Plotting Two Variables Simultaneously
The more tightly the points are clustered together the higher the correlation between the two variables and the higher the ability to predict one variable from another.
Correlation coefficients can take on any value between -1 and +1, with + and - 1 representing perfect correlations between the variables. And a correlation of zero representing no relationship between the variables.
A rule of thumb for interpreting correlation coefficients:
Corr Interpretation 0 to .1 trivial .1 to .3 small .3 to .5 moderate .5 to .7 large .7 to .9 very large
Correlations are interpreted by squaring the value of the correlation coefficient. The squared value represents the proportion of variance of one variace that is shared with the other variable, in other words, the proportion of the variance of one variable that can be predicted from the other variable.
corr n .10 617 .20 153 .30 68 .40 37 .50 22 .60 15 .70 10 .80 7 .90 5
Sources of Misleading Correlation Coefficients
Restriction of Range
Extreme Groups
Combining Groups
Outliers
Curvilinearity
Discuss Correlation & Causation
Of course, just because two variables are correlated it does not mean that they are causally related. Often a third variable, a lurking variable, that is not included in the analysis is responsible (causes) for the first two variables. A lurking variable is a variable that loiters in the background and affects both of the original variables
Other Correlation Coefficients
Spearman Example
Sub | xrank | yrank | d | d^{2} |
a | 1 | 3 | -2 | 4 |
b | 4 | 4 | 0 | 0 |
c | 5 | 8 | -3 | 9 |
d | 10 | 5 | 5 | 25 |
e | 8 | 2 | 6 | 36 |
f | 14 | 15 | -1 | 1 |
g | 7 | 9 | -2 | 4 |
h | 2 | 6 | -4 | 16 |
i | 12 | 14 | -2 | 4 |
j | 9 | 7 | 2 | 4 |
k | 15 | 13 | 2 | 4 |
l | 3 | 1 | 2 | 4 |
m | 13 | 12 | 1 | 1 |
n | 11 | 10 | 1 | 1 |
o | 6 | 11 | -5 | 25 |
Sum | 0 | 138 |
Stata Example
input xrank yrank 1 3 4 4 5 8 10 5 8 2 14 15 7 9 2 6 12 14 9 7 15 13 3 1 13 12 11 10 6 11 end corr (obs=15) | xrank yrank ---------+------------------ xrank | 1.0000 yrank | 0.7536 1.0000
Another Stata Example
input y x 100 135 120 105 160 155 220 175 110 105 140 145 200 185 260 195 130 145 110 105 180 175 210 165 200 175 170 145 120 145 end egen xrank = rank(x) egen yrank = rank(y) list y x xrank yrank 1. 100 135 4 1 2. 110 105 2 2.5 3. 110 105 2 2.5 4. 120 145 6.5 4.5 5. 120 105 2 4.5 6. 130 145 6.5 6 7. 140 145 6.5 7 8. 160 155 9 8 9. 170 145 6.5 9 10. 180 175 12 10 11. 200 185 14 11.5 12. 200 175 12 11.5 13. 210 165 10 13 14. 220 175 12 14 15. 260 195 15 15 corr x y xrank yrank (obs=15) | y x xrank yrank ---------+------------------------------------ y | 1.0000 x | 0.8768 1.0000 xrank | 0.9118 0.9853 1.0000 yrank | 0.9821 0.8753 0.9073 1.0000 spearman x y Number of obs = 15 Spearman's rho = 0.9073 Test of Ho: x and y independent Pr > |t| = 0.0000
Point Biserial Correlation
Point Biserial Example
input y x 100 0 120 1 160 0 220 1 110 0 140 0 200 1 260 1 130 0 110 1 180 0 210 1 200 1 170 1 120 0 end corr x y (obs=15) | x y ---------+------------------ x | 1.0000 y | 0.5541 1.0000
Fourfold Correlation - Phi Coefficient
Y | ||||
1 | 0 | |||
X | 1 | (a) 12 | (b) 16 | |
0 | (c) 14 | (d) 9 |
Stata Example
input x y w 0 0 9 0 1 14 1 0 16 1 1 12 end corr x y [fw=w] (obs=51) | x y ---------+------------------ x | 1.0000 y | -0.1793 1.0000
tab x y [fw=w], all | y x | 0 1 | Total -----------+----------------------+---------- 0 | 9 14 | 23 1 | 16 12 | 28 -----------+----------------------+---------- Total | 25 26 | 51 Pearson chi2(1) = 1.6394 Pr = 0.200 likelihood-ratio chi2(1) = 1.6495 Pr = 0.199 Cramer's V = -0.1793 gamma = -0.3494 ASE = 0.252 Kendall's tau-b = -0.1793 ASE = 0.138
When analyzing two-by-two tables, the value of Cramer's V is actually phi. Cramer's V is a generalization of the phi coefficient that can be used in tables larger than two-by-two.
Linear Statistical Models Course