Pearson correlation - overview
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Pearson correlation | Marginal Homogeneity test / Stuart-Maxwell test | McNemar's test |
You cannot compare more than 3 methods |
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Variable 1 | Independent variable | Independent variable | |
One quantitative of interval or ratio level | 2 paired groups | 2 paired groups | |
Variable 2 | Dependent variable | Dependent variable | |
One quantitative of interval or ratio level | One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with 2 independent groups | |
Null hypothesis | Null hypothesis | 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. | H0: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$ | Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options:
Other formulations of the null hypothesis are:
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Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
H1 two sided: $\rho \neq \rho_0$ H1 right sided: $\rho > \rho_0$ H1 left sided: $\rho < \rho_0$ | H1: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group. | The alternative hypothesis H1 is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are:
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Assumptions of test for correlation | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
Test statistic for testing H0: $\rho = 0$:
| Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand. | $X^2 = \dfrac{(b - c)^2}{b + c}$
Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0. | |
Sampling distribution of $t$ and of $z$ if H0 were true | Sampling distribution of the test statistic if H0 were true | Sampling distribution of $X^2$ if H0 were true | |
Sampling distribution of $t$:
| Approximately the chi-squared distribution with $J - 1$ degrees of freedom | If $b + c$ is large enough (say, > 20), approximately the chi-squared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$. | |
Significant? | Significant? | Significant? | |
$t$ Test two sided:
| If we denote the test statistic as $X^2$:
| For test statistic $X^2$:
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Approximate $C$% confidence interval for $\rho$ | n.a. | n.a. | |
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 | n.a. | n.a. | |
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Equivalent to | n.a. | Equivalent to | |
OLS regression with one independent variable:
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Example context | Example context | Example context | |
Is there a linear relationship between physical health and mental health? | Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best? | Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders? | |
SPSS | SPSS | SPSS | |
Analyze > Correlate > Bivariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
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Jamovi | n.a. | Jamovi | |
Regression > Correlation Matrix
| - | Frequencies > Paired Samples - McNemar test
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Practice questions | Practice questions | Practice questions | |