Pearson correlation - overview

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:

1. First score of pair is 0, second score of pair is 0
2. First score of pair is 0, second score of pair is 1 (switched)
3. First score of pair is 1, second score of pair is 0 (switched)
4. First score of pair is 1, second score of pair is 1

Other formulations of the null hypothesis are:

• H₀: $\pi_1 = \pi_2$, where $\pi_1$ is the population proportion of ones for the first paired group and $\pi_2$ is the population proportion of ones for the second paired group
• H₀: for each pair of scores, P(first score of pair is 1) = P(second score of pair is 1)

• H₀: the population median for group 1 is equal to the population median for group 2

• H₀: the population scores in group 1 are not systematically higher or lower than the population scores in group 2

• H₀: P(an observation from population 1 exceeds an observation from population 2) = P(an observation from population 2 exceeds observation from population 1)

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:

• H₁: $\pi_1 \neq \pi_2$
• H₁: for each pair of scores, P(first score of pair is 1) $\neq$ P(second score of pair is 1)

• H₁ two sided: the population median for group 1 is not equal to the population median for group 2
• H₁ right sided: the population median for group 1 is larger than the population median for group 2
• H₁ left sided: the population median for group 1 is smaller than the population median for group 2

• H₁ two sided: the population scores in group 1 are systematically higher or lower than the population scores in group 2
• H₁ right sided: the population scores in group 1 are systematically higher than the population scores in group 2
• H₁ left sided: the population scores in group 1 are systematically lower than the population scores in group 2

• H₁ two sided: P(an observation from population 1 exceeds an observation from population 2) $\neq$ P(an observation from population 2 exceeds an observation from population 1)
• H₁ right sided: P(an observation from population 1 exceeds an observation from population 2) > P(an observation from population 2 exceeds an observation from population 1)
• H₁ left sided: P(an observation from population 1 exceeds an observation from population 2) < P(an observation from population 2 exceeds an observation from population 1)

• In the population, the two variables are jointly normally distributed (this covers the normality, homoscedasticity, and linearity assumptions)
• Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another

• Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another

• Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another

• $t = \dfrac{r \times \sqrt{N - 2}}{\sqrt{1 - r^2}} $
  where $r$ is the sample correlation $r = \frac{1}{N - 1} \sum_{j}\Big(\frac{x_{j} - \bar{x}}{s_x} \Big) \Big(\frac{y_{j} - \bar{y}}{s_y} \Big)$ and $N$ is the sample size

• $z = \dfrac{r_{Fisher} - \rho_{0_{Fisher}}}{\sqrt{\dfrac{1}{N - 3}}}$
  • $r_{Fisher} = \dfrac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg )$, where $r$ is the sample correlation
  • $\rho_{0_{Fisher}} = \dfrac{1}{2} \times \log\Bigg( \dfrac{1 + \rho_0}{1 - \rho_0} \Bigg )$, where $\rho_0$ is the population correlation according to H0

• $W$ = sum of ranks in group 1

• $U = W - \dfrac{n_1(n_1 + 1)}{2}$

• $t$ distribution with $N - 2$ degrees of freedom

• Approximately the standard normal distribution

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 $b$.

Sampling distribution of $W$:
For large samples, $W$ is approximately normally distributed with mean $\mu_W$ and standard deviation $\sigma_W$ if the null hypothesis were true. Here $$ \begin{aligned} \mu_W &= \dfrac{n_1(n_1 + n_2 + 1)}{2}\\ \sigma_W &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}} \end{aligned} $$ Hence, for large samples, the standardized test statistic $$ z_W = \dfrac{W - \mu_W}{\sigma_W}\\ $$ follows approximately the standard normal distribution if the null hypothesis were true. Note that if your $W$ value is based on group 2, $\mu_W$ becomes $\frac{n_2(n_1 + n_2 + 1)}{2}$.

Sampling distribution of $U$:
For large samples, $U$ is approximately normally distributed with mean $\mu_U$ and standard deviation $\sigma_U$ if the null hypothesis were true. Here $$ \begin{aligned} \mu_U &= \dfrac{n_1 n_2}{2}\\ \sigma_U &= \sqrt{\dfrac{n_1 n_2(n_1 + n_2 + 1)}{12}} \end{aligned} $$ Hence, for large samples, the standardized test statistic $$ z_U = \dfrac{U - \mu_U}{\sigma_U}\\ $$ follows approximately the standard normal distribution if the null hypothesis were true.

For small samples, the exact distribution of $W$ or $U$ should be used.

• Check if $t$ observed in sample is at least as extreme as critical value $t^*$ or
• Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$

• Check if $t$ observed in sample is equal to or larger than critical value $t^*$ or
• Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$

• Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or
• Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$

• Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
• Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$

• Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
• Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$

• Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
• Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$

• Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or
• Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$

• Check if $b$ observed in sample is in the rejection region or
• Find two sided $p$ value corresponding to observed $b$ and check if it is equal to or smaller than $\alpha$

• Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
• Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$

• Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
• Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$

• Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
• Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$

• $lower_{Fisher} = r_{Fisher} - z^* \times \sqrt{\dfrac{1}{N - 3}}$
• $upper_{Fisher} = r_{Fisher} + z^* \times \sqrt{\dfrac{1}{N - 3}}$

• lower bound = $\dfrac{e^{2 \times lower_{Fisher}} - 1}{e^{2 \times lower_{Fisher}} + 1}$
• upper bound = $\dfrac{e^{2 \times upper_{Fisher}} - 1}{e^{2 \times upper_{Fisher}} + 1}$

• The Pearson correlation coefficient is a measure for the linear relationship between two quantitative variables.
• The Pearson correlation coefficient squared reflects the proportion of variance explained in one variable by the other variable.
• The Pearson correlation coefficient can take on values between -1 (perfect negative relationship) and 1 (perfect positive relationship). A value of 0 means no linear relationship.
• The absolute size of the Pearson correlation coefficient is not affected by any linear transformation of the variables. However, the sign of the Pearson correlation will flip when the scores on one of the two variables are multiplied by a negative number (reversing the direction of measurement of that variable).
  For example:
• the correlation between $x$ and $y$ is equivalent to the correlation between $3x + 5$ and $2y - 6$.
• the absolute value of the correlation between $x$ and $y$ is equivalent to the absolute value of the correlation between $-3x + 5$ and $2y - 6$. However, the signs of the two correlation coefficients will be in opposite directions, due to the multiplication of $x$ by $-3$.
• The Pearson correlation coefficient does not say anything about causality.
• The Pearson correlation coefficient is sensitive to outliers.

• $b_1 = r \times \frac{s_y}{s_x}$
• Results significance test ($t$ and $p$ value) testing $H_0$: $\beta_1 = 0$ are equivalent to results significance test testing $H_0$: $\rho = 0$

• Stuart-Maxwell test, with a categorical dependent variable consisting of two independent groups
• Cochran's Q test, with two related groups
• Two sided sign test: $b = W$, and $X^2 = z^2$

• Put your two variables in the box below Variables

• Put the two paired variables in the boxes below Variable 1 and Variable 2
• Under Test Type, select the McNemar test

• Put your dependent variable in the box below Test Variable List and your independent (grouping) variable in the box below Grouping Variable
• Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
• Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
• Continue and click OK

• Put your two variables in the white box at the right
• Under Correlation Coefficients, select Pearson (selected by default)
• Under Hypothesis, select your alternative hypothesis

• Put one of the two paired variables in the box below Rows and the other paired variable in the box below Columns

• Put your dependent variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
• Under Tests, select Mann-Whitney U
• Under Hypothesis, select your alternative hypothesis