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 ($I \geqslant 2$)
2 paired groups
Dependent variable
Dependent variable
One quantitative of interval or ratio level
One categorical with 2 independent groups
Null hypothesis
Null hypothesis
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 $h$
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
Null hypothesis 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 :
$\pi_1 = \pi_2$, where $\pi_1$ is the population proportion of ones in the first paired group and $\pi_2$ is the population proportion of ones in the second paired group
For each pair of scores, P(first score of pair is 1) = P(second score of pair is 1)
Alternative hypothesis
Alternative hypothesis
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: $\mu_g \neq \mu_h$
Alternative hypothesis 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 that, for each pair of scores:
$\pi_1 \neq \pi_2$
For each pair of scores, P(first score of pair is 1) $\neq$ P(second score of pair is 1)
Assumptions
Assumptions
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
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
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.
$X^2 = \dfrac{(b  c)^2}{b + c}$
$b$ 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
Pooled standard deviation
n.a.
$
\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 $i$
$F$ distribution with $I  1$ (df between, numerator) and $N  I$ (df error, denominator) degrees of freedom
Sampling distribution of $t$:
$t$ distribution with $N  I$ degrees of freedom
If $b + c$ is large enough (say, > 20), approximately a chisquared 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?
$F$ 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
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 $\alpha$
$C\%$ confidence interval for $\Psi$, for $\mu_g  \mu_h$, and for $\mu_i$
n.a.
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 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.
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)
Marginal homogeneity test, with a categorical dependent variable consisting of two independent groups
Is the average mental health score different between people from a low, moderate, and high economic class?
Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?
SPSS
SPSS
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)