correlation

Examples of correlation in the following topics:

• Properties of Pearson's r

  • State the relationship between the correlation of Y with X and the correlation of X with Y
  • A correlation of -1 means a perfect negative linear relationship, a correlation of 0 means no linear relationship, and a correlation of 1 means a perfect positive linear relationship.
  • Pearson's correlation is symmetric in the sense that the correlation of X with Y is the same as the correlation of Y with X.
  • For example, the correlation of Weight with Height is the same as the correlation of Height with Weight.
  • For instance, the correlation of Weight and Height does not depend on whether Height is measured in inches, feet, or even miles.

• Correlation and Causation

  • A correlation can be positive/direct or negative/inverse.
  • Ice cream consumption is positively correlated with incidents of crime.
  • It is important to not confound a correlation with a cause/effect relationship.
  • This diagram illustrates the difference between correlation and causation, as ice cream consumption is correlated with crime, but both are dependent on temperature.
  • Thus, the correlation between ice cream consumption and crime is spurious.

• Rank Correlation

  • To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers $(x, y)$:
  • This means that we have a perfect rank correlation and both Spearman's correlation coefficient and Kendall's correlation coefficient are 1.
  • This graph shows a Spearman rank correlation of 1 and a Pearson correlation coefficient of 0.88.
  • In contrast, this does not give a perfect Pearson correlation.
  • Define rank correlation and illustrate how it differs from linear correlation.

• Describing linear relationships with correlation

  • We denote the correlation by R.
  • Figure 7.10 shows eight plots and their corresponding correlations.
  • The correlation is intended to quantify the strength of a linear trend.
  • Sample scatterplots and their correlations.
  • Sample scatterplots and their correlations.

• Correlational Research

  • The attributes of correlations include strength and direction.
  • Direction refers to whether the correlation is positive or negative.
  • A correlation of 0 indicates no relationship between the variables.
  • It is extremely rare to find a perfect correlation between two variables, but the closer the correlation is to -1 or +1, the stronger the correlation is.
  • Always remember that correlation does not imply causation.

• Other Types of Correlation Coefficients

  • Other types of correlation coefficients include intraclass correlation and the concordance correlation coefficient.
  • Thus, if we are correlating $X$ and $Y$, where, say, $Y=2X+1$, the Pearson correlation between $X$ and $Y$ is 1: a perfect correlation.
  • The concordance correlation coefficient is nearly identical to some of the measures called intraclass correlations.
  • Comparisons of the concordance correlation coefficient with an "ordinary" intraclass correlation on different data sets will find only small differences between the two correlations.
  • Distinguish the intraclass and concordance correlation coefficients from previously discussed correlation coefficients.

• Values of the Pearson Correlation

  • Give the symbols for Pearson's correlation in the sample and in the population
  • The Pearson product-moment correlation coefficient is a measure of the strength of the linear relationship between two variables.
  • It is referred to as Pearson's correlation or simply as the correlation coefficient.
  • The symbol for Pearson's correlation is "$\rho$" when it is measured in the population and "r" when it is measured in a sample.
  • Because we will be dealing almost exclusively with samples, we will use r to represent Pearson's correlation unless otherwise noted.

• 95% Critical Values of the Sample Correlation Coefficient Table

• An Intuitive Approach to Relationships

  • Correlation refers to any of a broad class of statistical relationships involving dependence.
  • These are all examples of a statistical factor known as correlation.
  • Correlation refers to any of a broad class of statistical relationships involving dependence.
  • Familiar examples of dependent phenomena include the correlation between the physical statures of parents and their offspring and the correlation between the demand for a product and its price.
  • This graph shows a positive correlation between world population and total carbon emissions.

• Coefficient of Correlation

  • The most common coefficient of correlation is known as the Pearson product-moment correlation coefficient, or Pearson's $r$.
  • Pearson's correlation coefficient when applied to a sample is commonly represented by the letter $r$ and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient.
  • If $r=1$, there is perfect positive correlation.
  • If $r=-1$, there is perfect negative correlation.
  • Put the summary statistics into the correlation coefficient formula and solve for $r$, the correlation coefficient.