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
The population proportions in each of the $J$ conditions are $\pi_1$, $\pi_2$, $\ldots$, $\pi_J$
or equivalently
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 $\pi_J$
$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis
The scores in any of the related groups are not systematically higher or lower than the scores in any of the other related groups
Note: usually, the related groups are the different measurement points
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.
ANOVA $F$ test:
$\mu_1 = \mu_2 = \ldots = \mu_I$
$\mu_1$ is the unknown mean in population 1; $\mu_2$ is the unknown mean in population 2; $\mu_I$ is the unknown mean in population $I$
$t$ Test for contrast:
$\Psi = 0$
$\Psi$ is a contrast in the population, defined as $\Psi = \sum a_i\mu_i$. Here $\mu_i$ is the unknown mean in population $i$ and $a_i$ is the coefficient for $\mu_i$. The coefficients $a_i$ sum to 0.
$t$ Test multiple comparisons:
$\mu_g = \mu_h$
$\mu_g$ is the unknown mean in population $g$; $\mu_h$ is the unknown mean in population $h$
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
Alternative hypothesis
$F$ test for the complete regression model:
not all population regression coefficients are 0 or equivalenty
The variance explained by all the independent variables together (the complete model) is larger than 0 in the population: $\rho^2 > 0$
$t$ test for individual $\beta_k$:
Two sided: $\beta_k \neq 0$
Right sided: $\beta_k > 0$
Left sided: $\beta_k < 0$
The population proportions are not all as specified under the null hypothesis
or equivalently
The probabilities of drawing an observation from each of the conditions are not all as specified under the null hypothesis
Two sided: $\mu \neq \mu_0$
Right sided: $\mu > \mu_0$
Left sided: $\mu < \mu_0$
The scores in some of the related groups are systematically higher or lower than the scores in other related groups
ANOVA $F$ test:
Not all population means are equal
$t$ Test for contrast:
Two sided: $\Psi \neq 0$
Right sided: $\Psi > 0$
Left sided: $\Psi < 0$
$t$ Test multiple comparisons:
Usually two sided: $\mu_g \neq \mu_h$
Assumptions
Assumptions
Assumptions
Assumptions
Assumptions
In the population, the residuals are normally distributed at each combination of values of the independent variables
In the population, the standard deviation $\sigma$ of the residuals is the same for each combination of values of the independent variables (homoscedasticity)
In the population, the relationship between the independent variables and the mean of the dependent variable $\mu_y$ is linear. If this linearity assumption holds, the mean of the residuals is 0 for each combination of values of the independent variables
The residuals are independent of one another
Often ignored additional assumption:
Variables are measured without error
Also pay attention to:
Multicollinearity
Outliers
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
Scores are normally distributed in the population
Population standard deviation $\sigma$ is known
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
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
Test statistic
Test statistic
Test statistic
Test statistic
Test statistic
$F$ test for the complete regression model:
$
\begin{aligned}[t]
F &= \dfrac{\sum (\hat{y}_j  \bar{y})^2 / K}{\sum (y_j  \hat{y}_j)^2 / (N  K  1)}\\
&= \dfrac{\mbox{sum of squares model} / \mbox{degrees of freedom model}}{\mbox{sum of squares error} / \mbox{degrees of freedom error}}\\
&= \dfrac{\mbox{mean square model}}{\mbox{mean square error}}
\end{aligned}
$
where $\hat{y}_j$ is the predicted score on the dependent variable $y$ of subject $j$, $\bar{y}$ is the mean of $y$, $y_j$ is the score on $y$ of subject $j$, $N$ is the total sample size, and $K$ is the number of independent variables
$t$ test for individual $\beta_k$:
$t = \dfrac{b_k}{SE_{b_k}}$
If only one independent variable: $SE_{b_1} = \dfrac{\sqrt{\sum (y_j  \hat{y}_j)^2 / (N  2)}}{\sqrt{\sum (x_j  \bar{x})^2}} = \dfrac{s}{\sqrt{\sum (x_j  \bar{x})^2}}$, with $s$ the sample standard deviation of the residuals, $x_j$ the score of subject $j$ on the independent variable $x$, and $\bar{x}$ the mean of $x$. For models with more than one independent variable, computing $SE_{b_k}$ becomes complicated
Note 1: mean square model is also known as mean square regression; mean square error is also known as mean square residual
Note 2: if only one independent variable ($K = 1$), the $F$ test for the complete regression model is equivalent to the two sided $t$ test for $\beta_1$
$X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
where 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
$z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to H0, $\sigma$ is the population standard deviation,
$N$ is the sample size.
Here $N$ is the number of 'blocks' (usually the subjects  so if you have 4 repeated measurements for 60 subjects, $N$ equals 60), $k$ is the number of related groups (usually the number of repeated measurements), and $R_i$ is the sum of ranks in group $i$.
Remember that multiplication precedes addition, so first compute $\frac{12}{N \times k(k + 1)} \times \sum R^2_i$ and then subtract $3 \times N(k + 1)$.
Note: if ties are present in the data, the formula for $Q$ is more complicated.
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; 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.
Sample standard deviation of the residuals $s$
n.a.
n.a.
n.a.
Pooled standard deviation
$\begin{aligned}
s &= \sqrt{\dfrac{\sum (y_j  \hat{y}_j)^2}{N  K  1}}\\ &= \sqrt{\dfrac{\mbox{sum of squares error}}{\mbox{degrees of freedom error}}}\\ &= \sqrt{\mbox{mean square error}}
\end{aligned}
$



$
\begin{aligned}
s_p &= \sqrt{\dfrac{(n_1  1) \times s^2_1 + (n_2  1) \times s^2_2 + \ldots + (n_I  1) \times s^2_I}{N  I}}\\
&= \sqrt{\dfrac{\sum\nolimits_{subjects} (\mbox{subject's score}  \mbox{its group mean})^2}{N  I}}\\
&= \sqrt{\dfrac{\mbox{sum of squares error}}{\mbox{degrees of freedom error}}}\\
&= \sqrt{\mbox{mean square error}}
\end{aligned}
$
where $s^2_i$ is the variance in group $i$
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
$C\%$ confidence interval for $\beta_k$ and for $\mu_y$; $C\%$ prediction interval for $y_{new}$
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$C\%$ confidence interval for $\mu$
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$C\%$ confidence interval for $\Psi$, for $\mu_g  \mu_h$, and for $\mu_i$
Confidence interval for $\beta_k$:
$b_k \pm t^* \times SE_{b_k}$
If only one independent variable: $SE_{b_1} = \dfrac{\sqrt{\sum (y_j  \hat{y}_j)^2 / (N  2)}}{\sqrt{\sum (x_j  \bar{x})^2}} = \dfrac{s}{\sqrt{\sum (x_j  \bar{x})^2}}$
Confidence interval for $\mu_y$, the population mean of $y$ given the values on the independent variables:
$\hat{y} \pm t^* \times SE_{\hat{y}}$
If only one independent variable:
$SE_{\hat{y}} = s \sqrt{\dfrac{1}{N} + \dfrac{(x^*  \bar{x})^2}{\sum (x_j  \bar{x})^2}}$
Prediction interval for $y_{new}$, the score on $y$ of a future respondent:
$\hat{y} \pm t^* \times SE_{y_{new}}$
If only one independent variable:
$SE_{y_{new}} = s \sqrt{1 + \dfrac{1}{N} + \dfrac{(x^*  \bar{x})^2}{\sum (x_j  \bar{x})^2}}$
In all formulas, the critical value $t^*$ is the value under the $t_{N  K  1}$ distribution with the area $C / 100$ between $t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20).

$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
where $z^*$ 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)
$c \pm t^* \times s_p\sqrt{\sum \dfrac{a^2_i}{n_i}}$
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 for group $i$, $n_i$ is the sample size for 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.
Effect size
n.a.
Effect size
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Effect size
Complete model:
Proportion variance explained $R^2$:
Proportion variance of the dependent variable $y$ explained by the sample regression equation (the independent variables):
$$
\begin{align}
R^2 &= \dfrac{\sum (\hat{y}_j  \bar{y})^2}{\sum (y_j  \bar{y})^2}\\ &= \dfrac{\mbox{sum of squares model}}{\mbox{sum of squares total}}\\
&= 1  \dfrac{\mbox{sum of squares error}}{\mbox{sum of squares total}}\\
&= r(y, \hat{y})^2
\end{align}
$$
$R^2$ is the proportion variance explained in the sample by the sample regression equation. It is a positively biased estimate of the proportion variance explained in the population by the population regression equation, $\rho^2$. If there is only one independent variable, $R^2 = r^2$: the correlation between the independent variable $x$ and dependent variable $y$ squared.
Wherry's $R^2$ / shrunken $R^2$:
Corrects for the positive bias in $R^2$ and is equal to
$$R^2_W = 1  \frac{N  1}{N  K  1}(1  R^2)$$
$R^2_W$ is a less biased estimate than $R^2$ of the proportion variance explained in the population by the population regression equation, $\rho^2$
Stein's $R^2$:
Estimates the proportion of variance in $y$ that we expect the current sample regression equation to explain in a different sample drawn from the same population. It is equal to
$$R^2_S = 1  \frac{(N  1)(N  2)(N + 1)}{(N  K  1)(N  K  2)(N)}(1  R^2)$$
Per independent variable:
Correlation squared $r^2_k$: the proportion of the total variance in the dependent variable $y$ that is explained by the independent variable $x_k$, not corrected for the other independent variables in the model
Semipartial correlation squared $sr^2_k$: the proportion of the total variance in the dependent variable $y$ that is uniquely explained by the independent variable $x_k$, beyond the part that is already explained by the other independent variables in the model
Partial correlation squared $pr^2_k$: the proportion of the variance in the dependent variable $y$ not explained by the other independent variables, that is uniquely explained by the independent variable $x_k$

Cohen's $d$:
Standardized difference between the sample mean and $\mu_0$:
$$d = \frac{\bar{y}  \mu_0}{\sigma}$$
Indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0$

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 $h$:
$$d_{g,h} = \frac{\bar{y}_g  \bar{y}_h}{s_p}$$
Indicates how many standard deviations $s_p$ two sample means are removed from each other
OLS regression with one, categorical independent variable transformed into $I  1$ code variables:
$F$ test ANOVA equivalent to $F$ test regression model
$t$ test for contrast $i$ 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
Can mental health be predicted from fysical health, economic class, and gender?
Is the proportion of people with a low, moderate, and high social economic status in the population different from $\pi_{low}$ = .2, $\pi_{moderate}$ = .6, and $\pi_{high}$ = .2?
Is the average mental health score of office workers different from $\mu_0$ = 50? Assume that the standard deviation of the mental health scores in the population is $\sigma$ = 3.
Is there a difference in depression level between measurement point 1 (preintervention), measurement point 2 (1 week postintervention), and measurement point 3 (6 weeks postintervention)?
Is the average mental health score different between people from a low, moderate, and high economic class?
SPSS
SPSS
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SPSS
SPSS
Analyze > Regression > Linear...
Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Independent(s)
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 the Friedman 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)
Jamovi
Jamovi
n.a.
Jamovi
Jamovi
Regression > Linear Regression
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'
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 > 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