Options for identification
Among the options for identifying factor means are:
- Fixing the first items intercept to a constant (in most cases: zero)
- Fixing the latent mean to a constant (in most cases: zero)
- Forcing the intercepts to sum to a constant (in most cases: zero)
The second one is not suitable if you expect latent means to differ, and are interested in latent mean or intercept differences. The first and third one are more suitable in such a case, but offer some challenges as well. The third offers a challenge, because I don't know how to do this in LISREL. The first offers a challenge, because item intercepts have to be interpreted as some kind of deviation from the item with intercept fixed to a constant.
Identification by fixing the first item's intercept to zero
In this case, factor means and item intercepts are very dependent on the item chosen for identification. This is an important consideration if one wants to test an hypothesis concerning factor means, for example in multiple group comparisons. It almost amounts to a test concerning observed means for the items used for identification.
Formulae
Let i denote an item used for identification
Let j denote any other item
Let k denote a factor
Let y_ denote an observed item mean
Let alpha denote a factor mean
Let lambda denote a factor loading
Let tau denote an item intercept
alpha(k) = lambda(i) * y_(i)
tau(j) = y_(j) - lambda(j) * alpha(k)
When the model is identified by constraining the first items intercept to 0, and the first items loading to 1 is used, this simplifies to:
alpha(k) = y_(i)
tau(j) = y_(j) - lambda(j) * y_(i)
In words
The factor mean equals the observed mean of the identification item. Other items intercepts are determined by the items factor loading, times the identification items observed mean. An illustration is provided below, for a model of 3 factors fitted to 2 groups.
Examples
Values are taken from LISREL in- and output provided below. Small discrepancies arise from rounding.
1)
The first item is used as an identification item for the first factor.
In the first group, the item intercept for the second item (item 2) of the first factor is calculated as follows:
y_(item1) = 0.425 = alpha(factor1)
tau(item2) = 0.10 = y_(item2) - lambda(item2)*y_(item1) = 0.511 - 0.98*0.425
2)
The fourth item is used as an identification item for the second factor. In the second group, the item intercept for the second item of factor 2 is calculated as follows:
y_(item1) = 0.673 = alpha(factor2)
tau(item11) = -0.01 = y_(item11) - lambda(item11)*y_(item4) = 0.511 - 0.71*0.673
LISREL input
Observed means (y_; item 1-21):
G1: 0.425 0.511 0.488 0.618 0.380 0.320 0.611 0.582 0.182 0.222 0.406 0.389 0.387 0.602 0.530 0.570 0.588 0.092 0.042 0.206 0.375
G2: 0.566 0.459 0.475 0.673 0.289 0.275 0.590 0.559 0.247 0.288 0.474 0.463 0.343 0.553 0.422 0.603 0.494 0.202 0.023 0.217 0.642
Factor pattern (item 1-21):
item factor
1 1
2 1
3 1
4 2
5 1
6 1
7 1
8 1
9 1
10 1
11 2
12 2
13 2
14 1
15 2
16 3
17 2
18 3
19 3
20 2
21 3
LISREL output
Estimated factor means (alpha, factor 1-3):
G1: 0.425 0.618 0.570
G2: 0.566 0.673 0.603
Estimated item intercepts (tau):
item G1 G2
factor 1
1 0.00 0.00
2 0.10 -0.14
3 0.01 -0.23
5 0.03 -0.10
6 0.01 -0.34
7 0.11 -0.26
8 0.19 -0.40
9 0.05 -0.20
10 0.01 -0.13
14 0.16 -0.14
factor 2
4 0.00 0.00
11 0.05 -0.01
12 0.06 0.04
13 0.11 -0.21
15 0.09 -0.09
17 0.00 0.10
20 0.11 0.14
factor 3
16 0.00 0.00
18 0.23 -0.21
19 0.21 -0.04
21 0.16 -0.15
estimated factor loadings (lambda):
item G1 G2
factor 1
1 1.00 1.00
2 0.98 1.07
3 1.12 1.24
5 0.83 0.68
6 0.78 1.09
7 1.18 1.50
8 0.92 1.70
9 0.54 0.79
10 0.51 0.74
14 1.03 1.23
factor 2
4 1.00 1.00
11 0.57 0.71
12 0.72 0.63
13 0.81 0.82
15 1.00 0.75
17 0.96 0.58
20 0.15 0.12
factor 3
16 1.00 1.00
18 0.56 0.69
19 0.45 0.11
21 0.94 1.31
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