# Regression (OLS) - overview

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Regression (OLS)
Goodness of fit test
One sample $z$ test for the mean
Kruskal-Wallis test
$z$ test for the difference between two proportions
Two way ANOVA
Chi-squared test for the relationship between two categorical variables
Regression (OLS)
One way ANOVA
Independent variablesIndependent variableIndependent variableIndependent/grouping variableIndependent/grouping variableIndependent/grouping variablesIndependent /column variableIndependent variablesIndependent/grouping variable
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variablesNoneNoneOne categorical with $I$ independent groups ($I \geqslant 2$)One categorical with 2 independent groupsTwo categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$)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 variablesOne categorical with $I$ independent groups ($I \geqslant 2$)
Dependent variableDependent variableDependent variableDependent variableDependent variableDependent variableDependent /row variableDependent variableDependent variable
One quantitative of interval or ratio levelOne categorical with $J$ independent groups ($J \geqslant 2$)One quantitative of interval or ratio levelOne of ordinal levelOne categorical with 2 independent groupsOne quantitative of interval or ratio levelOne categorical with $J$ independent groups ($J \geqslant 2$)One quantitative of interval or ratio levelOne quantitative of interval or ratio level
Null hypothesisNull hypothesisNull hypothesisNull hypothesisNull hypothesisNull hypothesisNull hypothesisNull hypothesisNull hypothesis
$F$ test for the complete regression model:
• H0: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
or equivalenty
• H0: the variance explained by all the independent variables together (the complete model) is 0 in the population, i.e. $\rho^2 = 0$
$t$ test for individual regression coefficient $\beta_k$:
• H0: $\beta_k = 0$
in the regression equation $\mu_y = \beta_0 + \beta_1 \times x_1 + \beta_2 \times x_2 + \ldots + \beta_K \times x_K$. Here $x_i$ represents independent variable $i$, $\beta_i$ is the regression weight for independent variable $x_i$, and $\mu_y$ represents the population mean of the dependent variable $y$ given the scores on the independent variables.
• 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, and $\mu_0$ is the population mean according to 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:
• 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.
H0: $\pi_1 = \pi_2$

Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2.
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: there is no association between the row and column variable

More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
• H0: the distribution of the dependent variable is the same in each of the $I$ populations
If there is one random sample of size $N$ from the total population:
• H0: the row and column variables are independent
$F$ test for the complete regression model:
• H0: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
or equivalenty
• H0: the variance explained by all the independent variables together (the complete model) is 0 in the population, i.e. $\rho^2 = 0$
$t$ test for individual regression coefficient $\beta_k$:
• H0: $\beta_k = 0$
in the regression equation $\mu_y = \beta_0 + \beta_1 \times x_1 + \beta_2 \times x_2 + \ldots + \beta_K \times x_K$. Here $x_i$ represents independent variable $i$, $\beta_i$ is the regression weight for independent variable $x_i$, and $\mu_y$ represents the population mean of the dependent variable $y$ given the scores on the independent variables.
ANOVA $F$ test:
• H0: $\mu_1 = \mu_2 = \ldots = \mu_I$
$\mu_1$ 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:
• H0: $\Psi = 0$
$\Psi$ 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:
• H0: $\mu_g = \mu_h$
$\mu_g$ is the population mean for group $g$; $\mu_h$ is the population mean for group $h$
Alternative hypothesisAlternative hypothesisAlternative hypothesisAlternative hypothesisAlternative hypothesisAlternative hypothesisAlternative hypothesisAlternative hypothesisAlternative hypothesis
$F$ test for the complete regression model:
• H1: not all population regression coefficients are 0
or equivalenty
• H1: the variance explained by all the independent variables together (the complete model) is larger than 0 in the population, i.e. $\rho^2 > 0$
$t$ test for individual regression coefficient $\beta_k$:
• H1 two sided: $\beta_k \neq 0$
• H1 right sided: $\beta_k > 0$
• H1 left sided: $\beta_k < 0$
• 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 $g$)
H1 two sided: $\pi_1 \neq \pi_2$
H1 right sided: $\pi_1 > \pi_2$
H1 left sided: $\pi_1 < \pi_2$
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: there is an association between the row and column variable

More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
• H1: the distribution of the dependent variable is not the same in all of the $I$ populations
If there is one random sample of size $N$ from the total population:
• H1: the row and column variables are dependent
$F$ test for the complete regression model:
• H1: not all population regression coefficients are 0
or equivalenty
• H1: the variance explained by all the independent variables together (the complete model) is larger than 0 in the population, i.e. $\rho^2 > 0$
$t$ test for individual regression coefficient $\beta_k$:
• H1 two sided: $\beta_k \neq 0$
• H1 right sided: $\beta_k > 0$
• H1 left sided: $\beta_k < 0$
ANOVA $F$ test:
• H1: not all population means are equal
$t$ Test for contrast:
• H1 two sided: $\Psi \neq 0$
• H1 right sided: $\Psi > 0$
• H1 left sided: $\Psi < 0$
$t$ Test multiple comparisons:
• H1 - usually two sided: $\mu_g \neq \mu_h$
AssumptionsAssumptionsAssumptionsAssumptionsAssumptionsAssumptionsAssumptionsAssumptionsAssumptions
• 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
• Variables are measured without error
Also pay attention to:
• Multicollinearity
• Outliers
• 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
• 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
• 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 size is large enough for $z$ to be approximately normally distributed. Rule of thumb:
• Significance test: number of successes and number of failures are each 5 or more in both sample groups
• Regular (large sample) 90%, 95%, or 99% confidence interval: number of successes and number of failures are each 10 or more in both sample groups
• Plus four 90%, 95%, or 99% confidence interval: sample sizes of both groups are 5 or more
• 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
• 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)
• Sample size is large enough for $X^2$ to be approximately chi-squared distributed under the null hypothesis. Rule of thumb:
• 2 $\times$ 2 table: all four expected cell counts are 5 or more
• Larger than 2 $\times$ 2 tables: average of the expected cell counts is 5 or more, smallest expected cell count is 1 or more
• There are $I$ independent simple random samples from each of $I$ populations defined by the independent variable, or there is one simple random sample from the total population
• 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
• Variables are measured without error
Also pay attention to:
• Multicollinearity
• Outliers
• 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 statisticTest statisticTest statisticTest statisticTest statisticTest statisticTest statisticTest statisticTest 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 $SE_{b_k}$ is more complicated.
Note 1: mean square model is also known as mean square regression, and mean square error is also known as mean square residual.
Note 2: if there is only one independent variable in the model ($K = 1$), the $F$ test for the complete regression model is equivalent to the two sided $t$ test for $\beta_1.$
$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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.
$z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size.

The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ 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.
$z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2.
Note: we could just as well compute $p_2 - p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$
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.
$X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells.
$F$ 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 $SE_{b_k}$ is more complicated.
Note 1: mean square model is also known as mean square regression, and mean square error is also known as mean square residual.
Note 2: if there is only one independent variable in the model ($K = 1$), the $F$ test for the complete regression model is equivalent to the two sided $t$ test for $\beta_1.$
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.
Sample standard deviation of the residuals $s$n.a.n.a.n.a.n.a.Pooled standard deviationn.a.Sample standard deviation of the residuals $s$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{\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} -\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} Here s^2_i is the variance in group i. Sampling distribution of F and of t if H0 were trueSampling distribution of X^2 if H0 were trueSampling distribution of z if H0 were trueSampling distribution of H if H0 were trueSampling distribution of z if H0 were trueSampling distribution of F if H0 were trueSampling distribution of X^2 if H0 were trueSampling distribution of F and of t if H0 were trueSampling distribution of F and of t if H0 were true Sampling distribution of F: • F distribution with K (df model, numerator) and N - K - 1 (df error, denominator) degrees of freedom Sampling distribution of t: • t distribution with N - K - 1 (df error) degrees of freedom Approximately the chi-squared distribution with J - 1 degrees of freedomStandard normal distribution 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. Approximately the standard normal distributionFor 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. Approximately the chi-squared distribution with (I - 1) \times (J - 1) degrees of freedomSampling distribution of F: • F distribution with K (df model, numerator) and N - K - 1 (df error, denominator) degrees of freedom Sampling distribution of t: • t distribution with N - K - 1 (df error) degrees of freedom Sampling distribution of F: • 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 Significant?Significant?Significant?Significant?Significant?Significant?Significant?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 t Test two sided: t Test right sided: t Test left sided: • 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: Right sided: Left sided: 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 Two sided: Right sided: Left sided: • 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 • 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 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 t Test two sided: t Test right sided: t Test left sided: 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: t Test for contrast right sided: t Test for contrast left sided: 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}n.a.C\% confidence interval for \mun.a.Approximate C\% confidence interval for \pi_1 - \pi_2n.a.n.a.C\% confidence interval for \beta_k and for \mu_y, C\% prediction interval for y_{new}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 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).

The confidence interval for $\mu$ can also be used as significance test.
-Regular (large sample):
• $(p_1 - p_2) \pm z^* \times \sqrt{\dfrac{p_1(1 - p_1)}{n_1} + \dfrac{p_2(1 - p_2)}{n_2}}$
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)
With plus four method:
• $(p_{1.plus} - p_{2.plus}) \pm z^* \times \sqrt{\dfrac{p_{1.plus}(1 - p_{1.plus})}{n_1 + 2} + \dfrac{p_{2.plus}(1 - p_{2.plus})}{n_2 + 2}}$
where $p_{1.plus} = \dfrac{X_1 + 1}{n_1 + 2}$, $p_{2.plus} = \dfrac{X_2 + 1}{n_2 + 2}$, and 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)
--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).
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.
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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
• Semi-partial 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 $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$
--
• 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}
-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
• Semi-partial 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 $x_k$
• 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.
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Regression equations with: -----Regression equations with: -
ANOVA tablen.a.n.a.n.a.n.a.ANOVA tablen.a.ANOVA tableANOVA table
---- -
Click the link for a step by step explanation of how to compute the sum of squares.
n.a.n.a.n.a.n.a.Equivalent toEquivalent ton.a.n.a.Equivalent to
----When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levels.OLS regression with two categorical independent variables and the interaction term, transformed into $(I - 1)$ + $(J - 1)$ + $(I - 1) \times (J - 1)$ code variables.--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 contextExample contextExample contextExample contextExample contextExample contextExample contextExample contextExample 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 $\pi_{low} = 0.2,$ $\pi_{moderate} = 0.6,$ and $\pi_{high} = 0.2$?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 $\sigma = 3.$Do people from different religions tend to score differently on social economic status? Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.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 there an association between economic class and gender? Is the distribution of economic class different between men and women?Can mental health be predicted from fysical health, economic class, and gender?Is the average mental health score different between people from a low, moderate, and high economic class?
SPSSSPSSn.a.SPSSSPSSSPSSSPSSSPSSSPSS
Analyze > Regression > Linear...
• Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Independent(s)
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 > 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
SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:

Analyze > Descriptive Statistics > Crosstabs...
• Put your independent (grouping) variable in the box below Row(s), and your dependent variable in the box below Column(s)
• Click the Statistics... button, and click on the square in front of Chi-square
• Continue and click OK
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 > Descriptive Statistics > Crosstabs...
• Put one of your two categorical variables in the box below Row(s), and the other categorical variable in the box below Column(s)
• Click the Statistics... button, and click on the square in front of Chi-square
• Continue and click OK
Analyze > Regression > Linear...
• Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Independent(s)
Analyze > Compare Means > One-Way 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)
JamoviJamovin.a.JamoviJamoviJamoviJamoviJamoviJamovi
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 > 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
Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:

Frequencies > Independent Samples - $\chi^2$ test of association
• Put your independent (grouping) variable in the box below Rows, and your dependent variable in the box below Columns
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
Frequencies > Independent Samples - $\chi^2$ test of association
• Put one of your two categorical variables in the box below Rows, and the other categorical variable in the box below Columns
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'
ANOVA > ANOVA
• Put your dependent (quantitative) variable in the box below Dependent Variable and your independent (grouping) variable in the box below Fixed Factors
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