Transcript Mixture modelling of continuous variables

Mixture modelling of continuous
variables
Mixture modelling
• So far we have dealt with mixture modelling for a
selection of binary or ordinal variables collected
at a single time point (c/s) or longitudinally
across time
• The simplest example of a mixture model
consists of a single continuous manifest variable
• The multivariate extension to this simple model
is known as Latent Profile Analysis
Single continuous variable
• An underlying latent grouping might present
itself as a multi-modal distribution for the
continuous variable
Height
Single continuous variable
• An underlying latent grouping might present
itself as a multi-modal distribution for the
continuous variable
Females
Height
Single continuous variable
• An underlying latent grouping might present
itself as a multi-modal distribution for the
continuous variable
Males
Height
Single continuous variable
• But the distance between modes may be small
or even non-existent
• Depends on the variation in the item being
measured and also the sample in which the
measurement is taken (e.g. clinical or general
population)
Single continuous variable
Figure taken from: Muthén, B. (2001). Latent variable mixture modeling. In G. A.
Marcoulides & R. E. Schumacker (eds.), New Developments and Techniques in
Structural Equation Modeling (pp. 1-33). Lawrence Erlbaum Associates.
Single continuous variable
Figure taken from: Muthén, B. (2001). Latent variable mixture modeling. In G. A.
Marcoulides & R. E. Schumacker (eds.), New Developments and Techniques in
Structural Equation Modeling (pp. 1-33). Lawrence Erlbaum Associates.
Single continuous variable
• We assume that the manifest variable is
normally distributed within each latent class
GHQ Example
Data:
File is "ego_ghq12_id.dta.dat" ;
Define:
sumodd = ghq01 +ghq03 +ghq05 +ghq07 +ghq09 +ghq11;
sumeven = ghq02 +ghq04 +ghq06 +ghq08 +ghq10 +ghq12;
ghq_sum = sumodd + sumeven;
Variable:
Names are
ghq01 ghq02 ghq03 ghq04 ghq05 ghq06
ghq07 ghq08 ghq09 ghq10 ghq11 ghq12
f1 id;
Missing are all (-9999) ;
usevariables = ghq_sum;
Analysis:
Type = basic ;
plot:
type is plot3;
Here we derive a single
sum-score from the 12
ordinal GHQ items
The syntax shows that
variables can be created in
the define statement which
are not then used in the
final model
Examine the distribution of the scale
Scale appears unimodal, although there is a long upper-tail
Examine the distribution of the scale
By changing from the default number of bins we see secondary modes appearing
Fit a 2-class mixture
Variable:
<snip>
classes = c(2);
Analysis:
type = mixture ;
proc = 2 (starts);
starts = 100 20;
stiterations = 20;
stscale = 15;
model:
%overall%
%c#1%
[ghq_sum];
ghq_sum
(equal_var);
%c#2%
[ghq_sum];
ghq_sum
(equal_var);
Fit a 2-class mixture
Variable:
<snip>
classes = c(2);
Analysis:
type = mixture ;
proc = 2 (starts);
starts = 100 20;
stiterations = 20;
stscale = 15;
This funny set of symbols refers to the first
class
model:
%overall%
Means are referred to using square
brackets.
%c#1%
[ghq_sum];
ghq_sum
%c#2%
[ghq_sum];
ghq_sum
Variances are bracket-less.
(equal_var);
(equal_var);
Here we have constrained the variances
to be equal between classes by having the
same bit of text in brackets at the end of
the two variance lines
Means will be freely estimated.
Model results
TESTS OF MODEL FIT
Loglikelihood
H0 Value
H0 Scaling Correction Factor
for MLR
-3624.960
1.078
Information Criteria
Number of Free Parameters
Akaike (AIC)
Bayesian (BIC)
Sample-Size Adjusted BIC
(n* = (n + 2) / 24)
4
7257.921
7278.002
7265.297
Model results
FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES
BASED ON THE ESTIMATED MODEL
Latent classes
1
200.36980
2
918.63020
CLASSIFICATION QUALITY
Entropy
0.17906
0.82094
0.829
A smaller class of 18% has
emerged, consistent with the
expected behaviour of the GHQ
in this sample from primary care
Entropy is high
CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP
Latent classes
1
2
189
930
0.16890
0.83110
Average Latent Class Probabilities for Most Likely Latent Class Membership (Row)
by Latent Class (Column)
1
2
1
0.889
0.035
2
0.111
0.965
Class-specific entropies are both good
Model results
Two-Tailed
P-Value
Estimate
S.E.
Est./S.E.
Means
GHQ_SUM
37.131
0.574
64.737
0.000
Variances
GHQ_SUM
18.876
1.016
18.581
0.000
Latent Class 1
Latent Class 2
Huge separation in means
since SD = 4.3 (i.e. sqrt(18.88))
Means
GHQ_SUM
23.618
0.202
117.046
0.000
Variances
GHQ_SUM
18.876
1.016
18.581
0.000
0.118
-12.947
0.000
Categorical Latent Variables
Means
C#1
-1.523
Examine within-class distributions
Examine within-class distributions
What have we done?
• We have effectively done a t-test backwards.
• Rather than obtaining a manifest binary variable,
assuming equality of variances and testing for
equality of means
• we have derived a latent binary variable based
on the assumption of a difference in means (still
with equal variance)
What next?
• The bulk of the sample now falls into a class with a
GHQ distribution which is more symmetric than the
sample as a whole
• There appear to be additional modes within the
smaller class
• The ‘optimal’ number of classes can be assessed in
the usual way using aBIC, entropy and the bootstrap
LRT.
• In the univariate case, residual correlations are not
an issue, but when moving to a multivariate
example, these too will need to be assessed.
What next?
• As before, posterior probabilities can be
exported and modelled in a weighted regression
analysis
• A logistic regression analysis using a latent
binary variable derived from the data is likely to
be far more informative than a linear-regression
analysis using the manifest continuous variable
What if we had not constrained the variances?
Variable:
<snip>
classes = c(2);
Analysis:
type = mixture ;
proc = 2 (starts);
starts = 100 20;
stiterations = 20;
stscale = 15;
Commented out
model:
%overall%
%c#1%
[ghq_sum];
ghq_sum
; ! (equal_var);
%c#2%
[ghq_sum];
ghq_sum
; ! (equal_var);
TESTS OF MODEL FIT
Loglikelihood
H0 Value
H0 Scaling Correction Factor
for MLR
Information Criteria
Number of Free Parameters
Akaike (AIC)
Bayesian (BIC)
Sample-Size Adjusted BIC
-3610.586
0.932
5
7231.172
7256.273
7240.392
BIC is lower!
FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES
BASED ON THE ESTIMATED MODEL
Latent
Classes
1
2
456.72641
662.27359
0.40816
0.59184
CLASSIFICATION QUALITY
Entropy
0.479
Classes are
more equal
Entropy is
poor
Means are closer, variances differ greatly
MODEL RESULTS
Two-Tailed
P-Value
Estimate
S.E.
Est./S.E.
Means
GHQ_SUM
31.493
0.816
38.616
0.000
Variances
GHQ_SUM
45.570
2.439
18.686
0.000
Means
GHQ_SUM
22.275
0.328
67.823
0.000
Variances
GHQ_SUM
11.144
1.605
6.943
0.000
Latent Class 1
Latent Class 2
Distribution far from symmetric
Distribution far from symmetric