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Measurement model 1
Measurement model 2
Structural model
Ambient air
temperature
Thermometer
reading

Metabolic rate
of animal
Thermocouple
reading

Rate of CO2
increase in
chamber

Rate of O2
decrease in
chamber

Heat production
of animal

L
X1

Z=X+Y
X=3
What is the value
of z?
L=N(0,)
=N(0,)
X1 = aL + 
i.e. 3 free
parameters
BUT...
Z = 3 if Y = 0
Z=6 if Y = 3
and so on for
an infinite series
of possibilities
The equation is underidentified
Only 1 element in the
covariance matrix (S11)
The system is underidentified
It is impossible to find single values for the
three free parameters given only S11
L
X1
1
Fixing the scale of a
latent variable
L
X2
2
L=N(0, ) 1=N(0,)
2=N(0,)
X1=a1L+ 1 X2=a2L+ 2
5 free parameters
3 observed values
(S11, S12, S22)
X1
1
X2
X3
2
3
L=N(0, ) 1=N(0,)
2=N(0,) 3=N(0,)
X1=a1L+ 1 X2=a2L+ 2
X3=a3L+ 3
7 free parameters
6 observed values
(S11, S12, S13, S22, S23, S33)
1. Chose an observed
variable with the same scale
and fix the path from the latent
to the variable to 1
(therefore 1 less free parameter
to estimate).
2. Fix the variance of the latent
to a constant (eg. 1) - this fixes
its scale to SD units.
A
B
C
L
L
L
*
*
*
*
*
x1
x2
x1
x2
*
*
*
*
D
x1
*
*
x2
*
x3
*
E
*
F
*
L
L
*
x1
*
*
x2
*
*
*
*
x3
*
x1
*
* *
x2
x3
x4
*
*
*
*
*
L1
*
L2
*
*
*
x4
*
x1
x2
x3
x4
*
*
*
*
Model identification becomes quite complicated as the number of latent variables
increases.
One should carefully develop the measurement models - preferably based on good
biological knowledge and a step-by-step process, before trying to evaluate a complicated
structural model.
What is the nature of the latent variable that I want to model?
What would be good indirect measures of this - variables that are not also being
caused by other latents that will also be in my model?
Keep it as simple as possible!
Problem: the probabilities calculated using maximum likelihood methods are only
asymptotically unbiased. What does this mean?
The calculated probabilities are only exact as sample size reaches infinity!!!
In practice, the minimum sample size needed for good probability estimates
depends on the ratio of observations to free parameters that have to be estimated.
If the data are normally distributed, then 5 times as many observations as free
parameters is fine...
As the data become increasingly non-normal, the minimum sample size must increase
accordingly.
If the sample size is not large enough, then the estimated probabilities will usually be
smaller than they should be - so you reject models more often than you should.
The d-sep test does not have this defect and so small sample sizes are okay.
A simulation example...
3
X1
0.5
0.5
X4
4
X5
5
X3
X2
0.5
0.5
100
sample size of 100
0
0
20
20
40
40
60
60
80
80
100
sample size of 10
10
20
30
MLX2
40
10
20
MLX2
30
40
negative skew
Problems for
estimating
means
(ANOVA etc)
0.0
0.0
density
0.5 1.0 1.5
density
0.5 1.0 1.5
2.0
2.0
positive skew
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
t-distribution
0.6
0.8
1.0
negative kurtosis
normal
 distribution
normal
Problems for
estimating
variances
(SEM)
0
density
1
2
3
density
0.0 0.1 0.2 0.3 0.4 0.5
positive kurtosis
0.4
-3
-2
-1
0
1
2
3
0.0
0.2
0.4
0.6
0.8
1.0
Hypothesis: I think that these two variables are independent, and therefore the
correlation coefficient must be zero...
Ideal case: if the hypothesis is true then my statistical test will never report a low
probability value (i.e. below the significance level - say p=0.05)
Hypothesis never rejected.
if the hypothesis is false then my statistical test will always report a low
probability value (i.e. below the significance level)
Hypothesis always rejected.
100
% times that
the hypothesis
is rejected.
0
0 0.1 0.2 0.3
Correlation coefficient
1.0
100
80
60
Sample size
N=100
20
40
N=50
N=10
Rejection rate if =0
0
Number of times Ho rejected out of 100
N=500
0.0
0.1
0.2
0.3
True correlation between X and Y ()
0.4
0.5
Null hypothesis in testing a structural equation model:
The data were generated by the structural equations given in my model
- there are no causal relationships in my model that are wrong
- there are no causal relationships that are missing in my model
- the relationships between the variables are exactly linear
- the data are normally distributed or else there are lots of them...
As sample size increases, statistical power increases.
Therefore, even very small errors in the above assumptions will cause the model
to be rejected.