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If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
The median in population 1 is equal to the median in population 2
Else:
Formulation 1:
The scores in population 1 are not systematically higher or lower than the scores in population 2
Formulation 2:
P(an observation from population 1 exceeds an observation from population 2) = P(an observation from population 2 exceeds observation from population 1)
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.
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.
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: $\beta_k < 0$
If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Two sided: the median in population 1 is not equal to the median in population 2
Right sided: the median in population 1 is larger than the median in population 2
Left sided: the median in population 1 is smaller than the median in population 2
Else:
Formulation 1:
Two sided: The scores in population 1 are systematically higher or lower than the scores in population 2
Right sided: The scores in population 1 are systematically higher than the scores in population 2
Left sided: The scores in population 1 are systematically lower than the scores in population 2
Formulation 2:
Two sided: P(an observation from population 1 exceeds an observation from population 2) $\neq$ P(an observation from population 2 exceeds an observation from population 1)
Right sided: P(an observation from population 1 exceeds an observation from population 2) > P(an observation from population 2 exceeds an observation from population 1)
Left sided: P(an observation from population 1 exceeds an observation from population 2) < P(an observation from population 2 exceeds an observation from population 1)
The scores in some of the related groups are systematically higher or lower than the scores in other related groups
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
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 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another
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 $\beta_1$
Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
The second type of test statistic is the MannWhitney $U$ statistic:
$U = W  \dfrac{n_1(n_1 + 1)}{2}$
where $n_1$ is the sample size of group 1
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2.
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.
Sample standard deviation of the residuals $s$
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$\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}
$
Sampling distribution of $W$ and of $U$ if H0 were true
Sampling distribution of $Q$ 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
Sampling distribution of $W$:
For large samples, $W$ is approximately normally distributed with mean $\mu_W$ and standard deviation $\sigma_W$ if the null hypothesis were true. Here
$$
\begin{aligned}
\mu_W &= \dfrac{n_1(n_1 + n_2 + 1)}{2}\\
\sigma_W &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}}
\end{aligned}
$$
Hence, for large samples, the standardized test statistic
$$
z_W = \dfrac{W  \mu_W}{\sigma_W}\\
$$
follows approximately a standard normal distribution if the null hypothesis were true. Note that if your $W$ value is based on group 2, $\mu_W$ becomes $\frac{n_2(n_1 + n_2 + 1)}{2}$.
Sampling distribution of $U$:
For large samples, $U$ is approximately normally distributed with mean $\mu_U$ and standard deviation $\sigma_U$ if the null hypothesis were true. Here
$$
\begin{aligned}
\mu_U &= \dfrac{n_1 n_2}{2}\\
\sigma_U &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}}
\end{aligned}
$$
Hence, for large samples, the standardized test statistic
$$
z_U = \dfrac{U  \mu_U}{\sigma_U}\\
$$
follows approximately a standard normal distribution if the null hypothesis were true.
For small samples, the exact distribution of $W$ or $U$ should be used.
Note: the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated if ties are present in the data.
If the number of blocks $N$ is large, approximately the chisquared distribution with $k  1$ degrees of freedom.
For small samples, the exact distribution of
$Q$ should be used.
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:
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 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 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 $\alpha$
For large samples, the table for standard normal probabilities can be used:
Two sided:
Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
Right sided:
Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
Left sided:
Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
$C\%$ confidence interval for $\beta_k$ and for $\mu_y$; $C\%$ prediction interval for $y_{new}$
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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).


Effect size
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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
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 $x_k$


ANOVA table
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Equivalent to
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If no ties in the data: two sided MannWhitneyWilcoxon test is equivalent to KruskalWallis test with an independent variable with 2 levels ($I = 2$)

Example context
Example context
Example context
Can mental health be predicted from fysical health, economic class, and gender?
Do men tend to score higher on social economic status than women?
Is there a difference in depression level between measurement point 1 (preintervention), measurement point 2 (1 week postinterventiom), and measurement point 3 (6 weeks postintervention)?
SPSS
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 dependent variable in the box below Test Variable List 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 > 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
Jamovi
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'
TTests > Independent Samples TTest
Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
Under Tests, select MannWhitney U
Under Hypothesis, select your alternative hypothesis
ANOVA > Repeated Measures ANOVA  Friedman
Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures