Statistical Association and Correlation Analysis

Association Between Two Variables

• Association tests are used to explore if two variables, x and y, are related.

  • Examples:

    • Body length and mass

    • Concentration of ozone and NOx in the atmosphere

    • Daylight duration and oxygen progression in plants

    • Relative humidity and average temperature

Correlation

• Examines the direction and strength of the association between two numeric variables.

• Evaluates the direction and strength of the linear relationship between two numeric variables x and y.

• Assumptions:

  • Both variables are continuous (Pearson product moment correlation).

  • One variable continuous and one ordinal (Spearman correlation).

  • Related pairs: Each observation should have a value for both variables $(x<em>1, y</em>1; x<em>2, y</em>2; … x<em>n, y</em>n)$

  • Linearity

  • No outliers

Linear Regression

• Examines how well the line describing the association can be used to explain the variation in Y.

• As $x$ goes up, does $y$ change at the same rate (either goes up or down)?

Chi-square for Contingency Tables

• Examines the association or independence between two categorical variables.

Correlation Analysis

Step 1: Plot the Data Pairs

• Visualize the data to inspect for outliers and linearity using a scatterplot.

• Example: As PM10 increases in the 1st station, it also increases in the 2nd station.

  • Visually, it seems to be linearity.

  • The association is definitely positive.

Step 2: Estimate the Correlation Coefficient

• Scale-invariant numeric measure of the linear association between two numeric variables.

• Standardize the pairs of data $(x, y)$ using z-scores: $z = frac{observation - mean}{standard \ deviation}$

• Provides information about the directionality of the relationship: positive or negative.

  • Positive correlation: As the values of $x$ increase, the values of $y$ increase too (r is positive).

  • Negative correlation: As the values of $x$ increase, the values of $y$ decrease (r is negative).

• Provides information about the strength of the relationship.

  • The closer $r$ is to either -1 or +1, the stronger the relationship.

  • The closer $r$ is to 0, the weaker the relationship.

• Formula: $r = \frac{1}{n-1} \sum<em>{i=1}^{n} \frac{x</em>i - \bar{x}}{SD<em>x} * \frac{y</em>i - \bar{y}}{SD<em>y} = \frac{1}{n-1} \sum</em>{i=1}^{n} Z-Score(x) * Z-Score(y)$

• Example calculation:

  • Given sum of $Z(x) * Z(y) = -4.65$ and $n = 6$

  • Degrees of freedom = $n - 1 = 5$

  • $r = \frac{1}{6-1} * -4.65 = -0.93$

Step 3: Determine if the Observed Relationship Can Be Due to Chance (Hypothesis Testing)

• Step 3.1: State the statistical null hypothesis and set the confidence level.

  • $H_0: \rho = 0$ | There is no linear relationship between RH (x) and Avg. temperature (y).

  • $r$ is the sample correlation, an estimate of the population correlation $\rho$.

• Step 3.2: Calculate the test statistic and p-value.

  • $t = r * \sqrt{\frac{n-2}{1-r^2}}$

  • Example: $t = -1.987$, p-value = $7.13 \times 10^{-6}$

Interpreting a Correlation Analysis

• Step 3: Decision and report of results.

  • If p-value < $\alpha$ (significance level), we reject $H_0$.

  • Example: p-value = 0.00000713 < $\alpha$; we reject $H_0$ | The negative linear correlation between RH and avg. temperature is statistically significant.

  • RH showed a relatively high negative correlation to the avg. temperature ($r = -0.93$). This correlation is statistically significant ($t = 1.97$; $df = 5$; $p = 0.00000713$).

Caution

• A non-significant correlation means that the strength ($r$ value) could be explained by chance.

• A significant correlation does not mean a strong correlation.

• Small sample sizes can yield a non-significant correlation when in reality it is significant (Error type II).

• Large sample sizes can yield a significant correlation when in reality it is weak (Error type I).

• Not having a linear relation does not mean not having a relation; non-linear relations exist.