coursework Introduction
Prior followed a logical sequence: EDMS 645 t-tests correlation (X, Y) EDMS 646 ANOVA (X, Y)
EDMS 651 Multiple regression (X1, X2, ...., Y)
Factor analysis is very different all X variables.
Everything you ever wanted to know about correlation, but were afraid to ask
MY DEAR ADLE, I am 4 years old and I can read any English book. I can say all the Latin Substantives and Adjectives and active verbs besides 52 lines of Latin poetry. I can cast up any sum in addition and multiply by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. I can also say the pence table. I read French a little and I know the clock. FRANCIS GALTON Febuary 15, 1827
Francis Galton (1822-1911)
Correlation
Correlation
Parent height (X) Son's height (Y)
negative
positive
Galton did derive an index of "co-relation" (labeled r, for "regression"), but not the index we use today.
Correlation
Galton's student, Karl Pearson, formalized the mathematics of the correlation coefficient. He named it the "Galton function" or "Coefficient of correlation."
Covariance (Positive)
X
64 69 72 65 70 68
Y X-X Y-Y (X-X)(Y-Y)
66 69 71 67 72 69
means covariance
-4 1 4 -3 2
-3 0 2 -2 3
12 0 8 6 6 6.4
73 72 71 70 69 68 67 66 65 62 64 66 68 X 70 72 74 Y
Covariance = average deviation cross-product
Cov(X,Y)
Covariance ("Small")
X
64 69 72 65 70 68
Y X-X Y-Y (X-X)(Y-Y)
72 69 71 66 67 69
means covariance
-4 1 4 -3 2
3 0 2 -3 -2
-12 0 8 9 -4 0.20
73 71 72 70 Y 69 68 67 66 65 62 64 66 68 X 70 72 74
Covariance (Negative)
X
64 69 72 65 70 68
Y X-X Y-Y (X-X)(Y-Y)
72 66 67 71 69 69
means covariance
-4 1 4 -3 2
3 -3 -2 2 0
-12 -3 -8 -6 0 -5.8
73 72 71 70 69 68 67 66 65 62
Y
64
66
68 X
70
72
74
Covariance:
Populations and samples
Population
XY =
Sample
( X - X )(Y - Y ) N pop
s XY =
( X - X )(Y - Y )
n -1
Problems with Covariance
Tied to the units of measurement Tied to the variability of a variable No familiar or easily interpreted scale
Sample Covariance
X
64 69 72 65 70 68
Y
66 69 71 67 72 69
X-X
-4 1 4 -3 2
Y-Y
-3 0 2 -2 3
(X-X)(Y-Y)
12 0 8 6 6 8.0
73 72 71 70 69 68 67 66 65 62 64 66 68 X 70 72 74 Y
means
covariance
Cov(X,Y)
Sample Covariance Correlation
zX = (X - X ) sX zY = (Y - Y ) sY
X
64 69 72 65 70
Y
66 69 71 67 72
zX
zY
(zX)(zY)
1.39 0 .92 .69 .70 .925
73 72 71 70 69 68 67 66 65 62 64...