Calculating T-Value for indirect effects: a*b/sd(ai*bi) ?

Questions about the implementation and application of the PLS-SEM method, that are not related to the usage of the SmartPLS software.
nils_hachmeister
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Calculating T-Value for indirect effects: a*b/sd(ai*bi) ?

Post by nils_hachmeister »

Hello everybody,

i am trying to calculate the t-value for an indirect effect in my model. The number of participants in my experiment is quite small (n=104). Therefore, I can't compute the t-value using the Sobel-Test because the results will be biased if "n" is less than 200.
As far as I know from the forum, my only option is to calculate a t-value by bootstrapping it. There have been several posts about how to do it. (Run a bootstrap, calculate the indirect effect ai*bi, calculate the standard deviation of ai*bi, calculate the t-value).
However, I stumbled across a post stating that according to "Henseler/Ringle, (p. 151)" the correct formula to calculate a t-value by bootstrapping is:

a*b/sd(ai*bi)

a and b are supposed to be the original values and ai and bi are the bootstrapped ones. Sadly, no further information about where to find the Henseler/Ringle article can be found was given...

Can anybody please confirm whether this formula is correct or not? / Cite the article correctly?

This is where I found the citation:

viewtopic.php?t=1954&highlight=bootstrap+mediation
christian.nitzl
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Post by christian.nitzl »

Hi,

The formula is right. You can find the basis formula on page 306. The PLS results samples provide the mean value and standard error for each path model coefficient with all bootstraps. This information permits a student’s (pseudo) t-test to be performed for the significance of path model relationships.

a/sd(a)

You can easly translate the given formula for the t-test of an indirect effect.

a*b/sd(ai*bi)

Greetings,
Christian
bambacm
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Mediation

Post by bambacm »

Hi Christian,

You said:
The formula is right. You can find the basis formula on page 306. The PLS results samples provide the mean value and standard error for each path model coefficient with all bootstraps. This information permits a student’s (pseudo) t-test to be performed for the significance of path model relationships.

a/sd(a)

You can easly translate the given formula for the t-test of an indirect effect.

a*b/sd(ai*bi)

This is what i wanted. The problem I have when trasporting the data toan excel sheet to work out the bootstrap t-statistic, I am not sure how to work out the 'a'. Is it the mean of all the sample cases of the path coefficients? Please let me now how to work out the 'a' and 'b'.

Cheers
Mary
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nils_hachmeister
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Post by nils_hachmeister »

Hey Mary,

you don't need to actually "work out" the "a"s and "b"s. As I stated above, a and b are the path coefficients of the original sample. You can find them in your Bootstrap-Report under: Bootstrapping -> Path Coefficients (Mean, STDEV, T-Values) -> The first column on this table gives you the "Original Sample (O)" for a and b.

The only thing you actually need to calculate (in excel) is the value of your indirect effect ai*bi plus its standarddeviation.

Regards

Nils
bambacm
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Calculating T-Value for indirect effects: a*b/sd(ai*bi)

Post by bambacm »

Thank you so much Nils for your help. That is now crystal clear.

Another question about the Variance accounted for (VAF). When the equation asks for a*b/a*b+c..... Do I use the values as you mentioned above?

Cheers

Mary
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christian.nitzl
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Post by christian.nitzl »

Hi,

For completion here comes the citation of the mentioned article above:

Henseler, J. / Ringle, C. M. / Sinkovics, R. R.: The use of partial least squares path modeling in international marketing, in: Sinkovics, R. R. / Ghauri, P. N. (eds.), Advances in International Marketing, Vol. 20, Bingley 2009, pp. 277-320.
http://php.portals.mbs.ac.uk/Portals/49 ... cs-PLS.pdf

@ Mary: Yes, you use also the path coefficient values which you get from smartpls.

Greetings,
Christian
nils_hachmeister
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Post by nils_hachmeister »

Yes Mary, you do use the values mentioned aboved (original sample).

And thanks again Christian for the reference.
bambacm
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Calculating T-Value for indirect effects: a*b/sd(ai*bi)

Post by bambacm »

Thank you so much Christian and Nils.... Your assistance has been wonderful.

Cheers
Mary
Mary Bambacas
bambacm
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Interpreting the strength of mediation using the VAF

Post by bambacm »

Another question about the Variance accounted for (VAF). When the results is 0.43 what does it mean?

Cheers

Mary
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christian.nitzl
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Post by christian.nitzl »

…that 43% of the effect from an independent variable on a depended variable are mediated by the mediator variable.
Greetings,
Christian
bambacm
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Post by bambacm »

Thanks Christian... That is very helpful.

cheers

Mary
Mary Bambacas
Katja G
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Post by Katja G »

Thanks for helpful tips. One more question, I have calculated a t-value but how do I define if the effect is significant?

Would appreciate your response.
christian.nitzl
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Post by christian.nitzl »

Hi Katja,
For a sample size greater than 30 observations the normal critical values can be used determining the significance levels. Thus, if the t-value is above 1.96, you can assume that the path coefficient is significantly different from zero at a significance level of 5% (error term = 0.05; two tailed test) or 2.57 for 1% or 1.65 for 10%.
Greetings,
Christian
Katja G
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Post by Katja G »

Dear Christian,

Thanks for your fast response! I've read in Preacher and Hayes (2008) that confidence intervals should be reported. If so, for the calculation of CI (1) should I take a original sample parameter estimate and the corresponding standard error (not the ones from resampling) and (2) can I use the t-value based on the number of resamples and desired alpha or I have to calculate it myself? The article also mentioned bootstrapping bias correction, is it also applicable to this method of CI calculation?

Greetings,

Katja
christian.nitzl
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Post by christian.nitzl »

Hi Katja,

With help of the bootstraps results from a PLS-SEM program you can easily calculate the CI via assistance of a spread-sheet application, such as Microsoft Excel. That means the results table of the bootstrap subsamples needs to be copied and pasted into a spreadsheet application. To produce a ci% confidence interval in PLS-SEM the k, e.g. 5,000, values from the bootstrapping of a path have to be sorted from the smallest to largest. Thus, the lower bound of the bootstrap confidence interval of the path is in the k×(.5-ci%/2)th ordinal position of the ordered list, what is e.g. the 125th position for the lower bound if k=5,000 and a 95% confidence interval. The upper bound of the bootstrap confidence interval of the path in contrast can be found in the k×(.5+ci%/2))th ordinal position of the ordered list additional 1, what is e.g. the 4,876th position for the upper bound if k=5,000 and a 95% confidence interval. Because the mean of the bootstrapped distribution for a path is often not exactly equal to the estimated path in the PLS-SEM a correction for the bias have also to be made. This bias correction can calculate by the difference between the estimated path from the path model in PLS and the mean of the paht from the bootstrap sample. Since the bootstrap confidence intervals and the bias-corrected bootstrap intervals usually do not differ much in PLS-SEM, a researcher may simple use the percentile bootstrap confidence interval (Hair et al. 2013).
Greetings,
Christian
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