Sample Size - Science topic

The minimum sample size for pilot testing of a 12-item test depends on various factors, including the level of confidence desired, the margin of error acceptable, and the variability within your target population.

However, usually for piloting, it should be 10–20% of your actual study sample size.

Both null and alternate hypothesis can be formulated In a quasi-experimental design, But the type of hypothesis used should align with the research question and the structure of the design. Typically, a non-equivalent groups design is employed, where the researcher hypothesizes that the intervention or treatment will lead to a specific outcome compared to the control or comparison group. This hypothesis is directional and predicts the expected difference or change due to the intervention. The reason for using this type of hypothesis is that, unlike true experiments, quasi-experiments do not use random assignment, making it challenging to control for all confounding variables. Therefore, the hypothesis must be carefully constructed to account for these potential confounds and to assert a cause-and-effect relationship as confidently as possible within the constraints of the design.

Ravindra Pratap Singh How?

Product P chart for non-conforming products within one month, one sample will be taken every day within one month

Samples were taken and the number of non-conforming products was checked, with a total of 26 days of data collected.

The data is as follows:

Inspection quantity Unqualified quantity (nP)

158 11

140 11

140 8

155 6

160 4

144 7

139 10

151 11

163 9

148 5

150 2

153 7

149 7

145 8

160 6

165 15

136 18

153 10

150 9

148 5

135 0

165 12

143 10

138 8

144 14

161 20

MARBIL

Folks may find it difficult to answer your query as are many possible differences possible between two assessment methods (of blood dilution methods in this case). And each requires a different statistical – power-analysis (SPA) to determine sample sizes — so, suggest you to be more specific as to which is of interest ( or your purpose).

In this regard, two obvious basic concerns (IMO) —and arguably both — would be 1) respective reliabilities and their difference, and 2) respective means (and potential bias Of new dilution method vs. old- control).

“Sample-size“ also has several possible meanings depending on possibility of dividing a single persons sample into more than two subsamples , e.g. say 10 where alternating between dilution methods ( for relative independence) 5 are each would be ultimately produced. “Sample size’ also could mean the number of persons who provide a sample ( which might be divided as we just described).

Nonetheless: Given how you posed the question (sans specifics): One might guess you are interested in a quick eval where real ifocus is in use of new — maybe much faster or cheaper dilution method -- where WBC variation is “real interest.” In such a case, one might suggest selecting (~15) stratified sampled individuals ( animal or human) with WBCs over range of interest and then dividing into multiple even numbered sub samples ( GE. 6 , but 10 better).

Justification for the above could be provided, if it is on target. Most likely it is off in some significant way(s) , and to better advise would require a better statement of your underlying interest.

ALVAH

Alvah C Bittner, PhD, CPE

@

The Kish Leslie formula for survey sampling is:

\[ f = \frac{N}{n} \]

Where \( f \) is the sampling interval, \( N \) is the population size, and \( n \) is the desired sample size.

You could just simply use n = at least 30 to fulfil the CLT :)

EFA is less restrictive than CFA, as EFA allows all of the crossloadings. In contrast, most if not all crossloadings are typically fixed to zero in CFA. It is therefore rather typical for CFA to fit worse than EFA.

In terms of sample size, N = 140 is rather on the low side for both EFA and CFA. If anything, such a low sample size would typically result in a lack of statistical power to detect model misfit. Therefore, if you see misfit, I would definitely pay attention to it, especially in a small sample.

Just to be clear, Mamun Ur Rashid, I think you are not talking about a p-value, but the margin of error you are considering. Is that correct?

Bruce Weaver A useful app but it appears that its focus (currently) is exclusively on power. I believe that a simulation that is run "from scratch" provides greater flexibility and yields a lot more information than just on power. For example, you can simulate the effects of non-normal and missing data and evaluate the performance of fit statistics, as well as parameter and standard error bias in addition to estimating power. Power estimations can be misleading when your SEM parameters or standard errors are biased due to, for example, an insufficient sample size.

If your study compares groups, 10-20 animals per group is acceptable. This number can balance logistical convenience and data collection to guide the full-scale study design.

Thank you for clarifying, Bahar Ysr. PASS has routines for both logistic regression and the Wilcoxon-Mann-Whitney test. So I would estimate the needed sample sizes for both and then choose the larger of the two.

Esta en función de la superficie del campo a muestrear y de la homogeneidad en cuanto a características topográficas y composicion físico - química del suelo.

It is highly unlikely that the "statistical validity" of your study will be affected by discarding 5 participants from you sample. Yes, you will see a small drop in power, but it is also highly unlikely it will be enough to create a meaningful difference.

Alternatively the is a high likelihood that this reply was generated by ChatGPT, rather than by someone who actually knows this field.

This article link above might help you. Otherwise, the links below may help you perform the calculation. I wish you all the best. Vicki

Chuck A Arize Thank you for the quick response. I want to examine a positive linear relation between two continuous variables. Additionally, I use 6 control variables (mostly continuous).

One of the commonly used formulas for calculating the ideal sample size from a given universe is the Krejcie and Morgan formula (1970). In fact, one does not even have to calculate it as there is a pre-calculated table that allows the researcher to identify the ideal sample size simply by looking at the size of the univerise.

Reference: Krejcie, R.V., & Morgan, D.W., (1970). Determining Sample Size for Research Activities. Educational and Psychological Measurement.

Hello Rainer Duesing and Bruce Weaver

thank you for sharing your perspectives. I think I have only been introduced to G*Power. But it was interesting to hear this perspective. Thank you Bruce for the reading resource as well which I will definitely read.

Thank you Dr. Morgan.

You probably know this publication, but just in case you do not (Auspurg & Hinz 2015. Factorial Survey Experiments).

Do you plan on randomly drawing two vignettes from the vignette population for each participant, so that you end up with ratings for all vignettes after all? If that is the case, is it feasible (in terms of the time to complete your survey) that each respondent rates more than two vignettes?

Soper’s sample size calculator may help you decide on the right sample size for your research. This calculation method, which was developed for studies applying structural equation modeling, has recently been accepted by top-tier journals (e.g., https://doi.org/10.1080/13683500.2023.2301458).

Please follow the reference and the link below to benefit from the calculator: Soper, D. (2018). A-priori sample size calculator for structural equation models. https://www.danielsoper.com/statcalc/calculator.aspx?id=89

Hi,

Before your study, estimate the effect size for your mixed ANOVA by reviewing similar research or using Cohen's conventions if no prior studies exist. Then, determine your sample size using G*Power, considering your desired power and significance level.

Hope this helps.

If you know what analysis you are going to perform on the data, then you should perform a power analysis to tell you what sample size is needed.

There are multiple tools available to conduct a power analysis, but you need to find one that fits your planned regression. You can start by looking at the G*Power program and its capabilities (https://www.psychologie.hhu.de/arbeitsgruppen/allgemeine-psychologie-und-arbeitspsychologie/gpower ), but there are others available.

Hello Arya M. A.

Here are some strategies for selecting geographical locations of samples (households) using GIS tools and other systematic methods:

In fact, a combination of methods may be most effective. GIS tools, such as ArcGIS or QGIS, can assist in visualizing data, creating spatial layers, and analyzing patterns. Additionally, consider collaborating with experts in geography, sociology, and demography to refine your sampling strategy!

Best Regards,

Ali YOUNES

okay, yes I'll do it, thank youu!!

Hello,

Yes, techniques like bootstrapping or Bayesian regression can be used for multiple regression with small sample sizes, providing robust estimates even with limited data.

Hope this helps.

Hello,

Your study is suitable for a simulation-based power analysis approach, leveraging packages like simr or powerlmm in R. These tools allow for the flexibility needed to model your specific study design and outcomes accurately.

Hope this helps 

In terms of providing reliable statistical results, conducting a census of the entire population is ideal. By contrast, using a statistical sample is subject to sampling error. We sample for practical reasons: e.g., to limit the cost, to limit the burden on respondents, and because a complete list of the entire population often is not available. Also, only a relatively small sample is needed to produce reliable results, generally providing little reason to go to the extra time and expense of conducting a census.

One way would be to:

(1) Define your question

(2) Develop the most appropriate method to answer your research question

(3) Specify the minimum size of the effect/ difference between groups you wish to be able to detect

(4) Determine the level of statistical significance you wish for your test (p < 0.05 is common)

(5) Determine the level of statistical power you wish your test to have (80% or 90% power is normal, less than 60% would be of very little use)

(6) Use all of the above to carry out a power calculation to determine the required minimum sample size to meet all of the above. (There are online statistical power calculators and resources, for example: https://stats.oarc.ucla.edu/other/gpower/)

If you are not too sure what these steps involve I would take each one in turn, and speak to your PhD supervisor and/ or read up on each step.

Or you could plug lots of assumptions into a sample size calculator (many such sample size calculators are online), but you still need to think about all of the above.

Dursa Hussein Thank you ChatGPT

The experimental and control groups should be roughly equal in size, but it's not essential that they are exactly equal.

The more important question is whether the residuals are roughly normally distributed in each group - as most traditional NHST tests make this assumption.

A ROC curve is typically a good start. For sample size determination, you need some preliminary data regarding assay specificity/sensitivity, and use that data in eg GPower to determine sample size

Sure, you can conduct using Area Sampling or Cluster Sampling:

Steps

1. Define the study Area - this is geographical boundaries of your study area.

2. Divide the Area into Clusters e.g city blocks, streets, neighborhoods, based on the size and nature of your study area.

As far as I know, nothing about the small sample size makes the effect size estimate invalid, but you might calculate confidence intervals for it. The interval might be quite wide.

If you already have a sample collected and know the sample size, what would you want to compute in G*Power? Also, what type of statistical analysis are you looking at?

I have found ANCOVA a bit of a pain to do power calculations for. In the past I have used simulations, which is flexible but time consuming. More recently I've used Superpower in R:

Good afternoon, I have only used Revman once a long time ago to add a study done in Guinea Bassau to the cochrane review on antibiotics for the prevention of the complications associated with measles. I can't recall a lot of this now - it is a long time ago & now that review has been redone so my name was knocked off as co-author [I didn't assist with last review update]. However, my thoughts on your question from my limited knowledge of statistics is I wouldn't reduce sample to 20. Even though the same study group, but different amount of time after, it is more a repeat measure but changing one part of how your measure. I don't really know the answer to the question - sorry.

I suggest you look into the concept of statistical power (I.e., the ability to detect an effect), and more specifically into the g*power program which translates your desired level of power into a sample size.

Yes, the Krejcie and Morgan Formula is still widely used, although there are a few software packages /sites that offer one the sample size based on the size of the universe.

Studies that have been published as recently as January, 2024 have used this formula-

@ Vaisakh Venu

Thanks a lot for your help. If normalization is done for the non parametric dependent parameter but two parametric parameter (out of 10) are not homogenous; can multiple linear regression, result may be reliable?

Thanks once again Vaisakh Venu.

I am not sure a cross-sectional study makes much sense with a specific rare disease. The quantification will always require an inordinate number of people. If you go after general diseases in aggregate, it would make more sense as we know that about 5% of the population is stricken.

Regarding the specific disease you mention, here is a pointer to an association who deals with it in Pakistan:

I have used the sample size charts (and desired power) from charts in this book (Chapter 13 and appendices):

Applied Statistics for the Behavioral Sciences (5th Edition)

by Dennis E. Hinkle, William Wiersma and Stephen G. Jurs.

Hope this is helpful.

Here is a general approach:

Hope fully this article helps:

Controlling bias is also possible and is directly related to the researcher.

Conduct a random lottery to ensure that there is no bias in principle.

When conducting data collection, adhere to impartiality and scientific honesty.

There will be many references on the Internet.

That said, depends on what you want the sample for---and as posted above, other factors concerned with your research project.

Checking the normality of your data is important when using parametric statistical tests like t-tests and ANOVA. These tests make certain assumptions about the distribution of the data, and if those assumptions are violated, it can affect the validity of your results.

For t-tests and ANOVA, normality is particularly important when dealing with smaller sample sizes. However, these tests are somewhat robust against violations of normality, especially with larger sample sizes (typically, n > 30 is considered reasonably robust).

If normality is a concern, you might consider non-parametric tests like the Mann-Whitney U test (for two groups) or the Kruskal-Wallis test (for multiple groups) as alternatives.

In summary, while it's a good practice to check for normality, the impact of deviations from normality depends on your sample size and the specific assumptions of the statistical test you're using. If in doubt, consulting with a statistician or using non-parametric tests may be appropriate.

Yes, if you have to expect so much missing data, you would have to oversample massively to retain sufficient power given the 80% non-response rate.

That being said, I would be very concerned if I had to expect an 80% non-response rate. Why would so many responses be missing? What would the mechanism of (so much) missingness be? It is difficult to imagine that your remaining sample would be representative of the target population or that the missing cases would be "missing at random" or "missing completely at random."

I am not sure of the meaning of "overlapping" term you mention. From my perspective, it seems that the same patients could be used twice as separate study findings? This type of problem is depicted in Cochrane training book 6.2.4 section (Chapter 6: Choosing effect measures and computing estimates of effect | Cochrane Training).

This situation could be accounted for in the meta-analysis and would have an effect on the standard error of the estimate (with hierarchical model accounting for repeated measurement it would increase the standard error of your pooled estimate). Practical info should be retrieved in (Chapter 10 “Multilevel” Meta-Analysis | Doing Meta-Analysis in R (bookdown.org)) using metafor package in R environment.

Hi,

These papers note the strengths and limitations of GEE and nonparametric methods for analysing repeated measures in clinical trials. Paper 2 emphasises GEE's data optimisation and power but notes its reliance on covariance assumptions, which is problematic in small samples. Paper 1 presents nonparametric analysis as less assumption-dependent and effective for detecting specific effects, though generally less efficient than GEE. While GEE excels in efficiency and power, nonparametric methods are preferable for small samples where assumptions may fail. Using simulations to mirror actual data can guide the choice between efficiency and power and assumption dependency.

Hope this helps.

If you want to have separate experimental and control groups, you can use "random assignment" from the original sample. But be sure your overall sample size is large enough to ensure that you have the power to detect significance. If you are not familiar with assessing the power of a test, the most widely used tool is g*Power

Calculating the sample size for a cross-sectional survey study involves several factors, including the desired level of precision, the confidence level, the estimated variability in the population, and the margin of error you find acceptable. The formula commonly used for estimating the sample size in a cross-sectional survey is based on the formula for calculating the sample size in a population proportion, and it's given by:

What is the formula for the sample size of a population proportion?

n = N*X / (X + N – 1), where, X = Zα/22 *p*(1-p) / MOE2, and Zα/2 is the critical value of the Normal distribution at α/2 (e.g. for a confidence level of 95%, α is 0.05 and the critical value is 1.96), MOE is the margin of error, p is the sample proportion, and N is the population size.

Here's a step-by-step guide on how to use this formula:

Keep in mind that this formula assumes a simple random sample and that the population size is large enough relative to the sample size (typically when the population is at least 10 times larger than the sample size). If your population is small, you may want to consider using the finite population correction.

Also, if your study involves multiple subgroups or strata, you may need to calculate sample sizes for each stratum and then combine them.

Remember, calculating the sample size is just one step in the planning process. Other considerations, such as budget constraints, feasibility, and the potential for non-response, should also be taken into account in the design of your survey.

Andrew Paul McKenzie Pegman great advice, why bothering with sample size calculations, power considerations, taking into account model complexity or sampling strategies. One size fits all. Shame on all reasearchers who dared to have larger sample sizes, what a waste of ressources and especially money. *irony off*

I have no idea where people are getting the now common recommendation of 3-6 participants for the the qualitative strand of an explanatory sequential design. Yes, 3-6 is standard for IPA, but explanatory sequential designs are explicitly based on using the quantitative findings as a pre-existing foundation for the qualitative study, so I think that approach would seldom make IPA a good choice.

The classic standard for the size of a qualitative study is to achieve saturation, i.e., the point at which new cases no longer add any information. This is difficult to predict in advance, too planning for too few cases can be a serious problem.

Hi Raid Amin. As I understood it, Mariano Gallo Ruelas has no raw data for any of the studies. For many of them, he has only the median and Q₃-Q₁, but not the values of Q₁ and Q₃. And as I understood it, the two possible approaches he mentioned (Wan 2014; McGrath 2021) require the exact values of Q₁ and Q₃, not just the difference between them.

Mariano Gallo Ruelas, I have no firsthand experience with the problem you described, but a Google search took me to the documentation for this R-package:

Notice that some more articles are mentioned in the Description section. Some of them may be helpful. See also the Reference list.

HTH.

DISCLAIMER: I do not consider myself an expert on analysis of complex survey data, and I had no direct experience with NHANES. However, my Google search turned up this page, which you might find helpful.

What I read there makes me wonder if your Full.sample.etc. variable in the image you uploaded is the one that is called WTINT2YR on the CDC web page. I hope this helps move things along. And I hope that someone with relevant direct knowledge of NHANES jumps in!

By "calculated sample size required," I assume you refer to a power analysis done before data collection to calculate the minimum sample size required for obtaining statistical significance with a given probability of Type 1 error and a given minimum effect size. If so, then yes, you can proceed with the analysis, while acknowledging that the shortfall in desired sample size will necessarily increase the probability of Type 2 error.

You should also up-weight the respondents in low-response strata to compensate for under-representing the subpopulations in those strata. Unless your sampling involved strictly proportionate stratification, you would have to do some weighting even if all strata had the same response rate.

Of greater concern is whether the low response rate in some strata reflects a non-random bias in non-response within those strata. If you can obtain subpopulation figures for each stratum from other sources on key demographics (like age, gender, etc.), you can adjust (to some degree) for such bias by doing post-stratification weighting within each stratum, to make your sample demographic marginal percentages match the known subpopulation values. If you only have demographics for the whole population, you could do post-stratification weighting for the sample as a whole rather than within each stratum, although the within-stratum approach would be better.

By the way, with the population being students from a single school, unless that school has a huge enrollment, both your power analysis and your statistical inference should include the Finite Population Correction (as you may already know).

You say your N = 10. If that is actually the population size (i.e., the total number of possible observations), that is a problem because it is virtually impossible to obtain statistical significance with such a small number.

This article could help you select a tool. Please take a look. The sample size has to be estimated based on the prevalence rate. As a rough guidance, a sample of size around 300 will be adequate. Please calculate to determine it.

Thank you

For ready reference, here it is:

4.2. Sample size adequacy

Four of the 395 completed surveys were removed from the analysis for being completed too rapidly or filled inconsiderately. Moreover, six of the 59 items were deleted (see 5.2 for details). Therefore, the sample-to-item ratio was 391/53 = 7.4, i.e., there were 7.4 subjects per item. Regarding sample size requirements, “no simple rule of thumb about sample size works across all studies” (Kline, 2023, p. 16). In the absence of a sampling frame (non-probability sampling), the sample size issue remains “ambiguous,” and “there are no rules” (Saunders et al., 2019, p. 315).

In structural equation modeling (SEM), a factor analysis-based technique, there are at least two perspectives, “entrenched camps” arguing to look at total sample size (minimum sample sizes) or the ratios (number of cases required per item, N:p ratio) (Kline, 2023, Osborne and Costello, 2004). There is widespread consensus in the first camp that a sample of 100 or less is “untenable” or “poor,” and for a sample of less than 200, journals “routinely reject for publication” (Comrey and Lee, 1992, Kline, 2023). Traditionally, “more is always better” (Osborne & Costello, 2004, p. 8). In contrast, researchers believe that “more is not always better.” (Wolf et al., 2013, p. 14). Sekaran and Bougie (2016, p. 264) said that “too large a sample size (say, over 500) could become a problem” due to the possibility of Type II errors.” They went on to say that “neither too large nor too small sample sizes help research projects.” (Sekaran & Bougie, 2016, p. 264). The minimum sample size of 250 is acceptable (Hoyle, 1995, p. 186). For many, a sample size of 300 or above is acceptable/appropriate/good (Comrey and Lee, 1992; Floyd and Widaman, 1995; Tabachnick and Fidell, 1996).

In the second camp, N:p of 10:1 has been advocated for ages (Everitt, 1975, Nunnally, 1978). Osborne and Costello (2004, p. 2) said this “recommendation was not supported by published research.” Streiner (1994, p. 140) suggests the ratio should be at least 5:1, provided “there are at least 100 subjects. If there are fewer than 100, the ratio should be closer to 10:1.” Several authors consider a 5:1 ratio acceptable (e.g., Bentler and Chou, 1987; Comrey and Lee, 1992; Gorsuch, 1983; Hatcher, 1994; Tabachnick and Fidell, 1996). Rather than a threshold ratio, Cattell (1978) suggested a 3 to 6. Not one ratio is likely to work in all situations. According to Bentler and Chou (1987, p. 91), “when there are many indicators of latent variables and the associated factor loadings are large,” the ratio may go as low as 5:1. In other words, more indicators and loading are critical to deciding optimal sample size. MacCallum et al. (1999,p. 96) concluded N:p ratio depends upon some aspects of variables and design, “most importantly, level of communality plays a critical role.” Where communality (squared factor loadings) represents the “squared multiple correlations among variables” (Tabachnick & Fidell, 2019, p. 481). MacCallum et al., (2001,p. 636) summarized that “samples somewhat smaller than traditionally recommended are likely sufficient when communalities are high.” In a nutshell, for this study, a sample size of 391 is not only sufficiently large but all communalities (ranges from 0.5 to 0.9) are also high. Therefore, N:p ratio of 7.4:1 is adequate for this study.

Zubaida Abdul Sattar Thanks a lot for sharing detailed information.

Dear Mesay Dechasa Gudeta

See the attached link

Christopher Muhambuwa and Laith Faisal Lazem Thank you very much

I have a hard time imagining how increasing your sample size could be "unethical." The simple answer for increasing the sample size is to "increase the power of your test" (i.e., the ability of the analysis to detect an effect).

What do you consider to be "difficult" about this type of survey? You seem to be concerned about sample sizes.

If I may make an assumption here: Surely you do not, or even can not, expect every subject who enters the questionnaire to be able to answer all questions, thus maintaining the same number of responses to all questions. Even if it was a 50:50 split between "yes" and "no" answers to Q1, you are forcing a subdivision in Q2 amongst different sub-groups. It's built into your questionnaire design.

Calculate some percentages: 50% of respondents "like fruit" (or whatever it is). Of these, 25% equally indicated a preference for the 3 fruits suggested and 25% suggested "other fruit". Respondents who did not like fruit preferred "bananas" 90% of the time.

Perhaps there are some qualitative statistical methods you can employ. Go talk to your local statistician, I'm sure you can get some useful advice from them.

If you cannot find a directly relevant study, you need to widen your conceptual framework. Many communities have been affected by flooding, and many farming communities have been affected by weather and climate events that have caused hardship, loss and death. It might help to think of your research as a study in this wider context.

One difficulty is the absence of a baseline measurement, which is why I think your plan to do qualitative research is a good one. Key informant interviews and focus groups can produce rich data, not just about the mental health impacts but also on the way the community mobilised its resources in the face of adversity. It avoids painting the community as purely a victim, and paves the way for understanding factors that promote community resilience – the ability to learn from adversity so as to be better able to deal with future adversity.

Hi Laura,

I think the sample size 40 is big enough for doing the analyses you want. It would not be called small sample size once it's over 30.

So many studies did the analyses you mentioned but only has sample size less than 30.

Thank you sir

And bear in mind that standardized effect sizes are not always the best choice.

As Mark said, it depends on various specifics. Basically there are methods based on probability of selection, the simplest being simple random sampling, and there are methods based on models, generally regression models, also called "prediction." There are also model-assisted randomized methods (or model-assisted design-based approaches). There are many variants. But generally they assume that bias is under control, and you just want to calculate what sample size, under your conditions, will hold variance below some desirable level. This also depends on inherent variability. (Please research the difference between standard deviation and standard error.)

I found it interesting that although a simple model-based ("prediction-based" - but not forecasting) method that is helpful in Official Statistics where sample surveys and census surveys are repeated to monitor markets, and simple random sampling are very different theoretically, I obtained a formula for the former that is in the same format as what W.G. Cochran developed for the latter in his book, Sampling Techniques, 3rd ed, 1977, Wiley, in Chapter 4. See Knaub, J.R., Jr. (2013), “Projected Variance for the Model-Based Classical Ratio Estimator: Estimating Sample Size Requirements,” JSM Proceedings, Survey Research Methods Section, American Statistical Association, pp. 2885-2889, https://www.researchgate.net/publication/261947825_Projected_Variance_for_the_Model-based_Classical_Ratio_Estimator_Estimating_Sample_Size_Requirements

Note that stratification is often a good way to increase efficiency. If you are only interested in overall results (not those from each stratum), it can be used to reduce sample size.

If you search the internet on specific terms of interest, and include "Pennsylvania State University" in your search, you may find that they have helpful material available online for you on statistics.

Ceren Bektaş-Aydın, please report n, mean, and SD for each of your groups. Thanks.

Since this is not a pilot study, the small numbers are not relevant.

You need to base your sample size on the minimum size of the relationship between your predictors and predicted variables that you have adequate power to detect (90% or better).

If you have a look at the RCSI sample size handbook, it gives sample sizes for studies using correlations or regression on page 51. Worst case, where the variables only share 10% of their variation (correlation 0·32), is a sample size of 99, while with a really high correlation (0·71) you could need as few as 20.

You require n = 30 to fulfil the CLT and add a few more for drop-outs :)

You are right. It is odd that Fritz and MacKinnon (2007) do not seem to provide the specified alpha (Type I) error rate explicitly in their Methods or Results sections. In the Discussion section of that paper, they mention "a Type I error of .05", so I would assume that they used .05 in their simulations (it is the most commonly used Type-I error rate), but it is not entirely clear. Perhaps send an email to Matthew Fritz (https://cehs.unl.edu/edpsych/faculty/matthew-fritz/) to confirm.

Probably the best way to figure out the sample size requirements for a CFA is by using a Monte Carlo simulation. A simulation allows you to consider your exact model (i.e., the precise factor structure and expected parameter estimates) and to simulate realistic conditions (e.g., missing data, non-normal data, different estimators) instead of relying on rules of thumb that may not apply to you specific scenario. Simulations allow you to examine potential parameter estimate bias, standard error bias, statistical power, accuracy of fit statistics, and more.

See:

Muthén, L. K., & Muthén, B. O. (2002). How to use a Monte Carlo study to decide on sample size and determine power. Structural Equation Modeling, 9(4), 599-620.

For complex mediation (path analytic) models, Monte Carlo simulation techniques are probably your best bet. See:

Thoemmes, F., MacKinnon, D. P., & Reiser, M. R. (2010). Power analysis for complex mediational designs using Monte Carlo methods. Structural Equation Modeling, 17(3), 510-534.

I offer a free mini-course on simulation of path models in Mplus that you can find here: