The Stata Journal Editor H. Joseph Newton Department of Statistics Texas A&M University College Station, Texas 77843 979-845-8817; fax 979-845-6077 jnewton@stata-journal.com Associate Editors Christopher F. Baum Boston College Nathaniel Beck New York University Rino Bellocco Karolinska Institutet, Sweden, and University of Milano-Bicocca, Italy Maarten L. Buis Tübingen University, Germany A. Colin Cameron University of California Davis Mario A. Cleves Univ. of Arkansas for Medical Sciences William D. Dupont Vanderbilt University David Epstein Columbia University Allan Gregory Queen s University James Hardin University of South Carolina Ben Jann University of Bern, Switzerland Stephen Jenkins University of Essex Ulrich Kohler WZB, Berlin Frauke Kreuter University of Maryland College Park Stata Press Editorial Manager Stata Press Copy Editors Editor Nicholas J. Cox Department of Geography Durham University South Road Durham DH1 3LE UK n.j.cox@stata-journal.com Peter A. Lachenbruch Oregon State University Jens Lauritsen Odense University Hospital Stanley Lemeshow Ohio State University J. Scott Long Indiana University Roger Newson Imperial College, London Austin Nichols Urban Institute, Washington DC Marcello Pagano Harvard School of Public Health Sophia Rabe-Hesketh University of California Berkeley J. Patrick Royston MRC Clinical Trials Unit, London Philip Ryan University of Adelaide Mark E. Schaffer Heriot-Watt University, Edinburgh Jeroen Weesie Utrecht University Nicholas J. G. Winter University of Virginia Jeffrey Wooldridge Michigan State University Lisa Gilmore Deirdre Patterson and Erin Roberson
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The Stata Journal (2010) 10, Number 3, pp. 496 499 Stata tip 89: Estimating means and percentiles following multiple imputation Peter A. Lachenbruch Oregon State University Corvallis, OR peter.lachenbruch@oregonstate.edu 1 Introduction In a statistical analysis, I usually want some basic descriptive statistics such as the mean, standard deviation, extremes, and percentiles. See, for example, Pagano and Gauvreau (2000). Stata conveniently provides these descriptive statistics with the summarize command s detail option. Alternatively, I can obtain percentiles with the centile command. For example, with auto.dta, we have. sysuse auto (1978 Automobile Data). summarize price, detail Price Percentiles Smallest 1% 3291 3291 5% 3748 3299 10% 3895 3667 Obs 74 25% 4195 3748 Sum of Wgt. 74 50% 5006.5 Mean 6165.257 Largest Std. Dev. 2949.496 75% 6342 13466 90% 11385 13594 Variance 8699526 95% 13466 14500 Skewness 1.653434 99% 15906 15906 Kurtosis 4.819188 However, if I have missing values, the summarize command is not supported by mi estimate or by the user-written mim command (Royston 2004, 2005a,b, 2007; Royston, Carlin, and White 2009). 2 Finding means and percentiles when missing values are present For a general multiple-imputation reference, see Stata 11 Multiple-Imputation Reference Manual (2009). By recognizing that a regression with no independent variables estimates the mean, I can usemi estimate: regress to get multiply imputed means. If I wish to get multiply imputed quantiles, I can use mi estimate: qreg or mi estimate: sqreg for this purpose. c 2010 StataCorp LP st0205
P. A. Lachenbruch 497 I now create a dataset with missing values of price:. clonevar newprice = price. set seed 19670221. replace newprice =. if runiform() <.4 (32 real changes made, 32 to missing) The following commands were generated from the multiple-imputation dialog box. I used 20 imputations. Before Stata 11, this could also be done with the user-written commands ice and mim (Royston 2004, 2005a,b, 2007; Royston, Carlin, and White 2009).. mi set mlong. mi register imputed newprice (32 m=0 obs. now marked as incomplete). mi register regular mpg trunk weight length. mi impute regress newprice, add(20) rseed(3252010) Univariate imputation Imputations = 20 Linear regression added = 20 Imputed: m=1 through m=20 updated = 0 Observations per m Variable complete incomplete imputed total newprice 42 32 32 74 (complete + incomplete = total; imputed is the minimum across m of the number of filled in observations.). mi estimate: regress newprice Multiple-imputation estimates Imputations = 20 Linear regression Number of obs = 74 Average RVI = 1.3880 Complete DF = 73 DF: min = 19.46 avg = 19.46 DF adjustment: Small sample max = 19.46 F( 0,.) =. Within VCE type: OLS Prob > F =. newprice Coef. Std. Err. t P> t [95% Conf. Interval] _cons 5693.489 454.9877 12.51 0.000 4742.721 6644.258 From this output, we see that the estimated mean is 5,693 with a standard error of 455 (rounded up) compared with the complete data value of 6,165 with a standard error of 343 (also rounded up). However, we do not have estimates of quantiles. This could also have been done using mi estimate: mean newprice (the mean command is near the bottom of the estimation command list for mi estimate).
498 Stata tip 89 We can apply the same principle using qreg. For the 10th percentile, type. mi estimate: qreg newprice, quantile(10) Multiple-imputation estimates Imputations = 20.1 Quantile regression Number of obs = 74 Average RVI = 0.2901 Complete DF = 73 DF: min = 48.05 avg = 48.05 DF adjustment: Small sample max = 48.05 F( 0,.) =. Prob > F =. newprice Coef. Std. Err. t P> t [95% Conf. Interval] _cons 3495.635 708.54 4.93 0.000 2071.058 4920.212 Compare the value of 3,496 with the value of 3,895 from the full data. We can use the simultaneous estimates command for the full set:. mi estimate: sqreg newprice, quantiles(10 25 50 75 90) reps(20) Multiple-imputation estimates Imputations = 20 Simultaneous quantile regression Number of obs = 74 Average RVI = 0.6085 Complete DF = 73 DF adjustment: Small sample DF: min = 23.19 avg = 26.97 max = 31.65 newprice Coef. Std. Err. t P> t [95% Conf. Interval] q10 q25 q50 q75 q90 _cons 3495.635 533.5129 6.55 0.000 2408.434 4582.836 _cons 4130.037 237.1932 17.41 0.000 3642.459 4617.614 _cons 5200.238 441.294 11.78 0.000 4292.719 6107.757 _cons 6620.232 778.8488 8.50 0.000 5025.49 8214.974 _cons 8901.985 1417.022 6.28 0.000 5971.962 11832.01 3 Comments and cautions The qreg command does not give the same result as the centile command when you have complete data. This is because the centile command uses one observation, while the qreg command uses a weighted combination of the observations. It will have somewhat shorter confidence intervals, but with large datasets, the difference will be
P. A. Lachenbruch 499 small. A second caution is that comparing two medians can be tricky: the difference of two medians is not the median difference of the distributions. I have found it useful to use percentiles because there is a one-to-one relationship between percentiles if data are transformed. In our case, there is plentiful evidence that price is not normally distributed, so it would be good to look for a transformation and impute those values. This method of using regression commands without an independent variable can provide estimates of quantities that otherwise would be difficult to obtain. For example, it is much faster than finding 20 imputed percentiles and then combining them with Rubin s rules, and it is much less onerous and prone to error. 4 Acknowledgment This work was supported in part by a grant from the Cure JM Foundation. References Pagano, M., and K. Gauvreau. 2000. Principles of Biostatistics. 2nd ed. Belmont, CA: Duxbury. Royston, P. 2004. Multiple imputation of missing values. Stata Journal 4: 227 241.. 2005a. Multiple imputation of missing values: Update. Stata Journal 5: 188 201.. 2005b. Multiple imputation of missing values: Update of ice. Stata Journal 5: 527 536.. 2007. Multiple imputation of missing values: Further update of ice, with an emphasis on interval censoring. Stata Journal 7: 445 464. Royston, P., J. B. Carlin, and I. R. White. 2009. Multiple imputation of missing values: New features for mim. Stata Journal 9: 252 264. StataCorp. 2009. Stata 11 Multiple-Imputation Reference Manual. College Station, TX: Stata Press.