Pearson correlation - overview
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Pearson correlation | $z$ test for the difference between two proportions | Chi-squared test for the relationship between two categorical variables |
You cannot compare more than 3 methods |
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Variable 1 | Independent/grouping variable | Independent /column variable | |
One quantitative of interval or ratio level | One categorical with 2 independent groups | One categorical with $I$ independent groups ($I \geqslant 2$) | |
Variable 2 | Dependent variable | Dependent /row variable | |
One quantitative of interval or ratio level | One categorical with 2 independent groups | 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: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2. | 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:
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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 two sided: $\pi_1 \neq \pi_2$ H1 right sided: $\pi_1 > \pi_2$ H1 left sided: $\pi_1 < \pi_2$ | 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:
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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: $\rho = 0$:
| $z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2 - p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$ | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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. | |
Sampling distribution of $t$ and of $z$ if H0 were true | Sampling distribution of $z$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | |
Sampling distribution of $t$:
| Approximately the standard normal distribution | Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | |
Significant? | Significant? | Significant? | |
$t$ Test two sided:
| Two sided:
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Approximate $C$% confidence interval for $\rho$ | Approximate $C\%$ confidence interval for $\pi_1 - \pi_2$ | n.a. | |
First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
| Regular (large sample):
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Properties of the Pearson correlation coefficient | n.a. | n.a. | |
| - | - | |
Equivalent to | Equivalent to | n.a. | |
OLS regression with one independent variable:
| When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levels. | - | |
Example context | Example context | Example context | |
Is there a linear relationship between physical health and mental health? | Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic. | Is there an association between economic class and gender? Is the distribution of economic class different between men and women? | |
SPSS | SPSS | SPSS | |
Analyze > Correlate > Bivariate...
| SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
| Analyze > Descriptive Statistics > Crosstabs...
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Jamovi | Jamovi | Jamovi | |
Regression > Correlation Matrix
| Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples - $\chi^2$ test of association
| Frequencies > Independent Samples - $\chi^2$ test of association
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Practice questions | Practice questions | Practice questions | |