Statistical Modeling

1
Statistical Modeling

• Matthew Dirk Wiers

2
Purpose

• Statistical modeling is a mathematical technique
  used to verify and quantify associations between
  one or more quantitative and/or qualitative
  predictor variables (x1, x2, ), and a single
  quantitative or qualitative response variable
  (y), or multiple multivariate normal response
  variables (y1, y2, ). E.g., the association
  between income (x1), whether or not someone at
  home cooks (x2), and the number of dinners in the
  last k eaten outside the home (y).

3
Components

• Probability Model f (y, ?)
• Discrete Bernoulli, Binomial, Poisson,
  Multinomial
• Continuous Normal, Weibull, Multivariate
  Normal
• Linear Model ß0 ß1x1i ß2x2i
• Link ?i g (ß0 ß1x1i ß2x2i )

4
Components

• Maximum Likelihood Estimation
• Likelihood Ratio Tests

5
Probability Models

• Suppose there is a 6 week experiment with 15
  animals in treatment group A and 15 animals in
  treatment group B. Consider the following
  measurements on each animal
• Whether or not there were malignant tumors.
• The number of tumors that were malignant.
• The number of tumors.
• The average size of the tumors.
• The time to the first tumor.
• The number of tumors that were malignant,
  benign, or other.
• The average size and average weight of the
  tumors.
• The corresponding probability models are
  Bernoulli, Binomial, Poisson, Normal, Weibull,
  Multinomial, and Multivariate Normal.

6
Statistical Modeling

• Bernoulli Modeling
• Binomial Modeling
• Binomial Probit Modeling
• Binomial C-Log-Log Modeling
• Poisson Modeling
• Poisson Rate Modeling
• Multinomial Modeling
• Multinomial Ordinal Modeling

7
Statistical Modeling

• Normal Modeling
• Weibull Modeling
• Weibull Censor Modeling
• Multivariate Normal Modeling
• Multivariate Normal RM Modeling

8
Bernoulli Modeling

• Probability Model

9
Bernoulli Modeling

• Link

10
Bernoulli Modeling

• NLL

11
Bernoulli Modeling
12
Bernoulli Modeling
13
Bernoulli Modeling
14
Bernoulli Modeling
15
Bernoulli Modeling
16
Binomial Modeling

• Probability Model

17
Binomial Modeling

• Link

18
Binomial Modeling

• NLL

19
Binomial Modeling
20
Binomial Modeling
21
Binomial Modeling
22
Binomial Modeling
23
Binomial Probit Modeling

• Link
• NLL

24
Binomial Probit Modeling
25
Binomial Probit Modeling
26
Binomial Probit Modeling
27
Binomial C-Log-Log Modeling

• Link
• NLL

28
Binomial C-Log-Log Modeling
29
Binomial C-Log-Log Modeling
30
Binomial C-Log-Log Modeling
31
Poisson Modeling

• Probability Model

32
Poisson Modeling

• Link
• NLL

33
Poisson Modeling
34
Poisson Modeling
35
Poisson Modeling
36
Poisson Modeling
37
Poisson Modeling
38
Poisson Modeling
39
Poisson Modeling
40
Poisson Modeling
41
Poisson Modeling
42
Poisson Rate Modeling

• Link
• NLL

43
Poisson Rate Modeling
44
Poisson Rate Modeling
45
Poisson Rate Modeling
46
Poisson Rate Modeling
47
Poisson Rate Modeling
48
Poisson Rate Modeling
49
Poisson Rate Modeling
50
Poisson Rate Modeling
51
Multinomial Modeling

• Probability Model

52
Multinomial Modeling

• Linear Model
• Link

53
Multinomial Modeling

• Link

54
Multinomial Modeling

• NLL

55
Multinomial Modeling
56
Multinomial Modeling
57
Multinomial Modeling
58
Multinomial Modeling
59
Multinomial Ordinal Modeling

• Linear Model
• Link

60
Multinomial Ordinal Modeling

• Link

61
Multinomial Ordinal Modeling

• NLL

62
Multinomial Ordinal Modeling
63
Multinomial Ordinal Modeling
64
Multinomial Ordinal Modeling
65
Multinomial Ordinal Modeling
66
Normal Modeling

• Probability Model

67
Normal Modeling

• Link
• NLL
• LRS

68
Normal Modeling
69
Normal Modeling
70
Normal Modeling
71
Normal Modeling
72
Weibull Modeling

• Probability Model

73
Weibull Modeling

• Link
• NLL

74
Weibull Modeling
75
Weibull Modeling
76
Weibull Modeling
77
Weibull Modeling
78
Weibull Censor Modeling

• NLL

79
Weibull Censor Modeling
80
Weibull Censor Modeling
81
Weibull Censor Modeling
82
Weibull Censor Modeling
83
Weibull Censor Modeling
84
Multivariate Normal Modeling

• Probability Model

85
Multivariate Normal Modeling

• Linear Model Link

86
Multivariate Normal Modeling

• NLL
• LRS

87
Multivariate Normal Modeling
88
Multivariate Normal Modeling
89
Multivariate Normal Modeling
90
Multivariate Normal Modeling
91
Multivariate Normal RM Modeling
92
Multivariate Normal RM Modeling
93
Multivariate Normal RM Modeling
94
Multivariate Normal RM Modeling
95
Multivariate Normal RM Modeling
96
Multivariate Normal RM Modeling
97
Multivariate Normal RM Modeling
98
Multivariate Normal RM Modeling
99
Multivariate Normal RM Modeling
100
References

• Statistics for Business and Economics (9th
  Edition) by Andersen, Sweeney, and Williams (ISBN
  0-324-20082-X).
• Applied Linear Statistical Models (4th Edition)
  by Neter, Kutner, Nachtsheim, and Wasserman (ISBN
  0-256-11736-5).
• Applied Multiple Regression/Correlation Analysis
  (3rd Edition) by Cohen, Cohen, West, and Aiken
  (ISBN 0-8058-2223-2).
• An Introduction to Categorical Data Analysis by
  Agresti (ISBN 0-471-11338-7).
• Categorical Data Analysis (2nd Edition) by
  Agresti (ISBN 0-471-36093-7).
• Statistical Models and Methods for Lifetime Data
  (2nd Edition) by Lawless (ISBN 0-471-37215-3).
• Applied Multivariate Statistical Analysis (5th
  Edition) by Johnson and Wichern (ISBN
  0-13-092553-5).
• Generalized Linear Models (2nd Edition) by
  McCullagh and Nelder (ISBN 0-412-31760-5).