• The Big Picture
• Measuring Correlation
  

• Measuring Correlation (A Recap)
• Interpreting Correlation (contd.)
  

Note:

• The sign of the coefficient represents the direction of the relationship
• The absolute value of the coefficient represents the strength of the relationship

• Strong correlations = significant correlations
• However, statistical Correlation does not always mean meaningful correlation
• Strong versus Meaningful relationships

• A low Pearson r does not always indicate a insignificant correlation
• If your sample is large enough, even a weak correlation is statistically significant
  

• A high Pearson r does not always mean a meaningful correlation
• For example, a significant correlation between ice cream consumption and crime in a city might correlate highly, but the relationship itself is suspect
  

• Just “eyeballing” the correlation coefficient is not enough
• There are other, more sound ways of judging the meaningfulness of a correlation
1. The coefficient of determination
2. Hypothesis testing
   

1. The coefficient of determination (rXY2) is the amount of variance that is accounted for in one variable by another variable
• It allows you to estimate the amount of variance that can be accounted for in one variable by examining the variance in another variable
• Example: Recall from last class that the correlation between education level and level of prejudice is very strong (rXY = -.92)  
• But is it meaningful? 
• The coefficient of determination would be (-.92)2 = .8464 is approximately .85

• This means that:
• 85% of the variance in one variable is explained by the variance in the other variable
• 15% (or 100% - 85%) of the variance is unexplained
• This portion of unexplained variance is often referred to as the coefficient of alienation  
• The more variance explained, the better
• The less variance left unexplained, the better
• Therefore, the correlation between education and prejudice is a meaningful one because it the variance in one education explains most of the variance in prejudice and vice-versa (this leaves little variance between them unexplained)
  

2. Do a hypothesis test to determine whether the correlation between X and Y is a meaningful one

• The claim we are testing is: “There is a significant correlation”
• You want to determine whether: 
• The association between X and Y exists in the population (true correlation)
• Or whether:
• The correlation is merely due to sampling error (false correlation)
  

• H0: There is no relationship between X and Y
• The null states that the population correlation between X and Y (rhoXY) is zero
• H0: rho = 0
  

• H1: A relationship exists between X and Y
• The alternative hypothesis states that the population correlation between X and Y is not zero (H1: rho is not equal to 0)
• The correlation is either positive (H1: rho > 0) or negative (H1: rho < 0)
  

• To do hypothesis tests, you need to answer the following questions:
1. What are the degrees of freedom?
• df = the number of observation that are free to vary
• As a general equation, df = N -K
• Where:
  
  • N = # of observations
  • K = # of parameters to be estimated
    
• Degrees of freedom, explained
• If we take a sample of N observations, then they are free to vary in any way (take on any values)
• However, if we use the sample of N observations to calculate the standard deviation, we use the sample mean as an estimate of the population mean
• Therefore, we hold one parameter constant
• With this parameter fixed, only N – 1 observations are free to vary
• Now, since we got the means for two variables (X and Y), there are two parameters being held constant
• If 2 parameters are fixed, then there are only N – 2 observations that are free to vary
• Therefore, when doing bivariate correlation, we use the following formula to calculate the degrees of freedom: df = N – 2
• From the education/prejudice example, we know that N = 10, so N – 2 = 8
  

2. How confident do you want your test to be? 
• Usually, when people test hypotheses, they want to be at least 95% confident that they got it right
• The level of significance (represented by "alpha") of a hypothesis test tells you how confident you can be about being right
• Most people test their hypotheses at the significance level of .05 (alpha = .05)
• This means that we only have a 5% chance (100% - 95% =  5%) of making a Type I error (rejecting the null hypothesis when it is true)
• With a .05 alpha level, the odds of being wrong about whether X relates to Y are 20 to 1
• Now that we have the necessary information (based on the above data), we can actually do the hypothesis test
• rXY = -.92
• N = 10
• df = 8
• alpha = .05
• Most statistics textbooks have a table where you can find the a list of significant values of Pearson’s r for the .05 and .01 levels of significance with the number of degrees of freedom
• Based on the Appendix table, the critical r = .6319
• In order to reject the null that rho = 0 at the alpha = .05 level, our calculated Pearson’s r must exceed .6319
• Since our Pearson’s r is |.92| (disregarding the negative sign), we can reject the null
  

• Levin, Jack and James Alan Fox. 2003. Elementary Statistics in Social Research, 9th Edition. Boston, MA: Pearson Education Group, Inc.
• Salkind, Neil J. 2003. Exploring Research, 5th Edition. Upper Saddle River, NJ: Prentice Hall.
• Kranzler, John H. 2003. Statistics for the Terrified, 3rd Edition. Upper Saddle River, NJ: Prentice Hall.