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Units, Constants and Useful Formulas
Pearson Correlation
-------------------
Linear regression is similar to correlation in that the purpose
is to measure to what extent there is a linear relationship
between two variables. The major difference between the two
is that correlation makes no distinction between independent
and dependent variables while linear regression does.
ρ = population correlation coefficient.
ρ = (1/N)Σ[(xi - μx)/σx][(yi - μy)/σy]
Pearson r = sample correlation coefficient.
_ _
r = (1/(n - 1))Σ[(xi - x)/sx][(yi - y)/sy]
SSxy
= -------------------------------------
(n - 1)√[(SSxx/(n - 1)]√[SSyy/(n - 1)]
SSxy
= ----------
√(SSxxSSyy)
Alternatively,
r = √(SSM/Total SS)
Where:
_ _
SSxy = Σ(xi - x)(yi - y)
_
SSxx = Σ(xi - x)2
_
SSyy = Σ(yi - y)2
Total SS = Model SS + Error SS
= SSM + SSE
_ _
Σ(yi - y)2 = Σ(yp - y)2 + Σ(yi - yp)2
_
= Σ(yp - y)2 + Σ(yi - (βxi + α))2
Coefficient of Determination:
r2 = Proportion of the variance of the dependent variable (y)
that is explained by the independent variable (x).
= (SSyy - SSE)/SSyy
Hypotheses:
If there is a significant correlation between the variables,
r will not equal zero.
H0: ρ = 0
H1: ρ ≠ 0
Test statistic: t test = r√[(n - 2)/(1 - r2)]
= β/sβ
= β/(s/√SSxx)
The test has n - 2 df.
General rule of thumb:
ρ: 0.10 - 0.29 small correlation
0.30 - 0.49 medium correlation
0.50 - 1.0 strong correlation
A simple example:
_ _ _ _
xi yi (xi - x)2 (yi - y)2 (xi - x)(yi - y)
- --- ------- ------- --------------
1 1.9 2.25 0.8556 1.3875
2 2.3 0.25 0.2756 0.2625
3 2.8 0.25 0.0006 -0.0125
4 4.3 2.25 2.1756 2.2125
- --- ---- ------ ------
Σ 5.00 3.3075 3.8500
_
x = 2.5000
_
y = 2.8250
SSxx = 5.0000
SSxy = 3.8500
SSyy = 3.3075
β = SSxy/SSxy = 3.8500/5.0000 = 0.7700
SSE = SSyy - βSSxy = 3.3075 - 0.7700*3.8500 = 0.3430
= Σ(yi - yp)2 = 0.3430
_
yi yp (yi - yp)2 (yp - y)2
--- ---- ------ ------
1.9 1.67 0.0529 1.3340
2.3 2.44 0.0196 0.1482
2.8 3.21 0.1681 0.1482
4.3 3.98 0.1024 1.3340
------ ------
Σ 0.3430 2.9645
Total SS = SSM + SSE
_ _
Σ(yi - y)2 = Σ(yp - y)2 + Σ(yi - yp)2
3.3075 = 2.9645 + 0.3430
Coefficient of Correlation:
r = √(SSM/Total SS)
= √(2.9645/3.3075)
= 0.9466
= SSxy/√(SSxxSSyy)
= 3.8500/√(5.0000*3.3075)
= 0.9466
Coefficient of Determination:
r2 = (SSyy - SSE)/SSyy
= (3.3075 - 0.3430)/3.3075
= 0.8963
s2 = SSE/(n - 2) = 0.3430/2 = 0.1715 ∴ s = 0.4141
t = β/(s/√SSxx) = 0.77/(0.4141/√5.0000) = 4.158
t test = r√[(n - 2)/(1 - r2)]
= 0.9466√[2/(1 - 0.8962)]
= 4.155
t0.05/2 for 4 - 2 = 2 df is 4.303 (2-tailed). Therefore,
reject H0.
SPSS: Analyze>Correlate>Bivariate
Multiple Correlation
--------------------
Because we now have multiple variables, we have more than one
correlation value to consider => partial correlation coefficient.
PCC is the measure of the correlation between the dependent
variable and one independent variable holding constant the
influence of all other independent variables.
Correlation matrix:
x1 x2 x3 x4
x2 1 ? ?
x3 ? 1 ?
x4 ? ? 1
Test statistic: One-sample t test = r√[(n - 2)/(1 - r2)]
SPSS: Analyze>Correlate>Bivariate