Pearson correlation - overview
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Pearson correlation | Chi-squared test for the relationship between two categorical variables | Marginal Homogeneity test / Stuart-Maxwell test |
You cannot compare more than 3 methods |
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Variable 1 | Independent /column variable | Independent variable | |
One quantitative of interval or ratio level | One categorical with $I$ independent groups ($I \geqslant 2$) | 2 paired groups | |
Variable 2 | Dependent /row variable | Dependent variable | |
One quantitative of interval or ratio level | One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with $J$ independent groups ($J \geqslant 2$) | |
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: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
| 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 $j.$ | |
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: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
| H1: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group. | |
Assumptions of test for correlation | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
Test statistic for testing H0: $\rho = 0$:
| $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells. | 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. | |
Sampling distribution of $t$ and of $z$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | Sampling distribution of the test statistic if H0 were true | |
Sampling distribution of $t$:
| Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | Approximately the chi-squared distribution with $J - 1$ degrees of freedom | |
Significant? | Significant? | Significant? | |
$t$ Test two sided:
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| If we denote the test statistic as $X^2$:
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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. | n.a. | |
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? | Is there an association between economic class and gender? Is the distribution of economic class different between men and women? | 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? | |
SPSS | SPSS | SPSS | |
Analyze > Correlate > Bivariate...
| Analyze > Descriptive Statistics > Crosstabs...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
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Jamovi | Jamovi | n.a. | |
Regression > Correlation Matrix
| Frequencies > Independent Samples - $\chi^2$ test of association
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Practice questions | Practice questions | Practice questions | |