Correlation Analysis. These two forms of analysis are closely related.
Coefficient of determination (which is squared value of correlation
coefficient) is used as a measure of how good the regression equation fits the
data.
Measures of Linear Relationship Between Two Variables
Definition
The correlation coefficient (r) is an index which expresses the strength of linear
relationship/association between two variables (X and Y). It takes on values between –1
to +1, with –1 implying perfect negative relationship and +1 indicating perfect positive
relationship. The middle of the range is of course zero which represents no linear
relationship between the two variables.
Negative r values
Negative relationship implies that as values for X move in one direction, the values for Y
move in the opposite direction. For example if home values and crime are negatively
correlated, one would expect that if values for 100 randomly selected homes were
recorded, and the crime index for neighborhoods where the homes are located are also
recorded, you would expect that the higher priced homes would have lower crime index
and vise versa (i.e., values for the two variables move in opposite directions). In this
example, you have 100 paired observations (100 X values and 100 associated Y values).
By putting these into the correlation formula, the calculated r would have a negative sign,
indicating negative relationship between the variables.
Positive r values
Positive relationship implies that the two variables move in the same direction: as X
increases, Y increases in value, and as X decreases in value, Y also decreases in value.
For example, on a survey respondents could be asked to use a scale of 1 to 10 to answer
two questions concerning their job satisfaction (low=1, high=10) and supervisor’s
managerial style (autocratic=1, democratic=10). Assuming boss’s style is the only
variable to describe job satisfaction, one would expect that highly-satisfied employees
have managers that are democratic in their style, and that low-satisfied employees would
have autocratic managers. In this example, the two variables would go in same direction:
as job satisfaction rating goes up, the supervisor rating also goes up, and vise versa. So,
computing r in this case would result in a positive value.
Magnitude of r values
Values of correlation coefficient between zero to approximately 0.3 are considered low
positive correlation, between 0.3 to about 0.7 is considered moderate positive correlation,
and from 0.7 and higher is considered high positive correlation, with +1 being perfect
positive correlation.
Same statements can be made for the range between 0 to –1, but you
must state them as negative, of course.
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