1. Discrete Multivariate Analysis
Analysis of Multivariate Categorical Data

2. References
Fienberg, S. (1980), Analysis of Cross-Classified Data , MIT Press, Cambridge, Mass.
Fingelton, B. (1984), Models for Category Counts , Cambridge University Press.
Alan Agresti (1990) Categorical Data Analysis, Wiley, New York.

3. Example 1
In this study we examine n = 1237 individuals measuring X, Systolic Blood Pressure and Y, Serum Cholesterol

4. Example 2
The following data was taken from a study of parole success involving 5587 parolees in Ohio between 1965 and 1972 (a ten percent sample of all parolees during this period).

5. The study involved a dichotomous response Y
Success (no major parole violation) or
Failure (returned to prison either as technical violators or with a new conviction)
based on a one-year follow-up.
The predictors of parole success included are:
type of committed offence (Person offense or Other offense),
Age (25 or Older or Under 25),
Prior Record (No prior sentence or Prior Sentence), and
Drug or Alcohol Dependency (No drug or Alcohol dependency or Drug and/or Alcohol dependency).

6. The data were randomly split into two parts
The data were randomly split into two parts. The counts for each part are displayed in the table, with those for the second part in parentheses.
The second part of the data was set aside for a validation study of the model to be fitted in the first part.

7. Table

8. Multiway Frequency Tables
Two-Way A

9. Three -Way B A C

10. Three -Way C B A

11. four -Way B A C D

12. Analysis of a Two-way Frequency Table:

13. Frequency Distribution (Serum Cholesterol and Systolic Blood Pressure)

14. Joint and Marginal Distributions (Serum Cholesterol and Systolic Blood Pressure)
The Marginal distributions allow you to look at the effect of one variable, ignoring the other. The joint distribution allows you to look at the two variables simultaneously.

15. Conditional Distributions ( Systolic Blood Pressure given Serum Cholesterol )
The conditional distribution allows you to look at the effect of one variable, when the other variable is held fixed or known.

16. Conditional Distributions (Serum Cholesterol given Systolic Blood Pressure)

17. GRAPH: Conditional distributions of Systolic Blood Pressure given Serum Cholesterol

18. Notation:
Let xij denote the frequency (no. of cases) where X (row variable) is i and Y (row variable) is j.

19. Different Models The Multinomial Model:
Here the total number of cases N is fixed and xij follows a multinomial distribution with parameters pij

20. The Product Multinomial Model:
Here the row (or column) totals Ri are fixed and for a given row i, xij follows a multinomial distribution with parameters pj|i

21. The Poisson Model:
In this case we observe over a fixed period of time and all counts in the table (including Row, Column and overall totals) follow a Poisson distribution. Let mij denote the mean of xij.

22. Independence

23. Multinomial Model
if independent
and
The estimated expected frequency in cell (i,j) in the case of independence is:

24. The same can be shown for the other two models – the Product Multinomial model and the Poisson model
namely
The estimated expected frequency in cell (i,j) in the case of independence is:
Standardized residuals are defined for each cell:

25. The Chi-Square Statistic
The Chi-Square test for independence Reject H0: independence if

26. Table Expected frequencies, Observed frequencies, Standardized Residuals
c2 = (p = )

27. Example
In the example N = 57,407 cases in which individuals were victimized twice by crimes were studied.
The crime of the first victimization (X) and the crime of the second victimization (Y) were noted.
The data were tabulated on the following slide

28. Table 1: Frequencies

29. Table 2: Expected Frequencies (assuming independence)

30. Table 3: Standardized residuals

31. Table 3: Conditional distribution of second victimization given the first victimization (%)

32. Log Linear Model

33. Recall, if the two variables, rows (X) and columns (Y) are independent then

34. In general let
then
(1)
where
Equation (1) is called the log-linear model for the frequencies xij.

35. Note: X and Y are independent if
In this case the log-linear model becomes

36. Comment: The log-linear model for a two-way frequency table: is similar to the model for a two factor experiment

37. Three-way Frequency Tables

38. Example
Data from the Framingham Longitudinal Study of Coronary Heart Disease (Cornfield [1962])
Variables
Systolic Blood Pressure (X)
< 127, , , 167+
Serum Cholesterol
<200, , , 260+
Heart Disease
Present, Absent
The data is tabulated on the next slide

39. Three-way Frequency Table

40. Log-Linear model for three-way tables
Let mijk denote the expected frequency in cell (i,j,k) of the table then in general
where

41. Hierarchical Log-linear models for categorical Data
For three way tables
The hierarchical principle:
If an interaction is in the model, also keep lower order interactions and main effects associated with that interaction

42. 1. Model: (All Main effects model)
ln mijk = u + u1(i) + u2(j) + u3(k)
i.e. u12(i,j) = u13(i,k) = u23(j,k) = u123(i,j,k) = 0.
Notation:
[1][2][3]
Description:
Mutual independence between all three variables.

43. 2. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j)
i.e. u13(i,k) = u23(j,k) = u123(i,j,k) = 0.
Notation:
[12][3]
Description:
Independence of Variable 3 with variables 1 and 2.

44. 3. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u13(i,k)
i.e. u12(i,j) = u23(j,k) = u123(i,j,k) = 0.
Notation:
[13][2]
Description:
Independence of Variable 2 with variables 1 and 3.

45. 4. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u23(j,k)
i.e. u12(i,j) = u13(i,k) = u123(i,j,k) = 0.
Notation:
[23][1]
Description:
Independence of Variable 3 with variables 1 and 2.

46. 5. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u13(i,k)
i.e. u23(j,k) = u123(i,j,k) = 0.
Notation:
[12][13]
Description:
Conditional independence between variables 2 and 3 given variable 1.

47. 6. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u23(j,k)
i.e. u13(i,k) = u123(i,j,k) = 0.
Notation:
[12][23]
Description:
Conditional independence between variables 1 and 3 given variable 2.

48. 7. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u13(i,k) + u23(j,k)
i.e. u12(i,j) = u123(i,j,k) = 0.
Notation:
[13][23]
Description:
Conditional independence between variables 1 and 2 given variable 3.

49. 8. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u13(i,k) u23(j,k)
i.e. u123(i,j,k) = 0.
Notation:
[12][13][23]
Description:
Pairwise relations among all three variables, with each two variable interaction unaffected by the value of the third variable.

50. 9. Model: (the saturated model)
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u13(i,k) u23(j,k) + u123(i,j,k)
Notation:
[123]
Description:
No simplifying dependence structure.

51. Hierarchical Log-linear models for 3 way table
Description
[1][2][3]
Mutual independence between all three variables.
[1][23]
Independence of Variable 1 with variables 2 and 3.
[2][13]
Independence of Variable 2 with variables 1 and 3.
[3][12]
Independence of Variable 3 with variables 1 and 2.
[12][13]
Conditional independence between variables 2 and 3 given variable 1.
[12][23]
Conditional independence between variables 1 and 3 given variable 2.
[13][23]
Conditional independence between variables 1 and 2 given variable 3.
[12][13] [23]
Pairwise relations among all three variables, with each two variable interaction unaffected by the value of the third variable.
[123]
The saturated model

52. Maximum Likelihood Estimation
Log-Linear Model

53. For any Model it is possible to determine the maximum Likelihood Estimators of the parameters
Example
Two-way table – independence – multinomial model or

54. Log-likelihood
where
With the model of independence

55. and
with
also

56. Let
Now

57. Since

58. Now or

59. Hence
and
Similarly
Finally

60. Hence
Now
and

61. Hence
Note or

62. Comments
Maximum Likelihood estimates can be computed for any hierarchical log linear model (i.e. more than 2 variables)
In certain situations the equations need to be solved numerically
For the saturated model (all interactions and main effects), the estimate of mijk… is xijk… .

63. Discrete Multivariate Analysis
Analysis of Multivariate Categorical Data

64. Multiway Frequency Tables
Two-Way A

65. four -Way B A C D

66. Log Linear Model

67. Two- way table
where
The multiplicative form:

68. Log-Linear model for three-way tables
Let mijk denote the expected frequency in cell (i,j,k) of the table then in general
where

69. Log-Linear model for three-way tables
Let mijk denote the expected frequency in cell (i,j,k) of the table then in general
or the multiplicative form

70. Comments
The log-linear model is similar to the ANOVA models for factorial experiments.
The ANOVA models are used to understand the effects of categorical independent variables (factors) on a continuous dependent variable (Y).
The log-linear model is used to understand dependence amongst categorical variables
The presence of interactions indicate dependence between the variables present in the interactions

71. Hierarchical Log-linear models for categorical Data
For three way tables
The hierarchical principle:
If an interaction is in the model, also keep lower order interactions and main effects associated with that interaction

72. 1. Model: (All Main effects model)
ln mijk = u + u1(i) + u2(j) + u3(k)
i.e. u12(i,j) = u13(i,k) = u23(j,k) = u123(i,j,k) = 0.
Notation:
[1][2][3]
Description:
Mutual independence between all three variables.

73. 2. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j)
i.e. u13(i,k) = u23(j,k) = u123(i,j,k) = 0.
Notation:
[12][3]
Description:
Independence of Variable 3 with variables 1 and 2.

74. 3. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u13(i,k)
i.e. u12(i,j) = u23(j,k) = u123(i,j,k) = 0.
Notation:
[13][2]
Description:
Independence of Variable 2 with variables 1 and 3.

75. 4. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u23(j,k)
i.e. u12(i,j) = u13(i,k) = u123(i,j,k) = 0.
Notation:
[23][1]
Description:
Independence of Variable 3 with variables 1 and 2.

76. 5. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u13(i,k)
i.e. u23(j,k) = u123(i,j,k) = 0.
Notation:
[12][13]
Description:
Conditional independence between variables 2 and 3 given variable 1.

77. 6. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u23(j,k)
i.e. u13(i,k) = u123(i,j,k) = 0.
Notation:
[12][23]
Description:
Conditional independence between variables 1 and 3 given variable 2.

78. 7. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u13(i,k) + u23(j,k)
i.e. u12(i,j) = u123(i,j,k) = 0.
Notation:
[13][23]
Description:
Conditional independence between variables 1 and 2 given variable 3.

79. 8. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u13(i,k) u23(j,k)
i.e. u123(i,j,k) = 0.
Notation:
[12][13][23]
Description:
Pairwise relations among all three variables, with each two variable interaction unaffected by the value of the third variable.

80. 9. Model: (the saturated model)
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u13(i,k) u23(j,k) + u123(i,j,k)
Notation:
[123]
Description:
No simplifying dependence structure.

81. Hierarchical Log-linear models for 3 way table
Description
[1][2][3]
Mutual independence between all three variables.
[1][23]
Independence of Variable 1 with variables 2 and 3.
[2][13]
Independence of Variable 2 with variables 1 and 3.
[3][12]
Independence of Variable 3 with variables 1 and 2.
[12][13]
Conditional independence between variables 2 and 3 given variable 1.
[12][23]
Conditional independence between variables 1 and 3 given variable 2.
[13][23]
Conditional independence between variables 1 and 2 given variable 3.
[12][13] [23]
Pairwise relations among all three variables, with each two variable interaction unaffected by the value of the third variable.
[123]
The saturated model

82. Goodness of Fit Statistics
These statistics can be used to check if a log-linear model will fit the observed frequency table

83. Goodness of Fit Statistics
The Chi-squared statistic
The Likelihood Ratio statistic:
d.f. = # cells - # parameters fitted
We reject the model if c2 or G2 is greater than

84. Example: Variables
Systolic Blood Pressure (B) Serum Cholesterol (C) Coronary Heart Disease (H)

85. Goodness of fit testing of Models
MODEL DF LIKELIHOOD- PROB PEARSON PROB RATIO CHISQ CHISQ
B,C,H
B,CH
C,BH
H,BC
BC,BH
BH,CH n.s.
CH,BC
BC,BH,CH n.s.
Possible Models: 1. [BH][CH] – B and C independent given H [BC][BH][CH] – all two factor interaction model

86. Model 1: [BH][CH] Log-linear parameters
Heart disease -Blood Pressure Interaction

87. Multiplicative effect
Log-Linear Model

88. Heart Disease - Cholesterol Interaction

89. Multiplicative effect

90. Model 2: [BC][BH][CH] Log-linear parameters
Blood pressure-Cholesterol interaction:

91. Multiplicative effect

92. Heart disease -Blood Pressure Interaction

93. Multiplicative effect

94. Heart Disease - Cholesterol Interaction

95. Multiplicative effect

96. Another Example In this study it was determined for N = 4353 males
Occupation category
Educational Level
Academic Aptidude

97. Occupation categories
Self-employed Business
Teacher\Education
Self-employed Professional
Salaried Employed
Education levels
Low
Low/Med
Med
High/Med
High

98. Academic Aptitude
Low
Low/Med
High/Med
High

99. Self-employed, Business Teacher
Education Education
Aptitude Low LMed HMed High Total Aptitude Low LMed HMed High Total
Low Low
LMed LMed
Med Med
HMed HMed
High High
Total Total
Self-employed, Professional Salaried Employed
Low Low
LMed LMed
Med Med
HMed HMed
High High
Total Total

101. This is similar to looking at all the bivariate correlations
It is common to handle a Multiway table by testing for independence in all two way tables.
This is similar to looking at all the bivariate correlations
In this example we learn that:
Education is related to Aptitude
Education is related to Occupational category
Can we do better than this?

102. Fitting various log-linear models
Simplest model that fits is: [Apt,Ed][Occ,Ed]
This model implies conditional independence between Aptitude and Occupation given Education.

103. Log-linear Parameters
Aptitude – Education Interaction

104. Aptitude – Education Interaction (Multiplicative)

105. Occupation – Education Interaction

106. Occupation – Education Interaction (Multiplicative)

107. Conditional Test Statistics

108. Suppose that we are considering two Log-linear models and that Model 2 is a special case of Model 1.
That is the parameters of Model 2 are a subset of the parameters of Model 1.
Also assume that Model 1 has been shown to adequately fit the data.

109. In this case one is interested in testing if the differences in the expected frequencies between Model 1 and Model 2 is simply due to random variation]
The likelihood ratio chi-square statistic that achieves this goal is:

110. Example

111. Goodness of Fit test for the all k-factor models
Conditional tests for zero k-factor interactions

112. Conclusions
The four factor interaction is not significant G2(3|4) = 0.7 (p = 0.705)
The all three factor model provides a significant fit G2(3) = 0.7 (p = 0.705)
All the three factor interactions are not significantly different from 0, G2(2|3) = 9.2 (p = 0.239).
The all two factor model provides a significant fit G2(2) = 9.9 (p = 0.359)
There are significant 2 factor interactions G2(1|2) = 33.0 (p =
Conclude that the model should contain main effects and some two-factor interactions

113. There also may be a natural sequence of progressively complicated models that one might want to identify.
In the laundry detergent example the variables are:
Softness of Laundry Used
Previous use of Brand M
Temperature of laundry water used
Preference of brand X over brand M

114. A natural order for increasingly complex models which should be considered might be:
[1][2][3][4]
[1][3][24]
[1][34][24]
[13][34][24]
[13][234]
[134][234]
The all-Main effects model Independence amongst all four variables
Since previous use of brand M may be highly related to preference for brand M, add first the 2-4 interaction
Brand M is recommended for hot water add 2nd the 3-4 interaction
brand M is also recommended for Soft laundry add 3rd the 1-3 interaction
Add finally some possible 3-factor interactions

115. Likelihood Ratio G2 for various models
d]f] G2
[1][3][24] 17
22.4
[1][24][34] 16 18
[13][24][34] 14
11.9
[13][23][24][34] 13
11.2
[12][13][23][24][34] 11
10.1
[1][234]
14.5
[134][24] 10
12.2
[13][234] 12
8.4
[24][34][123] 9
[123][234] 8
5.6

117. Discrete Multivariate Analysis
Analysis of Multivariate Categorical Data

118. Log-Linear model for three-way tables
Let mijk denote the expected frequency in cell (i,j,k) of the table then in general
where

119. Hierarchical Log-linear models for categorical Data
For three way tables
The hierarchical principle:
If an interaction is in the model, also keep lower order interactions and main effects associated with that interaction

120. Models for three-way tables

121. 1. Model: (All Main effects model)
ln mijk = u + u1(i) + u2(j) + u3(k)
i.e. u12(i,j) = u13(i,k) = u23(j,k) = u123(i,j,k) = 0.
Notation:
[1][2][3]
Description:
Mutual independence between all three variables.
Comment: For any model the parameters (u, u1(i) , u2(j) , u3(k)) can be estimated in addition to the expected frequencies (mijk) in each cell

122. 2. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j)
i.e. u13(i,k) = u23(j,k) = u123(i,j,k) = 0.
Notation:
[12][3]
Description:
Independence of Variable 3 with variables 1 and 2.

123. 3. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u13(i,k)
i.e. u12(i,j) = u23(j,k) = u123(i,j,k) = 0.
Notation:
[13][2]
Description:
Independence of Variable 2 with variables 1 and 3.

124. 4. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u23(j,k)
i.e. u12(i,j) = u13(i,k) = u123(i,j,k) = 0.
Notation:
[23][1]
Description:
Independence of Variable 3 with variables 1 and 2.

125. 5. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u13(i,k)
i.e. u23(j,k) = u123(i,j,k) = 0.
Notation:
[12][13]
Description:
Conditional independence between variables 2 and 3 given variable 1.

126. 6. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u23(j,k)
i.e. u13(i,k) = u123(i,j,k) = 0.
Notation:
[12][23]
Description:
Conditional independence between variables 1 and 3 given variable 2.

127. 7. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u13(i,k) + u23(j,k)
i.e. u12(i,j) = u123(i,j,k) = 0.
Notation:
[13][23]
Description:
Conditional independence between variables 1 and 2 given variable 3.

128. 8. Model:
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u13(i,k) u23(j,k)
i.e. u123(i,j,k) = 0.
Notation:
[12][13][23]
Description:
Pairwise relations among all three variables, with each two variable interaction unaffected by the value of the third variable.

129. 9. Model: (the saturated model)
ln mijk = u + u1(i) + u2(j) + u3(k) + u12(i,j) + u13(i,k) u23(j,k) + u123(i,j,k)
Notation:
[123]
Description:
No simplifying dependence structure.

130. Goodness of Fit Statistics
The Chi-squared statistic
The Likelihood Ratio statistic:
d.f. = # cells - # parameters fitted
We reject the model if c2 or G2 is greater than

131. Conditional Test Statistics

132. In this case one is interested in testing if the differences in the expected frequencies between Model 1 and Model 2 is simply due to random variation]
The likelihood ratio chi-square statistic that achieves this goal is:

133. Stepwise selection procedures
Forward Selection
Backward Elimination

134. Forward Selection:
Starting with a model that under fits the data, log-linear parameters that are not in the model are added step by step until a model that does fit is achieved.
At each step the log-linear parameter that is most significant is added to the model:
To determine the significance of a parameter added we use the statistic:
G2(2|1) = G2(2) – G2(1)
Model 1 contains the parameter.
Model 2 does not contain the parameter

135. Backward Elimination:
Starting with a model that over fits the data, log-linear parameters that are in the model are deleted step by step until a model that continues to fit the model and has the smallest number of significant parameters is achieved.
At each step the log-linear parameter that is least significant is deleted from the model:
To determine the significance of a parameter deleted we use the statistic:
G2(2|1) = G2(2) – G2(1)
Model 1 contains the parameter.
Model 2 does not contain the parameter

137. K = knowledge
N = Newspaper
R = Radio
S = Reading
L = Lectures

140. Continuing after 10 steps

141. The final step

142. The best model was found a the previous step
[LN][KLS][KR][KN][LR][NR][NS]

143. Modelling of response variables
Independent → Dependent

144. Logit Models
To date we have not worried whether any of the variables were dependent of independent variables.
The logit model is used when we have a single binary dependent variable.

146. The variables Type of seedling (T) Depth of planting (D)
Longleaf seedling
Slash seedling
Depth of planting (D)
Too low.
Too high
Mortality (M) (the dependent variable)
Dead
Alive

147. The Log-linear Model Note: mij1 = # dead when T = i and D = j.
mij2 = # alive when T = i and D = j.
= mortality ratio when T = i and D = j.

148. Hence
since

149. The logit model:
where

150. Thus corresponding to a loglinear model there is logit model predicting log ratio of expected frequencies of the two categories of the independent variable.
Also k +1 factor interactions with the dependent variable in the loglinear model determine k factor interactions in the logit model
k + 1 = constant term in logit model
k + 1 = 2, main effects in logit model

151. 1 = Depth, 2 = Mort, 3 = Type

152. Log-Linear parameters for Model: [TM][TD][DM]

153. Logit Model for predicting the Mortality

155. The best model was found by forward selection was
[LN][KLS][KR][KN][LR][NR][NS]
To fit a logit model to predict K (Knowledge) we need to fit a loglinear model with important interactions with K (knowledge), namely
[LNRS][KLS][KR][KN]
The logit model will contain
Main effects for L (Lectures), N (Newspapers), R (Radio), and S (Reading)
Two factor interaction effect for L and S

156. The Logit Parameters for the Model : LNSR, KLS, KR, KN
( Multiplicative effects are given in brackets, Logit Parameters = 2 Loglinear parameters)
The Constant term:
(0.798)
The Main effects on Knowledge:
Lectures Lect (1.307)
None (0.765)
Newspaper News (1.383)
None (0.723)
Reading Solid (1.405)
Not (0.712)
Radio Radio (1.162)
None (0.861)
The Two-factor interaction Effect of Reading and Lectures on Knowledge

157. Fitting a Logit Model with a Polytomous Response Variable

158. Example:
NA – Not available

159. The variables Race – white, black Age - < 22, ≥ 22
Father’s education – GS, some HS, HS grad, NA
Respondents Education - GS, some HS, HS grad – the response (dependent) variable

161. Techniques for handling Polytomous Response Variable Approaches
Consider the categories 2 at a time. Do this for all possible pairs of the categories.
Look at the continuation ratios
1 vs 2
1,2 vs 3
1,2,3 vs 4
etc

165. Causal or Path Analysis for Categorical Data

166. When the data is continuous, a causal pattern may be assumed to exist amongst the variables.
The path diagram
This is a diagram summarizing causal relationships.
Straight arrows are drawn between a variable that has some cause and effect on another variable X Y
Curved double sided arrows are drawn between variables that are simply correlated Y X

167. Job Stress Smoking Example 1
The variables – Job stress, Smoking, Heart Disease
The path diagram
Job Stress
Smoking
Heart Disease
In Path Analysis for continuous variables, one is interested in determining the contribution along each path (the path coefficents)

168. Job Stress Smoking Drinking Example 2
The variables – Job stress, Alcoholic Drinking, Smoking, Heart Disease
The path diagram
Job Stress
Smoking
Drinking
Heart Disease

169. In analysis of categorical data there are no path coefficients but path diagrams can point to the appropriate logit analysis
Example
In this example the data consists of a two wave, two variable panel data for a sample of n =3398 schoolboys.
It is looking at “membership” and “attitude towards” the leading crowd.

170. A B C D The path diagram: This suggest predicting B from A, then
C from A and B and finally
D from A, B and C.

174. Example 2 In this example we are looking at
Social Economic Status (SES)
Sex IQ
Parental Encouragement for Higher Education (PE)
College Plans(CP)

176. The Path Diagram
Sex
SES IQ PE CP

177. The path diagram suggests
Predicting Parental Encouragement from Sex, SocioEconomic status, and IQ, then
Predicting College Plans from Parental Encouragement, Sex, SocioEconomic status, and IQ.

179. Logit Parameters: Model [ABC][ABD][ACD][BCD]

180. Two factor Interactions

183. Logit Parameters for Predicting College Plans Using Model 9:
Logit Parameters for Predicting College Plans Using Model 9: [ABCD][BCE][AE][DE]