AMOS高级分析方法-多组比较

Multigroup Modeling

The basic LISREL model, which was originally formulated in

terms of variances and covariances, was extended to include

means and intercepts in 1974, based on work by Sorbom*. This

extension requires the consideration of 4 additional matrices

(potentially), one for the exogenous structural means, one for the endogenous structural intercepts, and two for the intercepts for

the exogenous and endogenous indicators (in latent variable

models). For observed variable models, this reduces to just two

matrices, one for the exogenous means (which are calculated

directly from the data) and the other for the intercepts of the

regression equations.

*Sorbom, D. (1974) A general method for studying differences in factor means and factor structures between groups. British J. Mathematical and Statistical

Multigroup Modeling (cont.) Multigroup modeling has a number of valuable scientific applications. It not only allows us to compare path coefficients between groups, it allows us to compare means and intercepts as well.

Amos has some nice features for implementing multigroup analyses, either with single models or multiple models. In this tutorial we will compare groups using a single model.

Example Data: The Effects of Grazing on Finnish Coastal Meadows*

*previously published in:

Jutila, H. (1997) Vascular plant species richness in grazed and ungrazed coastal meadows, SW Finland. -Ann. Bot. Fenn. 34:245-263.

Grace, J.B. and Jutila, H. (1999) The relationship between species density and community biomass in grazed and ungrazed coastal meadows. Oikos, 85:398-

View of Data in FinnishMeadows.xls

grazed = 0 is no, 1 is yes (this is our grouping variable)elev = elevation of the plot above mean sea level

stressmn = flood stress index derived from long-term flooding records dol = depth of litter layer in the plot par1 -par5 = different parent materials sol1 -sol5 = different soil types

mass, mass2, masslog = biomass in g/m2, square of biomass, and log biomass Data from 1-m2 plots arrayed along an elevation gradient in each of several paired grazed and ungrazed meadows in SW Finland.

How to set up a multigroup analysis in Amos 1. Data can be set up in either of two ways, with data for each group in separate excel spreadsheets, or with one spreadsheet including a grouping variable. We will use the first of these options in building our models

(from "FinnishMeadows.xls").

2. From the menus, select FILE and then NEW to create a new model.

3. Choose data files and then file name and spreadsheet to select the first dataset.

4. Drag variables to palette and create model.

5. Under menu "Model-Fit" select "Manage Groups".

6. Change name from "Group number 1" to a descriptive name ("ungrazed" in our case).

7. Select "New" on the "Manage Groups" box to create the second model and change the name from "Group number 2" to "grazed" and select "close".

8. Now you need to select the file that will be associated with the grazed model. Be sure the "grazed" option is selected (which is should be) and then select the file to go with it.

Then select "Close" twice.

9. You now have a single model diagram configured for two groups. You can tell this by the fact that the second little window on the palette has both "ungrazed" and "grazed" in it. Since none of the parameters are named, all parameters are allowed to have unique values for each group. If you were to toggle between ungrazed and grazed, the model would look the same for both. In the next set of slides, we will alternate between looking at model specifications and then looking at the results. The basic steps will be to set individual parameters equal across groups, look at the results, and then either leave the equality in place or remove equality depending on results.

Multigroup Model 1: All Parameters Free*

Objective is to compare the above

model across two groups, grazed

and ungrazed meadows.

Multigroup Model 1 Results*

750.78

elev

mass

rich

Multigroup Model 1 - Ungrazed

Comparing Grazed and Ungrazed Meadows

43567.99e11

14.38

e21

-.01

-3.27

Chi-square = .000

df = 0p = \p .09

769.91

elev

mass

rich

Multigroup Model 1 - Grazed

Comparing Grazed and Ungrazed Meadows

28057.59

e11

12.46

e21

.00

-1.20

Chi-square = .000

df = 0p = \p

.07

variances

unstd. path coefficients

*Unstandardized estimates are shown. Keep in mind that groups are

compared based on the unstandardized parameters.

Multigroup Model 1a: Coefficients equal across groups

Giving parameters names and making them apply to name of the variance

name of the coefficient (gamma11)

Multigroup Model 1a Results

Model degrees of freedom are created by setting parameters equal across groups.Note that model fit is very poor , indicating that some equality constraints are not 759.70

elev

mass

rich

Multigroup Model 1a - Ungrazed

Comparing Grazed and Ungrazed Meadows

coefs equal between groups

37150.49

e11

13.98

e21

-.01

-2.30

Chi-square = 29.704

df = 6p = .000

.08

759.70

elev

mass

rich

Multigroup Model 1a - Grazed

Comparing Grazed and Ungrazed Meadows

coefs equal across groups

37150.49

e11

13.98

e21

-.01

-2.30

Chi-square = 29.704

df = 6p = .000

.08

Examining Residuals to Locate Inequalities

Selecting "Residual moments"

from the Analysis Properties

dialog box will allow us to see

where the lack of fit in our

model is located.

Examining Residuals to Locate Inequalities (continued)

Standardized residuals are generally

easier to interpret. Here we see large

values for:

1) mass-rich covariance,

2) mass error variance,

3) rich variance,

4) elev-mass covariance, and

5) elev-rich covariance.

We generally will expect that if the mass-

rich covariances are unequal, then the

Model 1b: relax some constraints

Multigroup Model 1b Results

759.70

elev

mass

rich

Multigroup Model 1b - Grazed

Comparing Grazed and Ungrazed Meadow

modification 1

28057.59

e11

13.98

e21

-.01

-1.20

Chi-square = 13.593

df = 4p = .009

.08

759.70

elev

mass

rich

Multigroup Model 1b - Ungrazed

Comparing Grazed and Ungrazed Meadows

modification 1

43567.99e11

13.98

e21

-.01

-3.27

Chi-square = 13.593

df = 4p = .009.08

Model fit is improved, but still inadequate, suggesting additional constraints to relax.

Further Examination of Residuals

Standardized residuals now indicate

largest value for rich error variance, so

that will be the next constraint to relax.

Note that in most cases we will want

to relax one parameter constraint at a

time.

Model 1c: relax another constraint

759.70

elev

mass

rich

Multigroup Model 1b - Grazed

Comparing Grazed and Ungrazed Meadow

modification 1

28057.59e11

13.98

e21

-.01

-1.20

Chi-square = 13.593

df = 4p = .009.08

Chi-square drop only 0.554, so rich error variance not different between groups. We will reinstate that equality constraint and evaluate the constraint on the path from mass to rich. 759.70

elev

mass

rich

Multigroup Model 1c - Ungrazed

Comparing Grazed and Ungrazed Meadows

modification 2

43567.99

e11

14.74

e21

.00

-3.27

Chi-square = 13.039

df = 3p = .005

.08

759.70

elev

mass

rich

Multigroup Model 1b - Grazed

Comparing Grazed and Ungrazed Meadow

modification 1

28057.59e11

13.98

e21

-.01

-1.20

Chi-square = 13.593

df = 4p = .009.08

Chi-square drop nearly 12 points, so path from mass to rich different between groups. We will now examine output to see if our model appears to be adequate based on all available information. 759.70

elev

mass

rich

Multigroup Model 1d - Grazed

Comparing Grazed and Ungrazed Meadow

modification 3

28057.59

e11

13.52

e21

.00

-1.20

Chi-square = 1.867

df = 3p = .601

.08

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