## Sample size calculation using G*power Analysis

Frequently asked questions about PLS path modeling.
reve1122
PLS Junior User
Posts: 2
Joined: Mon Sep 21, 2015 12:54 pm
Real name and title: Shiaw Tong Ha

### Sample size calculation using G*power Analysis

Dear all,

My model contains 4 independent variables, 1 moderator, and 2 dependent variables.

I plan to use g*power analysis to calculate the sample size. May I know which test should I use?

I appreciate any feedback. Thanks.

Hosam
PLS Junior User
Posts: 10
Joined: Thu Dec 13, 2012 8:43 am
Real name and title:

### Re: Sample size calculation using G*power Analysis

You may use the table in page 21 of the book "A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM)" which calculate sample size based on power of analysis.

References

Hair, J. F., Hult, G. T. M., Ringle, C. M., & Sarstedt, M. (2013). A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM). Thousand Oaks: Sage.

Hosam
PLS Junior User
Posts: 10
Joined: Thu Dec 13, 2012 8:43 am
Real name and title:

### Re: Sample size calculation using G*power Analysis

However, if you want to use Gpower you could get sample size of 138 respondent as the following (please correct me if I'm mistaken)

F tests - Linear multiple regression: Fixed model, R² increase
Analysis: A priori: Compute required sample size
Input: Effect size f² = 0.15
α err prob = 0.05
Power (1-β err prob) = 0.95
Number of tested predictors = 5
Total number of predictors = 5
Output: Noncentrality parameter λ = 20.7000000
Critical F = 2.2828562
Numerator df = 5
Denominator df = 132
Total sample size = 138
Actual power = 0.9507643
Attachments samplesize.jpg (173.46 KiB) Viewed 15201 times

jmbecker
SmartPLS Developer
Posts: 1032
Joined: Tue Mar 28, 2006 11:09 am
Real name and title: Dr. Jan-Michael Becker

### Re: Sample size calculation using G*power Analysis

The questions is, which test is the one you are interested in.

The test above is for the F test of a regression. It assesses the significance of the R² at the given effect size level (e.g., effects size 0.15 = R²). Hence, with 138 cases you will find a model with an R² of 0.15 significant at 95% of the time.
Many researchers are more interested in the significance of single effects instead of the variance explained by the overall regression equation. Hence, you might want to choose the "Linear multiple regression: Fixed model, single regression coefficient" as the method in G*Power.
There you have to set the effect size f², which is the same effect size that is reported in SmartPLS under f². It is the contribution to the R² by the predictor (R²inlcuded - R²excluded) / (1 - R²included).
Unfortunately, this effect size is not very straightforward to determine. The default of 0.15 is relatively high. It would require a standardized coefficient of roughly 0.275 if your overall R² is at about 0.5 and a standardized coefficient of 0.336 if your overall R² is at 0.25.
If you have an expectation for the standardized coefficient of 0.2 and an overall R² of 0.25, you would get an effect size f² of 0.053. With five predictors, this would require you to have a sample size of 248 for a power of 95% to find the 0.2 coefficient significant.
Dr. Jan-Michael Becker, University of Cologne, SmartPLS Developer
Researchgate: https://www.researchgate.net/profile/Jan_Michael_Becker

reve1122
PLS Junior User
Posts: 2
Joined: Mon Sep 21, 2015 12:54 pm
Real name and title: Shiaw Tong Ha

### Re: Sample size calculation using G*power Analysis

Thank you very much for your valuable information and support. I really appreciated.

indox2
PLS Junior User
Posts: 2
Joined: Tue May 16, 2017 12:19 pm
Real name and title: indoxx

### Re: Sample size calculation using G*power Analysis

Dear All,

My model consists of 9 independent variables, 2 dependent variables, 2 moderating variables and 3 control variables.

I also want to use gpower analysis in order calculate the sample size. Could you pls enlighten me on how to tackle this ?

Thank you

Piddzilla
PLS Junior User
Posts: 7
Joined: Tue Nov 22, 2016 10:39 am
Real name and title: Peter Bergwall, Doctoral student

### Re: Sample size calculation using G*power Analysis

Hi everyone!

Great to see that there is a thread on G*power here already.

I have a problem relating to the statistical power and critical t boundaries for f square values (effect size).

My dataset has 98 respondents and I have 7 constructs predicting one specific dependent variable.

I am interested in the statistical power of the predictors' effects on the dependent variable.

I use the following settings in G*Power 3.1.9.2:

Test family: t tests
Statistical test: Linear multiple regression: Fixed model, single regression coefficient
Type of power analysis: Sensitivity: Compute required effect size - given alpha, power, and sample size
Tail(s): Two
Alpha err prob: 0.05
Power (1 - beta err prob): 0.80
Total sample size: 98
Number of predictors: 7

Hence, the output parameters are:

Noncentrality parameter (delta): 2.83
Critical t: 1.99
Df: 90
Effect size (f square): 0.082

Running the PLS algorithm in SmartPLS v.3.2.6 results in four paths with f square values above 0.082 (namely, 0.27, 0.25, 0.23, and 0.10). Based on the previous G*power calculation, I interpret these results as statistically significant and with sufficient statistical power based on the criteria I've specified.

However, when I bootstrap the f square values in SmartPLS (500 subsamples), the t statistics are not even close to the critical t boundary. The one largest effect (f square = 0.27) displays a t-statistic of 1.563 and a p-value of 0.119.

Can anybody tell me how to interpret these differences between G*power and SmartPLS? Is the t statistic (in relation to the critical t value) more "important" than the G*power calculation of the power of the effect sizes?

/ Peter

Piddzilla
PLS Junior User
Posts: 7
Joined: Tue Nov 22, 2016 10:39 am
Real name and title: Peter Bergwall, Doctoral student

### Re: Sample size calculation using G*power Analysis

Okay, after sleeping on it I realized a possible way to go about the problem described above.

Commonly, the relation between type 1/alpha errors ("false positives") and type 2/beta errors ("false negatives") is viewed as a trade-off. If you accept a more liberal boundary for the one you should take on a more conservative approach for the other. At least, that's my understanding of power analysis.

In general, 0.05/0.80 (Alpha err prob/1-Beta err prob) are used as conventional criteria in the social sciences. Consequently, if I change my settings in G*power to, say:

Alpha err prob: 0.20
1-Beta err prob: 0.99

then my results makes more sense, compared to the bootstrapping results in SmartPLS:

Noncentrality parameter (delta): 3.62
Critical t: 1.291
Df: 90
Effect size (f square): 0.13

So, there is a 20% probability that I will erroneously reject the null hypothesis (=no effect) for effects as low as f square=0.13. On the other hand, there is a 1% probability that I will fail to detect an effect as low as f square=0.13, if the effect is indeed present.

Going back to the t statistics generated by the bootstrapping procedure. Applying these adjusted specifications, the three predictors showing the largest f square values now have acceptable t values and p values (t values above 1.291 and p values below 0.20). However, the predictor displaying a f square value of around 0.10 still displays insignificant t and p values.

I attribute the higher alpha error probability to the small sample size (n=98). But since it is a pilot study, I still think there are some interesting findings that are worth looking into with a larger sample size (the full scale study).

jmbecker
SmartPLS Developer
Posts: 1032
Joined: Tue Mar 28, 2006 11:09 am
Real name and title: Dr. Jan-Michael Becker

### Re: Sample size calculation using G*power Analysis

I think your analysis is ok, but I would usually run a post hoc test with the effect sizes in my model. Given your results, your lowest effect size is 0.10 and you have a sample size of 98. This results in a power of 92.8% for that effect size. You find an insignificant effect, thus you may have an effect that is really insignificant (zero) or you are in the remaining 7.2% of false negatives.

For the effect size of 0.27 you have a power of 99.9%. If this effect size is not significant it is unlikely that this is due to a lack of power. Hence, the effect is indeed not significant.
Dr. Jan-Michael Becker, University of Cologne, SmartPLS Developer
Researchgate: https://www.researchgate.net/profile/Jan_Michael_Becker

yblieck
PLS Junior User
Posts: 1
Joined: Fri Aug 04, 2017 11:23 am
Real name and title: Yves Blieck (PhD student)

### Re: Sample size calculation using G*power Analysis

Hello all

I want to calculate the required sample size to validate a model:
• the measurement model (wich has already been validated in a Delphi study)
• the structural model has a lot af relations (we have several models designed a priori)
• IPMA analysis to determine what latent variables and indicators are important to ‘influence’ dependent latent variable.

The model has 8 latent variables: 7 independent (1 exogenous and 6 endogenous) and 1 dependent variable (2 dimensions).

There are in total 47 variables in the measurement model (the variable with the largest number of indicators is 16). The dependent variable has 11 indicators.`

I use G*Power for the power analysis, the settings are:
• F test (linear multiple regression: fixed model. R2 deviation from zero)
• Type of power analysis " a priori" compute required sample size
• F2 = 0,35 or 0,15
• Alpha error = 0,05
• Power 0,80
• Number of predictors: ?

My question questions are:
1. Are my settings correct?
2. what should I input as number of predictors:
a. the largest number of indicators connected to a latent variable (n=16)?
b. the number variables in the measurement model (n=47)?
c. the number of indirect latent variables (n=7)?

The 10 times rule confuses me and I do not find straightforward information on this topic.

I computed all possibilities and my current sample size N=+/-191 is only insufficient in the most severe scenario (F2 = 0,15 number of predictors=47).

jmbecker
SmartPLS Developer
Posts: 1032
Joined: Tue Mar 28, 2006 11:09 am
Real name and title: Dr. Jan-Michael Becker

### Re: Sample size calculation using G*power Analysis

1. Are my settings correct?
It depends!!! It actually depends on quite many decisions. For example, it depends on what you want to focus (test). If you read the prior discussions you would notice that one other common power analysis method appropriate in this case (and often the one that people actually want to use) would be under T-Test the "Linear multiple regression: Fixed model, single regression coefficient".
2. what should I input as number of predictors:
It depends on what you want to test and whether you have formative or reflective measurement models. If your n=16 indicators latent variable is a formative latent variable and you are interested in testing the weights, then you should use 16 (and the above mentioned t-test method). If it is a reflective latent variable and/or you are concerned with the R² increase of the final dependent variable, you would use n=7 for the number of predictors of that variable.

However, you would never use n=47. PLS (the name partial least squares says it already) considers only partial models and thus you are interested in powering each partial model to the required level.
Dr. Jan-Michael Becker, University of Cologne, SmartPLS Developer
Researchgate: https://www.researchgate.net/profile/Jan_Michael_Becker

tlh12345
PLS Junior User
Posts: 1
Joined: Tue Aug 29, 2017 4:17 am
Real name and title: tan lek hok

### Re: Sample size calculation using G*power Analysis

Dear all,

My model contains 1 independent variables, 1 mediator, and 1 dependent variables in which a moderator in between mediator and dependent variable.

I have 5 dimensions(a,b,c,d,e) for the independent variable which have (5 questions for a , 5 questions for b, 4 questions for c , 5 questions for d, and 4 questions for e).

The mediator has 6 questions. The dependent variable has 5 questions. The moderator has 4 questions. The demographic has 7 questions. There are 45 questions in total.

I plan to use g*power analysis to calculate the sample size. May I know which test should I use?

I appreciate any feedback. Thank you very much.

Regards

Tan

kamellia.ch
PLS Junior User
Posts: 10
Joined: Thu May 18, 2017 2:47 pm
Real name and title: kamelia chaichi

### Re: Sample size calculation using G*power Analysis

I have a question too. My model contains 9 moderators and three variable going to one construct, totally 12 paths (arrows ) to one construct, but the table in PLS book demonstrated until the number of max 10 arrows, how can I calculate sample size? can anyone guide me, please? thank you

PLS Junior User
Posts: 1
Joined: Sat Sep 01, 2018 7:19 am
Real name and title: ahmad abbasi

### Re: Sample size calculation using G*power Analysis

hi
i have 3 variable ( endogenous latent variable ) and every 3 variable has 6 ( Exogenous latent variable ) .
how to use Gpower I stimate the minimuem and total size of my community?
I want to use smart pls 3 for figure out of my thesis.

jmbecker
SmartPLS Developer
Posts: 1032
Joined: Tue Mar 28, 2006 11:09 am
Real name and title: Dr. Jan-Michael Becker

### Re: Sample size calculation using G*power Analysis

I think the former descriptions describe the process quite well. The most important aspect is to determine the expected effect size. It is central in determining the power. The number of variables is only peripheral.
Dr. Jan-Michael Becker, University of Cologne, SmartPLS Developer
Researchgate: https://www.researchgate.net/profile/Jan_Michael_Becker