An introduction to simulating Cronbach’s Alpha values with the help of simAlpha© algorithm

An introduction to simulating Cronbach’s Alpha values with the help of simAlpha© algorithm | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Short Report An introduction to simulating Cronbach’s Alpha values with the help of simAlpha© algorithm Dimitris Papadopoulos This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7726096/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract simAlpha algorithm was designed in 2023 from Basispap {to simulate Cronbach's Alpha and uses R code to do so}. It has been used professionally on several occasions where a simulated study on Cronbach's Alpha was needed. Additionally, it can provide prefixed Likert scale mean values prior to the study as well as number of participants on demand. It has been a useful tool to reduce time and money needs. In this paper a brief presentation of the algorithm is given as well as potential and weaknesses. The results of the simulation show improved behavior for large number of simulated participants and replications (N > 100,000) but the number of simulated questions does remain satisfactory even in the case of small numbers such as 10 questions. The flexibility of the algorithm needs to be improved for alpha values lower than 0.7. This means that an unexperienced user might find it difficult to perform simulations for requested alpha values lower than 0.7 although a more experienced user can easily adjust requested alpha in order to obtain the exact needed value. Applied Statistics Cronbach’s alpha Simulation simAlpha Application Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Introduction The reliability of a questionnaire is the main concern of a researcher when constructing a data collection tool assuring that this tool can assure it repeatability as well as the reassurance of valid research. Validity and reliability are quite often used in the same conceptual framework but Tavakol and Dennick (Tavakol and Dennick 2011) emphasize that an instrument cannot be valid unless it is reliable. Still, the importance of reliability is undeniable since it is an indicator of good practice in research. 2. Cronbach’s Alpha The most widely known and used indicator of reliability is Cronbach’s Alpha (Cronbach 1951 ). This index incorporates the variance of each observation and each variable under the formula $$\:a=\:\frac{k}{k-1}\left[\frac{{\sigma\:}_{\tau\:}^{2}-\sum\:_{\iota\:=1}^{\kappa\:}{\sigma\:}_{i}^{2}}{{\sigma\:}_{\tau\:}^{2}}\right]$$ 1 where \(\:{\sigma\:}_{i}^{2}\) is the variance of each variable denoted as column of \(\:n\times\:k\) dataset of n observations and k variables (n ; k > 1). The incorporation of each row is done with the help of \(\:{\sigma\:}_{\tau\:}^{2}\) which denotes row variance. Although alpha can show negative values, it produces informative results when it lies between 0 and 1 and shows a reliable result when it is greater than 0.7. 3. Past simulation attempts Although not expected, not many attempts at simulating alpha values have been made in the past. Leontitsis and Page (Leontitsis and Pagge 2007 ) used a simulation of 20 subjects with 6 items under 1,000 replication, did not produce alpha values grater than 0.5. The comparison of their results to other attempts (Hakstian and Whalen 1976 ; Cortina 1993 ; Charter 2000 ; Duhachek and Iacobucci 2004 ) presented wide ranges of alpha values with the lowest being that of Cortina (Cortina 1993 ) and equal to 0.20 [0.40–0.60]. Methodology In order to simulate an alpha value, one needs to know the number participants and the research tool. simAlpha allows the use of the mean value per question – variable of the questionnaire as well. In total simAlpha can use 4 parameters. These are, number of participants [2, ∞), number of questions [2, ∞), mean value and standard deviation per item – question. In what follows simulation results are presented on a 5-point Likert scale with N participants, Q items and Rep number of replications. The purpose of this simulation is to present its capabilities and its performance on requested alpha values. This simulation was done with R programming language Version 4.2.3 in Rstudio environment. Results Effect of replications The first part of the results shows the distribution and behavior of simAlpha algorithm on variable parameters. Figure 1 shows the distribution of simulated alpha values for a hypothetical questionnaire of 10 items. All items are supposed to have mean value near 3 and standard deviation near 1. We should note that simAlpha can also use different mean values and standard deviations per item. The use of common mean and SD values is done only for comparison reasons. The first comparison uses variable replication numbers ranging from 10 to 1,000,000 replications. According to the results of table 1 and figure 1 the increment in replications does not contribute to the goal alpha value of 0.95 especially after 1,000 replications. Still, it is worth noting that increment in replication shifts the distribution to the right with lower min and increased max values. This shift is depicted in figure 1. Table 1. Effect of replication number to alpha approximation (N = 200, Q = 10, Alpha = 0.95) Replication Mean Min Max Elapsed time 10 0.9421422 0.9293807 0.9522801 < 1 second 100 0.9451850 0.9242415 0.9550180 1 second 1,000 0.9464342 0.9228699 0.9609187 2 seconds 10,000 0.9464284 0.9164307 0.9626803 23 seconds 100,000 0.9463887 0.9164307 0.9636376 231 seconds 1,000,000 0.9463948 0.9112057 0.9661475 2223 seconds Effect of participants The next part was to investigate the behavior of simAlpha algorithm to a variable number of participants. The range used was between 10 to 1,000,000 participants and the results for 1,000 replications are shown in table 2 and figure 2. According to the results, 100 participants are enough for a good approximation of the 0.95 targeted alpha value. In addition, we can see that more participants narrow down the min-max interval of simulated alpha values. Finally, figure 2 shows that 1,000 participants produce a more symmetrical distribution of alpha values compared to a larger number of participants. Table 2. Effect of participant number to alpha approximation (Rep = 1,000, Q = 10, Alpha = 0.95) Participants Mean Min Max Elapsed time 10 0.9324295 0.5054251 0.9879183 1 second 100 0.9457435 0.9129090 0.9667423 2 seconds 1,000 0.9468957 0.9369180 0.9535305 4 seconds 10,000 0.9468373 0.9438134 0.9494351 21 seconds 100,000 0.9468561 0.9460229 0.9475808 260 seconds 1,000,000 0.9468629 0.9466537 0.9470779 2286 seconds Effect of number of items The last part of the results included algorithm performance under various item number. According to the results of table 3, there is an overestimation of alpha for questionnaires with more than 10 questions. This overestimation increases along with the increment of the number of questions but it distribution does not show improvement, that is a more symmetrical behavior as shown in figure. Table 3. Effect of participant number to alpha approximation (Rep = 1,000, N = 200, Alpha = 0.95) Questions Mean Min Max Elapsed time 10 0.9464342 0.9228699 0.9609187 2 seconds 20 0.9725333 0.9621068 0.9801231 4 seconds 30 0.9814750 0.9734824 0.9863947 5 seconds 40 0.9859774 0.9796366 0.9897918 7 seconds 50 0.9888061 0.9844060 0.9915331 8 seconds 100 0.9943914 0.9922301 0.9956507 2223 seconds Benchmarking The last step was to check deviations from the requested alpha value. Figure 4 shows mean values of calculated alpha values along with upper and lower limits expressed with the maximum and minimum values. This simulation used 0.001 to 0.999 alpha value approximation with 0.001 step, under 200 hypothetical participants, on a 10 time research tool. According to figure 4 there is a deviance which varies from 0.1 to 0.5 for requested alpha values from 0.1 to 0.9. The largest deviation (see also figure 5) is shown in 0.2 to 0.4 requested alpha value. In the case of the lower critical alpha value, that is α = 0.7 the mean deviation is near 0.25 . These results clearly show the need for improvement of the target alpha value. Although a rule of thumbs can be, and is, used to approximate specific alpha values even up to the third decimal place, this improvement is necessary in order to present a more user-friendly algorithm, that is an algorithm that anyone can use regardless of previous background in computing. In addition, since this is only an instance of the possible parameters can enter the simulation, more parameters should be used to compare the algorithms behavior. Conclusions simAlpha algorithm can cope quite well with the requested task, that is to simulate a complete study with a questionnaire under a specific Alpha value. Still, it needs improvement on the requested alpha value for an inexperienced user or a user who is looking for an absolute Alpha value without deviation on the second or third decimal number. For a more lenient or a more experienced user it can be a helpful asset for pilot studies or any kind of simulation study of a new or well-established research tool. Computation time is low, to tolerable for studies that want to involve a large number of hypothetical subjects. simAlpha can support the usual parameters of a hypothetical study i.e. N = 200 participants, Q = 20 items on 1,000 replications with small deviations from the requested alpha. The time needed on these parameters is less than 4 seconds. Therefore simAlpha algorithm is expected to aim academics on their simulated study attempts, with the expectation of future improvement in the future on Alpha accuracy to lie below the third decimal place. References Charter RA (2000) Confidence Interval Formulas for Split-Half Reliability Coefficients. Psychol Rep 86:1168–1170. https://doi.org/10.2466/pr0.2000.86.3c.1168 Cortina JM (1993) What is coefficient alpha? An examination of theory and applications. J Appl Psychol 78:98–104. https://doi.org/10.1037/0021-9010.78.1.98 Cronbach LJ (1951) Coefficient Alpha and the Internal Structure of Tests. Psychometrika 16:297–334. https://doi.org/10.1007/BF02310555 Duhachek A, Iacobucci D (2004) Alpha’s Standard Error (ASE): An Accurate and Precise Confidence Interval Estimate. J Appl Psychol 89:792–808. https://doi.org/10.1037/0021-9010.89.5.792 Hakstian AR, Whalen TE (1976) A K-Sample Significance Test for Independent Alpha Coefficients. Psychometrika 41:219–231. https://doi.org/10.1007/BF02291840 Leontitsis A, Pagge J (2007) A simulation approach on Cronbach’s alpha statistical significance. Math Comput Simul 73:336–340. https://doi.org/10.1016/j.matcom.2006.08.001 Tavakol M, Dennick R (2011) Making sense of Cronbach’s alpha. Int J Med Educ 2:53–55. https://doi.org/10.5116/ijme.4dfb.8dfd Additional Declarations The authors declare no competing interests. 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2","display":"","copyAsset":false,"role":"figure","size":6305,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of alpha values for 10 to 1,000,000 participants\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-7726096/v1/79ca0504fe0cabd1f5d91770.png"},{"id":92491578,"identity":"5373f8b1-69d3-488d-8b2f-bc48e86616b7","added_by":"auto","created_at":"2025-09-30 09:40:46","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":6353,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of alpha values for 10 to 100 items\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-7726096/v1/e1dafe6ca4b50d9c1a96c8d5.png"},{"id":92491580,"identity":"bd58e942-2165-4782-8ef2-f3712cb517b5","added_by":"auto","created_at":"2025-09-30 09:40:46","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":9403,"visible":true,"origin":"","legend":"\u003cp\u003eSimulated Alpha values under Ν = 200, Q = 10, Rep = 100, Step = 0.001.\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-7726096/v1/1a561cb1201c12278c98b598.png"},{"id":92491586,"identity":"3ed99206-4406-4ae3-bac8-71bfa441402d","added_by":"auto","created_at":"2025-09-30 09:40:46","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":10489,"visible":true,"origin":"","legend":"\u003cp\u003eDeviation and absolute deviation from requested Alpha values under Ν = 200, Q = 10, Rep = 100, Step = 0.001.\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-7726096/v1/9fed89197e59356d893f1dad.png"},{"id":92494716,"identity":"71684753-c7ed-4bcf-8484-b9aa32d4d826","added_by":"auto","created_at":"2025-09-30 10:12:51","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":379712,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7726096/v1/ad633374-77e1-4130-890c-0cd6bc0ce8ca.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eAn introduction to simulating Cronbach’s Alpha values with the help of simAlpha© algorithm\u003c/p\u003e","fulltext":[{"header":"Introduction ","content":"\u003cp\u003eThe reliability of a questionnaire is the main concern of a researcher when constructing a data collection tool assuring that this tool can assure it repeatability as well as the reassurance of valid research. Validity and reliability are quite often used in the same conceptual framework but Tavakol and Dennick (Tavakol and Dennick 2011) emphasize that an instrument cannot be valid unless it is reliable. Still, the importance of reliability is undeniable since it is an indicator of good practice in research.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Cronbach\u0026rsquo;s Alpha\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe most widely known and used indicator of reliability is Cronbach\u0026rsquo;s Alpha (Cronbach \u003cspan class=\"CitationRef\"\u003e1951\u003c/span\u003e). This index incorporates the variance of each observation and each variable under the formula\u003c/p\u003e\n\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$$\\:a=\\:\\frac{k}{k-1}\\left[\\frac{{\\sigma\\:}_{\\tau\\:}^{2}-\\sum\\:_{\\iota\\:=1}^{\\kappa\\:}{\\sigma\\:}_{i}^{2}}{{\\sigma\\:}_{\\tau\\:}^{2}}\\right]$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\sigma\\:}_{i}^{2}\\)\u003c/span\u003e\u003c/span\u003eis the variance of each variable denoted as column of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:n\\times\\:k\\)\u003c/span\u003e\u003c/span\u003e dataset of n observations and k variables (n ; k\u0026thinsp;\u0026gt;\u0026thinsp;1). The incorporation of each row is done with the help of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\sigma\\:}_{\\tau\\:}^{2}\\)\u003c/span\u003e\u003c/span\u003e which denotes row variance. Although alpha can show negative values, it produces informative results when it lies between 0 and 1 and shows a reliable result when it is greater than 0.7.\u003c/p\u003e\n\u003ch3\u003e3. Past simulation attempts\u0026nbsp;\u003c/h3\u003e\n\u003cp\u003eAlthough not expected, not many attempts at simulating alpha values have been made in the past. Leontitsis and Page (Leontitsis and Pagge \u003cspan class=\"CitationRef\"\u003e2007\u003c/span\u003e) used a simulation of 20 subjects with 6 items under 1,000 replication, did not produce alpha values grater than 0.5. The comparison of their results to other attempts (Hakstian and Whalen \u003cspan class=\"CitationRef\"\u003e1976\u003c/span\u003e; Cortina \u003cspan class=\"CitationRef\"\u003e1993\u003c/span\u003e; Charter \u003cspan class=\"CitationRef\"\u003e2000\u003c/span\u003e; Duhachek and Iacobucci \u003cspan class=\"CitationRef\"\u003e2004\u003c/span\u003e) presented wide ranges of alpha values with the lowest being that of Cortina (Cortina \u003cspan class=\"CitationRef\"\u003e1993\u003c/span\u003e) and equal to 0.20 [0.40\u0026ndash;0.60].\u003c/p\u003e"},{"header":"Methodology","content":"\u003cp\u003eIn order to simulate an alpha value, one needs to know the number participants and the research tool. simAlpha allows the use of the mean value per question – variable of the questionnaire as well. In total simAlpha can use 4 parameters. These are, number of participants [2, ∞), number of questions [2, ∞), mean value and standard deviation per item – question. In what follows simulation results are presented on a 5-point Likert scale with N participants, Q items and Rep number of replications. The purpose of this simulation is to present its capabilities and its performance on requested alpha values. This simulation was done with R programming language Version 4.2.3 in Rstudio environment.\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003e\u003cem\u003eEffect of replications\u0026nbsp;\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThe first part of the results shows the distribution and behavior of simAlpha algorithm on variable parameters. Figure 1 shows the distribution of simulated alpha values for a hypothetical questionnaire of 10 items. All items are supposed to have mean value near 3 and standard deviation near 1. We should note that simAlpha can also use different mean values and standard deviations per item. The use of common mean and SD values is done only for comparison reasons. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe first comparison uses variable replication numbers ranging from 10 to 1,000,000 replications. According to the results of \u0026nbsp; table 1 and figure 1 the increment in replications does not contribute to the goal alpha value of 0.95 especially after 1,000 replications. Still, it is worth noting that increment in replication shifts the distribution to the right with lower min and increased max values. This shift is depicted in figure 1.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTable 1. Effect of replication number to alpha approximation (N = 200, Q = 10, Alpha = 0.95)\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003eReplication\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003eMean\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003eMin\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003eMax\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003eElapsed time\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9421422\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9293807\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9522801\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e\u0026lt; 1 second\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e100\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9451850\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9242415\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9550180\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e1 second\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e1,000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9464342\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9228699\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9609187\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e2 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e10,000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9464284\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9164307\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9626803\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e23 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e100,000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9463887\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9164307\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9636376\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e231 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e1,000,000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9463948\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9112057\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9661475\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e2223 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cem\u003eEffect of participants\u0026nbsp;\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThe next part was to investigate the behavior of simAlpha algorithm to a variable number of participants. The range used was between 10 to 1,000,000 participants and the results for 1,000 replications are shown in table 2 and figure 2. According to the results, 100 participants are enough for a good approximation of the 0.95 targeted alpha value. In addition, we can see that more participants narrow down the min-max interval of simulated alpha values. Finally, figure 2 shows that 1,000 participants produce a more symmetrical distribution of alpha values compared to a larger number of participants.\u003c/p\u003e\n\u003cp\u003eTable 2. Effect of participant number to alpha approximation (Rep = 1,000, Q = 10, Alpha = 0.95)\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003eParticipants\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003eMean\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003eMin\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003eMax\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003eElapsed time\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9324295\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.5054251\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9879183\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e\u0026nbsp;1 second\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e100\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9457435\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9129090\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9667423\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e2 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e1,000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9468957\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9369180\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9535305\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e4 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e10,000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9468373\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9438134\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9494351\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e21 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e100,000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9468561\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9460229\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9475808\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e260 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e1,000,000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9468629\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9466537\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9470779\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e2286 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cem\u003eEffect of number of items\u0026nbsp;\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThe last part of the results included algorithm performance under various item number. According to the results of table 3, there is an overestimation of alpha for questionnaires with more than 10 questions. This overestimation increases along with the increment of the number of questions but it distribution does not show improvement, that is a more symmetrical behavior as shown in figure.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTable 3. Effect of participant number to alpha approximation (Rep = 1,000, N = 200, Alpha = 0.95)\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003eQuestions\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003eMean\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003eMin\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003eMax\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003eElapsed time\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9464342\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9228699\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9609187\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e2 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9725333\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9621068\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9801231\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e4 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9814750\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9734824\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9863947\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e5 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9859774\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9796366\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9897918\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e7 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9888061\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9844060\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9915331\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e8 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 112px;\"\u003e\n \u003cp\u003e100\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9943914\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\n \u003cp\u003e0.9922301\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 113px;\"\u003e\n \u003cp\u003e0.9956507\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 115px;\"\u003e\n \u003cp\u003e2223 seconds\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cem\u003eBenchmarking\u0026nbsp;\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThe last step was to check deviations from the requested alpha value. Figure 4 shows mean values of calculated alpha values along with upper and lower limits expressed with the maximum and minimum values. This simulation used 0.001 to 0.999 alpha value approximation with 0.001 step, under 200 hypothetical participants, on a 10 time research tool.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAccording to figure 4 there is a deviance which varies from 0.1 to 0.5 for requested alpha values from 0.1 to 0.9. The largest deviation (see also figure 5) is shown in 0.2 to 0.4 requested alpha value. In the case of the lower critical alpha value, that is \u0026alpha; = 0.7 the mean deviation is near 0.25 . These results clearly show the need for improvement of the target alpha value. Although a rule of thumbs can be, and is, used to approximate specific alpha values even up to the third decimal place, this improvement is necessary in order to present a more user-friendly algorithm, that is an algorithm that anyone can use regardless of previous background in computing. In addition, since this is only an instance of the possible parameters can enter the simulation, more parameters should be used to compare the algorithms behavior.\u0026nbsp;\u003c/p\u003e"},{"header":"Conclusions","content":"\u003cp\u003esimAlpha algorithm can cope quite well with the requested task, that is to simulate a complete study with a questionnaire under a specific Alpha value. Still, it needs improvement on the requested alpha value for an inexperienced user or a user who is looking for an absolute Alpha value without deviation on the second or third decimal number. For a more lenient or a more experienced user it can be a helpful asset for pilot studies or any kind of simulation study of a new or well-established research tool. Computation time is low, to tolerable for studies that want to involve a large number of hypothetical subjects. simAlpha can support the usual parameters of a hypothetical study i.e. N\u0026thinsp;=\u0026thinsp;200 participants, Q\u0026thinsp;=\u0026thinsp;20 items on 1,000 replications with small deviations from the requested alpha. The time needed on these parameters is less than 4 seconds. Therefore simAlpha algorithm is expected to aim academics on their simulated study attempts, with the expectation of future improvement in the future on Alpha accuracy to lie below the third decimal place.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eCharter RA (2000) Confidence Interval Formulas for Split-Half Reliability Coefficients. Psychol Rep 86:1168\u0026ndash;1170. https://doi.org/10.2466/pr0.2000.86.3c.1168\u003c/li\u003e\n \u003cli\u003eCortina JM (1993) What is coefficient alpha? An examination of theory and applications. J Appl Psychol 78:98\u0026ndash;104. https://doi.org/10.1037/0021-9010.78.1.98\u003c/li\u003e\n \u003cli\u003eCronbach LJ (1951) Coefficient Alpha and the Internal Structure of Tests. Psychometrika 16:297\u0026ndash;334. https://doi.org/10.1007/BF02310555\u003c/li\u003e\n \u003cli\u003eDuhachek A, Iacobucci D (2004) Alpha\u0026rsquo;s Standard Error (ASE): An Accurate and Precise Confidence Interval Estimate. J Appl Psychol 89:792\u0026ndash;808. https://doi.org/10.1037/0021-9010.89.5.792\u003c/li\u003e\n \u003cli\u003eHakstian AR, Whalen TE (1976) A K-Sample Significance Test for Independent Alpha Coefficients. Psychometrika 41:219\u0026ndash;231. https://doi.org/10.1007/BF02291840\u003c/li\u003e\n \u003cli\u003eLeontitsis A, Pagge J (2007) A simulation approach on Cronbach\u0026rsquo;s alpha statistical significance. Math Comput Simul 73:336\u0026ndash;340. https://doi.org/10.1016/j.matcom.2006.08.001\u003c/li\u003e\n \u003cli\u003eTavakol M, Dennick R (2011) Making sense of Cronbach\u0026rsquo;s alpha. Int J Med Educ 2:53\u0026ndash;55. https://doi.org/10.5116/ijme.4dfb.8dfd\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"BasisPap","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Cronbach’s alpha, Simulation, simAlpha, Application","lastPublishedDoi":"10.21203/rs.3.rs-7726096/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7726096/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003esimAlpha algorithm was designed in 2023 from Basispap {to simulate Cronbach's Alpha and uses R code to do so}. It has been used professionally on several occasions where a simulated study on Cronbach's Alpha was needed. Additionally, it can provide prefixed Likert scale mean values prior to the study as well as number of participants on demand. It has been a useful tool to reduce time and money needs. In this paper a brief presentation of the algorithm is given as well as potential and weaknesses. The results of the simulation show improved behavior for large number of simulated participants and replications (N \u0026gt; 100,000) but the number of simulated questions does remain satisfactory even in the case of small numbers such as 10 questions. The flexibility of the algorithm needs to be improved for alpha values lower than 0.7. This means that an unexperienced user might find it difficult to perform simulations for requested alpha values lower than 0.7 although a more experienced user can easily adjust requested alpha in order to obtain the exact needed value.\u003c/p\u003e","manuscriptTitle":"An introduction to simulating Cronbach’s Alpha values with the help of simAlpha© algorithm","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-09-30 09:40:42","doi":"10.21203/rs.3.rs-7726096/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"57f0572e-b1a0-4aa6-9265-c1e22b4eb78a","owner":[],"postedDate":"September 30th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":55425169,"name":"Applied Statistics"}],"tags":[],"updatedAt":"2025-09-30T09:40:42+00:00","versionOfRecord":[],"versionCreatedAt":"2025-09-30 09:40:42","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7726096","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7726096","identity":"rs-7726096","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}