A simulation study of Pathmox with nonnormal data

1
A simulation study of Pathmox with non-normal
data

• Gastón Sánchez, Tomàs Aluja-Banet

based on Ph.D. Gaston Sánchez. PATHMOX approach
Segementation tress in PLS-PM
Laboratory of Information Analysis and Modeling
Universitat Politècnica de Catalunya
2
Outline

• Heterogeneity in PLS-PM
• PATHMOX Approach
• Simulation Studies
• Conclusions

3
Heterogeneity
Standard application of PLS Path Model One model
for the whole population
We assume the same model for all individuals
(i.e. we are assuming that the satisfaction
model for mobile phone operators Is the same
regardless the individual is a teenager or an
adult)
4
Heterogeneity
Different sets of individuals with particular
behavior
5
Assignable sources of heterogeneity
Heterogeneity with observed segmentation variables
Segments defined by observable segmentation
variables Group Information (age, gender,
ethnicity, religion, etc) Number of segments
based on the levels (categories) of segmentation
variables Multi-group approaches
Heterogeneity without segmentation variables
Unknown variables causing heterogeneity No Group
Information Unknown number of segments Cluster-bas
ed / Latent Class approaches
6
Heterogeneity in PATHMOX
Where the data come from?
Socio-demographic
Segmentation Variables
Survey data
Psycho-demographic
Consumption
It is interesting to detect the segments with
different PLS-PM models
We are searching for groups of individuals where
the relation between image and satisfaction (for
instance) is different for one group to
another. We perform this search to define the
different groups using the external segmentation
variables.
7
Heterogeneity in PATHMOX
How to use the segmentation variables?
For each segmentation variable, we can define
binary partitions
Global Model
X11
?1
X1k
Y31
h3
Y32
X21
Y3p
?2
X2q
Z1 Zk Zm Segmentation variables
8
The PATHMOX Approach
Segmentation Tree of PLS Path Models
Root Node Global Model
Child Node
MOX gtMOXEXELOA From Nahuatl (Aztec language)
which means  Divide into groups 
Leaf Node
9
Split criterion
Parent Model
A B
Child Models
A
B
10
Hypothesis test
Equality of Coefficients
BA1
BB1
B1
VS
BA2
BB2
B2
A B
Under H0
Under H1
E(z1) N(0, s2In) , E(z2 ) N(0, s2In)
Assuming a common variance
F-statistic for multiple regression (Lebart et
al, 1979)
F distribution with p1p2, 2n-2(p1p2) d.f.
Best split with most significant F-statistic
11
Stopping criterion
1. Minimum number of elements inside one node
2. p-value gt threshold
p value gt a
3. Specifying a growth depth-level
12
Simulation studies
Sensitivity of the F-test respect to

• Skewness of data
• Distance between path coefficients
• Sample Sizes
• Levels of noise of the endogenous term
• Levels of noise of indicators
• Unbalanced segments
• Variances of endogenous residuals

13
Experimental conditions
Node A (fixed inner model)
Node B
l11
l11
x11
e11
x11
e11
l12
x1
l12
z
z
x12
e12
x1
x12
e12
x31
e31
l31
b31
x13
e13
x31
e31
l31
l13
x13
e13
l13
b31
l32
x3
l32
x3
x32
e32
x32
e32
l33
l33
b32
l21
x21
e21
b32
x21
l21
e21
x33
e33
x33
e33
l22
x2
l22
x22
e22
x2
x22
e22
l23
x23
e23
l23
x23
e23
Simulation according to the following
experimental conditions

• Data Distributions Normal (Sanchez Aluja
  PLS07), Non-normal
• Path Coefficients
• Sample size 100, 200, 500, 1000
• Balancing proportions 60, 70, 80, 90
• Noise levels of z 10, 30, 50
• Noise levels of e 10, 30, 50

14
Path coefficients
9 Pairs of path coefficients
Gradually changed values
Fixed values
15
Data distributions
Examples of Distributions
Beta (6,6)
Beta (9,4)
Beta (9,1)
16
Symmetric distribution b (6,6)
17
Moderate skew distribution b (9,4)
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High skew distribution b (9,1)
19
Global results
20
Influence of b distance by distribution
21
Influence of sample size by distribution
22
Influence of noise of LVs
23
Influence of noise of MVs
24
Unbalanced Segments (normal data)
25
Unbalanced Segments b (9,4)
26
Influence of different variances
Different Variances of endogenous error terms
x1
x1
0.7
0.7
h2
h2
0.8
z2
0.8
z2
h1
h1
z1
z1
0.5
0.5
E(z1) N(0, s12In) , E(z2 ) N(0, s22In)
Four types of different variances
27
Influence of different variances
28
Conclusions

• Non normality of distributions doesnt affect the
  results of the test statistic
• Splits with unbalanced children nodes delivers
  less sensitive p-values of the statistic.
  F-statistic favors balanced splits.
• Unequal variances of endogenous latent variables
  render less reliable the test statistic and hence
  the tree.
• The F test is used to discover unexpected
  segments by ordering the splits for a given node,
  as a data mining tool.

29
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