Understanding Correlation

Understanding Correlation

• Correlation is a statistical measure describing the relationship between two variables
• It indicates that a change in one variable tends to cause a specific directional change in the other
• Real-life examples include correlations between income and expenditure, and supply and demand

Types of Correlation

• Correlations are categorized by sign, which can be positive, negative, or neutral/zero
• Positive Correlation: Variables move in the same direction (both increase or both decrease)
• Negative Correlation: Variables move in opposite directions (one increases as the other decreases, and vice versa)
• Neutral Correlation: Variables exhibits no relationship to one another

Correlation Form and Visualization

• Correlations can be linear, non-linear, or monotonic
• Linear Correlation: Variables change at a constant rate and fit the equation Y = aX + b, graphing as a straight line
• Non-Linear Correlation: Variables don't change at a constant rate, graphing as a curved pattern like a parabola or hyperbola
• Scatter plots are useful for visually identifying correlations between variables
• Numerical quantification of correlation requires calculating the correlation coefficient

Pearson's Correlation Coefficient (r)

• Pearson's r is the most common type of correlation coefficient
• Use when variables exhibit a normal distribution between and a linear relationship
• Non-parametric tests like Kendall and Spearman should be used otherwise
• Pearson correlation measures the linear relationship between two continuous variables
• It assumes variables are normally distributed with equal variances
• It determines the strength and direction of a linear relationship between two variables
• The value of r ranges from -1 to 1
• A correlation of -1 indicates a perfect negative correlation
• A correlation of 1 indicates a perfect positive correlation
• A correlation of 0 indicates no relationship

Testing for Significance

• Step 1 involves testing for the significance of the correlation using a null hypothesis (ρ = 0) versus an alternative hypothesis (ρ ≠ 0)
• Step 2 involves the T-test to test an assumption applicable to a population and to generalize to an entire population
• T-test value calculation aids in determining the repeatability of the sample correlation between x and y for the entire population

Interpreting P-Values in Correlation

• Reject the null hypothesis if the P-value is less than the significance level (α = 0.05)
• Indicates a statistically significant correlation and a linear relationship between x and y in the population
• A p-value represents the probability that the correlation between x and y happened by chance
• Fail to reject the null hypothesis if the P-value is bigger than the significance level (α = 0.05)
• Indicates the correlation is not statistically significant

Spearman's Rank Correlation (ρ)

• Spearman correlation measures the relationship between two variables using their rank order instead of actual values
• It is used when variables are not normally distributed or when the relationship between them is nonlinear
• Spearman correlation assesses the strength and direction of the monotonic association between continuous or ordinal variables
• Monotonicity indicated the degree to which a relationship is increasing or decreasing, without necessarily being linear
• The Spearman correlation coefficient (r) ranges from -1 to 1
• A value of -1 indicates a perfect negative correlation
• A value of +1 indicates a perfect positive correlation

Calculating Spearman Correlation

• Assign each value of each variable a rank starting from 1 for the smallest value and increasing to the highest value
• Tied values are assigned an average rank
• Calculate the difference between the ranks of each pair of observations
• Sum the squares of the computed differences
• Calculate the Spearman correlation coefficient using the formula
• r = 1 - (6 * sum of squared differences) / (n * (n^2 - 1)), where n is the number of observations.
• If the p-value is less than the chosen level of significance (usually 0.05), then the correlation is considered statistically significant

Point-Biserial Correlation

• Point-Biserial measures the relationship between a continuous variable and a dichotomous variable
• This type of correlation is computed by comparing the mean of the continuous variable for the two groups defined by the dichotomous variable
• The correlation coefficient (rpb) ranges from -1 to +1,
• rpb indicates a perfect negative correlation at -1
• rpb indicates a perfect positive correlation at +1
• rpb indicates no correlation at 0

Interpretation of the Point-Biserial Correlation Coefficient

• The point-biserial correlation relates to the Pearson correlation coefficient
• A positive rpb indicates that higher values of the continuous variable are associated with the 1 category of the dichotomous variable, and vice versa
• A negative rpb indicates the opposite
• The magnitude of rpb indicates association strength

Point-Biserial Formula and Components

• The formula for point-biserial correlation is rpb = (M1 - M0) / (SD x sqrt(p x q))

• M1 is the mean of the continuous variable for the 1 category of the dichotomous variable

• M0 is the mean of the continuous variable for the 0 category of the dichotomous variable

• SD is the standard deviation of the continuous variable

• p is the proportion of cases in the 1 category of the dichotomous variable

• q is the proportion of cases in the 0 category of the dichotomous variable

Causation

• Causation signifies that one event is directly responsible for causing another
• Establishing causation requires controlled experiments, strong evidence, and ruling out confounding factors
• An example of causation is that smoking causes lung cancer because this relationship has been confirmed through rigorous medical studies

Correlation vs. Causation

Identifying Causation

• Confirming causation requires several methods
• Randomized Controlled Trials (RCTs) are considered the gold standard
• Time Sequence dictates that the cause happens before the effect
• Confounders should be eliminated using statistical methods like regression analysis or propensity score matching
• Plausible Mechanism means that there should be a logical explanation for the cause-and-effect relationship

Spurious Correlation

• Two unrelated variables that appear correlated purely by coincidence
• Per capita cheese consumption is correlated with the number of people who are dying by becoming tangled in bedsheets, but the variables are unrelated
• Correlation indicates a relationship, while causation proves one variable drives another
• Correlation alone should never be treated as proof of causation without additional investigation

Examples of Correlations in Health Psychology

• Stress and physical health are positively correlated to risks of heart disease, diabetes, and other chronic illnesses
• Unhealthy behaviors are negatively correlated to health outcomes while the opposite is true of healthy behaviors
• Social support and health are positively correlated with positive health outcomes, such as better mental health and lower mortality rates
• Adherence to medical treatment and health are positively correlated, with positive health outcomes
• Health beliefs and health behaviors are positively correlated where individuals who believe their health behaviors are important and within their control will engage in healthy behaviors

Phi and Cramer’s V Correlations

• The Phi coefficient measures relationships between two dichotomous variables
• Cramer's V measures the strength of association between two nominal variables
• Select correlation method depends on the nature of the variables and the research question

Phi and Cramer V Calculations

• Both parameters are measures of association between two nominal variables and determine relationships between two categorical variables
• The Phi coefficient is a measure of association for a 2x2 contingency table
• It can be described by the equation
• phi = (ad - bc) / sqrt((a+b)(c+d)(a+c)(b+d))
• where a, b, c, and d are the frequencies of the four possible combinations of the two nominal variables
• This ranges from -1 to +1
• -1 indicates perfect negative association
• +1 indicates perfect positive association
• 0 indicates no association

Cramer’s V Calculations

• Cramer's V is a measure of association for contingency tables larger than 2x2
• It can be described by the equation
• V = sqrt(X^2 / (N x (min(r,c)-1)))
• X^2 is the chi-squared statistic
• N is the total number of observations
• r is the number of rows
• c is the number of columns in the contingency table
• The computed value ranges from 0 to 1
• 0 indicates no association
• 1 indicates a perfect association
• Cramer's V is preferred over Phi when the contingency table is larger than 2x2

Partial Correlation

• It measures the strength of the relationship between two variables while controlling for the addition of influence of other variables
• It can be described by the equation
• r_xy.z = (r_xy - r_xz*r_yz) / sqrt((1-r_xz^2)(1-r_yz^2))
• r_xy.z represents the partial correlation between variables x and y while controlling for the influence of variable z
• r_xy is the simple correlation between variables x and y
• r_xz and r_yz are the simple correlations between variables x and z

Example of Partial Correlation in Health Psychology

• To investigate the relationship between sleep quality and mental health while controlling for the effects of stress.
• Find that there is a significant correlation between sleep quality and mental health symptoms, suggesting that better sleep quality is associated with better mental health outcomes.
• This approach helps to identify the unique contribution of sleep quality to mental health outcomes, while controlling for the effects of other variables

Multiple Correlation

• It is a statistical method used to measure the relationship between a dependent variable and two or more independent variables
• It indicates how much of the variation in the dependent variable can be explained by the independent variables
• the equation for multiple correlation: R = sqrt(R^2_y1y2...yn)
• R represents the multiple correlation coefficient
• Measures the strength of the relationship between the dependent variable y and the independent variables x1, x2,..., xn
• R_y1y2...yn represents the multiple correlation coefficient for all the independent variables together
• The multiple correlation coefficient can be interpreted as the proportion of variance in the dependent variable that can be explained by the independent variables

Example of Multiple Correlation in Health Psychology

• It can be used in health psychology to examine the relationships between multiple predictor variables and a health outcome variable
• For example, researchers may want to study the factors that predict physical activity levels among individuals with chronic conditions such as heart disease or diabetes
• partial and multiple correlations are useful statistical tools
• Helps control for the effects of other variables that may influence the outcome of interest