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1 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 Psychology. 27:229-239.

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2 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.

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3 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- 408.

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4 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 rich, rich2, richlog = species richness per m2, square of richness and log richness Data from 1-m2 plots arrayed along an elevation gradient in each of several paired grazed and ungrazed meadows in SW Finland.

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5 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. select OK twice to accept this spreadsheet for your first model.

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6 How to set up a multigroup analysis (cont.) 4. Drag variables to palette and create model.

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7 How to set up a multigroup analysis (cont.) 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.

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8 How to set up a multigroup analysis (cont.) 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.

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9 Multigroup Model 1: All Parameters Free * Objective is to compare the above model across two groups, grazed and ungrazed meadows. *By "free" we mean parameters are free to be different for each group.

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10 Multigroup Model 1 Results* variances unstd. path coefficients * Unstandardized estimates are shown. Keep in mind that groups are compared based on the unstandardized parameters.

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11 Multigroup Model 1a: Coefficients equal across groups Giving parameters names and making them apply to all groups results in a single estimate for both groups. name of the variance name of the coefficient (gamma11)

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12 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 supported by the data.

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13 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.

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14 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 mass error variances will be unequal as well. We will relax these constraints first.

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15 Model 1b: relax some constraints

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16 Multigroup Model 1b Results Model fit is improved, but still inadequate, suggesting additional constraints to relax.

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17 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.

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18 Model 1c: relax another constraint

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19 Multigroup Model 1c Results compared to Model 1b 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.

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20 Multigroup Model 1d Results compared to Model 1b 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.

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21 Examination of Parameters Since all parameters appear to be significant and our model fit is good, we conclude that this is the correct model.

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22 Final Results So, we can see that in grazed meadows biomass has lower variance, relates to elevation differently, and has a different effect on richness.

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23 Multigroup Model 2: Means/Intercepts Analysis Now we select the estimate means and intercepts option.

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24 Multigroup Model 2 Results Our model chi-square remains the same, but now means are shown. All of these means/intercepts are different among groups. Would we declare them significantly different? Lets start by testing whether grazing has shifted the mean elevations of the meadows. elev mean mass intercept rich intercept

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25 Multigroup Model 2a Results Our model chi-square increased by nearly 22 points, therefore the means for elev are different among meadows. The biological reason for this result is because cattle reduce the trees that otherwise set the upslope limits of the meadows. So, lets remove the equality constraint for elevation and set one for mass.

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26 Multigroup Model 2b Results Our model chi-square is very elevated with the mass at zero elevation equal across groups, so we reject this equality also. We will remove this equality and test the rich intercept.

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27 Multigroup Model 2c Results Our model chi-square is again very elevated, so we reject this equality also. We conclude that our results from Model 2 are the ones we accept.

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28 Discussion In this example, prolonged cattle grazing appears to have had a number of important effects on the vegetation. In grazed meadows: 1) vegetation is shifted upslope due to effects of cattle on woody plants, 2) biomass is reduced for any given elevation, 3) the ability of biomass to respond to increased water supply at lower elevations is suppressed, 4) richness is lower for any given biomass value (presumably due to selective loss of species), and 5) richness is suppressed by elevated biomass when grazing is absent (due to a release from competition by grazing). The multigroup analysis permits us to examine a great number of specific differences between groups and is particularly useful in examining group by process interactions.

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