Pearson correlation - overview
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Pearson correlation |
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Variable 1 | |
One quantitative of interval or ratio level | |
Variable 2 | |
One quantitative of interval or ratio level | |
Null 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. | |
Alternative hypothesis | |
H1 two sided: $\rho \neq \rho_0$ H1 right sided: $\rho > \rho_0$ H1 left sided: $\rho < \rho_0$ | |
Assumptions of test for correlation | |
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Test statistic | |
Test statistic for testing H0: $\rho = 0$:
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Sampling distribution of $t$ and of $z$ if H0 were true | |
Sampling distribution of $t$:
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Significant? | |
$t$ Test two sided:
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Approximate $C$% confidence interval for $\rho$ | |
First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
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Properties of the Pearson correlation coefficient | |
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Equivalent to | |
OLS regression with one independent variable:
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Example context | |
Is there a linear relationship between physical health and mental health? | |
SPSS | |
Analyze > Correlate > Bivariate...
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Jamovi | |
Regression > Correlation Matrix
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Practice questions | |