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
Regression (OLS)
One sample $t$ test for the mean
Chisquared test for the relationship between two categorical variables
$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis
There is no association between the row and column variable
More precise statement:
If there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable: 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: The row and column variables are independent
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: $\beta_k < 0$
Two sided: $\mu \neq \mu_0$
Right sided: $\mu > \mu_0$
Left sided: $\mu < \mu_0$
There is an association between the row and column variable More precise statement:
If there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable: 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: The row and column variables are dependent
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
Scores are normally distributed in the population
Sample is a simple random sample from the population. That is, observations are independent of one another
Sample size is large enough for $X^2$ to be approximately chisquared 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
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 $\beta_1$
$t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to H0, $s$ is the sample standard deviation,
$N$ is the sample size.
$X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
where 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
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}
$
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 $y_{new}$
$C\%$ confidence interval for $\mu$
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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).
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N1}$ distribution with the area $C / 100$ between $t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20)
Proportion variance explained $R^2$:
Proportion variance of the dependent variable $y$ explained by the sample regression equation (the independent variables):
$$
\begin{align}
R^2 &= \dfrac{\sum (\hat{y}_j  \bar{y})^2}{\sum (y_j  \bar{y})^2}\\ &= \dfrac{\mbox{sum of squares model}}{\mbox{sum of squares total}}\\
&= 1  \dfrac{\mbox{sum of squares error}}{\mbox{sum of squares total}}\\
&= r(y, \hat{y})^2
\end{align}
$$
$R^2$ is the proportion variance explained in the sample by the sample regression equation. It is a positively biased estimate of the proportion variance explained in the population by the population regression equation, $\rho^2$. If there is only one independent variable, $R^2 = r^2$: the correlation between the independent variable $x$ and dependent variable $y$ squared.
Wherry's $R^2$ / shrunken $R^2$:
Corrects for the positive bias in $R^2$ and is equal to
$$R^2_W = 1  \frac{N  1}{N  K  1}(1  R^2)$$
$R^2_W$ is a less biased estimate than $R^2$ of the proportion variance explained in the population by the population regression equation, $\rho^2$
Stein's $R^2$:
Estimates the proportion of variance in $y$ that we expect the current sample regression equation to explain in a different sample drawn from the same population. It is equal to
$$R^2_S = 1  \frac{(N  1)(N  2)(N + 1)}{(N  K  1)(N  K  2)(N)}(1  R^2)$$
Per independent variable:
Correlation squared $r^2_k$: the proportion of the total variance in the dependent variable $y$ that is explained by the independent variable $x_k$, not corrected for the other independent variables in the model
Semipartial correlation squared $sr^2_k$: the proportion of the total variance in the dependent variable $y$ that is uniquely explained by the independent variable $x_k$, beyond the part that is already explained by the other independent variables in the model
Partial correlation squared $pr^2_k$: the proportion of the variance in the dependent variable $y$ not explained by the other independent variables, that is uniquely explained by the independent variable $x_k$
Cohen's $d$:
Standardized difference between the sample mean and $\mu_0$:
$$d = \frac{\bar{y}  \mu_0}{s}$$
Indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0$

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Visual representation
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ANOVA table
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Example context
Example context
Example context
Can mental health be predicted from fysical health, economic class, and gender?
Is the average mental health score of office workers different from $\mu_0$ = 50?
Is there an association between economic class and gender? Is the distribution of economic class different between men and women?
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)
Analyze > Compare Means > OneSample T Test...
Put your variable in the box below Test Variable(s)
Fill in the value for $\mu_0$ in the box next to Test Value
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 Chisquare
Continue and click OK
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 > One Sample TTest
Put your variable in the box below Dependent Variables
Under Hypothesis, fill in the value for $\mu_0$ in the box next to Test Value, and select your alternative hypothesis
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