# Pearson correlation - overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

Pearson correlation
Paired sample $t$ test
Variable 1Independent variable
One quantitative of interval or ratio level2 paired groups
Variable 2Dependent variable
One quantitative of interval or ratio levelOne quantitative of interval or ratio level
Null hypothesisNull hypothesis
H0: $\rho = \rho_0$

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.
H0: $\mu = \mu_0$

Here $\mu$ is the population mean of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.
Alternative hypothesisAlternative hypothesis
H1 two sided: $\rho \neq \rho_0$
H1 right sided: $\rho > \rho_0$
H1 left sided: $\rho < \rho_0$
H1 two sided: $\mu \neq \mu_0$
H1 right sided: $\mu > \mu_0$
H1 left sided: $\mu < \mu_0$
Assumptions of test for correlationAssumptions
• 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.
• 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
Test statisticTest 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 $\rho = 0$:
• $z = \dfrac{r_{Fisher} - \rho_{0_{Fisher}}}{\sqrt{\dfrac{1}{N - 3}}}$
• $r_{Fisher} = \dfrac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg )$, 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
$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores).

The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.
Sampling distribution of $t$ and of $z$ if H0 were trueSampling distribution of $t$ if H0 were true
Sampling distribution of $t$:
• $t$ distribution with $N - 2$ degrees of freedom
Sampling distribution of $z$:
• Approximately the standard normal distribution
$t$ distribution with $N - 1$ degrees of freedom
Significant?Significant?
$t$ Test two sided:
$t$ Test right sided:
$t$ Test left sided:
$z$ Test two sided:
$z$ Test right sided:
$z$ Test left sided:
Two sided:
Right sided:
Left sided:
Approximate $C$% confidence interval for $\rho$$C\% confidence interval for \mu First compute the approximate C% confidence interval for \rho_{Fisher}: • lower_{Fisher} = r_{Fisher} - z^* \times \sqrt{\dfrac{1}{N - 3}} • upper_{Fisher} = r_{Fisher} + z^* \times \sqrt{\dfrac{1}{N - 3}} 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: • lower bound = \dfrac{e^{2 \times lower_{Fisher}} - 1}{e^{2 \times lower_{Fisher}} + 1} • upper bound = \dfrac{e^{2 \times upper_{Fisher}} - 1}{e^{2 \times upper_{Fisher}} + 1} \bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}} where the critical value t^* is the value under the t_{N-1} distribution with the area C / 100 between -t^* and t^* (e.g. t^* = 2.086 for a 95% confidence interval when df = 20). The confidence interval for \mu can also be used as significance test. Properties of the Pearson correlation coefficientEffect size • 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. Cohen's d: Standardized difference between the sample mean of the difference scores and \mu_0:$$d = \frac{\bar{y} - \mu_0}{s}$$Cohen's$d$indicates how many standard deviations$s$the sample mean of the difference scores$\bar{y}$is removed from$\mu_0.$n.a.Visual representation - Equivalent toEquivalent to OLS regression with one independent variable: •$b_1 = r \times \frac{s_y}{s_x}$• Results significance test ($t$and$p$value) testing$H_0$:$\beta_1 = 0$are equivalent to results significance test testing$H_0$:$\rho = 0$• One sample$t$test on the difference scores. • Repeated measures ANOVA with one dichotomous within subjects factor. Example contextExample context Is there a linear relationship between physical health and mental health?Is the average difference between the mental health scores before and after an intervention different from$\mu_0 = 0\$?
SPSSSPSS
Analyze > Correlate > Bivariate...
• Put your two variables in the box below Variables
Analyze > Compare Means > Paired-Samples T Test...
• Put the two paired variables in the boxes below Variable 1 and Variable 2
JamoviJamovi
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
T-Tests > Paired Samples T-Test
• 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
Practice questionsPractice questions