Revisitng standardized coefficients and reference for the effect size

[Blain Waan]

Müller, R. et al. (2015) reported that the values of the standardized path coefficients (β) greater than 0.50 indicate a large effect, values around 0.30 a medium effect, and values around 0.10 a small effect in SEM. I have a cross-lagged SEM where all the endogenous variables (emp and SA) are binary. Moreover, all variables are observed variables. I ran the following codes to get the standardized coefficients. 

dp$emp1 <- ordered(dp$Emp1F_1C)
is.ordered(dp$emp1)
attributes(dp$emp1)

dp$emp2 <- ordered(dp$Emp1F_2C)
is.ordered(dp$emp2)
attributes(dp$emp2)

dp$emp5 <- ordered(dp$Emp1F_5C)
is.ordered(dp$emp5)
attributes(dp$emp5)

dp$SA1 <- ordered(dp$SA1)
is.ordered(dp$SA1)
attributes(dp$SA1)

dp$SA2 <- ordered(dp$SA2)
is.ordered(dp$SA2)
attributes(dp$SA2)

dp$SA5 <- ordered(dp$SA5)
is.ordered(dp$SA5)
attributes(dp$SA5)

full_clpm1 <- '
# synchronous covariances
SA1 ~~ emp1
SA2 ~~ emp2
SA5 ~~ emp5
# autoregressive + cross-lagged paths
emp1 ~ AGE + sex + ISS_Trauma
SA1 ~ AGE + sex + ISS_Trauma
emp2 ~ AGE + sex + ISS_Trauma + emp1 + SA1
SA2 ~ AGE + sex + ISS_Trauma + emp1 + SA1
emp5 ~ AGE + sex + ISS_Trauma + emp1 + SA1 + emp2 + SA2
SA5 ~ AGE + sex + ISS_Trauma + emp1 + SA1 + emp2 + SA2
'
# fit the model
fit1 <- sem(full_clpm1, data=dp)
summary(fit1, standardized=T, rsquare=T)

My question is: how do I interpret the unstandardized and standardized coefficients (for example, for ISS_trauma)? Is the interpretation of an unstandardized coefficient going to be like "the amount of increase/decrease in the quantile of the standard normal that underlies the endogenous variable for 1 unit increase in ISS_trauma"? How about the standardized coefficients? Is it suggested that I use standardized coefficients in my path diagrams? Or I should report the unstandardized coefficients in the path diagrams? I have read in this forum that we should use the unstandardized coefficients for description of the effects, can someone explain why? 

[Nabil Awan]

I forgot to mention why I was confused by the reference. Please see part of the SEM output below:

	Estimate	Std. Err	z-value	P(>|z|)	Standardized coefficient
Employment at year 1 ~
Age	-0.016	0.004	-3.870	<0.001	-0.163
Sex (Male vs. Female)	0.144	0.098	1.469	0.142	0.048
ISS	-0.008	0.004	-2.214	0.027	-0.074
Marital status (Married vs. Single)	0.207	0.112	1.849	0.064	0.079
Marital status (Divorced/Widowed vs. Single)	-0.037	0.130	-0.283	0.777	-0.011
Education (‘<=11 years’ vs. ‘Worked towards associates or higher’)	-0.726	0.109	-6.658	<0.001	-0.261
Education (‘HS diploma’ vs. ‘Worked towards associates or higher’)	-0.326	0.091	-3.584	<0.001	-0.121
Pre-injury SA (Yes vs. No)	-0.300	0.112	-2.689	0.007	-0.094
Rehospitalization (Yes vs. No)	-0.264	0.094	-2.803	0.005	-0.091
TBI severity (Severe vs. Moderate)	-0.425	0.127	-3.349	0.001	-0.115
Preinjury employment (Yes vs. No)	1.250	0.135	9.290	<0.001	0.410

Here ISS is significant with a very small unstandardized and standardized (both) coefficients. The same is observed for some other highly significant coefficients. 

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[Terrence Jorgensen]

Is the interpretation of an unstandardized coefficient going to be like "the amount of increase/decrease in the quantile of the standard normal that underlies the endogenous variable for 1 unit increase in ISS_trauma"?

Yes, just like regular regression coefficients, except the units of the outcome refer to the latent item response, not the 0-1 scale of the observed outcome.

https://doi.org/10.1080/10705510701758406

You can also Google for sources on probit regression for more information.

How about the standardized coefficients?

Same, but in units of SD for the predictor and the outcome's latent item response.

Is it suggested that I use standardized coefficients in my path diagrams? Or I should report the unstandardized coefficients in the path diagrams? I have read in this forum that we should use the unstandardized coefficients for description of the effects, can someone explain why? 

Generally unstd are for inference, std are used as measures of effects size.  These aren't software questions, so you can ask stuff like this on SEMNET:

http://www2.gsu.edu/~mkteer/semnet.html

Terrence D. Jorgensen

Assistant Professor, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

http://www.uva.nl/profile/t.d.jorgensen

[Christopher Bratt]

Many psychologists use std as measures of effect sizes. Others will criticise that due problems with standardisation. The unstandardised are also effect sizes (how much will the score on the DV increase if the IV increases with 1?) and the unstandardised are more informative. If possible, I would avoid standardis since for regression weights (but I would still include R-square since it helps understand how predictive the model is). I'm not at the computer, but will send references later.

Christopher
School of Psychology
University of Kent, UK

[Christopher Bratt]

Here are a few references
Baguley, Thom. (2009). Standardized or simple effect size: What should be reported? British Journal of Psychology, 100(3), 603–617. https://doi.org/10.1348/000712608x377117  

Baguley, Thomas. (2010). When correlations go bad. The Psychologist, 23(2), 122–123. (Brief and enjoyable.)

Greenland, S., Maclure, M., Schlesselman, J. J., Poole, C., & Morgenstern, H. (1991). Standardized Regression Coefficients. Epidemiology, 2(5), 387–392. https://doi.org/10.1097/00001648-199109000-00015  

Kim, J.-O., & Ferree, D. G. (1981). Standardization in Causal Analysis. Sociological Methods & Research, 10(2), 187–210. https://doi.org/10.1177/004912418101000203 

[Christopher Bratt]

Also, note problems with standardisation when using dichotomous variables: What is the standard deviation of, let's say gender? Unstandardised estimates are much clearer, even as effect sizes: Going from 0 to 1 on the independent gives what effect on the dependent?

Christopher

[Dan Laxman]

Some programs given an option of standardizing only the outcome variable so that, in this situation, the standardized coefficient indicates how much the continuous outcome differs in standard deviations for one group vs. the other (i.e., "going from 0 to 1 on the [categorical] independent"). I've used this option in the past. A footnote in the table would probably be required to clarify which estimates are based on standardizing only the outcome.

I don't think lavaan has this option. I think I've seen it in Mplus as an additional column next to the other standardized estimates.

-Dan

Daniel J. Laxman, PhD
Independent Scholar and Data Analyst
Human Development and Family Studies
Dan.J....@gmail.com

On 3/9/2019 11:19 AM, Christopher Bratt wrote:

Also, note problems with standardisation when using dichotomous variables: What is the standard deviation of, let's say gender? Unstandardised estimates are much clearer, even as effect sizes: Going from 0 to 1 on the independent gives what effect on the dependent?

Christopher

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[Blain Waan]

Thanks Christopher and Dan for raising the point. I believe "std.lv=T" gives you the option to standardize the latent continuous variable so that you can explain the SD change in latent DV for going from 0 to 1 for a binary explanatory variable. Could Terrence confirm this? 

I have some binary explanatory variables, so I guess it's better for me to use the "std.lv=T" option instead of "std.all=T" to avoid explanation based on 1 SD change of the binary explanatory variable. But I'm not sure if the cutoffs mentioned in the Müller, R. et al. (2015) reference still the same for when I just standardize the latent continuous variable.  

[Terrence Jorgensen]

I believe "std.lv=T" gives you the option to standardize the latent continuous variable so that you can explain the SD change in latent DV for going from 0 to 1 for a binary explanatory variable. Could Terrence confirm this? 

It standardizes the latent variables in the Psi matrix (typically common factors, growth factors, method factors), but not the latent item responses underlying discrete observed outcomes.  Setting std.lv=TRUE is redundant with the "std.lv" column of standardized output (like STDX in Mplus, if you only have latent exogenous variables).  

The slope for a binary outcome is not telling you about change from 0 to 1 on the observed outcome, but rather change on the latent item response's scale, which is determined arbitrarily by the parameterization= argument.  Unless you have multiple groups and use group.equal="thresholds", setting parameterization="delta" (the default) will be equivalent to standardizing the outcome (like STDY in Mplus).

I have some binary explanatory variables, so I guess it's better for me to use the "std.lv=T" option instead of "std.all=T" 

There is no std.all=TRUE argument.  std.lv=TRUE is only used to choose that identification method for latent common factors.  You can review standardized slopes in regression literature, the same principles apply.  If your outcome is a z score and the predictor is 0-1, then its slope is a standardized group mean difference (like Cohen's d, unless you also control for other effects).  If the 0-1 variable is also standardized, then its slope is more like a point biserial correlation (again, unless you control for other effects).

[Nabil Awan]

Terrence, this is not related to the package itself but could you refer/ direct me to some materials where I can see the math behind these choices?  

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[Terrence Jorgensen]

could you refer/ direct me to some materials where I can see the math behind these choices?  

About scaling latent item responses:

https://doi.org/10.1080/10705510701758406

https://www.statmodel.com/download/webnotes/CatMGLong.pdf

https://doi.org/10.1007/s11336-016-9506-0

About standardizing slopes (in any context, not just SEM):

http://www.jstor.org/stable/2684719

https://doi.org/10.1080/10705510903438872

[Nabil Awan]

Thanks!