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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