BTC 801: Correlation Analysis

• BTC 801 studies correlation and regression.

Correlation

• Correlation is a statistical method to analyze the relationship between two or more variables.
• It checks if two variables are related or independent.
• It identifies the relationship type (positive or negative).
• It determines the extent of variation in one variable due to changes in another.
• The dependent variable Y is the variable of interest
• The independent variable X influences Y
• The correlation assumes a linear relationship
• Scatter diagrams are used to visualize this relationship.
• The correlation calculation works best for straight-line relationships.
• Correlation does not prove causation, only relationship
• There are three types of correlation: positive, negative, and none.
• Positive means an increase in one variable corresponds with an increase in the other variable.
• Negative correlation means an increase in one variable corresponds with a decrease in the other
• No correlation means there is no relationship
• Correlation values range from -1 to +1:
  • 1 means a perfect positive correlation
  • 0 means no correlation
  • -1 means a perfect negative correlation

Estimation of Coefficient of Correlation (r)

• Calculation for "r" was proposed by Karl Pearson around 1900.
• The coefficient describes the strength of the relationship between variables.
• The coefficient values range from -1.00 to +1.00.
• Values near +/- 1.00 indicate a high correlation.
• Two methods to calculate correlation:
  • Direct use of the equation
  • Make a table of five columns to find: X, Y, X², XY, then Y²
• After calculating "r": the denominator is always positive, numerator can be positive or negative
• Ranging from -1.00 to +1.00
• "r" has no unit and is a measure of intensity
• It does not determine cause and effect relationships.

Coefficient of Determination

• "r" does not ascribe concrete meaning to the estimated relationship, but the coefficient of determination (r²) does.
• It measures the amount of variability in one variable accounted for by the other, expressed as a percentage.
• r values close to 1 indicate a strong correlation: with r = 0.87, r² = 0.76; therefore 76% of the variation in abdominal length is accounted for by wing length

Testing Significance of Correlation Coefficient

• "r" and "r²" may not always be correct, especially with small units and the null hypothesis must be tested.
• Use a t-test if the number of paired observations is less than 50.
• Use a z-test if the number of observations is greater than 50.
• Z = r/1 divided by the square root of n-1

T-Test of Correlation

• H0: P = 0 (population correlation is zero).
• H1: P ≠ 0 (population correlation is not zero).
• Use t = r * sqrt(n-2) / sqrt(1-r²).
• Check the t-table at a 0.05 confidence interval and 11 degrees of freedom.
• Reject the null hypothesis and accept H1 at a 0.05 level when t calculated is greater than t.
• Software available Excel (CORREL function), LibreOffice Calc, and SPSS.

Regression analysis

• It is a statistical process for estimating relationships among variables.
• It models and analyzes several variables and focuses on the relationship between a dependent variable and one or more independent variables.
• The line reveals how the typical value of the 'criterion variable' changes when any one of the independent variables is varied, while the other independent variables are held fixed.
• Estimates the conditional expectation of the dependent variable given the independent variables when the independent variables are fixed, and is used for prediction and forecasting with usage of a line of best fit.

Line of Best Fit calculation

• Use least squares regression to calculate values m (slope) and b (y-intercept) in the equation of a line.
• y = mx + b, where y = how far up, x = how far along, m = slope/gradient, b = the Y Intercept.
• To find the line of best fit for N points:
  • Calculate x² and xy for each (x,y) point .
  • Sum all x, y, x² and xy (∑ means "sum up").
  • Calculate Slope m: m = (NΣ(xy) – ΣxΣy) / (NΣ(x²) – (Σx)²).
  • Calculate Intercept b: b = Σy – mΣx
  • Assemble the equation of a line y = mx + b
• Total square of the errors is small ("least squares").
• Straight lines minimize the sum of squared errors
• Each point is connected to a straight bar by springs.
• Least square calculator apps assist in best fit usage.

Explore correlation analysis, a statistical method for quantifying relationships between variables. Learn about positive, negative, and zero correlation. Understand how correlation is visualized using scatter plots and identify the types of relationships. Note that correlation does not mean causation.