Structural equation modelling



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STRUCTURAL EQUATION MODELLING

Winnifred R. Louis, School of Psychology, University of Queensland



w.louis@psy.uq.edu.au
You can distribute the following freely for non-commercial use provided you retain the credit to me and periodically send me appreciative e-mails.

What is SEM?


I think of it as a powerful extension of regression that allows you to predict a DV (path analysis) and/or multiple DVs and/or look at the factor structure of a set of data (confirmatory factor analysis – measurement models). In social psych we normally use it to model predictive paths for one or more DVs, so that’s what we’ll focus on today.
Technically it’s called ‘path analysis’ when all the variables in the model are measured scales. It’s called ‘SEM’ when there’s an unmeasured “latent” variable that is imagined to underlie some of the scales. We can ignore this distinction for our purposes and call it all SEM.
Writing up SEM
This whole field is only 10-15 years old and the conventions are still evolving. At the moment though, you can safely use the following:

  • A write-up involving fit statistics and path coefficients – analogous to R2 and betas in regression, only more complex.

  • Fit stats - usually several are reported. These always include the chi-square & significance – this is supposed to be NS to be good, but never is for large N, so freely report sig chi-squares as long as the other fit statistics are good. Usually also the GFI [Goodness of Fit index] and AGFI [Adjusted GFI] or GFI and CFI [comparative fit index] –all should be in the 90s to be good. Nowadays also usually the RMSEA [Root Mean Square error of approximation]- should be <.08 to be reasonable >.10 not good <.05 good.

  • If you are comparing non-nested models, you also report the AIC [Akaike’s information criterion] – the smaller the better. There are some spin-offs lately of this stat, but none have become accepted widely, whereas the AIC is well known.

  • Coefficients – in the text, you may report sig betas (use standardized coefficients by default, as in regression – only use unstandardized if there is some special and meaningful scale to report). Also may report significant indirect effects. Alternatively, refer reader to a figure.

How to do this in SPSS




  1. You can’t do it in SPSS – but you can do it in AMOS, an SEM package which is ‘bundled’ with SPSS. Our dept licences AMOS and you can ask (I believe) even as a postgrad to have it put on your machine.

  2. Before you begin AMOS, go through a three-step preparation in SPSS. (a) Save the data file as a new file ‘data no mv’ [no missing values]. (b) Look at the variables (c) Deal with missing values.

  3. NB – Every time you make changes in the data file, you must resave before AMOS will recognise the changes.

  4. Open Start > Programs > Amos 4 > AMOS Graphics

  5. Create a model and check it.

  6. Run the model and look whether the fit is ok and there are no recommended M.I. [Modification Indices].

  7. Adapt model if necessary and re-run.

  8. Report fit in text. Report paths and/or create figure.




  1. Use analyse > descriptive > frequencies to get descriptive statistics and histograms for the data. Have a look for errors and violations of assumptions. Never skip this step. As noted above, SEM is vulnerable to all the skew, bimodality, & outlier issues of regression. But you are also looking at the proportion of missing values. You want something < 5%. As it gets higher, your results become more unstable.

FREQUENCIES

VARIABLES=iv mediator control1 control2 gender group dv1 dv2

/STATISTICS=STDDEV MINIMUM MAXIMUM SEMEAN MEAN MEDIAN SKEWNESS SESKEW

KURTOSIS SEKURT

/HISTOGRAM

/ORDER= ANALYSIS .


  1. Check out the inter-correlations among the IVs now and save yourself some trouble. The correlations should be consistent with the proposed model – IVs correlated with DVs, mediators, etc.. (NB under some circs you don’t need the zero-order correlation to be sig – i.e. if you hypothesize some IV -> DV when other variables are controlled.)

1. Analyze > Correlate > Bivariate

2. enter all ivs and DVs

3. click options > “Exclude cases listwise” and in the same window “Means and standard deviations” > continue

4. click paste
CORRELATIONS

/VARIABLES= iv mediator control1 control2 gender group dv1 dv2

/PRINT=TWOTAIL NOSIG

/STATISTICS DESCRIPTIVES

/MISSING=LISTWISE .
Run this syntax. In SEM as well as regression, you can use the means and standard deviations and inter-correlations to form in Table 1. Often Table 1 also contains the scale reliabilities in the diagonal. You get this from earlier reliability analyses when you created the scales. NB for SEM some journals omit Table 1, but it would be in all theses.

3. Centering and recoding for meaningful zeroes is optional for SEM. It is a good habit to get into, but where the constant is almost never reported (as in these models) it won’t make a difference to your results. You know how to do this already, in any case.


4. Deal with missing values.

  • You can delete all cases with MVs but this lowers your power and biases the sample if the MVs are non-random. Not recommended unless you have almost no MVs (e.g. < 1%).

  • Another technique is to “impute” the MV by looking at the correlations among a set of variables for the other participants and constructing a regression equation that you use to predict the MV for the participant(s) where it’s missing. This does not reduce your power and if anything over-capitalises on chance (inflates alpha). It is the accepted technique in some subdisciplines.

  • Most social psychologists use mean substitution – this lowers your power in regression and biases the sample as well, but less horribly. Double check to make sure you have saved the data file under a new name.

A not recommended way:

  • Click on transform > recode > into same variable

    • Enter all variables

    • Click on old and new variables

    • Click on system or user missing in ‘old’

    • Enter the mean in ‘new’ from the frequency above.

    • Hit paste

You get syntax that looks like this:


RECODE

posdesc (MISSING=[Mean]) .

EXECUTE .
This is inefficient and dangerous. You have to do it separately for each variable and if you make a mistake, you’ve over-written your original variables.
Better is Transform > Replace Missing values.

Enter all the variables into the box – in SPSS13, it will automatically create new variable names with _1 at the end. In earlier versions it truncates to keep the name < 8 characters. The point is new variables are created with missing values replaced by the ‘series mean’. Hit paste. You get:


RMV

/posdesc_1=SMEAN(posdesc) /negdesc_1=SMEAN(negdesc) /candyt1_1=SMEAN(candyt1).


Save the date file.
Open Start > Programs > Amos 4 > AMOS Graphics

It will come up with the last working model. Go to file > new


Create a model:

Drawing:


  • Use rectangle to create a rectangle for all the observed variables.

  • Use oval to create an oval for any imaginary ‘latent’ variables.

  • Use copy to create more rectangles and ovals as needed, so everything’s the same size.

  • Use the truck to move boxes around on the graph.

Labelling:

  • Double click on a box and click on the text tab. Where it says variable name, write the variable name exactly as it appears in SPSS. Don’t forget to use the names for the variables with no MV.

  • The variable label can be anything.

Modelling:

  • Use single-headed arrows to connect the boxes for predictive paths. Variables with no arrows into them are called “exogenous” (they come from outside the model – i.e., IVs). Variables with arrows into them are called “endogenous” (they come from inside the model – mediators and DVs).

  • The IVs have no variance being modelled (all IV variance is assumed to be true variance with no error), but all mediators and DVs do. For every box which has an arrow to it, click on the box and circle icon (beside the double-headed arrow). This creates a circle with an arrow into your mediator/DV. You’ll see the arrow has 1 beside it, meaning it has a regression weight of 1. (You can also draw a circle, draw an arrow to your dv/mediator box, and double click on the arrow, click on the parameters tag, and put 1 as the regression weight – but it takes longer). Meanwhile click on the circle and label it e# (e.g., e1).

  • Use double-headed arrows to connect the boxes for variables that are modelled as correlated.

  • You can’t have any feedback loops in your model.

  • You can’t have all the possible paths included – at least one correlation or path has to be omitted.

  • Where you have latent variables, at least 1 of the regression weights between the observed scales and the latent variable has to be set to 1.

  • Go to file > data files, click on file name and specify the appropriate SPSS file. (Remember you must have saved the SPSS file before this step or AMOS will not recognise the changes.)

  • Click on View > Analysis Properties. Click on the bootstrap tab. Click on perform bootstrap (leave 200 iterations), confidence intervals, bias-corrected confidence intervals, and bootstrap ML. Click on the output tab. Click on standardized effects, modification indices and direct, total and indirect effects.

Running & interp:

  • Click on the piano keys to run.

  • When it has run, click on the path icon with the upward red arrow to see the output. Click on standardized coefficients to see the output with standardized coefficients (this is normally what you report).

  • View Table Output > Notes for model. Look at the number of parameters estimated. Ponder the adequacy of your N. (Should be 15/parameter – at least 200 people – otherwise low power & instability – violations of this are common in social.)

  • View Table Output > Fit > Fitmeasures 1.

  • As noted above, Fit stats - usually several are reported. These always include the chi-square & significance – this is supposed to be NS to be good, but never is for large N, so freely report sig chi-squares as long as the other fit statistics are good. Usually also the GFI [Goodness of Fit index] and AGFI [Adjusted GFI] or GFI and CFI [comparative fit index] –all should be in the 90s to be good. Nowadays also usually the RMSEA [Root Mean Square error of approximation]- should be <.08 to be reasonable >.10 not good <.05 good. With non-nested models to be compared also report AIC – smaller is better.

  • If the model is crappy or adequate instead of good, you also want to pay attention to the modification indices. Click on table outputs > Modification indices. MI > 4 means it will benefit your model to include a particular parameter. The larger MI the more benefit to your model. Adding parameters based on MI has a huge potential to overcapitalise on chance. You always want to be theory driven if you can. Sometimes you may prefer to add one parameter before another one with larger MI because the first one has more theoretical meaning.

  • Add parameters to create ‘nested’ models, usually 1 at a time. When you do this, if you take the chi-square for the first model as output in the Fit measures 1 table, and subtract the chi-square for the second model from its fit measures 1 table, this # can be reported as a chi-square change statistic with 1 df [the # of parameters added]. If it is significant (look up chi square table in textbook or online) it means it improves the model fit / variance accounted for to add this parameter – like R2 ch in regression.

  • When you have an ok model, you can go to the standardized output, highlight all with the open hand icon, copy, go to word, and paste. This figure can be used in your thesis / ms.

  • Report significant coefficients (view >table output > standardized regression weights) and significant indirect effects where you have mediators (nb you get the effect size from “Standardized indirect effects” | “Estimates” and then you have to go down and click on “Two-tailed significance” to get the p values). A significant indirect effect says your IV is acting through your mediators on the DV. But if you have multiple mediators, it does not say which specifically are significant actors, only that somewhere there is an effect. You then have to use regressions and Sobels to laboriously compare the alternative paths.


SEM is highly unstable and sensitive to the particular IVs included and the paths. Even though it is technically better for inter-correlated IVs than regression, many social psychology editors and reviewers consider SEM an exercise in ‘smoke and mirrors’ and will prefer regression. It depends a lot on the area. E.g. in health psych, SEM is more common.


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