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
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables
Variable 2
Dependent variable
One quantitative of interval or ratio level
One categorical with 2 independent groups
Null hypothesis
Null hypothesis
H_{0}: $\rho = \rho_0$
Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level.
Model chisquared test for the complete regression model:
H_{0}: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
Wald test for individual regression coefficient $\beta_k$:
H_{0}: $\beta_k = 0$
or in terms of odds ratio:
H_{0}: $e^{\beta_k} = 1$
Likelihood ratio chisquared test for individual regression coefficient $\beta_k$:
H_{0}: $\beta_k = 0$
or in terms of odds ratio:
H_{0}: $e^{\beta_k} = 1$
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 hypothesis
Alternative hypothesis
H_{1} two sided: $\rho \neq \rho_0$
H_{1} right sided: $\rho > \rho_0$
H_{1} left sided: $\rho < \rho_0$
Model chisquared test for the complete regression model:
H_{1}: not all population regression coefficients are 0
Wald test for individual regression coefficient $\beta_k$:
H_{1}: $\beta_k \neq 0$
or in terms of odds ratio:
H_{1}: $e^{\beta_k} \neq 1$
If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested:
H_{1} right sided: $\beta_k > 0$
H_{1} left sided: $\beta_k < 0$
Likelihood ratio chisquared test for individual regression coefficient $\beta_k$:
H_{1}: $\beta_k \neq 0$
or in terms of odds ratio:
H_{1}: $e^{\beta_k} \neq 1$
Assumptions of test for correlation
Assumptions
In the population, the two variables are jointly normally distributed (this covers the normality, homoscedasticity, and linearity assumptions)
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Note: these assumptions are only important for the significance test and confidence interval, not for the correlation coefficient itself. The correlation coefficient just measures the strength of the linear 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 statistic
Test statistic
Test statistic for testing H0: $\rho = 0$:
$t = \dfrac{r \times \sqrt{N  2}}{\sqrt{1  r^2}} $
where $r$ is the sample correlation $r = \frac{1}{N  1} \sum_{j}\Big(\frac{x_{j}  \bar{x}}{s_x} \Big) \Big(\frac{y_{j}  \bar{y}}{s_y} \Big)$ and $N$ is the sample size
Test statistic for testing values for $\rho$ other than $\rho = 0$:
$r_{Fisher} = \dfrac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1  r} \Bigg )$, where $r$ is the sample correlation
$\rho_{0_{Fisher}} = \dfrac{1}{2} \times \log\Bigg( \dfrac{1 + \rho_0}{1  \rho_0} \Bigg )$, where $\rho_0$ is the population correlation according to H0
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.
where $r_{Fisher} = \frac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1  r} \Bigg )$ 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).
Then transform back to get the approximate $C$% confidence interval for $\rho$:
$b_k \pm z^* \times SE_{b_k}$
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).
Properties of the Pearson correlation coefficient
Goodness of fit measure $R^2_L$
The Pearson correlation coefficient is a measure for the linear relationship between two quantitative variables.
The Pearson correlation coefficient squared reflects the proportion of variance explained in one variable by the other variable.
The Pearson correlation coefficient can take on values between 1 (perfect negative relationship) and 1 (perfect positive relationship). A value of 0 means no linear relationship.
The absolute size of the Pearson correlation coefficient is not affected by any linear transformation of the variables. However, the sign of the Pearson correlation will flip when the scores on one of the two variables are multiplied by a negative number (reversing the direction of measurement of that variable). For example:
the correlation between $x$ and $y$ is equivalent to the correlation between $3x + 5$ and $2y  6$.
the absolute value of the correlation between $x$ and $y$ is equivalent to the absolute value of the correlation between $3x + 5$ and $2y  6$. However, the signs of the two correlation coefficients will be in opposite directions, due to the multiplication of $x$ by $3$.
The Pearson correlation coefficient does not say anything about causality.
The Pearson correlation coefficient is sensitive to outliers.
$R^2_L = \dfrac{D_{null}  D_K}{D_{null}}$
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.
Results significance test ($t$ and $p$ value) testing $H_0$: $\beta_1 = 0$ are equivalent to results significance test testing $H_0$: $\rho = 0$

Example context
Example context
Is there a linear relationship between physical health and mental health?
Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?
SPSS
SPSS
Analyze > Correlate > Bivariate...
Put your two variables in the box below Variables
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
Regression > Correlation Matrix
Put your two variables in the white box at the right
Under Correlation Coefficients, select Pearson (selected by default)
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'