correlation coefficient

Examples of correlation coefficient in the following topics:

• 95% Critical Values of the Sample Correlation Coefficient Table

• Hypothesis Tests with the Pearson Correlation

  • We need to look at both the value of the correlation coefficient $r$ and the sample size $n$, together.
  • We decide this based on the sample correlation coefficient $r$ and the sample size $n$.
  • If the test concludes that the correlation coefficient is significantly different from 0, we say that the correlation coefficient is "significant."
  • If the test concludes that the correlation coefficient is not significantly different from 0 (it is close to 0), we say that correlation coefficient is "not significant. "
  • Our null hypothesis will be that the correlation coefficient IS NOT significantly different from 0.

• Rank Correlation

  • Rank correlation coefficients, such as Spearman's rank correlation coefficient and Kendall's rank correlation coefficient, measure the extent to which as one variable increases the other variable tends to increase, without requiring that increase to be represented by a linear relationship .
  • However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than the Pearson product-moment correlation coefficient.
  • This means that we have a perfect rank correlation and both Spearman's correlation coefficient and Kendall's correlation coefficient are 1.
  • In the same way, if $y$ always decreases when $x$ increases, the rank correlation coefficients will be $-1$ while the Pearson product-moment correlation coefficient may or may not be close to $-1$.
  • This graph shows a Spearman rank correlation of 1 and a Pearson correlation coefficient of 0.88.

• 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 population is commonly represented by the Greek letter $\rho$ (rho) and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient.
  • 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.
  • This fact holds for both the population and sample Pearson correlation coefficients.
  • Put the summary statistics into the correlation coefficient formula and solve for $r$, the correlation coefficient.

• Testing the Significance of the Correlation Coefficient

  • The sample correlation coefficient, r, is our estimate of the unknown population correlation coefficient.
  • If the test concludes that the correlation coefficient is significantly different from 0, we say that the correlation coefficient is "significant".
  • If the test concludes that the correlation coefficient is not significantly different from 0 (it is close to 0), we say that correlation coefficient is "not significant".
  • The test statistic t has the same sign as the correlation coefficient r.
  • Suppose you computed the following correlation coefficients.

• Other Types of Correlation Coefficients

  • Other types of correlation coefficients include intraclass correlation and the concordance correlation coefficient.
  • Whereas Pearson's correlation coefficient is immune to whether the biased or unbiased version for estimation of the variance is used, the concordance correlation coefficient is not.
  • 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.

• Student Learning Outcomes

• 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.
  • If the relationship between the variables is not linear, then the correlation coefficient does not adequately represent the strength of the relationship between the variables.
  • The symbol for Pearson's correlation is "$\rho$" when it is measured in the population and "r" when it is measured in a sample.

• The Correlation Coefficient r

  • Use the correlation coefficient as another indicator (besides the scatterplot) of the strength of the relationship between x and y.
  • The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is a numerical measure of the strength of association between the independent variable x and the dependent variable y.
  • If r = 1, there is perfect positive correlation.
  • We say "correlation does not imply causation."
  • The correlation coefficient r is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions).

• Rank Correlation

  • A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them.
  • A rank correlation coefficient can measure that relationship, and the measure of significance of the rank correlation coefficient can show whether the measured relationship is small enough to be likely to be a coincidence.
  • Spearman developed a method of measuring rank correlation known as Spearman's rank correlation coefficient.
  • There are three cases when calculating Spearman's rank correlation coefficient:
  • Kendall's $\tau$ and Spearman's $\rho$ are particular cases of a general correlation coefficient.