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
$\pi = \pi_0$
$\pi$ is the population proportion of "successes"; $\pi_0$ is the population proportion of successes according to the null hypothesis
$\mu = \mu_0$
$\mu$ is the unknown population mean of the difference scores; $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0
Model chisquared test for the complete regression model:
$\beta_1 = \beta_2 = \ldots = \beta_K = 0$
Wald test for individual regression coefficient $\beta_k$:
$\beta_k = 0$
or in terms of odds ratio:
$e^{\beta_k} = 1$
Likelihood ratio chisquared test for individual regression coefficient $\beta_k$:
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: $\pi \neq \pi_0$
Right sided: $\pi > \pi_0$
Left sided: $\pi < \pi_0$
Two sided: $\mu \neq \mu_0$
Right sided: $\mu > \mu_0$
Left sided: $\mu < \mu_0$
Model chisquared test for the complete regression model:
not all population regression coefficients are 0
Wald test for individual $\beta_k$:
$\beta_k \neq 0$
or in terms of odds ratio:
$e^{\beta_k} \neq 1$
If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested:
right sided: $\beta_k > 0$
left sided: $\beta_k < 0$
Likelihood ratio chisquared test for individual $\beta_k$:
$\beta_k \neq 0$
or in terms of odds ratio:
$e^{\beta_k} \neq 1$
Assumptions
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
Sample is a simple random sample from the population. That is, observations are independent of one another
Difference scores are normally distributed in the population
Sample of difference scores is a simple random sample from the population of difference scores. That is, difference scores are independent of one another
Population of difference scores can be conceived of as the difference scores we would find if we would apply our study (e.g., applying an intervention and measuring prepost scores) to all individuals in the population.
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 statistic
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 $\beta_1$
$X$ = number of successes in the sample
$t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to H0, $s$ is the sample standard deviation of the difference scores,
$N$ is the sample size (number of difference scores).
Model chisquared 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 chisquared test for individual $\beta_k$:
$X^2 = D_{K1}  D_K$
$D_{K1}$ 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.
Sample standard deviation of the residuals $s$
n.a.
n.a.
n.a.
$\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 $\alpha$
For the Wald test:
If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: same procedure as for the chisquared 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 $z$.
$C\%$ confidence interval for $\beta_k$ and for $\mu_y$; $C\%$ prediction interval for $y_{new}$
n.a.
$C\%$ confidence interval for $\mu$
Waldtype approximate $C\%$ confidence interval for $\beta_k$
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)
$b_k \pm z^* \times SE_{b_k}$
where $z^*$ 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 size
n.a.
Effect size
Goodness of fit measure $R^2_L$
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 $x_k$

Cohen's $d$:
Standardized difference between the sample mean of the difference scores and $\mu_0$:
$$d = \frac{\bar{y}  \mu_0}{s}$$
Indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0$
$R^2_L = \dfrac{D_{null}  D_K}{D_{null}}$
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.
Put your dichotomous variable in the box below Test Variable List
Fill in the value for $\pi_0$ in the box next to Test Proportion
Analyze > Compare Means > PairedSamples T Test...
Put the two paired variables in the boxes below Variable 1 and Variable 2
Analyze > Regression > Binary Logistic...
Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Covariate(s)
Jamovi
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'
Frequencies > 2 Outcomes  Binomial test
Put your dichotomous variable in the white box at the right
Fill in the value for $\pi_0$ in the box next to Test value
Under Hypothesis, select your alternative hypothesis
TTests > Paired Samples TTest
Put the two paired variables in the box below Paired Variables, one on the left side of the vertical line and one on the right side of the vertical line
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'