Pearson correlation  overview
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Pearson correlation  Two sample $t$ test  equal variances assumed 


Variable 1  Independent/grouping variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  
Variable 2  Dependent variable  
One quantitative of interval or ratio level  One quantitative of interval or ratio level  
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.  H_{0}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.  
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\rho \neq \rho_0$ H_{1} right sided: $\rho > \rho_0$ H_{1} left sided: $\rho < \rho_0$  H_{1} two sided: $\mu_1 \neq \mu_2$ H_{1} right sided: $\mu_1 > \mu_2$ H_{1} left sided: $\mu_1 < \mu_2$  
Assumptions of test for correlation  Assumptions  

 
Test statistic  Test statistic  
Test statistic for testing H0: $\rho = 0$:
 $t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1  \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1  \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2  \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.  
n.a.  Pooled standard deviation  
  $s_p = \sqrt{\dfrac{(n_1  1) \times s^2_1 + (n_2  1) \times s^2_2}{n_1 + n_2  2}}$  
Sampling distribution of $t$ and of $z$ if H_{0} were true  Sampling distribution of $t$ if H_{0} were true  
Sampling distribution of $t$:
 $t$ distribution with $n_1 + n_2  2$ degrees of freedom  
Significant?  Significant?  
$t$ Test two sided:
 Two sided:
 
Approximate $C$% confidence interval for $\rho$  $C\%$ confidence interval for $\mu_1  \mu_2$  
First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
 $(\bar{y}_1  \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
where the critical value $t^*$ is the value under the $t_{n_1 + n_2  2}$ 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 $\mu_1  \mu_2$ can also be used as significance test.  
Properties of the Pearson correlation coefficient  Effect size  
 Cohen's $d$: Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1  \bar{y}_2}{s_p}$$ Cohen's $d$ indicates how many standard deviations $s_p$ the two sample means are removed from each other.  
n.a.  Visual representation  
  
Equivalent to  Equivalent to  
OLS regression with one independent variable:
 One way ANOVA with an independent variable with 2 levels ($I$ = 2):
 
Example context  Example context  
Is there a linear relationship between physical health and mental health?  Is the average mental health score different between men and women? Assume that in the population, the standard deviation of mental health scores is equal amongst men and women.  
SPSS  SPSS  
Analyze > Correlate > Bivariate...
 Analyze > Compare Means > IndependentSamples T Test...
 
Jamovi  Jamovi  
Regression > Correlation Matrix
 TTests > Independent Samples TTest
 
Practice questions  Practice questions  