1. Multivariate Statistical Methods
Tests of Hypotheses of Means
by Jen-pei Liu, PhD
Division of Biometry, Department of Agronomy,
National Taiwan University
and
Division of Biostatistics and Bioinformatics
National Health Research Institutes
2019/2/27
Copyright by Jen-pei Liu, PhD

2. Tests of Hypotheses of Means
Introduction
Review of Univariate One-sample t Test
Multivariate One-sample T2 Test
Review of Univariate Two-sample t Test
Multivariate Two-sample T2 Test
Profile Analysis
Summary
2019/2/27
Copyright by Jen-pei Liu, PhD

3. Copyright by Jen-pei Liu, PhD
Introduction
Wechsler Adult Intelligence Scale scores of 101 elderly men and women (60-64, years old, Morrison, 2005)
Item Mean Variance Covariance
verbal
Performance
2019/2/27
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4. Copyright by Jen-pei Liu, PhD
Introduction
The mean scores for the population between 18 and 59 are 60 and 50 for verbal and performance items
Univariate question: Is the mean score of verbal item for elderly population different from that of the population between 18 and 59?
Multivariate question: Is the Wechsler Adult Intelligence Scale scores for elderly population different from that of the population between 18 and 59?
2019/2/27
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5. Copyright by Jen-pei Liu, PhD
Introduction
Survivors and non-survivors for Bumpus’s female sparrows
Survivors Nonsurvivors
Variables mean variance Mean Variance
Total length
Alar extent
Length of bead and head
Length of humerus
Length of keel of sterum
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Introduction
Univariate questions: Does difference exist in each of the five morphological measurements between the survivors and non-survivors?
Multivariate questions: Is there any difference in morphology between the survivors and nonsurvivors?
2019/2/27
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7. Review of Univariate One-sample t test
Hypothesis of mean:H0: v = o vs. Ha: v  o
2019/2/27
Copyright by Jen-pei Liu, PhD

8. Review of Univariate One-sample t test
Wechsler Adult Intelligence Scale (WAIS) verbal scores
2019/2/27
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9. Review of Univariate One-sample t test
Univaraite t2 follow a F distribution with 1 and n-1 d.f.
t and t2 are unitless
Example of verbal scores
t2 = (-4.76)2/(210.54/101)
= (-4.76)(210.54/101)-1(-4.76) =
> F0.01,1,100 = 6.76
2019/2/27
Copyright by Jen-pei Liu, PhD

10. Review of Univariate One-sample test
Properties
It is the uniformly most powerful unbiased test for mean H0: v = o vs. Ha: v  o under the normal assumption when the variance is unknown in sense the probability of rejecting H0 is less than or equal to  if v = o and is at least  if v  o
2019/2/27
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11. Multivariate One-sample T2 Test
Multivariate Hypothesis of One-sample Means
2019/2/27
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12. Multivariate One-sample T2 Test
2019/2/27
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13. Multivariate One-sample T2 Test
2019/2/27
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14. Multivariate One-sample T2 Test
2019/2/27
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15. Multivariate One-sample T2 Test
2019/2/27
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16. Multivariate One-sample T2 Test
Union-intersection principle
a’X ~ N1(a’, a’a)
Univariate hypothesis
H0(a): a’ = a’0
vs.
Ha(a): a’  a’0
2019/2/27
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17. Multivariate One-sample T2 Test
2019/2/27
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18. Multivariate One-sample T2 Test
2019/2/27
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19. Multivariate One-sample T2 Test
2019/2/27
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20. Multivariate One-sample T2 Test
2019/2/27
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21. Multivariate One-sample T2 Test
2019/2/27
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22. Multivariate One-sample T2 Test
2019/2/27
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23. Multivariate One-sample T2 Test
2019/2/27
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24. Multivariate One-sample T2 Test
2019/2/27
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25. Multivariate One-sample T2 Test
2019/2/27
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26. Multivariate One-sample T2 Test
2019/2/27
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27. Multivariate One-sample T2 Test
2019/2/27
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28. Multivariate One-sample T2 Test
Test Statistic
Hotelling T2 Statistic: a generalization of univariate t statistic by union-intersection principle
2019/2/27
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29. Multivariate One-sample T2 Test
Confidence Region and Simultaneous CIs
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30. Multivariate One-sample T2 Test
Wechsler Adult Intelligence Scale (WAIS) scores
2019/2/27
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31. Multivariate One-sample T2 Test
Wechsler Adult Intelligence Scale verbal scores
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32. Multivariate One-sample T2 Test
2019/2/27
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33. Multivariate One-sample T2 Test
2019/2/27
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34. Multivariate One-sample T2 Test
2019/2/27
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35. Review of Univariate Two-sample t Test
2019/2/27
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36. Review of Univariate Two-sample t Test
2019/2/27
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37. Review of Univariate Two-sample t Test
Example: The total length of female sparrows between survivors and nonsuvivors
Statistics Survivors Nonsurvivors
Sample size
Mean
Variance
2019/2/27
Copyright by Jen-pei Liu, PhD

38. Review of Univariate Two-sample t Test
Example: The total length of female sparrows between survivors and nonsuvivors
2019/2/27
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39. Multivariate Two-sample T2 Test
Multivariate Hypothesis of Two-sample Means
2019/2/27
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40. Multivariate Two-sample T2 Test
2019/2/27
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41. Multivariate Two-sample T2 Test
2019/2/27
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42. Multivariate Two-sample T2 Test
2019/2/27
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43. Multivariate Two-sample T2 Test
2019/2/27
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44. Multivariate Two-sample T2 Test
2019/2/27
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45. Multivariate Two-sample T2 Test
2019/2/27
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46. Multivariate Two-sample T2 Test
2019/2/27
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47. Multivariate Two-sample T2 Test
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48. Multivariate Two-sample T2 Test
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49. Multivariate Two-sample T2 Test
The Paired T2 Test
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50. Multivariate Two-sample T2 Test
The Paired T2 Test
Take the difference in the corresponding vectors of measurements within each object between the two conditions
Obtain the sample mean difference vector and sample covariance matrix
Apply one-sample T2 test
2019/2/27
Copyright by Jen-pei Liu, PhD

51. Multivariate Two-sample T2 Test
The Paired T2 Test – Example
Dataset: Measurement of uraic acid (X1) and total cholesterol level (X2) before and after the treatment for six patients
Before After Difference
Pat. No X x X X d d2
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52. Multivariate Two-sample T2 Test
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53. Multivariate Two-sample T2 Test
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54. Copyright by Jen-pei Liu, PhD
Profile Analysis
Forty-nine elderly men were classified into “senile factor present” (SFP, n1=12) and “no senile factor” (NSF, n2=37). The Wechsler Adult Intelligence Scale (WAIS) was administered to all subjects. The WAIS consists of four domains: information, similarity, arithmetic and picture completion. The sample means are given below (Morrison, 2005):
Group NSF SFP
Domain n1 = n2 = 12
information
similarity
arithmetic
picture completion
2019/2/27
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55. Copyright by Jen-pei Liu, PhD
Profile Analysis
2019/2/27
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56. Copyright by Jen-pei Liu, PhD
Profile Analysis
The two-sample Hotelling T2 statistic is > F0.05, 4,44 = 5.18
The 95% simultaneous CI
Domain CI
Information ( 0.12, 7.52)
Similarity ( 0.18, 8.30)
Arithmetic (-0.77, 6.75)
Picture Completion ( 0.56, 5.88)
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Profile Analysis
Assumptions:
The same battery of psychological tests
Measurements of continuous random variables
The responses are commensurable or in comparable units
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58. Copyright by Jen-pei Liu, PhD
Profile Analysis
2019/2/27
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59. Copyright by Jen-pei Liu, PhD
Profile Analysis
Three questions (in terms of priority):
Are the population mean profiles similar, in the sense that the line segments of adjacent tests are parallel? (Test for parallelism: response-by-group interaction)
If the two population profiles are indeed parallel, are they also at the same level? (equal group effects)
Again assuming parallelism, are the population means of the tests different?
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Profile Analysis
Parallelism: (response-by-group interaction)
The slopes of the population profile segments are the same under each condition
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Profile Analysis
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62. Copyright by Jen-pei Liu, PhD
Profile Analysis
2019/2/27
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Profile Analysis
If the null hypothesis of parallelism is not rejected at the 
significance level, we can test the hypothesis of the same level
2019/2/27
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64. Copyright by Jen-pei Liu, PhD
Profile Analysis
If the null hypothesis of parallelism is not rejected at the 
significance level, we can test the hypothesis of the equal response
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65. Copyright by Jen-pei Liu, PhD
Profile Analysis
Example: WAIS Dataset
Hypothesis of parallelism
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66. Copyright by Jen-pei Liu, PhD
Profile Analysis
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67. Copyright by Jen-pei Liu, PhD
Profile Analysis
Example: WAIS Dataset
Hypothesis of the same level
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68. Copyright by Jen-pei Liu, PhD
Profile Analysis
Example: WAIS Dataset
Hypothesis of the equal mean responses
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69. Copyright by Jen-pei Liu, PhD
Profile Analysis
Example: WAIS Dataset
Hypothesis of the equal mean responses
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70. Copyright by Jen-pei Liu, PhD
Assumptions
Multivariate normality
Equal covariance matrices
Independent samples
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71. Homogeneity of Covariance Matrix
Hypothesis of Equal Covariance Matrices of
m p-dimensional multivariate normal
Distributions
Ho: 1 = …= m
2019/2/27
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72. Homogeneity of Covariance Matrix
2019/2/27
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73. Homogeneity of Covariance Matrix
2019/2/27
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74. Homogeneity of Covariance Matrix
Reaction Times
32 male and 32 female normal subjects reacted to visual stimuli preceded by warning intervals of different lengths in 0.5 and 0.15 seconds
2019/2/27
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75. Homogeneity of Covariance Matrix
2019/2/27
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76. Multivariate Normality
Calculate squared Mahalanobis distance (MD2) of each observed vector of p variables from the sample mean vector
Order the MD2 from the largest to the largest, MD2(1) < MD2(2) <…< MD2(n)
For each ordered MD2, compute the
(i-0.5)/n percentile where i is the ith order observed vector
2019/2/27
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77. Multivariate Normality
The 2 values for the percentiles are obtained from the 2 with p d.f., which can be computed by CINV function in SAS
Plot MD2 vs. 2 (similar to Q-Q plot)
The plot should be linear
Deviation from linearity indicates non-normality
2019/2/27
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78. Multivariate Normality
Several tests available, see Seber (1984). However, methods are ad-hoc and are not implemented in most of statistical software
Transformations
Counts – square-root
Proportions – logit
Skewed and positive - logarithm
2019/2/27
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79. Multivariate Outliers
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80. Multivariate Outliers
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81. Multivariate Outliers
Hotelling T2 is invariant under any full- rank linear transformation. The joint distribution of {T2i,i=1,…,n} is independent of parameter  and 
Liu and Weng (1991, SIM) proposed a test procedure to identify multiple multivariate outliers
2019/2/27
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82. Multivariate Outliers
Let T2(1) ,…, T2(n) be the ordered statistics of T21 ,…, T2n and H0(1) ,…, H0(n) be the corresponding sub-hypothesis based on T2(i).
Let {W21 ,…, W2n} be a vector of n Hotelling T2 statistics computed from a sample of size n from a p-dimensional normal with mean 0 and covariance matrix Ip
2019/2/27
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83. Multivariate Outliers
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84. Copyright by Jen-pei Liu, PhD
Summary
Hotelling T2 statistics
One sample
Two independent samples (unpaired)
Paired samples
Confidence regions
Simultaneous confidence interval
Profile analysis
2019/2/27
Copyright by Jen-pei Liu, PhD