ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Conceptual Paper
Volume 2 Issue 3 - 2014
Clinical Trial Laboratory Data Nested Within Subject: Components of Variance, Sample Size and Cost
Borko D Jovanovic1*, Hariharan Subramanian2, Irene B Helenowski1, Hemant K Roy1 and Vadim Backman2
1Northwestern University, Feinberg School of Medicine, USA
3Boston University, USA
Received: March 06, 2015 | Published: April 15, 2015
*Corresponding author: Borko D Jovanovic, Feinberg School of Medicine, Northwestern University Department of Preventive Medicine, 680 N. Lake Shore Drive, Suite 1400, Chicago, IL 60611; Email:
Citation: Jovanovic DB, Subramanian H, Helenowski IB, Roy HK, Backman V (2014) Clinical Trial Laboratory Data Nested With in Subject: Components of Variance, Sample Size and Cost. Biom Biostat Int J 2(3): 00029. DOI: 10.15406/bbij.2015.02.00029

Abstract

Nesting of experimental factors is well established in statistical design literature related to agricultural, environmental and engineering studies. It is perhaps not sufficiently discussed in biological and laboratory experiments stemming from the use of human bio-specimens, where sample size considerations are often provided a priori on subject level, but there is little advice regarding the needed number of units at lower levels. Motivated by an example from spectroscopic microscopy and lung cancer, we revisit the experimental nesting frame work and discuss how variability, cost of sampling and sample size at lower levels may be coherently utilized. We show how the number of subjects may have to be adjusted to account for inadequate sampling decisions made at lower levels.

Keywords: Clinical trials, Sample size, Lung cancer, Spectroscopic microscopy, ANOVA

Introduction

In randomized clinical trials, the sample size (i.e. the number of subjects planned to be used) is carefully scrutinized, studied in statistics courses, and advised in this realm [1]. Quite contrary to that, the number of sampling units to be studied on sub-subject level is often ignored or chosen according to existing laboratory folklore: e.g. “we always do three repeats”. Most often only the subject and group level data are reported and considered in sample size calculations. The expected effect size is typically considered on a treatment group level as a result of an average or summary across all existing levels: sub-cell level, cell level, tissues level, per human subject, and per treatment group. Sample size calculations are then based on overall measure of variability considering the putative effect size that would make a clinically important difference. Possible knowledge of variability at lower nested levels may be available, but is rarely included in the planning of a trial. This makes the answer to the question ‘how many items should be measured at lower levels, left to budgetary limitations. In this paper, we revisit the nesting framework and discuss how effect size and sample size at various levels may be used in sample size calculations.

Motivating Example: A Lung Cancer Study

An observational study Roy et al. [2] was used as a template for in preparation for designing a randomized clinical trial example. Particularly, it involved collecting Ld measurements of “disorder strength of cell nano architecture” in saliva swab samples, based on partial wave spectroscopic microscopy. The population comprised of lung cancer patients, and three groups of controls: patients with COPD, smoking controls, and nonsmoking controls. Large values of Ld are in theory associated with disarray in cell nano architecture and suggest presence of cell stress, potentially leading to development of cancer.

In the initial study measurements were recorded for each of 135 subjects (cancer, COPD, smokers, non-smokers), with approximately 20-30 cells per subject, and within each cell, data were obtained from approximately 100,000-200,000 pixels per cell, each providing a measure of Ld. Such large number of pixels was provided by a machine which visually recorded the entire cell structure, as a part of a separate project. A summary of results is provided in Figure 1 below. Cancer patients have the largest average level of Ld, followed by COPD patients, smoker-controls, and finally by non-smoking controls. The ROC curves were formed and AUC (ROC) was observed to be in the 0.85 realm.

Figure 1: Summary of Roy et al. (2010) study results.

Importantly, measurements were structured such that pixels were nested within cells; cells were nested within subjects, while subjects were nested in several diagnosis groups as in Figure 1. The underlying working hypothesis was that Ld levels sufficiently differ among cancer and control groups so that a prediction rule may be developed and tested prospectively to detect yet undetected cancer cases. Alternatively a prophylactic prevention treatment could be applied to subjects at risk, subjects with high Ld, so that such measure of cell disarray would be brought to normal levels.

The original data were summarized and analyzed by averaging pixel intensity, providing values of the cell intensity, averaging over cells, thus providing a subject intensity and then, finally averaging subject intensities over groups of patients. The means of a “COPD smoker” group and the “COPD only” group were 4.8 +/- 2.1 and 4.0 +/- 2.3, respectively. In order to distinguish between the ‘high’ and ‘decreased’ level of cell disarray, it was felt that a decrease of 25% over the level seen in the ‘high’ group would adequately deem an intervention aimed at reducing Ld as effective. Thus, using the standard, two sided, two sample t-test formulas for sample size, with Type 1 error = 5%, different variances and a power of 80%, for each group, one would need: n=( 2.1 2 + 2.3 2 ) (1.96+0.84) 2 / (4.84.0) 2 =119 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2 da9iaacIcacaaIYaGaaiOlaiaaigdadaahaaWcbeqaaiaaikdaaaGc cqGHRaWkcaaIYaGaaiOlaiaaiodadaahaaWcbeqaaiaaikdaaaGcca GGPaGaaiikaiaaigdacaGGUaGaaGyoaiaaiAdacqGHRaWkcaaIWaGa aiOlaiaaiIdacaaI0aGaaiykamaaCaaaleqabaGaaGOmaaaakiaac+ cacaGGOaGaaGinaiaac6cacaaI4aGaeyOeI0IaaGinaiaac6cacaaI WaGaaiykamaaCaaaleqabaGaaGOmaaaakiabg2da9iaaigdacaaIXa GaaGyoaaaa@54F5@ Subjects per group.

The next question is: what sample sizes should be selected at lower levels below subject level? This question is related to the specific components of variance which we look into next.

Components of variance and averages across sampling levels

Here we make some simple assumptions. Let X=x be the measurement at the pixel level and assume that it is independent from other observations on the pixel level, with common finite variance σ pix 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadchacaWGPbGaamiEaaqaaiaaikdaaaaaaa@3B82@ . Then the averages across pixels in a cell have the variance given by:

Var( x ¯ pix )= σ pix 2 n p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GaamyyaiaadkhajuaGdaqadaGcbaqcLbsaceWG4bGbaebajuaGdaWg aaWcbaqcLbsacaWGWbGaamyAaiaadIhaaSqabaaakiaawIcacaGLPa aajugibiabg2da9Kqbaoaalaaakeaajugibiabeo8aZLqbaoaaDaaa leaajugOaiaadchacaWGPbGaamiEaaWcbaqcLbkacaaIYaaaaaGcba qcLbsacaWGUbqcfa4aaSbaaSqaaKqzGcGaamiCaaWcbeaaaaaaaa@50A1@ (1)

And higher, on the cell level:

σ cell 2 = σ Betweencells 2 + σ WithinCell 2 = σ Betweencells 2 + σ pix 2 n p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WCjuaGdaqhaaWcbaqcLbkacaWGJbGaamyzaiaadYgacaWGSbaaleaa jugOaiaaikdaaaqcLbsacqGH9aqpcqaHdpWCjuaGdaqhaaWcbaqcLb kacaWGcbGaamyzaiaadshacaWG3bGaamyzaiaadwgacaWGUbGaam4y aiaadwgacaWGSbGaamiBaiaadohaaSqaaKqzGcGaaGOmaaaajugibi abgUcaRiabeo8aZLqbaoaaDaaaleaajugOaiaadEfacaWGPbGaamiD aiaadIgacaWGPbGaamOBaiaadoeacaWGLbGaamiBaiaadYgaaSqaaK qzGcGaaGOmaaaajugibiabg2da9iabeo8aZLqbaoaaDaaaleaajugO aiaadkeacaWGLbGaamiDaiaadEhacaWGLbGaamyzaiaad6gacaWGJb GaamyzaiaadYgacaWGSbGaam4CaaWcbaqcLbkacaaIYaaaaKqzGeGa ey4kaSscfa4aaSaaaOqaaKqzGeGaeq4Wdmxcfa4aa0baaSqaaKqzGc GaamiCaiaadMgacaWG4baaleaajugOaiaaikdaaaaakeaajugibiaa d6gajuaGdaWgaaWcbaqcLbkacaWGWbaaleqaaaaaaaa@82EC@ (2)

Then the average across cells has variance:

Var( x ¯ cell )= σ cell 2 n c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GaamyyaiaadkhajuaGdaqadaGcbaqcLbsaceWG4bGbaebammaaBaaa leaajugOaiaadogacaWGLbGaamiBaiaadYgaaSqabaaakiaawIcaca GLPaaajugibiabg2da9Kqbaoaalaaakeaajugibiabeo8aZLqbaoaa DaaaleaajugOaiaadogacaWGLbGaamiBaiaadYgaaSqaaKqzGcGaaG OmaaaaaOqaaKqzGeGaamOBaKqbaoaaBaaaleaajugOaiaadogaaSqa baaaaaaa@5239@

σ cell 2 n c = 1 n c ( σ Betweencells 2 + σ pix 2 n p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aajugibiabeo8aZLqbaoaaDaaabaqcLbkacaWGJbGaamyzaiaadYga caWGSbaajuaGbaqcLbkacaaIYaaaaaqcfayaaKqzGeGaamOBaKqbao aaBaaabaqcLbkacaWGJbaajuaGbeaaaaqcLbsacqGH9aqpjuaGdaWc aaqaaKqzGeGaaGymaaqcfayaaKqzGeGaamOBaKqbaoaaBaaabaqcLb kacaWGJbaajuaGbeaaaaWaaeWaaeaajugibiabeo8aZLqbaoaaDaaa baqcLbkacaWGcbGaamyzaiaadshacaWG3bGaamyzaiaadwgacaWGUb Gaam4yaiaadwgacaWGSbGaamiBaiaadohaaKqbagaajugOaiaaikda aaqcLbsacqGHRaWkjuaGdaWcaaqaaKqzGeGaeq4Wdmxcfa4aa0baae aajugOaiaadchacaWGPbGaamiEaaqcfayaaKqzGcGaaGOmaaaaaKqb agaajugibiaad6gajuaGdaWgaaqaaKqzGcGaamiCaaqcfayabaaaaa GaayjkaiaawMcaaaaa@7264@ (3)

On the subject level:

σ Subject 2 = σ BetweenSubjects 2 + σ WithinSubject 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadofacaWG1bGaamOyaiaadQgacaWGLbGaam4yaiaadsha aeaacaaIYaaaaOGaeyypa0Jaeq4Wdm3aa0baaSqaaiaadkeacaWGLb GaamiDaiaadEhacaWGLbGaamyzaiaad6gacaWGtbGaamyDaiaadkga caWGQbGaamyzaiaadogacaWG0bGaam4CaaqaaiaaikdaaaGccqGHRa WkcqaHdpWCdaqhaaWcbaGaam4vaiaadMgacaWG0bGaamiAaiaadMga caWGUbGaam4uaiaadwhacaWGIbGaamOAaiaadwgacaWGJbGaamiDaa qaaiaaikdaaaaaaa@6045@ (4)

This can be estimated as:

Var( x ¯ subject )= σ subject 2 n s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GaamyyaiaadkhajuaGdaqadaGcbaqcfa4aa0aaaOqaaKqzGeGaamiE aaaajuaGdaWgaaWcbaqcLbsacaWGZbGaamyDaiaadkgacaWGQbGaam yzaiaadogacaWG0baaleqaaaGccaGLOaGaayzkaaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacqaHdpWCjuaGdaqhaaWcbaqcLbkacaWGZb GaamyDaiaadkgacaWGQbGaamyzaiaadogacaWG0baaleaajugOaiaa ikdaaaaakeaajugibiaad6gajuaGdaWgaaWcbaqcLbkacaWGZbaale qaaaaaaaa@589B@ (5)

Further giving us:

Var( x ¯ subject )= 1 n s ( σ BetweenSubjects 2 + σ WithinSubjects 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GaamyyaiaadkhajuaGdaqadaGcbaqcfa4aa0aaaOqaaKqzGeGaamiE aaaajuaGdaWgaaWcbaqcLbsacaWGZbGaamyDaiaadkgacaWGQbGaam yzaiaadogacaWG0baaleqaaaGccaGLOaGaayzkaaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaad6gajuaGdaWgaa WcbaqcLbkacaWGZbaaleqaaaaajuaGdaqadaGcbaqcLbsacqaHdpWC juaGdaqhaaWcbaqcLbkacaWGcbGaamyzaiaadshacaWG3bGaamyzai aadwgacaWGUbGaam4uaiaadwhacaWGIbGaamOAaiaadwgacaWGJbGa amiDaiaadohaaSqaaKqzGcGaaGOmaaaajugibiabgUcaRiabeo8aZL qbaoaaDaaaleaajugOaiaadEfacaWGPbGaamiDaiaadIgacaWGPbGa amOBaiaadofacaWG1bGaamOyaiaadQgacaWGLbGaam4yaiaadshaca WGZbaaleaajugOaiaaikdaaaaakiaawIcacaGLPaaaaaa@7723@ (6)

And since

σ WithinSubjects 2 = 1 n p σ pix 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WCjuaGdaqhaaWcbaqcLbkacaWGxbGaamyAaiaadshacaWGObGaamyA aiaad6gacaWGtbGaamyDaiaadkgacaWGQbGaamyzaiaadogacaWG0b Gaam4CaaWcbaqcLbkacaaIYaaaaKqbakabg2da9maalaaakeaajugi biaaigdaaOqaaKqzGeGaamOBaKqbaoaaBaaaleaajugOaiaadchaaS qabaaaaKqzGeGaeq4Wdmxcfa4aa0baaSqaaKqzGcGaamiCaiaadMga caWG4baaleaajugOaiaaikdaaaaaaa@58FE@

Var( x ¯ subject )= 1 n s ( σ BetweenSubjects 2 + 1 n p σ pix 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GaamyyaiaadkhajuaGdaqadaGcbaqcfa4aa0aaaOqaaKqzGeGaamiE aaaajuaGdaWgaaWcbaqcLbsacaWGZbGaamyDaiaadkgacaWGQbGaam yzaiaadogacaWG0baaleqaaaGccaGLOaGaayzkaaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaad6gajuaGdaWgaa WcbaqcLbkacaWGZbaaleqaaaaajuaGdaqadaGcbaqcLbsacqaHdpWC juaGdaqhaaWcbaqcLbkacaWGcbGaamyzaiaadshacaWG3bGaamyzai aadwgacaWGUbGaam4uaiaadwhacaWGIbGaamOAaiaadwgacaWGJbGa amiDaiaadohaaSqaaKqzGcGaaGOmaaaajugibiabgUcaRKqbaoaala aakeaajugibiaaigdaaOqaaKqzGeGaamOBaKqbaoaaBaaaleaajugO aiaadchaaSqabaaaaKqzGeGaeq4Wdmxcfa4aa0baaSqaaKqzGcGaam iCaiaadMgacaWG4baaleaajugOaiaaikdaaaaakiaawIcacaGLPaaa aaa@73DC@ (7)

Finally, the last expression simplifies to a result we will find useful:

Var( X ¯ subject )= σ BetweenSubjects 2 n s + σ BetweenCells 2 n s n c + σ BetweePixels 2 n s n c n p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaiaadg gacaWGYbGaaiikaiqadIfagaqeamaaBaaaleaacaWGZbGaamyDaiaa dkgacaWGQbGaamyzaiaadogacaWG0baabeaakiaacMcacqGH9aqpda Wcaaqaaiabeo8aZnaaDaaaleaacaWGcbGaamyzaiaadshacaWG3bGa amyzaiaadwgacaWGUbGaam4uaiaadwhacaWGIbGaamOAaiaadwgaca WGJbGaamiDaiaadohaaeaacaaIYaaaaaGcbaGaamOBamaaBaaaleaa caWGZbaabeaaaaGccqGHRaWkdaWcaaqaaiabeo8aZnaaDaaaleaaca WGcbGaamyzaiaadshacaWG3bGaamyzaiaadwgacaWGUbGaam4qaiaa dwgacaWGSbGaamiBaiaadohaaeaacaaIYaaaaaGcbaGaamOBamaaBa aaleaacaWGZbaabeaakiaad6gadaWgaaWcbaGaam4yaaqabaaaaOGa ey4kaSYaaSaaaeaacqaHdpWCdaqhaaWcbaGaamOqaiaadwgacaWG0b Gaam4DaiaadwgacaWGLbGaamiuaiaadMgacaWG4bGaamyzaiaadYga caWGZbaabaGaaGOmaaaaaOqaaiaad6gadaWgaaWcbaGaam4Caaqaba GccaWGUbWaaSbaaSqaaiaadogaaeqaaOGaamOBamaaBaaaleaacaWG Wbaabeaaaaaaaa@7D58@ (8)

Several things are worth noting here.

  1. First, the first summand in the formula above is usually used to estimate the entire expression.
  2. Second, however one determines ns, once it is determined other elements in the equation may be used to minimize, with appropriate constraints, the entire expression for variance.
  3. Finally, one can study the trade-off among three sample sizes above, total variance, and total cost of the experiment.

Sample size justification at lower levels as proportion of total variance of the mean

From established expression for variance of the overall mean across nssubjects, given in equation (8).
We can derive proportion of variability due to subjects so that sample sizes at lower levels guarantee that proportion of total variability due to lower levels is small, say 1% or smaller. This would translate to:

σ Betweensubjects 2 + σ Betweensubjects 2 n c + σ BetweenPixels 2 n c n p σ Betweensubjects 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHdpWCdaqhaaWcbaGaamOqaiaadwgacaWG0bGaam4DaiaadwgacaWG LbGaamOBaiaadohacaWG1bGaamOyaiaadQgacaWGLbGaam4yaiaads hacaWGZbaabaGaaGOmaaaakiabgUcaRmaalaaabaGaeq4Wdm3aa0ba aSqaaiaadkeacaWGLbGaamiDaiaadEhacaWGLbGaamyzaiaad6gaca WGZbGaamyDaiaadkgacaWGQbGaamyzaiaadogacaWG0bGaam4Caaqa aiaaikdaaaaakeaacaWGUbWaaSbaaSqaaiaadogaaeqaaaaakiabgU caRmaalaaabaGaeq4Wdm3aa0baaSqaaiaadkeacaWGLbGaamiDaiaa dEhacaWGLbGaamyzaiaad6gacaWGqbGaamyAaiaadIhacaWGLbGaam iBaiaadohaaeaacaaIYaaaaaGcbaGaamOBamaaBaaaleaacaWGJbaa beaakiaad6gadaWgaaWcbaGaamiCaaqabaaaaaGcbaGaeq4Wdm3aa0 baaSqaaiaadkeacaWGLbGaamiDaiaadEhacaWGLbGaamyzaiaad6ga caWGZbGaamyDaiaadkgacaWGQbGaamyzaiaadogacaWG0bGaam4Caa qaaiaaikdaaaaaaaaa@7EE0@ (9)

Notice that ns cancel out from the left hand side, giving the inflation ratio IR:

σ Betweensubjects 2 + σ Betweensubjects 2 n c + σ BetweenPixels 2 n c n p σ Betweensubjects 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHdpWCdaqhaaWcbaGaamOqaiaadwgacaWG0bGaam4DaiaadwgacaWG LbGaamOBaiaadohacaWG1bGaamOyaiaadQgacaWGLbGaam4yaiaads hacaWGZbaabaGaaGOmaaaakiabgUcaRmaalaaabaGaeq4Wdm3aa0ba aSqaaiaadkeacaWGLbGaamiDaiaadEhacaWGLbGaamyzaiaad6gaca WGZbGaamyDaiaadkgacaWGQbGaamyzaiaadogacaWG0bGaam4Caaqa aiaaikdaaaaakeaacaWGUbWaaSbaaSqaaiaadogaaeqaaaaakiabgU caRmaalaaabaGaeq4Wdm3aa0baaSqaaiaadkeacaWGLbGaamiDaiaa dEhacaWGLbGaamyzaiaad6gacaWGqbGaamyAaiaadIhacaWGLbGaam iBaiaadohaaeaacaaIYaaaaaGcbaGaamOBamaaBaaaleaacaWGJbaa beaakiaad6gadaWgaaWcbaGaamiCaaqabaaaaaGcbaGaeq4Wdm3aa0 baaSqaaiaadkeacaWGLbGaamiDaiaadEhacaWGLbGaamyzaiaad6ga caWGZbGaamyDaiaadkgacaWGQbGaamyzaiaadogacaWG0bGaam4Caa qaaiaaikdaaaaaaaaa@7EE0@ (10)

IR can be interpreted as the proportional increase in total variance due to lower (nested) levels, and should remain low.

Using σ BetweenSubjects 2 =0.308 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WCjuaGdaqhaaWcbaqcLbkacaWGcbGaamyzaiaadshacaWG3bGaamyz aiaadwgacaWGUbGaam4uaiaadwhacaWGIbGaamOAaiaadwgacaWGJb GaamiDaiaadohaaSqaaKqzGcGaaGOmaaaajugibiabg2da9iaaicda caGGUaGaaG4maiaaicdacaaI4aaaaa@4EF8@ , σ BetweenCells 2 =0.112 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaKqzGcGaamOqaiaadwgacaWG0bGaam4DaiaadwgacaWGLbGa amOBaiaadoeacaWGLbGaamiBaiaadYgacaWGZbaaleaajugOaiaaik daaaGccqGH9aqpcaaIWaGaaiOlaiaaigdacaaIXaGaaGOmaaaa@4A70@ and σ pix 2 =2.552 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaKqzGcGaamiCaiaadMgacaWG4baaleaajugOaiaaikdaaaGc cqGH9aqpcaaIYaGaaiOlaiaaiwdacaaI1aGaaGOmaaaa@4261@ , observed by Roy et al. [2], we numerically compute:

IR = (0.308 + 0.112/nc+2.552/ncnp)/0.308.
For various values of the two unknown sample sizes we can compute the inflation ratio (IR), the following table provides several values of IR of the two sample sizes at lower levels.

For example, if we chose 3 observations per each lower level, we will need to increase subject level sample size by 208%, or from n=119 to n*=250. With 10 observations per lower level we need to increase n by 23.8% or to n*=148, and with 100 observations per level this becomes less than 1%, a very tolerable increase to from n= 119 to n*=120. Cost difference between the processing of a cell and processing of a pixel may add to deciding on optimality discussed in the next section.

Sample Size Justification involving cost

Snedecor and Cochran [3] provide rationale for estimation of sample sizes on various levels using optimization via a cost function. Consider the cost of obtaining all of the samples on three levels as Cost = nscosts +nsnccosts +nsncnpcostp, along with equation (8).

Then, using advanced calculus in derivation, the product:
VC = Variance x Cost                                                                                                                          (11)
This can be minimized for:

n c = cos t s σ 2 BetweenCells cos t c σ 2 BetweenSubjects MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzGcGaam4yaaWcbeaajugibiabg2da9Kqbaoaa kaaakeaajuaGdaWcaaGcbaqcLbsaciGGJbGaai4BaiaacohacaWG0b qcfa4aaSbaaSqaaKqzGcGaam4CaaWcbeaajugibiabeo8aZLqbaoaa CaaaleqabaqcLbkacaaIYaaaaKqbaoaaBaaaleaajugOaiaadkeaca WGLbGaamiDaiaadEhacaWGLbGaamyzaiaad6gacaWGdbGaamyzaiaa dYgacaWGSbGaam4CaaWcbeaaaOqaaKqzGeGaci4yaiaac+gacaGGZb GaamiDaKqbaoaaBaaaleaajugOaiaadogaaSqabaqcLbsacqaHdpWC juaGdaahaaWcbeqaaKqzGcGaaGOmaaaajuaGdaWgaaWcbaqcLbkaca WGcbGaamyzaiaadshacaWG3bGaamyzaiaadwgacaWGUbGaam4uaiaa dwhacaWGIbGaamOAaiaadwgacaWGJbGaamiDaiaadohaaSqabaaaaa qabaaaaa@7183@ and n p = cos t c σ 2 pix cos t p σ 2 BetweenCells MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzGcGaamiCaaWcbeaajugibiabg2da9Kqbaoaa kaaakeaajuaGdaWcaaGcbaqcLbsaciGGJbGaai4BaiaacohacaWG0b qcfa4aaSbaaSqaaKqzGcGaam4yaaWcbeaajugibiabeo8aZLqbaoaa CaaaleqabaqcLbkacaaIYaaaaKqbaoaaBaaaleaajugOaiaadchaca WGPbGaamiEaaWcbeaaaOqaaKqzGeGaci4yaiaac+gacaGGZbGaamiD aKqbaoaaBaaaleaajugOaiaadchaaSqabaqcLbsacqaHdpWCjuaGda ahaaWcbeqaaKqzGcGaaGOmaaaajuaGdaWgaaWcbaqcLbkacaWGcbGa amyzaiaadshacaWG3bGaamyzaiaadwgacaWGUbGaam4qaiaadwgaca WGSbGaamiBaiaadohaaSqabaaaaaqabaaaaa@6695@ (12)

Where ns drops out from the equation:
In reality, it is either known beforehand, or found from the usual sample size considerations on the subject level.

To verify these expressions for our data, we use:

σ BetweenSubjects 2 =0.308 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WCjuaGdaqhaaWcbaqcLbkacaWGcbGaamyzaiaadshacaWG3bGaamyz aiaadwgacaWGUbGaam4uaiaadwhacaWGIbGaamOAaiaadwgacaWGJb GaamiDaiaadohaaSqaaKqzGcGaaGOmaaaajugibiabg2da9iaaicda caGGUaGaaG4maiaaicdacaaI4aaaaa@4EF8@ σ BetweenCells 2 =0.112 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WCjuaGdaqhaaWcbaqcLbkacaWGcbGaamyzaiaadshacaWG3bGaamyz aiaadwgacaWGUbGaam4qaiaadwgacaWGSbGaamiBaiaadohaaSqaaK qzGcGaaGOmaaaajugibiabg2da9iaaicdacaGGUaGaaGymaiaaigda caaIYaaaaa@4C12@ σ pix 2 =2.552 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WCjuaGdaqhaaWcbaqcLbkacaWGWbGaamyAaiaadIhaaSqaaKqzGcGa aGOmaaaajugibiabg2da9iaaikdacaGGUaGaaGynaiaaiwdacaaIYa aaaa@4403@, and the cost estimates provided below.

We take an educated guess that cost per subject = $1,000, cost per cell = $1, cost per pixel = $0.001. Simple application of formulas above provides:

n p = 1×2.552 0.001×0.112 =150.94 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzGcGaamiCaaWcbeaajugibiabg2da9Kqbaoaa kaaakeaajuaGdaWcaaGcbaqcLbsacaaIXaGaey41aqRaaGOmaiaac6 cacaaI1aGaaGynaiaaikdaaOqaaKqzGeGaaGimaiaac6cacaaIWaGa aGimaiaaigdacqGHxdaTcaaIWaGaaiOlaiaaigdacaaIXaGaaGOmaa aaaSqabaqcLbsacqGH9aqpcaaIXaGaaGynaiaaicdacaGGUaGaaGyo aiaaisdaaaa@541E@ (13)

And

n c = 1000×0.112 1×0.308 =19.01 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzGcGaam4yaaWcbeaajugibiabg2da9Kqbaoaa kaaakeaajuaGdaWcaaGcbaqcLbsacaaIXaGaaGimaiaaicdacaaIWa Gaey41aqRaaGimaiaac6cacaaIXaGaaGymaiaaikdaaOqaaKqzGeGa aGymaiabgEna0kaaicdacaGGUaGaaG4maiaaicdacaaI4aaaaaWcbe aajugibiabg2da9iaaigdacaaI5aGaaiOlaiaaicdacaaIXaaaaa@529A@ (14)

When these two values are used in (Table 1), as 150 and 20 approximately, we see that the total sample size on subject level has to be increased by about 4.2%.

nc

3

5

10

50

10

100

20

np

3

5

10

10

100

100

150

IR%

208

80.8

23.8

4.77

8.93

0.89

4.19

Table 1: Percent increase in the subject level sample size needed for a future study given components of variance from past study in Roy et al. (2010).

If the total sample size previously planned is n=119, the adjusted sample size would be about n*=124 to have similar power. This would translate into a $5,000 additional cost if approximate cost per subject is $1,000, for a total of $129,000 for subject recruitment. For lab work we have 20x$1 + 150x$0.001 =$ 20.15 per subject or 124 x $20.15 = $2,498.6 for all subjects, for the grand total cost of the trial of $131,498.60, assuming the trial drug or treatment is paid for from other resources.

One level of nesting only

In the context of the study described so far, we have cells nested in subjects and pixels nested in cells. Suppose now that pixel level does not exist but that an observation is made on each cell by some other means or some other technology. Then similar formulas follow and are applicable, as presented below.

Var( X ¯ subject )= σ BetweenSubjects 2 n s + σ BetweenCells 2 n s n c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb GaamyyaiaadkhacaGGOaGabmiwayaaraqcfa4aaSbaaSqaaKqzGcGa am4CaiaadwhacaWGIbGaamOAaiaadwgacaWGJbGaamiDaaWcbeaaju gibiaacMcacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaHdpWCjuaGdaqh aaWcbaqcLbkacaWGcbGaamyzaiaadshacaWG3bGaamyzaiaadwgaca WGUbGaam4uaiaadwhacaWGIbGaamOAaiaadwgacaWGJbGaamiDaiaa dohaaSqaaKqzGcGaaGOmaaaaaOqaaKqzGeGaamOBaKqbaoaaBaaale aajugOaiaadohaaSqabaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqz GeGaeq4Wdmxcfa4aa0baaSqaaKqzGcGaamOqaiaadwgacaWG0bGaam 4DaiaadwgacaWGLbGaamOBaiaadoeacaWGLbGaamiBaiaadYgacaWG ZbaaleaajugOaiaaikdaaaaakeaajugibiaad6gajuaGdaWgaaWcba qcLbkacaWGZbaaleqaaKqzGeGaamOBaKqbaoaaBaaaleaajugOaiaa dogaaSqabaaaaaaa@79E3@ (15)

The simplified expressions for cost still apply:
Cost = nscosts +nsnccostsand the sample size on lower level, conditional on sample size on higher level is

n c = cos t s σ 2 BetweenCells cos t c σ 2 BetweenSubjects MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaSqaaKqzGcGaam4yaaWcbeaajugibiabg2da9Kqbaoaa kaaakeaajuaGdaWcaaGcbaqcLbsaciGGJbGaai4BaiaacohacaWG0b qcfa4aaSbaaSqaaKqzGcGaam4CaaWcbeaajugibiabeo8aZLqbaoaa CaaaleqabaqcLbkacaaIYaaaaKqbaoaaBaaaleaajugOaiaadkeaca WGLbGaamiDaiaadEhacaWGLbGaamyzaiaad6gacaWGdbGaamyzaiaa dYgacaWGSbGaam4CaaWcbeaaaOqaaKqzGeGaci4yaiaac+gacaGGZb GaamiDaKqbaoaaBaaaleaajugOaiaadogaaSqabaqcLbsacqaHdpWC juaGdaahaaWcbeqaaKqzGcGaaGOmaaaajuaGdaWgaaWcbaqcLbkaca WGcbGaamyzaiaadshacaWG3bGaamyzaiaadwgacaWGUbGaam4uaiaa dwhacaWGIbGaamOAaiaadwgacaWGJbGaamiDaiaadohaaSqabaaaaa qabaaaaa@7183@ (16)

Discussion

The effect of nesting on experimental design has been a topic of interest in a vast range of equations in previous literature. Sokal and Rohlf [4] provide example of an experiment involving drugs, rats, rat livers and readings within livers. Quinn and Keough [5] provide an example of effect of grazing of sea urchins on percentage cover of filamentous algae. Snedecor and Cochran [3], provide an example of three stage sampling of turnip green plants: the first stage is plants, second stage is leaves within plans, and the third stage is determinations within one leaf. Underwood [6] provides an example of nested sampling via orchards, trees, branches and twigs. All these examples essentially provide the same solution to the questions raised in this article.

If total available cost of the experiment is provided, sample size on the subject level can be calculated to fit the cost constraints. In clinical trials, however, one usually starts with the sample size on subject level, and not the total cost allowable for the trial. Laboratory or pathology costs are calculated separately and are often unknown.

We then suggest that in designing a trial, one should first get an estimate of variability on each sampling level and calculate sample size on subject level first, obtaining ns. Next we recommend finding an optimal combination of sample sizes on lower levels, following arguments and methods provided in this paper. Finally, we should increase ns as needed to achieve previously planned power.

Conclusion

We have exemplified this method using the lung cancer study in Roy et al. 2010, calculating the sample at each level and cost required to deem a difference between groups as statistically significant. Our resulting power analyses give feasible sample size and cost estimates compliant with our study design.

Acknowledgement

This research was funded in part by grants: R01CA128641, N01CN35157, HHSN2612201200035I, 5P50CA090386-09 and 5P30CA60553-19

References

  1. Julious SA (2009) Sample Sizes for Clinical Trials. CRC Press, Boca Raton, FL, USA.
  2. Hemant KR, Hariharan S, Damania D, Hensing TA, Rom WN,  et al. (2010) Optical Detection of Buccal Epithelial Nanoarchitectural Alterations in Patients Harboring Lung Cancer: Implications for Screening. Cancer Research 70(20): 7748-7754.
  3. Snedecor GW, Cochran WG (1967) Statistical Methods, Sixth Edition. Iowa State University Press. Ames, IA, USA.
  4. Sokal RR, Rohlf FJ (2012) The Principles and Practice of Statistics in Biological Research, Fourth Edition. WH Freeman and Co. New York, NY, USA.
  5. Quinn GP, Keough MJ (2002) Experimental Design and Data Analysis for Biologists. Cambridge University Press. New York, NY, USA.
  6. Underwood AJ (1997) Experiments in Ecology. Their logical design and interpretation using analysis of variance. Cambridge University Press. Cambridge, UK.
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