Chapter 11 Correlation Coefficients

Chapter 11: Correlation Coefficients New Vocabulary: Correlation: a relationship between at least two variables Pearson’s correlation coefficient (r): the test used if you are looking for a relationship between two variables that are normally distributed and at the interval/ratio level Spearman’s correlation coefficient (p): the test used to determine whether there is a relationship between two variables that are ordinal, interval, or ratio level but don’t meet the full assumptions for the Pearson’s correlation coefficient Coefficient of determination: the square of Pearson’s r Percentage of variance: the amount of variance in one variable explained by the other variable that can be determined by multiplying the coefficient of determination by 100 Strength of relationship: determined by the absolute value of the correlation coefficient Homoscedasticity: equal spread of one variable around all levels of another variable Selecting the Best Correlation Test -​ Consider what level of measurement the variables are (NOIR) -​ If they are categorical variables at the nominal/ordinal level, use the Chi Square Test -​ If they are ordinal, interval, or ratio: -​ If the lowest level for at least one of the variables is at the ordinal level or is not normally distributed → use Spearman's p -​ If you have two variables at the interval/ratio level that are normally distributed → use Pearson's r Direction of the Relationship -​ Both Spearman and Pearson describe the direction and strength of a linear relationship between two variables -​ The direction of the relationship is either positive or negative -​ Positive: when one variable increases, the other also increases -​ The coefficient (p or r) will then be positive -​ Negative: when one variable increases, the other decreases -​ The coefficient (p or r) will then be negative -​ Correlation coefficients are always between -1 and +1 -​ A correlation of -1 indicates a perfect negative relationship -​ The closer the correlation is to 0, the weaker the relationship is -​ If the correlation is 0 there is no relationship at all -​ A correlation of +1 indicates a perfect positive relationship Sample Size -​ You must have at least 3 subjects for correlation coefficients -​ You need 2 variables to correlate at least 3 subjects in the sample Strength of the Relationship -​ The strength of the relationship or correlation is determined by the absolute value of the correlation coefficient -​ If the absolute value of the correlation coefficient is <0.3 the relationship is weak -​ If it is 0.3-0.5 the relationship is moderate -​ If it is >0.5 the relationship between the variables is strong -​ Eg. a correlation coefficient of 0.2 shows a weak positive relationship between the variables, and a correlation coefficient of -0.6 shows a strong negative relationship between the variables Statistical Significance -​ Don’t forget about the p-value! -​ You need to consider the corresponding p-value to determine whether or not your correlation coefficient is significant Appropriate use of Correlation Coefficients 1.​ The sample subjects should be independent and random 2.​ The level of measurement should be used to select the right correlation coefficient 3.​ Two variables must be compared, and a linear relationship must be present 4.​ If you wish to use the Pearson correlation coefficient the variables should be normally distributed and homoscedasticity should be present ●​ ●​ ●​ ●​ Two nominal variables: select Chi-Square Ordinal and ordinal/interval/ratio variables: select Spearmans Both variables are interval/ratio and meet Pearsons assumptions: select Pearsons Both variables are interval/ratio but do not meet Pearsons assumptions: select Spearmans More Uses of Pearson's r -​ If you square Pearson’s r it becomes the coefficient of determination -​ If you multiply the coefficient of determination by 100 it becomes the percentage of variance -​ The percentage of variance is the amount of variance in one variable that is explained by the other variable -​ Eg. Pearson's r = 0.7, coefficient of determination = 0.49, percentage of variance = 49% → differences in daily caloric intake explained 49% of the differences the found in total serum cholesterol -​ You can also use Pearson’s r to determine effect size when calculating sample size -​ When you have a strong correlation you have a large effect size and don’t need such a large sample to find a statistically significant difference -​ When the strength of the correlation is weak, the effect size is small, and you need a larger sample to detect statistically significant differences