What threshold should be set for standardized coefficients in a first difference model?

Threshold for what? Including or excluding a path or effect? To me, such thresholds are completely arbitrary and honestly not helpful. In most applications, a coefficient of .11 is in no way better than a coefficient of .09, given sampling fluctuations/error, which is considerable in most studies.

Relying on arbitrary cut off values for point estimates which are subject to considerable sampling error makes no sense to me. I would report all coefficients, preferably along with their standard errors and/or confidence intervals.

Hi Christian Geiser, thank you so much for your answer.

I have to specify my question: In hypothesis testing, I have often seen that hypotheses are accepted only if the sign (positive/negative) is as expected, if the result is significant AND if the standardized path coefficient is at least 0.1.

I can understand that, for example, a standardized coefficient of 0.00001 has no meaning (or indicates 'no influence' if the coefficient is still significant).

Is this threshold (for example 0.1) also to be used for models with first differences? As far as I know, the R² as well as (standarddized) coefficients are smaller here anyway, because only the changes (e.g. to the previous week) are analyzed. Is it possible to interpret standardized values smaller than 0.1 in a meaningful way? And isn't this also strongly influenced by the size of the sample?

Because the larger the sample, the smaller may be the standardized coefficients, but the more likely there are significant (but very weak) results? Or is this expectation wrong?

Thanks a lot!

As I wrote in my previous post, I don't find any kind of threshold such as .1 helpful and I would not use it. The interpretation of a path coefficient is meaningful, whether it is zero, .09, .1, or .5, etc. Effects may not be large or practically relevant but the coefficients are still interpretable. I would simply report the estimate of the path coefficient along with its SE and CI and then discuss whether the size of the effect does or does not have any practical relevance.

I second Christian Geiser : general cut off values are not at all useful. And I go one step further: using (and interpreting) standardized effects does not get you anywhere. They conflate signal and noise. The signal-to-nois ratio is certainly important to judge the statistical significance of your sample, but not to make any informed interpretation of the estimates. If the signal (the estimate) has no interpretable meaning to you, then the signal-to-noise ratio (a stadardized estimate) can neither be meangingful. Instead of asking questions about cut offs and standardized effects you should better do more research on the meaning of the variable you are actually analyzing.

PS: I don't know if this sounds offending in someone's ears - but it's not meant that way. I know there is a large body of literature and courses promoting the analysis of standardized effects. But to my experience following this line produces many papers that are not worth the paper they are written on. There are extremely rare exceptions that such models produce any conclusion that has been actually confirmed and/or proven useful. Most of then just stand and remain isolated. Neither the model nor the estimates are used in other studies to go further. I have not seen a single example where research built up on what was learned (supposedly) by such models. But I am happy to get counter-examples.

Jochen Wilhelm I agree very much with your statement that "you should better do more research on the meaning of the variable you are actually analyzing." I think that this is generally desirable for many studies. I also agree that there is a tendency in the social sciences to overemphasize standardized coefficients and to not even report unstandardized coefficients. That is very unfortunate in my opinion, as I believe both are important and have their place.

That being said, there are fields (mine included: psychology) where we are dealing with variables that simply do not have an intuitive metric. Many variables are based on test or questionnaire sum or average scores. People use different tests/questionnaires with different metrics/scoring rules in different studies. What does it mean when, for example, subjective well-being is expected to change by 2.57 for every one-unit change in self-rated health and by 1.24 for every one-unit change in extraversion when self rated health is measured on a 0 - 10 scale and extraversion ranges between 20 and 50?

Standardized estimates can give us a better sense for the "strength" of influence/association in the presence of other predictors than unstandardized coefficients when variables have such arbitrary and not widely agreed upon metrics. The interpretation in standard deviation (SD) units is not completely useless in my opinion, especially since we operate a lot with SD units also in the context of other effect size measures such as Cohen's d. It allows us (often, not always) to see fairly quickly which variables are relatively more important as predictors of an outcome--we may not care so much about the absolute/concrete interpretation or magnitude of a standardized coefficient, but it does matter whether it is .1 or .6.

In addition, in the context of latent variable regression or path models (i.e., structural equation models), unstandardized paths between latent variables often have an even more arbitrary interpretation as there are different ways to identify/fix latent variable scales (e.g., by using a reference indicator or by standardizing the latent variable to a variance of 1). Regardless of the scaling of the latent variables, the standardized coefficients will generally be the same.

This does not mean that I recommend standardized coefficients over unstandardized coefficients. Variance dependence and (non-)comparability across studies/different populations are important issues/drawbacks of standardized coefficients. Unstandardized coefficients should always be reported as well, and they are very useful when variables have clear/intuitive/known metrics such as, for example, income in dollar, age, number of siblings (or pretty much any count), IQ scores, etc. Unstandardized coefficients are also preferable for making comparisons across groups/populations/studies that used the same variables. I would always report both unstandardized and standardized coefficients along with standard errors and, if possible, confidence intervals.

I believe there are many examples of regression or path models in psychology for which standardized coefficients were reported and that did advance our knowledge regarding which variables are more important than others in predicting an outcome.

Christian Geiser , although I am not a phychologist I think I understand your argument. But I don't think it is the "intuitive understanding" of the variables being critical. If you look at it, most variables are not really or only seemingly understood "intuitively". Money is surely one example (the actual value of a monetary unit is hard to grasp and highly context-specific [you can count the number of dollars, yen or euros, but this is a very involved and indirect measure of what it is actually about]), but even very fundamental physical variables like space, time and energy are not really understood. Nobody can give you the meaning of a unit of these things. They are just defined within a theory to allow a mathematical formulation of quantitative relationships between observables. The key is that these variables have that external validity, wheras I don't see how standardized measures can be externally valid. Maybe I am on the wrong track, maybe I don't use the correct words. I'd be grateful to hear your opinion.

Dear Christian Geiser, dear Jochen Wilhelm,

thank you for such a helpful discussion!

In my analysis (in the field of economics), I have derived all variables very precisely and carefully and checked their substance. I use the difference from the previous period for all variables (first differences). In my output, I report both standardized and non-standardized effects. This is also necessary because I calculate and compare three models. These three models differ only in the dependent variable. I use this procedure to decompose market shares (as dependent variable). This results in three dependent variables (market share, market share by regular prices, market share by promotional prices). I then calculate three SEM where only (!) the dependent variable changes. For this reason alone, I also have to report the non-standardized coefficients in order to compare the results from the different models.

I now understand that cut-off values make not much sense. Nevertheless, I have to reject hypotheses if the effect found is significant but very small. But now I would argue this in terms of substance: the effect is too small to provide a relevant influence in terms of content.

By the way, your explanations do not seem offended at all, but are very helpful. Thank you so much for taking the time to write such detailed answers!

how can low and negative standardised beta coefficient values be improved?