微信公眾號：醫學統計與R語言
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在非常多的中文文獻中關於多重線回歸分析的樣本量是這樣描述的：
「根據Kendall粗糙確定樣本量原則，樣本量可取變量數的5-10倍，考慮到失訪和不合作情況，在原有樣本量的基礎上再擴大20%作為擬調查的樣本例數。」，本人花了一上午也沒找到這篇文獻的原始出處。國內文章引用的資料來源主要為《臨床流行病學 臨床科研設計、測量與評價》一書，在此書中有如下描述：
如有人能提供關於Kendall的參考文獻，請留言。
關於回歸分析中樣本量的經驗法則(Rules of Thumb)，可參考如下：
1.From Wikipedia. https://en.wikipedia.org/wiki/One_in_ten_ruleIn statistics, the one in ten rule is a rule of thumb for how many predictor parameters can be estimated from data when doing regression analysis (in particular proportional hazards models in survival analysis and logistic regression) while keeping the risk of overfitting low. The rule states that one predictive variable can be studied for every ten events.[1][2][3][4] For logistic regression the number of events is given by the size of the smallest of the outcome categories, and for survival analysis it is given by the number of uncensored events.[3]
For example, if a sample of 200 patients are studied and 20 patients die during the study (so that 180 patients survive), the one in ten rule implies that two pre-specified predictors can reliably be fitted to the total data. Similarly, if 100 patients die during the study (so that 100 patients survive), ten pre-specified predictors can be fitted reliably. If more are fitted, the rule implies that overfitting is likely and the results will not predict well outside the training data. It is not uncommon to see the 1:10 rule violated in fields with many variables (e.g. gene expression studies in cancer), decreasing the confidence in reported findings.[5]
ImprovementsA "one in 20 rule" has been suggested, indicating the need for shrinkage of regression coefficients, and a "one in 50 rule" for stepwise selection with the default p-value of 5%.[4][6] Other studies, however, show that the one in ten rule may be too conservative as a general recommendation and that five to nine events per predictor can be enough, depending on the research question.[7]
More recently, a study has shown that the ratio of events per predictive variable is not a reliable statistic for estimating the minimum number of events for estimating a logistic prediction model.[8] Instead, the number of predictor variables, the total sample size (events + non-events) and the events fraction (events / total sample size) can be used to calculate the expected prediction error of the model that is to be developed.[9] One can then estimate the required sample size to achieve an expected prediction error that is smaller than a predetermined allowable prediction error value.[9]
Alternatively, three requirements for prediction model estimation have been suggested: the model should have a global shrinkage factor of ≥ .9, an absolute difference of ≤ .05 in the model's apparent and adjusted Nagelkerke R2, and a precise estimation of the overall risk or rate in the target population.[10] The necessary sample size and number of events for model development are then given by the values that meet these requirements.[10]
[1] Harrell, F. E. Jr.; Lee, K. L.; Califf, R. M.; Pryor, D. B.; Rosati, R. A. (1984). "Regression modelling strategies for improved prognostic prediction". Stat Med. 3 (2): 143–52. doi:10.1002/sim.4780030207.
[2] Harrell, F. E. Jr.; Lee, K. L.; Mark, D. B. (1996). "Multivariable prognostic models: issues in developing models, evaluating assumptions and adequacy, and measuring and reducing errors" (PDF). Stat Med. 15 (4): 361–87. doi:10.1002/(sici)1097-0258(19960229)15:4<361::aid-sim168>3.0.co;2-4.
[3] Peduzzi, Peter; Concato, John; Kemper, Elizabeth; Holford, Theodore R.; Feinstein, Alvan R. (1996). "A simulation study of the number of events per variable in logistic regression analysis". Journal of Clinical Epidemiology. 49 (12): 1373–1379. doi:10.1016/s0895-4356(96)00236-3. PMID 8970487.
[4] "Chapter 8: Statistical Models for Prognostication: Problems with Regression Models". Archived from the original on October 31, 2004. Retrieved 2013-10-11.
[5] Ernest S. Shtatland, Ken Kleinman, Emily M. Cain. Model building in Proc PHREG with automatic variable selection and information criteria. Paper 206–30 in SUGI 30 Proceedings, Philadelphia, Pennsylvania April 10–13, 2005. http://www2.sas.com/proceedings/sugi30/206-30.pdf
[6] Steyerberg, E. W.; Eijkemans, M. J.; Harrell, F. E. Jr.; Habbema, J. D. (2000). "Prognostic modelling with logistic regression analysis: a comparison of selection and estimation methods in small data sets". Stat Med. 19 (8): 1059–1079. doi:10.1002/(sici)1097-0258(20000430)19:8<1059::aid-sim412>3.0.co;2-0.
[7] Vittinghoff, E.; McCulloch, C. E. (2007). "Relaxing the Rule of Ten Events per Variable in Logistic and Cox Regression". American Journal of Epidemiology. 165 (6): 710–718. doi:10.1093/aje/kwk052. PMID 17182981.
[8]van Smeden, Maarten; de Groot, Joris A. H.; Moons, Karel G. M.; Collins, Gary S.; Altman, Douglas G.; Eijkemans, Marinus J. C.; Reitsma, Johannes B. (2016-11-24). "No rationale for 1 variable per 10 events criterion for binary logistic regression analysis". BMC Medical Research Methodology. 16 (1): 163. doi:10.1186/s12874-016-0267-3. ISSN 1471-2288. PMC 5122171. PMID 27881078.
[9]van Smeden, Maarten; Moons, Karel Gm; de Groot, Joris Ah; Collins, Gary S.; Altman, Douglas G.; Eijkemans, Marinus Jc; Reitsma, Johannes B. (2018-01-01). "Sample size for binary logistic prediction models: Beyond events per variable criteria". Statistical Methods in Medical Research. 28: 962280218784726. doi:10.1177/0962280218784726. ISSN 1477-0334. PMID 29966490.
[10] Riley, Richard D.; Snell, Kym IE; Ensor, Joie; Burke, Danielle L.; Jr, Frank E. Harrell; Moons, Karel GM; Collins, Gary S. (2018). "Minimum sample size for developing a multivariable prediction model: PART II - binary and time-to-event outcomes". Statistics in Medicine. 0: 1276–1296. doi:10.1002/sim.7992. ISSN 1097-0258. PMC 6519266. PMID 30357870.
Although there are more complex formulae, the general rule of thumb is no less than 50 participants for a correlation or regression with the number increasing with larger numbers of independent variables (IVs). Green (1991)provides a comprehensive overview of the procedures used to determine regression sample sizes. He suggestsN > 50 + 8m (where m is the number of IVs) for testing the multiple correlation andN > 104 + mfor testing individual predictors (assuming a medium‐sized relationship). If testing both, use the larger sample size.
Although Greenʹs (1991) formula is more comprehensive,there are two other rules of thumb that could be used. With five or fewer predictors (this number would include correlations), a researcher can use Harrisʹs (1985) formula for yielding the absolute minimum number of participants.Harris suggests that the number of participants should exceed the number of predictors by at least 50 (i.e., total number of participants equals the number of predictor variables plus 50)‐‐a formula much the same as Greenʹs mentioned above.For regression equations using six or more predictors, an absolute minimum of 10 participants per predictor variable is appropriate. However, if the circumstances allow, a researcher would have better power to detect a small effect size with approximately 30 participants per variable. For instance, Cohen and Cohen (1975) demonstrate that with a single predictor that in the population correlates with the DV at .30, 124 participants are needed to maintain 80% power. With five predictors and a population correlation of .30, 187 participants would be needed to achieve 80% power. Larger samples are needed when the DV is skewed, the effect size expected is small, there is substantial measurement error, or stepwise regression is being used(Tabachnick & Fidell, 1996).
[1] Green S B. How many subjects does it take to do a regression analysis[J]. Multivariate behavioral research, 1991, 26(3): 499-510.
[2]VanVoorhis C R W, Morgan B L. Understanding power and rules of thumb for determining sample sizes[J]. Tutorials in quantitative methods for psychology, 2007, 3(2): 43-50.http://www.tqmp.org/RegularArticles/vol03-2/p043/p043.pdf
[3]Harris, R. J. (1985). A primer of multivariate statistics (2nd ed.).New York: Academic Press.
[4]https://www.youtube.com/watch?v=PD_xC3Xtqlw
[5]https://www.youtube.com/watch?v=v7YlRye7aMw
[6]https://books.google.com.sg/books?id=vlO2BQAAQBAJ&pg=PA212&lpg=PA212&dq=Rules+of+Thumb+++multiple++regression&source=bl&ots=XBb5fizvwZ&sig=ACfU3U3Y7649s2O4WpG4_8UOvR0gvecrNQ&hl=zh-CN&sa=X&redir_esc=y#v=onepage&q&f=false
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