# Two way ANOVA - overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

Two way ANOVA
Two sample $t$ test - equal variances assumed
Spearman's rho
Logistic regression
Independent/grouping variablesIndependent/grouping variableVariable 1Independent variables
Two categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$)One categorical with 2 independent groupsOne of ordinal levelOne or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables
Dependent variableDependent variableVariable 2Dependent variable
One quantitative of interval or ratio levelOne quantitative of interval or ratio levelOne of ordinal levelOne categorical with 2 independent groups
Null hypothesisNull hypothesisNull hypothesisNull hypothesis
ANOVA $F$ tests:
• H0 for main and interaction effects together (model): no main effects and interaction effect
• H0 for independent variable A: no main effect for A
• H0 for independent variable B: no main effect for B
• H0 for the interaction term: no interaction effect between A and B
Like in one way ANOVA, we can also perform $t$ tests for specific contrasts and multiple comparisons. This is more advanced stuff.
H0: $\mu_1 = \mu_2$

Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.
H0: $\rho_s = 0$

Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level.

In words, the null hypothesis would be:

H0: there is no monotonic relationship between the two variables in the population.
Model chi-squared test for the complete regression model:
• H0: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
Wald test for individual regression coefficient $\beta_k$:
• H0: $\beta_k = 0$
or in terms of odds ratio:
• H0: $e^{\beta_k} = 1$
Likelihood ratio chi-squared test for individual regression coefficient $\beta_k$:
• H0: $\beta_k = 0$
or in terms of odds ratio:
• H0: $e^{\beta_k} = 1$
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 hypothesisAlternative hypothesisAlternative hypothesisAlternative hypothesis
ANOVA $F$ tests:
• H1 for main and interaction effects together (model): there is a main effect for A, and/or for B, and/or an interaction effect
• H1 for independent variable A: there is a main effect for A
• H1 for independent variable B: there is a main effect for B
• H1 for the interaction term: there is an interaction effect between A and B
H1 two sided: $\mu_1 \neq \mu_2$
H1 right sided: $\mu_1 > \mu_2$
H1 left sided: $\mu_1 < \mu_2$
H1 two sided: $\rho_s \neq 0$
H1 right sided: $\rho_s > 0$
H1 left sided: $\rho_s < 0$
Model chi-squared test for the complete regression model:
• H1: not all population regression coefficients are 0
Wald test for individual regression coefficient $\beta_k$:
• H1: $\beta_k \neq 0$
or in terms of odds ratio:
• H1: $e^{\beta_k} \neq 1$
If defined as Wald $= \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested:
• H1 right sided: $\beta_k > 0$
• H1 left sided: $\beta_k < 0$
Likelihood ratio chi-squared test for individual regression coefficient $\beta_k$:
• H1: $\beta_k \neq 0$
or in terms of odds ratio:
• H1: $e^{\beta_k} \neq 1$
AssumptionsAssumptionsAssumptionsAssumptions
• Within each of the $I \times J$ populations, 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 $I \times J$ populations
• For each of the $I \times J$ groups, the sample is an independent and simple random sample from the population defined by that group. That is, within and between groups, observations are independent of one another
• Equal sample sizes for each group make the interpretation of the ANOVA output easier (unequal sample sizes result in overlap in the sum of squares; this is advanced stuff)
• 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 both populations: $\sigma_1 = \sigma_2$
• Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. 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
Note: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.
• 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 statisticTest statisticTest statisticTest statistic
For main and interaction effects together (model):
• $F = \dfrac{\mbox{mean square model}}{\mbox{mean square error}}$
For independent variable A:
• $F = \dfrac{\mbox{mean square A}}{\mbox{mean square error}}$
For independent variable B:
• $F = \dfrac{\mbox{mean square B}}{\mbox{mean square error}}$
For the interaction term:
• $F = \dfrac{\mbox{mean square interaction}}{\mbox{mean square error}}$
Note: mean square error is also known as mean square residual or mean square within.
$t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis.

The denominator $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1 - \bar{y}_2$ is removed from 0.

Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.
$t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}}$
Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.
Model chi-squared 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 chi-squared test for individual $\beta_k$:
• $X^2 = D_{K-1} - D_K$
$D_{K-1}$ 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.
Pooled standard deviationPooled standard deviationn.a.n.a.
\begin{aligned} s_p &= \sqrt{\dfrac{\sum\nolimits_{subjects} (\mbox{subject's score} - \mbox{its group mean})^2}{N - (I \times J)}}\\ &= \sqrt{\dfrac{\mbox{sum of squares error}}{\mbox{degrees of freedom error}}}\\ &= \sqrt{\mbox{mean square error}} \end{aligned} $s_p = \sqrt{\dfrac{(n_1 - 1) \times s^2_1 + (n_2 - 1) \times s^2_2}{n_1 + n_2 - 2}}$--
Sampling distribution of $F$ if H0 were trueSampling distribution of $t$ if H0 were trueSampling distribution of $t$ if H0 were trueSampling distribution of $X^2$ and of the Wald statistic if H0 were true
For main and interaction effects together (model):
• $F$ distribution with $(I - 1) + (J - 1) + (I - 1) \times (J - 1)$ (df model, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
For independent variable A:
• $F$ distribution with $I - 1$ (df A, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
For independent variable B:
• $F$ distribution with $J - 1$ (df B, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
For the interaction term:
• $F$ distribution with $(I - 1) \times (J - 1)$ (df interaction, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom
Here $N$ is the total sample size.
$t$ distribution with $n_1 + n_2 - 2$ degrees of freedomApproximately the $t$ distribution with $N - 2$ degrees of freedomSampling distribution of $X^2$, as computed in the model chi-squared test for the complete model:
• chi-squared distribution with $K$ (number of independent variables) degrees of freedom
Sampling distribution of the Wald statistic:
• If defined as Wald $= \dfrac{b_k^2}{SE^2_{b_k}}$: approximately the chi-squared distribution with 1 degree of freedom
• If defined as Wald $= \dfrac{b_k}{SE_{b_k}}$: approximately the standard normal distribution
Sampling distribution of $X^2$, as computed in the likelihood ratio chi-squared test for individual $\beta_k$:
• chi-squared distribution with 1 degree of freedom
Significant?Significant?Significant?Significant?
• 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 $\alpha$
Two sided:
Right sided:
Left sided:
Two sided:
Right sided:
Left sided:
For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$:
• 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 the Wald test:
• If defined as Wald $= \dfrac{b_k^2}{SE^2_{b_k}}$: same procedure as for the chi-squared 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 $z$.
n.a.$C\%$ confidence interval for $\mu_1 - \mu_2$n.a.Wald-type approximate $C\%$ confidence interval for $\beta_k$
-$(\bar{y}_1 - \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
where the critical value $t^*$ is the value under the $t_{n_1 + n_2 - 2}$ 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 $\mu_1 - \mu_2$ can also be used as significance test.
-$b_k \pm z^* \times SE_{b_k}$
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).
Effect sizeEffect sizen.a.Goodness of fit measure $R^2_L$
• Proportion variance explained $R^2$:
Proportion variance of the dependent variable $y$ explained by the independent variables and the interaction effect together:
\begin{align} R^2 &= \dfrac{\mbox{sum of squares model}}{\mbox{sum of squares total}} \end{align} $R^2$ 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 $\eta^2$:
Proportion variance of the dependent variable $y$ explained by an independent variable or interaction effect:
\begin{align} \eta^2_A &= \dfrac{\mbox{sum of squares A}}{\mbox{sum of squares total}}\\ \\ \eta^2_B &= \dfrac{\mbox{sum of squares B}}{\mbox{sum of squares total}}\\ \\ \eta^2_{int} &= \dfrac{\mbox{sum of squares int}}{\mbox{sum of squares total}} \end{align} $\eta^2$ 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:
\begin{align} \omega^2_A &= \dfrac{\mbox{sum of squares A} - \mbox{degrees of freedom A} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \\ \omega^2_B &= \dfrac{\mbox{sum of squares B} - \mbox{degrees of freedom B} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \\ \omega^2_{int} &= \dfrac{\mbox{sum of squares int} - \mbox{degrees of freedom int} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \end{align} $\omega^2$ is a better estimate of the explained variance in the population than $\eta^2$. Only for balanced designs (equal sample sizes).

• Proportion variance explained $\eta^2_{partial}$: \begin{align} \eta^2_{partial\,A} &= \frac{\mbox{sum of squares A}}{\mbox{sum of squares A} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,B} &= \frac{\mbox{sum of squares B}}{\mbox{sum of squares B} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,int} &= \frac{\mbox{sum of squares int}}{\mbox{sum of squares int} + \mbox{sum of squares error}} \end{align}
Cohen's $d$:
Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1 - \bar{y}_2}{s_p}$$ Cohen's $d$ indicates how many standard deviations $s_p$ the 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.
n.a.Visual representationn.a.n.a.
---
ANOVA tablen.a.n.a.n.a.
---
Equivalent toEquivalent ton.a.n.a.
OLS regression with two categorical independent variables and the interaction term, transformed into $(I - 1)$ + $(J - 1)$ + $(I - 1) \times (J - 1)$ code variables.One way ANOVA with an independent variable with 2 levels ($I$ = 2):
• two sided two sample $t$ test is equivalent to ANOVA $F$ test when $I$ = 2
• two sample $t$ test is equivalent to $t$ test for contrast when $I$ = 2
• two sample $t$ test is equivalent to $t$ test multiple comparisons when $I$ = 2
OLS regression with one categorical independent variable with 2 levels:
• two sided two sample $t$ test is equivalent to $F$ test regression model
• two sample $t$ test is equivalent to $t$ test for regression coefficient $\beta_1$
--
Example contextExample contextExample contextExample context
Is the average mental health score different between people from a low, moderate, and high economic class? And is the average mental health score different between men and women? And is there an interaction effect between economic class and gender?Is the average mental health score different between men and women? Assume that in the population, the standard deviation of mental health scores is equal amongst men and women.Is there a monotonic relationship between physical health and mental health?Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?
SPSSSPSSSPSSSPSS
Analyze > General Linear Model > Univariate...
• Put your dependent (quantitative) variable in the box below Dependent Variable and your two independent (grouping) variables in the box below Fixed Factor(s)
Analyze > Compare Means > Independent-Samples T Test...
• Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable
• Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
• Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
• Continue and click OK
Analyze > Correlate > Bivariate...
• Put your two variables in the box below Variables
• Under Correlation Coefficients, select Spearman
Analyze > Regression > Binary Logistic...
• Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Covariate(s)
JamoviJamoviJamoviJamovi
ANOVA > ANOVA
• Put your dependent (quantitative) variable in the box below Dependent Variable and your two independent (grouping) variables in the box below Fixed Factors
T-Tests > Independent Samples T-Test
• Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
• Under Tests, select Student's (selected by default)
• Under Hypothesis, select your alternative hypothesis
Regression > Correlation Matrix
• Put your two variables in the white box at the right
• Under Correlation Coefficients, select Spearman
• Under Hypothesis, select your alternative hypothesis
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'
Practice questionsPractice questionsPractice questionsPractice questions