1. Lecture 2- Alternate Correlation Procedures
EPSY 640
Texas A&M University

2. CORRELATION MEASURES FOR VARIOUS SCALES OF MEASUREMENTY
Nominal Level Nominal Level Ordinal Level Interval or ratio
X (dichotomous) (polychotomous) Level
Nominal Level Phi coefficient,
Dichotomous Yule's Q, Pearson rank-biserial point-biserial,
Goodman;s  Association, biserial, Nagelkirke
Lambda , Tschuprow’s T, R-square (logistic)
tetrachoric
Pearson
Nominal Level Association, reduce to
(Polychotomous) Tschiuprow’s T, dichotomous or R-square
Cramer’s C Kruskal-Wallis
based statistic
Ordinal Spearman, R-squared
Level Kendall’s tau
Interval/Ratio Pearson r

3. Dichotomous-Dichotomous Case- PHI COEFFICIENT
the phi coefficient can be written as
rphi = (ad – bc) / [(a+c)(b+d)(a+b)(c+d)]1/2
MINORITY STATUS
a b
c d
GENDER

4. Dichotomous-Dichotomous Case- PHI COEFFICIENT
Political affiliation
rphi = (7x10 – 2x2) / (7+2)(2+10)(7+2)(2+10)]1/2
= (70 - 4) / [9x12x9x12]1/2
= 66/108 = .611
Pearson r = .157/(.5071 x .5071) = .611
Row total 9 12
7 2
2 10
Gender
Column Total

5. 2 =   n.j (nij/n – ni./n)2 / ni./n i=1 j=1
CHI-SQUARE:
I J
2 =   n.j (nij/n – ni./n)2 / ni./n
i=1 j=1
= 9[7/21 – 9/21]2 /( 9/21) + 9[2/21 – 9/21] 2/( 9/21) + 12[2/21 – 12/21]2 /(12/21) [10/21 – 12/21]2 / (12/21)
= / / / /21
= 157/21 =
PEARSON ASSOCIATION: TSCHUPROW'S T:
P = {2 / 2 + n}1/2 T = {2/n[(r-c)(c-1)]1/2}1/2
= {7.476/22.476}1/ = {7.476/21 x [1 x 1]1/2}1/2
= = .597

6. SPSS Crosstabs procedure
Select “Analyze/ Descriptive Statistics/ Crosstabs”
Select “Row” and “Column” variables for the two nominal variables
Select under “Statistics” the options you want such as “Chi Square” and various “Nominal” association measures

9. TETRACHORIC ASSUMPTIONS- underlying normality of observed dichotomies
0,1
1,1 1
0,0
1,0 1
TETRACHORIC ASSUMPTIONS- underlying normality of observed dichotomies

10. ux = height of normal curve for proportion = 90/210 = U(.4290)
n11 = 70 n12 = 20 n21 = 20 n22 = 100
ux = height of normal curve for proportion = 90/210 = U(.4290)
The z-score for .429 = -.18
The ux for = U(.4290) (requires stat table or SPSS procedure Pdf.Norm)
= .3637
uy = height of normal curve for proportion = 90/210 = U(.4290)
rtet = [ (70 x 100) – (20 x 20) ] / [.3637 x x 2102 ]
= 6600/ (.132 x 2102)
= not a good estimate!
Table 3.4: Computation of tetrachoric correlation 70 20 20
100

11. y u
0.00
0.3989
0.02
0.52
0.3485
1.02
0.2371
1.52
0.1257
0.04
0.3986
0.54
0.3448
1.04
0.2323
1.54
0.1219
0.06
0.3982
0.56
0.3410
1.06
0.2275
1.56
0.1182
0.08
0.3977
0.58
0.3372
1.08
0.2227
1.58
0.1145
0.10
0.3970
0.60
0.3332
1.10
0.2179
1.60
0.1109
0.12
0.3961
0.62
0.3292
1.12
0.2131
1.62
0.1074
0.14
0.3951
0.64
0.3251
1.14
0.2083
1.64
0.1040
0.16
0.3939
0.66
0.3209
1.16
0.2036
1.66
0.1006
0.18
0.3925
0.68
0.3166
1.18
0.1989
1.68
0.0973
0.20
0.3910
0.70
0.3123
1.20
0.1942
1.70
0.0940
0.22
0.3894
0.72
0.3079
1.22
0.1895
1.72
0.0909
0.24
0.3876
0.74
0.3034
1.24
0.1849
1.74
0.0878
0.26
0.3857
0.76
0.2989
1.26
0.1804
1.76
0.0848
0.28
0.3836
0.78
0.2943
1.28
0.1758
1.78
0.0818
0.30
0.3814
0.80
0.2897
1.30
0.1714
1.80
0.0790
0.32
0.3790
0.82
0.2850
1.32
0.1669
1.82
0.0761
0.34
0.3765
0.84
0.2803
1.34
0.1626
1.84
0.0734
0.36
0.3739
0.86
0.2756
1.36
0.1582
1.86
0.0707
0.38
0.3712
0.88
0.2709
1.38
0.1539
1.88
0.0681
0.40
0.3683
0.90
0.2661
1.40
0.1497
1.90
0.0656
0.42
0.3653
0.92
0.2613
1.42
0.1456
1.92
0.0632
0.44
0.3621
0.94
0.2565
1.44
0.1415
1.94
0.0608
0.46
0.3589
0.96
0.2516
1.46
0.1374
1.96
0.0584
0.48
0.3555
0.98
0.2468
1.48
0.1334
1.98
0.0562
0.50
0.3521
1.00
0.2420
1.50
0.1295
2.00
0.0540 N O R M A L D E S I T Y H G

12. y u
2.02
0.0519
2.52
0.0167
3.02
0.0042
3.52
0.0008
2.04
0.0498
2.54
0.0158
3.04
0.0039
3.54
2.06
0.0478
2.56
0.0151
3.06
0.0037
3.56
0.0007
2.08
0.0459
2.58
0.0143
3.08
0.0035
3.58
2.10
0.0440
2.60
0.0136
3.10
0.0033
3.60
0.0006
2.12
0.0422
2.62
0.0129
3.12
0.0031
3.62
2.14
0.0404
2.64
0.0122
3.14
0.0029
3.64
0.0005
2.16
0.0387
2.66
0.0116
3.16
0.0027
3.66
2.18
0.0371
2.68
0.0110
3.18
0.0025
3.68
2.20
0.0355
2.70
0.0104
3.20
0.0024
3.70
0.0004
2.22
0.0339
2.72
0.0099
3.22
0.0022
3.72
2.24
0.0325
2.74
0.0093
3.24
0.0021
3.74
2.26
0.0310
2.76
0.0088
3.26
0.0020
3.76
0.0003
2.28
0.0297
2.78
0.0084
3.28
0.0018
3.78
2.30
0.0283
2.80
0.0079
3.30
0.0017
3.80
2.32
0.0270
2.82
0.0075
3.32
0.0016
3.82
2.34
0.0258
2.84
0.0071
3.34
0.0015
3.84
2.36
0.0246
2.86
0.0067
3.36
0.0014
3.86
0.0002
2.38
0.0235
2.88
0.0063
3.38
0.0013
3.88
2.40
0.0224
2.90
0.0060
3.40
0.0012
3.90
2.42
0.0213
2.92
0.0056
3.42
3.92
2.44
0.0203
2.94
0.0053
3.44
0.0011
3.94
2.46
0.0194
2.96
0.0050
3.46
0.0010
3.96
2.48
0.0184
2.98
0.0047
3.48
0.0009
3.98
0.0001
2.50
0.0175
3.00
0.0044
3.50 N O R M A L D E S I T Y H G

13. POINT-BISERIAL CORRELATION
Y = score on interval measure (eg. Test score)
x = 0 or 1 (grouping; eg. gender)
Y1. – Y ____________
rpb = _________  n1n0 / n(n-1) sy

14. Descriptive Statistics
Mean SD N Covariance
GENDER
READING COMPREHENSION
0 (boys)
1 (girls)
Total
_________________
rpb = –  [ 8 x 7 ] / [15 x 14 ]
15.32
= .233
Table 3.5: Calculation of point-biserial correlation coefficient for First Grade reading comprehension of boys and girls

15. POINT BISERIAL CORRELATIONY X
m = mean m X m F M X

16. Dichotomous (Normal)-Interval Case
biserial correlation
Y1. – Y0.
rbis = _________ . [n1n0 / uxn2] , sy
where u = height of normal curve for proportion n1/(n0 + n1)

17. rbis = _________ . [n1n0 / uxn2] , sy
Y1. – Y0.
rbis = _________ . [n1n0 / uxn2] , sy
= – [ 8 x 7 ] / [.3675 x 152 ]
15.32
= .306

18. BISERIAL CORRELATION Y X m X m F M X

19. RANK-RANK DATA 1. DATA ARE INTERVAL OR RATIO
Transformed to ranks because of odd distribution
2. DATA ARE ORDINAL, NO INTERVAL INFORMATION AVAILABLE
USE SPEARMAN CORRELATION (Pearson formula used on the ranks - no ties assumed)

20. Rank distribution of real estate price per square foot in Manhattan
Rank Price per foot
Battery Central Park Location
The relative position of the ranks above is only approximate, due to typeface limitations. All are ordered
correctly.
This results in the following ranks:
rank:
location:
price:
Computation of rank correlation coefficient:
rrank = sxy/sxsy
= -.647
rSpearman = (from SPSS, Version 13)
Table 3.7: Computation of rank correlation for Real Estate location in Manhattan with Price Per Square Foot

21. Least squares estimation
The best estimate will be one in which the sum of squared differences between each score and the estimate will be the smallest among all possible linear unbiased estimates (BLUES, or best linear unbiased estimate).

22. Least squares estimation
errors or disturbances. They represent in this case the part of the y score not predictable from x:
ei = yi – b1xi -b0x.
The sum of squares for errors follows: n
SSe =  e2i .
i=1

23. y e e e e e e e e
SSe = e2i x

24. Matrix representation of least squares estimation.
We can represent the regression model in matrix form:
y = X + e

25. Matrix representation of least squares estimation
y = X  e
y x1 e1 0
y x 1 e2
y x3 e3
y = x4 + e4

26. Matrix representation of least squares estimation
y = Xb + e
The least squares criterion is satisfied by the following matrix equation:
b = (X΄X)-1X΄y .
The term X΄ is called the transpose of the X matrix. It is the matrix turned on its side. When
X΄X is multiplied together, the result is a 2 x 2 matrix
n xi
xi x2i Note: all information here: sample size, mean (sum of scores), variance (squared scores)

27. SUMS OF SQUARES computational equivalents
SSe = (n – 2 )s2e
SSreg =  ( b1 xi – y. )2
SSy = SSreg + SSe

28. SUMS OF SQUARES-Venn Diagram
SSy
SSreg
SSx
SSe
Fig. 8.3: Venn diagram for linear regression with one predictor and one outcome measure

29. SUMS OF SQUARES- ANOVA Table
SOURCE df Sum of Squares Mean Square F
x 1 SSreg SSreg / 1 SSreg/ 1
SSe /(n-2)
e n-2 SSe SSe / (n-2)
Total n-1 SSy SSy / (n-1)
Table 8.1: Regression table for Sums of Squares

30. Rupley and Willson (1997) studied the relationship between word recognition and reading comprehension for 200 six- and seven-year olds using a national sample of students that mirrored the U.S. census of The mean for Word Recognition was 100, SD=15, and the mean for Reading Comprehension was 23.16, SD= The regression analysis is reported in the table below:
Dep. Var.: Reading Comprehension
SOURCE df Sum of Squares Mean Square F Prob. R2
Word recog
nition
error
total se =6.71

31. Two variable linear regression: Which direction?
Regression equations:
y = xb1x+ xb0
x = yb1y + yb0
Regression coefficients:
xb1 = rxy sy / sx
yb1 = rxy sx / sy

32. Two variable linear regression
y = b1x + b0
If the correlation coefficient is calculated, then b1 can be calculated from the equation above:
b1 = rxy sy / sx
The intercept, b0, follows by placing the means for x and y into the equation above and solving:
b0 = y. – [ rxysy/sx ] x.

33. Two variable linear regression.
yb1 = rxy sx / sy
xb1 = rxy sy / sx x y y x
Fig. 8.1: Slopes for two regression representations of Pearson correlation

34. Three variable linear regression
y = b1x1 + b2x2 + b0
Two predictors: all variables may be correlated with each other
Exact equations exist to compute b1 , b2 but not for more than two predictors, matrix form normal equations must be used

35. Three variable linear regression
Path model representation:unstandardized x1 b1 y e
12 x2 b2

36. Three variable linear regression
Path model representation:standardized x1 1 y e
r12 x2 2

37. SUMS OF SQUARES-Venn Diagram
SSx1
SSy
SSreg
SSe
SSx2
Fig. 8.3: Venn diagram for linear regression with two predictors and one outcome measure