Causality in Statistical Power: Isomorphic Properties of Measurement, Research Design, Effect Size, and Sample Size
My newest published article in Scientifica is now available for download online and on the Research Engineer website. The creation of the Statistical Power engine of Research Engineer led me to write the article. Click on the Download Article button below to download a .pdf directly from the website or click on the Statistical Power button to be taken to the aforementioned engine. Many thanks and regards to everyone that uses Research Engineer! -EH
Research Engineer is designed to get you to the correct research question, research design, sample size, database, and statistical test
Based on your decisions to the questions presented, you will get to right place
A few words on what I'm doing on here. I am a biostatistician, methodologist, psychometrician, and counselor. Everyday, the incredibly intelligent people I work with including physicians, residents, fellows, staff, and faculty feel anxiety when it comes to statistics and research. Research has shown that statistics can induce cognitive dissonance in an individual due to limited experiences and competencies. The collective unconscious has sequestered statistics and research into a dark corner and that's scary.
Research and statistics are the methods by which we, as scientists, analyze, synthesize, and evaluate our research findings in a manner that can be generalized to the appropriate audience. If our methods for communicating research findings causes cognitive dissonance, just because it relates to research and statistics, then how can one ever really be able to generalize the clinical literature and integrate it into clinical practice?
After seven years of being the one to induce cognitive dissonance in others related to research and statistics, I decided to make a useful tool for students and researchers that could alleviate some of the feelings of anxiety associated with research and statistics. I built Research Engineer.
Research Engineer is designed to get you to the correct research question, research design, sample size, database, statistical test, evidence-based medicine intervention, diagnostic calculation, epidemiological calculation, variables, surveys, psychometrics, and educational framework to answer your current question (and future questions).
I am trying to bring research and statistics out of the collective unconscious and into the conscious mind where it can be effectively communicated among researchers, scientists, and students by creating this decision engine. It is easy to get to the correct research or statistical component, just answer the questions that I present you in the webpages and click on the buttons with your answer in them. Also, the step-by-step methods for conducting and interpreting each statistical test in SPSS are presented on their respective webpages.
You can also contact me via phone, social media, and email at any time in you have questions. If you need some help conducting statistics for a research project, I have eight years of experience across thousands of individual projects and I would love to help you on your study. We can negotiate prices if you are an undergraduate or graduate researcher.
In conclusion, Research Engineer makes choosing research methods and statistical tests MUCH EASIER. Just answer the questions embedded in the various decision engines and get to the correct method or test, EVERY TIME.
Thanks for your continued support, dear friends and colleagues. And many thanks and salutations to the individuals that use Research Engineer. I am honored and humbled to have this great opportunity to create a very useful and unique website. You all are the ones that make it shine!
R. Eric Heidel, Ph.D.
Assistant Professor of Biostatistics
Affiliate Professor of Biomedical Engineering
Department of Surgery
Office of Medical Education, Research, and Development
University of Tennessee Graduate School of Medicine
Owner and Operator, Scale, LLC
Within-subjects designs increase statistical power
Each participant serves as their own control in within-subjects designs
Within-subjects designs increase statistical power. because participants serve as their own control. Between-subjects designs necessitate more observations of the outcome to be able to effectively compare independent groups on an outcome. Multivariate analyses further decrease statistical power in that many more observations of the outcome to detect significant effects. At least 20 -40 more observations of the outcome have to collected per variable entered into a simultaneous of hierarchial regression model in order to meet statistical power when trying to account for demographic, etiological, clinical, and confounding effects.
Within-subjects designs, when coupled with with continuous outcomes, large effect sizes, limited variance in the outcome and a large sample size, greatly increase statistical power. Small effect sizes are also easier to detect using within-subjects statistics because participants serve as their own control. Within-subjects design also provide more statistical power when small sample sizes are used.
Effects of categorical measurement
Decrease statistical power and increase sample size
Categorical variables are very prevalent in medicine. Measures like presence of comorbidities, mortality, and test results are categorical in nature. Here are some general caveats associated with categorical measurement and sample size:
1. Categorical outcomes will always DECREASE statistical power and INCREASE the needed sample size. This is due to the lack of precision and accuracy in categorical measurement.
2. The underlying algebra associated with calculating 95% confidence intervals of odds ratios and relative risk is 100% dependent upon the sample size. With smaller sample sizes, by default, wider and less precise 95% confidence intervals will be found. If one of the cells of a cross-tabulation table has fewer observations that the other cells, then the 95% confidence interval will be wider and potentially not truly interpretable. A 95% confidence interval will become narrower or more precise only with larger sample sizes.
3. When using categorical variables for diagnostic testing purposes, larger samples sizes will be needed to calculate precise measures of sensitivity, specificity, positive predictive value (PPV), negative predictive value (NPV). With smaller sample sizes in diagnostic studies, a change in one or two observations can have drastic effects on the diagnostic values.
This is especially true when there is a subjective rating used for purposes of diagnosing someone as "positive" or "negative" for a given disease state (radiologist reading an X-ray). Inter-rater reliability coefficients such as Kappa or ICC should be employed to ensure consistency and reliability among subsequent ratings and raters. Sensitivity, specificity, and PPV will be affected by inter-rater reliability. Receiver Operator Characteristic (ROC) curves can be used to find a given value where sensitivity and specificity of a test is maximized. ROC curves can also be used to compare the area under the curve (AUC) between several diagnostic tests at the same time so that the best can be chosen.
4. For each predictor categorical parameter (or variable) that you want to include in a multivariate model, you have to increase your sample size by at least 20-40 observations of the outcome. This due to the limited precision, accuracy, and statistical power associated with categorical measurement. Researchers HAVE to collect more observations in order to detect any potential significant multivariate associations.
In the case that a polychotomous variable is to be used in a model, create (a-1), where a is the number of categories, dichotomous variables with "0" as not being that category and "1" as being that category. For each level, 20-40 more observations of the outcome will be needed to have enough statistical power to detect differences amongst the multiple groups.
Small sample sizes can lead to Type II errors
Significant effects may not be able to be detected
In instances where a phenomenon or outcome is less prevalent in the population, scientists are forced to work small sample sizes. It is just the nature of the science, and the phenomenon or outcome.
1. When working with smaller sample sizes, adequate statistical power (and therefore statistical significance) is VERY hard to achieve.
2. There is limited precision and accuracy when using categorical or ordinal outcomes, which can further decreases statistical power.
3. When measuring for small effect sizes, small sample sizes cannot provide enough variance in the outcome to detect clinically meaningful, but small effects. This REALLY decreases your statistical power since inferential statistics depend upon variance in the mathematical sense.
With this being said, remember to interpret the p-values yielded from RCT level studies with small sample sizes in the context of the aforementioned points. If a treatment effect does not obtain statistical significance, but appears to be CLINICALLY SIGNIFICANT with a p-value approaching significance (Type II error), then perhaps more credence can be found in the effect.
If researchers run bivariate tests on 30 different outcomes with less than 20 observations and claim significance without a Bonferroni adjustment, throw the article out.
Feasible research in terms of scope, time, resources, and expertise
Changing the face of medicine versus completing a research study
I have conducted thousands of statistical consultations over the years and have worked with many novice resident researchers over that time. One cannot help but admire the spirit, energy, and motivation of young people wanting to make an impact on medicine through research. I enjoy the zeal and drive of bright people wanting to be physicians and researchers. This is a good thing!
That being said, I spend a lot of my time with novice researchers using deductive reasoning to hone down their research questions into something tangible and feasible. They come into the office with an idea that will change medicine forever and we will be cruising around the Caribbean in a year! This has never been researched before! No one has ever done this before! Trust me, I want all of these proclamations to be true and I also want to change the face of medicine. Yet, most times it just not feasible to do so given the time, resources, participants, competencies and environment associated with the study.
I focus on a few primary areas when it comes to feasible research questions with my consultees:
1. Participant pool - Are there enough participants available in the immediate clinical or empirical environment to achieve adequate statistical power for inferential analyses? How will you recruit the participants? What are your inclusion and exclusion criteria? Inclusion and exclusion criteria may need to be modified to increase sample size.
2. Effect size - Small effect sizes require large sample sizes.
3. Research design - Retrospective designs are always more feasible because the data already exists.
4. Communication - Research never occurs in isolation. Researchers should communicate and collaborate with their peers regarding their research projects. Attendings and academic physicians can give you ideas on how to feasibly conduct your research.
5. Time - What is the time frame for the study from inception to publication? How much time do you have to set aside for the research study? Does the completion of your research coincide with abstract deadlines of interest?
6. Power analysis - Conduct an a priori power anlaysis based on an evidence-based measure of effect to see if the study is feasible in regards to sample size needed to achieve power.
Evidence-based measures of effect
Use the empirical literature to your advantage
One of the most important things you can do when designing your study is to conduct an a priori power analysis. Doing so will tell you how many people that you will need in your sample size to detect the effect size or treatment effect in your study.
Without an a priori calculation, you could frivolously waste months or years of your life conducting a study only to find out that you only needed 100 in each group to achieve significance. Or, with the inverse, you conduct a study with only 50 patients and find out in a post hoc fashion that you would have needed 10,000 to prove your effect!
If you are using Research Engineer and G*Power to run your analyses, here are the things you will need:
1. An evidence-based measure of effect from the literature is the first thing you should seek out. Find a study that is theoretically, conceptually, or clinically similar to your own. Try to find a study that uses the same outcome you plan to use in your study.
2. Use the means, standard deviations, and proportions from these published studies as evidence-based measures of effect size to calculate how large of a sample size you will need. These values will be reported in body of the results section or in tables within the manuscript. It shows more empirical rigor on your part if you conduct an a priori power analysis based on a well-known study in the field.
3. Plug these values into G*Power using the steps published on the sample size page to find out how many people you will need to collect for your study.
Non-parametric statistics are robust to small sample sizes
The right way to conduct statistics
Mark Twain said it best, "There are lies, damn lies, and statistics." Statistics can be misleading from both the standpoint of the person conducting the statistics and the person that is interpreting the analyses. Many between-subjects studies have small sample sizes (n < 20) and statistical assumptions for parametric statistics cannot be met.
For basic researchers that operate day in and day out with small sample sizes, the answer is to use non-parametric statistics. Non-parametric statistical tests such as the Mann-Whitney U, Kruskal-Wallis, Wilcoxon, and Friedman's ANOVA are robust to violations of statistical assumptions and skewed distributions. These tests can yield interpretable medians, interquartile ranges, and p-values.
Non-parametric statistics are also useful in the social sciences due to the inherent measurement error associated with assessing human behaviors, thoughts, feelings, intelligence, and emotional states. The underlying algebra associated with psychometrics relies on intercorrelations amongst constructs or items. Correlations can easily be skewed by outlying observations and measurement error. Therefore, in concordance with mathematical and empirical reasoning, non-parametric statistics should be used often for between-subjects comparisons of surveys, instruments, and psychological measures.
G*Power is a necessary tool for every researcher's toolkit
Easy statistical power and sample size calculations
I'm trying to run an online business so I'm fully Google-integrated. I see that there many search queries of different derivations related to sample size calculation as it relates to behind-the-scenes tracking measures.
There is an open-source tool available to EVERYONE that allows you to calculate your own a priori and post hoc power analyses. It is called G*Power and as your personal statistical consultant, I highly suggest you go to the following web address and download Version 3.0 to your respective device:
The researchers that developed this program have made a great contribution to science. It is truly a great and FREE program that can run a litany of different power analyses. You can find out in minutes how large of a sample size that you need, given that you have an idea of the effect size that you are attempting to detect in your study.
Use means, proportions, and variance measures from published studies in your field to have the most empirically rigorous hypothesized effect. Enter these values into G*Power and the adjust the variance and magnitude of the effect size to see how the required sample size changes.
Click on the Sample Size button to access the methods of conducting and interpreting sample size calculations for ten different statistical tests.
Support your local statistician!
Seek out methodological and statistical consultation
If you have access to a statistical consultants or statisticians within your empirical or clinical environment, seek out their services in the preliminary phases of planning your study. Here is a list of things that I do for residents, fellows, faculty, physicians, pharmacists, nurses, and staff at an academic regional medical campus:
1. Sample Size - I conduct sample size calculations for at least of 80-85% of my first-time clients. They often want to know how many people they need to reach a significant p-value. We work through the process of acquiring an evidence-based measure of effect that reflects what their research question is trying to answer.
It feels good knowing that you have a good chance of detecting significance with a small sample size. Also, it is good to find out that you have to collect A LOT more observations than you thought you would. Post hoc power analyses should be run for any non-significant main effects that may be considered Type II errors (limited or small sample sizes).
2. Statistical analysis - Real biostatistical scientists and statisticians will conduct your statistical analyses in an objective and expeditious manner to help you answer your research questions. Please help them understand what your research question is and what research design you want to use to answer it to the best of your abilities. They will be able to help you choose the correct statistic given that you can tell them the scale of measurement for your primary outcome and what type of design (between-subjects, within-subjects, correlational, mixed, or multivariate) you want to use to answer your question. It is also important to know WHO or WHAT you want to include in your sample in terms of inclusion and exclusion criteria. Finally, know your content area. We may not know your knowledge/philosophical base and need to understand the entire picture, as much as you can tell us.
3. Database management - Go ahead and let us build your database in a basic Excel spreadsheet and send an accompanying code book in Word so that we are all on the same page. It helps us all know what is going on, what variables are being collected, what they mean, how they are measured, and how the analysis will work. Share it with all members of the research team. Use the code book when entering your data. Tell the rest of us if you make changes to the code book or database. These simple tasks and communicative efforts can mean the difference between your statistics being run in five minutes versus five weeks. SERIOUSLY.
4. Write-up of findings for publication - We will give you an annotated write-up of your findings with statistical outputs and give you basic and unbiased interpretations of the statistical results of your study. We can help you write up the statistical methods and results sections of your abstracts and manuscripts. We can even help you design tables and graphs that will make your study findings more aesthetically and visually appealing to your audience.
When it comes to authorship, if you feel that your statistical professional's contribution to the design, execution, and interpretation of your study warrants authorship, offer it to them. They will greatly appreciate it! However, YOU SHOULD NEVER BE REQUIRED TO GIVE US AUTHORSHIP JUST BECAUSE WE RAN YOUR STATISTICS FOR YOU. IT IS UNETHICAL FOR US TO REQUIRE AUTHORSHIP FOR DOING OUR JOB. THAT IS, IF OUR JOB IS TO RUN STATISTICS IN YOUR EMPIRICAL OR CLINICAL ENVIRONMENT.
Eric Heidel, Ph.D. is Owner and Operator of Scalë, LLC.