1. The researcher wants to examine whether three dimensions of sports team cohesion (correct task knowledge, ability evaluation, and emotional awareness) can predict competition ranking using discriminant analysis. 2. Competition ranking is the categorical dependent variable. 3. Discriminant analysis will be used to find the dimensions that discriminate between competition rankings and develop classification functions to predict group membership.

1. DISCRIMINANT
ANALYSIS
Wei-Jiun, Shen Ph. D.

2. Purpose
 To predict group membership from a set of
predictors
 Level of DV > 2
 What analytic strategy will you use?

3. Kinds of research questions
 Goals of discriminant analysis are to find the
dimension or dimensions along which groups differ,
and to find classification function to predict group
membership.
 Significance of prediction
 Number of significant discriminant functions
 Dimensions of discrimination
 Classification functions
 Adequacy of classification
 Effect size
 Importance of predictor variables
 Significance of prediction with covariates
 Estimation of group means

4. Limitations to discriminant analysis
 Theoretical issues
 Random assignment
 Generalizability
 Practical issues
 Unequal sample size, missing data, and power
 Multivariate normality
 Absence of outliers
 Homogeneity of variance-covariance matrices
 Linearity
 Absence of multicollinearity and singularity

5. Fundamental equation for discriminant analysis
 Derivation and test of discriminant functions
𝑆𝑡𝑜𝑡𝑎𝑙 = 𝑆 𝑏𝑔 + 𝑆 𝑤𝑔
Λ =
𝑆 𝑤𝑔
𝑆 𝑏𝑔 + 𝑆 𝑤𝑔
Approximate F = 𝑑𝑓1, 𝑑𝑓2 =
1−𝑦
𝑦
𝑑𝑓2
𝑑𝑓1
Λ
1
2 = 𝑦;

6. Fundamental equation for discriminant analysis
 Classification
 Equal group size
 Unequal group size
𝐶𝑗 = 𝑐𝑗0 +𝑐𝑗1 𝑋1 + 𝑐𝑗2 𝑋2 + ⋯ + 𝑐𝑖𝑝 𝑋 𝑝
𝐂𝑗 = 𝐖−1
𝐌𝑗 𝑐𝑗0 = −
1
2
𝐂𝑗
′
𝐌𝑗
𝐂𝑗 = 𝑐𝑗0 +
𝑖=1
𝑝
𝑐𝑗𝑖 𝑋𝑖 + 𝑙𝑛
𝑛𝑗
𝑁

7. Fundamental equation for discriminant analysis
 Projection
𝐶𝑗 = 𝑐𝑗0 +𝑐𝑗1 𝑋1 + 𝑐𝑗2 𝑋2 + ⋯ + 𝑐𝑖𝑝 𝑋 𝑝
Xp
Cj
θ

8. Fundamental equation for discriminant analysis
 Solution
B − ΛW K = 0
𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 = Λ =
𝑟𝑐1
2
⋯ ⋯
⋮ ⋱ ⋮
⋮ ⋯ 𝑟𝑐𝑖
2
𝑒𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟 = K = 𝑘1 ⋯ 𝑘 𝑞
Do you smell
something?

9. Discriminant analysis
 Prediction
1
1
N
P
N*P
X
1
2
3
2
1
3
2
1
N*1
Y

10. Discriminant function
 Special case of canonical analysis
1
1
N
P
N*P
X
1
2
3
1
2
3
1
2
N*1
Y

11. Discriminant function
 Special case of canonical analysis
1
1
N
P
N*P
X
1
N
N*(L-1)
…D1 D2 D3 DL-1
1 0 0 … 0
0 1 0 … 0
0 0 1 … 0
0 0 0 … 1

12. Discriminant function
 Special case of canonical analysis
χ1
χ2
X1
X2
X3
X4
X5
η1
η2
D1
D2
𝑟𝑐2
𝑟𝑐3
0
0

13. Example: 1 predictor
 DV
Y=0
Y=1
0 1 2 3 4 5 6 7 8 9 10
X
1 27
4
9
31013 5 6 12
11
3
14
15

14. Example: 2 predictors
 DV
Y=0
Y=1
Y=2
0 1 2 3 4 5 6 7
X1
12
4
3
5
6
7
8
9
10
12
11
13
14
15
0 1 2 3 4 5 6 7
X2
14
2
5
3
6
78
9
10
12
11
13
14
15

15. Example: 2 predictors
 DV
Y=0
Y=1
Y=2
0 1 2 3 4 5 6 7
1
2 4 3
5
6
7
8
9
10
1211 13 14
15
Discriminant function 2
= 0*X1+1*X2
Discriminant function 1
= 1*X1+0*X2
1
2
3
4
5
6
7
Number
Min (N of IV, df for DV)

16. Principle of discriminant function
 Various linear combination (projection)

17. Principle of discriminant analysis
 Maximize difference among levels
 Minimize variance within level
X1
X2
C1

18. Types of discriminant analysis
 Direct(standard) discriminant analysis
 Test for the significance
 Sequential discriminant analysis
 Mathematical resolution of combining variables
 Stepwise(statistical) discriminant analysis
 Mathematical resolution of combining variables

19. Criteria for overall statistical significance
 General
 Wilk’s lamda
 Hotelling’s trace
 Roy’s gcr
 Pillai’s criterion
 Distance-based (usually for stepwise)
 Mahalanobis’D2
 Rao’s V

20. Interpreting discriminant functions
 Discriminant function plots (F1 & F2; F2 & F3;…)
0 1 2 3 4 5 6 7
G1
Discriminant function 2
= 0*X1+1*X2
Discriminant function 1
= 1*X1+0*X2
1
2
3
4
5
6
7
G2
G3

21. Interpreting discriminant functions
 Discriminant weight (coefficient)
 Beta in regression analysis
 Discriminant loading
 Simple linear correlation between IV and discriminant
function
 >.33 (Tabachnick & Fidell, 2009)
 >.40 (Hair et al., 2010)

22. Evaluating predictor variables
 Classification function coefficient / loading
 Magnitude
 Mean difference
 Compare MGi and Mpooled(Gn-i)
 Inflation of type 1 error
 Adjusted alpha
 Potency index

23. Potency index
χn
X1
X2
X3
X4
X5
ηn
D1
D2
D3
D4
𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑐𝑛λ 𝑥𝑛1
λ 𝑥𝑛2
λ 𝑥𝑛3
λ 𝑥𝑛4
λ 𝑥𝑛5
λ 𝑦𝑛1
λ 𝑦𝑛2
λ 𝑦𝑛3
λ 𝑦𝑛4
Square
d
𝑃𝑜𝑡𝑒𝑛𝑐𝑦 𝑖𝑛𝑑𝑒𝑥 𝑥𝑛𝑖→y = λ 𝑥𝑛𝑖
2
×
𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒 𝑐𝑛
𝑡𝑜𝑡𝑎𝑙 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒
2

24. Effect size
 Total
 Partial η2
 Specific
 Eigenvalue (each discriminant function)
 Squared structure matric loading (each predictor)
𝑝𝑎𝑟𝑡𝑖𝑎𝑙 η2
= 1 − Λ
1
3

25. Use of classification procedure
 Generalization (hit ratio)
 Cross-validation and new case
 Jackknified classification
 Leave one out classification
 Evaluating improvement in classification
 Sequential
Early step
classification
Correct Incorrect
Later step
classification
Correct A B
Incorrect C D
χ2 1 =
𝐵 − 𝐶 − 1
𝐵 + 𝐶

26. Standards of hit ratio
 Proportion chance criterion
 Press’s Q
𝑃𝑟𝑒𝑠𝑠′ 𝑠 𝑄 =
𝑁 − 𝑛𝐾 2
𝑁(𝐾 − 1)
𝐶 𝑝𝑟𝑜 = 𝑝2 + (1 − 𝑝)2
ℎ𝑖𝑡 𝑟𝑎𝑡𝑖𝑜 > 1.25 × 𝐶 𝑝𝑟𝑜 𝑝2
χ2 > 3.84 (𝛼 = .05)
χ2 > 6.63 (𝛼 = .01)

27. Procedure
1. Research question
2. Designing a canonical analysis
3. Check the assumptions
4. Derive canonical analysis and assess overall fit
5. Interpret the canonical variate
6. Validation and diagnosis

28. PRACTICE

29. 過去學業表現與現在學業表現
 研究生倪之道想瞭解運動團隊默契對團隊表現的
影響。他的研究問題是，個體認為團隊有正確的
任務知識、能力評價與情緒覺察等三個運動團隊
默契的向度是否能預測比賽名次？其中，比賽名
次為類別變項。請以區別分析解答此問題。