the ANOVA,
$n_g$ is the sample size of group $g$, and $n_h$ is the sample size of group $h$.
Note that this $t$ statistic is very similar to the two sample $t$ statistic (assuming equal population standard deviations). The only difference is that we now base the pooled standard deviation on all the $I$ groups, which affects the $t$ value if $I \geqslant 3$. It also affects the corresponding degrees of freedom.
Model chisquared test for the complete regression model:
$X^2 = D_{null}  D_K = \mbox{null deviance}  \mbox{model deviance} $
$D_{null}$, the null deviance, is conceptually similar to the total variance of the dependent variable in OLS regression analysis. $D_K$, the model deviance, is conceptually similar to the residual variance in OLS regression analysis.
Wald test for individual $\beta_k$:
The wald statistic can be defined in two ways:
Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$
Wald $ = \dfrac{b_k}{SE_{b_k}}$
SPSS uses the first definition.
Likelihood ratio chisquared test for individual $\beta_k$:
$X^2 = D_{K1}  D_K$
$D_{K1}$ is the model deviance, where independent variable $k$ is excluded from the model. $D_{K}$ is the model deviance, where independent variable $k$ is included in the model.
Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
$F$ test:
Check if $F$ observed in sample is equal to or larger than critical value $F^*$ or
Find $p$ value corresponding to observed $F$ and check if it is equal to or smaller than $\alpha$ (e.g. .01 < $p$ < .025 when $F$ = 3.91, df between = 4, and df error = 20)
$t$ Test for contrast 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 $\alpha$
$t$ Test for contrast 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 $\alpha$
$t$ Test for contrast 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 $\alpha$
$t$ Test multiple comparisons two sided:
Check if $t$ observed in sample is at least as extreme as critical value $t^{**}$. Adapt $t^{**}$ according to a multiple comparison procedure (e.g., Bonferroni) or
Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$. Adapt the $p$ value or $\alpha$ according to a multiple comparison procedure
$t$ Test multiple comparisons right sided
Check if $t$ observed in sample is equal to or larger than critical value $t^{**}$. Adapt $t^{**}$ according to a multiple comparison procedure (e.g., Bonferroni) or
Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$. Adapt the $p$ value or $\alpha$ according to a multiple comparison procedure
$t$ Test multiple comparisons left sided
Check if $t$ observed in sample is equal to or smaller than critical value $t^{**}$. Adapt $t^{**}$ according to a multiple comparison procedure (e.g., Bonferroni) or
Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$. Adapt the $p$ value or $\alpha$ according to a multiple comparison procedure
For the model chisquared test for the complete regression model and likelihood ratio chisquared test for individual $\beta_k$:
Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
For the Wald test:
If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: same procedure as for the chisquared tests. Wald can be interpret as $X^2$
If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: same procedure as for any $z$ test. Wald can be interpreted as $z$.
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$C\%$ confidence interval for $\Psi$, for $\mu_g  \mu_h$, and for $\mu_i$
Waldtype approximate $C\%$ confidence interval for $\beta_k$



Confidence interval for $\Psi$ (contrast):
$c \pm t^* \times s_p\sqrt{\sum \dfrac{a^2_i}{n_i}}$
where the critical value $t^*$ is the value under the $t_{N  I}$ distribution with the area $C / 100$ between $t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). Note that $n_i$ is the sample size of group $i$, and $N$ is the total sample size, based on all the $I$ groups.
Confidence interval for $\mu_g  \mu_h$ (multiple comparisons):
$(\bar{y}_g  \bar{y}_h) \pm t^{**} \times s_p\sqrt{\dfrac{1}{n_g} + \dfrac{1}{n_h}}$
where $t^{**}$ depends upon $C$, degrees of freedom ($N  I$), and the multiple comparison procedure. If you do not want to apply a multiple comparison procedure, $t^{**} = t^* = $ the value under the $t_{N  I}$ distribution with the area $C / 100$ between $t^*$ and $t^*$. Note that $n_g$ is the sample size of group $g$, $n_h$ is the sample size of group $h$, and $N$ is the total sample size, based on all the $I$ groups.
Confidence interval for single population mean $\mu_i$:
$\bar{y}_i \pm t^* \times \dfrac{s_p}{\sqrt{n_i}}$
where $\bar{y}_i$ is the sample mean in group $i$, $n_i$ is the sample size of group $i$, and the critical value $t^*$ is the value under the $t_{N  I}$ distribution with the area $C / 100$ between $t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). Note that $n_i$ is the sample size of group $i$, and $N$ is the total sample size, based on all the $I$ groups.
$b_k \pm z^* \times SE_{b_k}$
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).
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Effect size
Goodness of fit measure $R^2_L$



Proportion variance explained $\eta^2$ and $R^2$:
Proportion variance of the dependent variable $y$ explained by the independent variable:
$$
\begin{align}
\eta^2 = R^2
&= \dfrac{\mbox{sum of squares between}}{\mbox{sum of squares total}}
\end{align}
$$
Only in one way ANOVA $\eta^2 = R^2.$ $\eta^2$ (and $R^2$) is the proportion variance explained in the sample. It is a positively biased estimate of the proportion variance explained in the population.
Proportion variance explained $\omega^2$:
Corrects for the positive bias in $\eta^2$ and is equal to:
$$\omega^2 = \frac{\mbox{sum of squares between}  \mbox{df between} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}$$
$\omega^2$ is a better estimate of the explained variance in the population than $\eta^2.$
Cohen's $d$:
Standardized difference between the mean in group $g$ and in group $h$:
$$d_{g,h} = \frac{\bar{y}_g  \bar{y}_h}{s_p}$$
Cohen's $d$ indicates how many standard deviations $s_p$ two sample means are removed from each other.
$R^2_L = \dfrac{D_{null}  D_K}{D_{null}}$
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.
Friedman test, with a categorical dependent variable consisting of two independent groups.
OLS regression with one categorical independent variable transformed into $I  1$ code variables:
$F$ test ANOVA is equivalent to $F$ test regression model
$t$ test for contrast $i$ is equivalent to $t$ test for regression coefficient $\beta_i$ (specific contrast tested depends on how the code variables are defined)

Example context
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 $\pi_{high} = 0.2$?
Do people from different religions tend to score differently on social economic status?
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?
Is the average mental health score different between people from a low, moderate, and high economic class?
Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?
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)
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 > 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 > Compare Means > OneWay ANOVA...
Put your dependent (quantitative) variable in the box below Dependent List and your independent (grouping) variable in the box below Factor
or
Analyze > General Linear Model > Univariate...
Put your dependent (quantitative) variable in the box below Dependent Variable and your independent (grouping) variable in the box below Fixed Factor(s)
Analyze > Regression > Binary Logistic...
Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Covariate(s)
Jamovi
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)
ANOVA > One Way ANOVA  KruskalWallis
Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
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
ANOVA > ANOVA
Put your dependent (quantitative) variable in the box below Dependent Variable and your independent (grouping) variable in the box below Fixed Factors
Regression > 2 Outcomes  Binomial
Put your dependent variable in the box below Dependent Variable and your independent variables of interval/ratio level in the box below Covariates
If you also have code (dummy) variables as independent variables, you can put these in the box below Covariates as well
Instead of transforming your categorical independent variable(s) into code variables, you can also put the untransformed categorical independent variables in the box below Factors. Jamovi will then make the code variables for you 'behind the scenes'