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Title stata.com sdtest — Variance-comparison tests Syntax Remarks and examples Also see Menu Stored results Description Methods and formulas Options References Syntax One-sample variance-comparison test sdtest varname == # if in , level(#) Two-sample variance-comparison test using groups sdtest varname if in , by(groupvar) level(#) Two-sample variance-comparison test using variables sdtest varname1 == varname2 if in , level(#) Immediate form of one-sample variance-comparison test sdtesti # obs # mean | . # sd # val , level(#) Immediate form of two-sample variance-comparison test sdtesti # obs,1 # mean,1 | . # sd,1 # obs,2 # mean,2 | . # sd,2 , level(#) Robust tests for equality of variances robvar varname if in , by(groupvar) by is allowed with sdtest and robvar; see [D] by. Menu sdtest Statistics > Summaries, tables, and tests > Classical tests of hypotheses > Variance-comparison test > Summaries, tables, and tests > Classical tests of hypotheses > Variance-comparison test calculator > Summaries, tables, and tests > Classical tests of hypotheses > Robust equal-variance test sdtesti Statistics robvar Statistics 1 2 sdtest — Variance-comparison tests Description sdtest performs tests on the equality of standard deviations (variances). In the first form, sdtest tests that the standard deviation of varname is #. In the second form, sdtest performs the same test, using the standard deviations of the two groups defined by groupvar. In the third form, sdtest tests that varname1 and varname2 have the same standard deviation. sdtesti is the immediate form of sdtest; see [U] 19 Immediate commands. Both the traditional F test for the homogeneity of variances and Bartlett’s generalization of this test to K samples are sensitive to the assumption that the data are drawn from an underlying Gaussian distribution. See, for example, the cautionary results discussed by Markowski and Markowski (1990). Levene (1960) proposed a test statistic for equality of variance that was found to be robust under nonnormality. Then Brown and Forsythe (1974) proposed alternative formulations of Levene’s test statistic that use more robust estimators of central tendency in place of the mean. These reformulations were demonstrated to be more robust than Levene’s test when dealing with skewed populations. robvar reports Levene’s robust test statistic (W0 ) for the equality of variances between the groups defined by groupvar and the two statistics proposed by Brown and Forsythe that replace the mean in Levene’s formula with alternative location estimators. The first alternative (W50 ) replaces the mean with the median. The second alternative replaces the mean with the 10% trimmed mean (W10 ). Options level(#) specifies the confidence level, as a percentage, for confidence intervals of the means. The default is level(95) or as set by set level; see [U] 20.7 Specifying the width of confidence intervals. by(groupvar) specifies the groupvar that defines the groups to be compared. For sdtest, there should be two groups, but for robvar there may be more than two groups. Do not confuse the by() option with the by prefix; both may be specified. Remarks and examples stata.com Remarks are presented under the following headings: Basic form Immediate form Robust test Basic form sdtest performs two different statistical tests: one testing equality of variances and the other testing that the standard deviation is equal to a known constant. Which test it performs is determined by whether you type a variable name or a number to the right of the equal sign. Example 1: One-sample test of variance We have a sample of 74 automobiles. For each automobile, we know the mileage rating. We wish to test whether the overall standard deviation is 5 mpg: sdtest — Variance-comparison tests 3 . use http://www.stata-press.com/data/r13/auto (1978 Automobile Data) . sdtest mpg == 5 One-sample test of variance Variable Obs Mean mpg 74 21.2973 Std. Err. Std. Dev. .6725511 5.785503 sd = sd(mpg) Ho: sd = 5 [95% Conf. Interval] 19.9569 c = chi2 = degrees of freedom = Ha: sd < 5 Pr(C < c) = 0.9717 Ha: sd != 5 2*Pr(C > c) = 0.0565 22.63769 97.7384 73 Ha: sd > 5 Pr(C > c) = 0.0283 Example 2: Variance ratio test We are testing the effectiveness of a new fuel additive. We run an experiment on 12 cars, running each without and with the additive. The data can be found in [R] ttest. The results for each car are stored in the variables mpg1 and mpg2: . use http://www.stata-press.com/data/r13/fuel . sdtest mpg1==mpg2 Variance ratio test Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] mpg1 mpg2 12 12 21 22.75 .7881701 .9384465 2.730301 3.250874 19.26525 20.68449 22.73475 24.81551 combined 24 21.875 .6264476 3.068954 20.57909 23.17091 ratio = sd(mpg1) / sd(mpg2) Ho: ratio = 1 Ha: ratio < 1 Pr(F < f) = 0.2862 Ha: ratio != 1 2*Pr(F < f) = 0.5725 f = degrees of freedom = 0.7054 11, 11 Ha: ratio > 1 Pr(F > f) = 0.7138 We cannot reject the hypothesis that the standard deviations are the same. In [R] ttest, we draw an important distinction between paired and unpaired data, which, in this example, means whether there are 12 cars in a before-and-after experiment or 24 different cars. For sdtest, on the other hand, there is no distinction. If the data had been unpaired and stored as described in [R] ttest, we could have typed sdtest mpg, by(treated), and the results would have been the same. Immediate form Example 3: sdtesti Immediate commands are used not with data, but with reported summary statistics. For instance, to test whether a variable on which we have 75 observations and a reported standard deviation of 6.5 comes from a population with underlying standard deviation 6, we would type 4 sdtest — Variance-comparison tests . sdtesti 75 . 6.5 6 One-sample test of variance x Obs Mean 75 . Std. Err. Std. Dev. .7505553 6.5 sd = sd(x) Ho: sd = 6 [95% Conf. Interval] . c = chi2 = degrees of freedom = Ha: sd < 6 Pr(C < c) = 0.8542 Ha: sd != 6 2*Pr(C > c) = 0.2916 . 86.8472 74 Ha: sd > 6 Pr(C > c) = 0.1458 The mean plays no role in the calculation, so it may be omitted. To test whether the variable comes from a population with the same standard deviation as another for which we have a calculated standard deviation of 7.5 over 65 observations, we would type . sdtesti 75 . 6.5 65 . 7.5 Variance ratio test Obs Mean Std. Err. Std. Dev. x y 75 65 . . .7505553 .9302605 6.5 7.5 . . . . combined 140 . . . . . ratio = sd(x) / sd(y) Ho: ratio = 1 [95% Conf. Interval] f = degrees of freedom = Ha: ratio < 1 Pr(F < f) = 0.1172 Ha: ratio != 1 2*Pr(F < f) = 0.2344 0.7511 74, 64 Ha: ratio > 1 Pr(F > f) = 0.8828 Robust test Example 4: robvar We wish to test whether the standard deviation of the length of stay for patients hospitalized for a given medical procedure differs by gender. Our data consist of observations on the length of hospital stay for 1778 patients: 884 males and 894 females. Length of stay, lengthstay, is highly skewed (skewness coefficient = 4.912591) and thus violates Bartlett’s normality assumption. Therefore, we use robvar to compare the variances. . use http://www.stata-press.com/data/r13/stay . robvar lengthstay, by(sex) sex W0 Summary of Length of stay in days Mean Std. Dev. Freq. male female 9.0874434 8.800671 9.7884747 9.1081478 884 894 Total 8.9432508 9.4509466 1778 = 0.55505315 df(1, 1776) Pr > F = 0.45635888 W50 = 0.42714734 df(1, 1776) Pr > F = 0.51347664 W10 = 0.44577674 df(1, 1776) Pr > F = 0.50443411 sdtest — Variance-comparison tests 5 For these data, we cannot reject the null hypothesis that the variances are equal. However, Bartlett’s test yields a significance probability of 0.0319 because of the pronounced skewness of the data. Technical note robvar implements both the conventional Levene’s test centered at the mean and a median-centered test. In a simulation study, Conover, Johnson, and Johnson (1981) compare the properties of the two tests and recommend using the median test for asymmetric data, although for small sample sizes the test is somewhat conservative. See Carroll and Schneider (1985) for an explanation of why both mean- and median-centered tests have approximately the same level for symmetric distributions, but for asymmetric distributions the median test is closer to the correct level. Stored results sdtest and sdtesti store the following in r(): Scalars r(N) r(p l) r(p u) r(p) r(F) r(sd) r(sd 1) r(sd 2) r(df) r(df 1) r(df 2) r(chi2) number of observations lower one-sided p-value upper one-sided p-value two-sided p-value F statistic standard deviation standard deviation for first variable standard deviation for second variable degrees of freedom numerator degrees of freedom denominator degrees of freedom χ2 robvar stores the following in r(): Scalars r(N) r(w50) r(p w50) r(w0) r(p w0) r(w10) r(p w10) r(df 1) r(df 2) number of observations Brown and Forsythe’s F statistic (median) Brown and Forsythe’s p-value Levene’s F statistic Levene’s p-value Brown and Forsythe’s F statistic (trimmed mean) Brown and Forsythe’s p-value (trimmed mean) numerator degrees of freedom denominator degrees of freedom Methods and formulas See Armitage et al. (2002, 149 – 153) or Bland (2000, 171–172) for an introduction and explanation of the calculation of these tests. The test for σ = σ0 is given by χ2 = (n − 1)s2 σ02 which is distributed as χ2 with n − 1 degrees of freedom. 6 sdtest — Variance-comparison tests The test for σx2 = σy2 is given by F = s2x s2y which is distributed as F with nx − 1 and ny − 1 degrees of freedom. Let Xij be the j th observation of X for the ith group. Let Zij = |Xij − X i |, where X i is the mean of X in the ith group. Levene’s test statistic is P ni (Z i − Z)2 /(g − 1) W0 = P P i P 2 i j (Zij − Z i ) / i (ni − 1) where ni is the number of observations in group i and g is the number of groups. W50 is obtained by replacing X i with the ith group median of Xij , whereas W10 is obtained by replacing X i with the 10% trimmed mean for group i. References Armitage, P., G. Berry, and J. N. S. Matthews. 2002. Statistical Methods in Medical Research. 4th ed. Oxford: Blackwell. Bland, M. 2000. An Introduction to Medical Statistics. 3rd ed. Oxford: Oxford University Press. Brown, M. B., and A. B. Forsythe. 1974. Robust tests for the equality of variances. Journal of the American Statistical Association 69: 364–367. Carroll, R. J., and H. Schneider. 1985. A note on Levene’s tests for equality of variances. Statistics and Probability Letters 3: 191–194. Cleves, M. A. 1995. sg35: Robust tests for the equality of variances. Stata Technical Bulletin 25: 13–15. Reprinted in Stata Technical Bulletin Reprints, vol. 5, pp. 91–93. College Station, TX: Stata Press. . 2000. sg35.2: Robust tests for the equality of variances update to Stata 6. Stata Technical Bulletin 53: 17–18. Reprinted in Stata Technical Bulletin Reprints, vol. 9, pp. 158–159. College Station, TX: Stata Press. Conover, W. J., M. E. Johnson, and M. M. Johnson. 1981. A comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data. Technometrics 23: 351–361. Gastwirth, J. L., Y. R. Gel, and W. Miao. 2009. The impact of Levene’s test of equality of variances on statistical theory and practice. Statistical Science 24: 343–360. Levene, H. 1960. Robust tests for equality of variances. In Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling, ed. I. Olkin, S. G. Ghurye, W. Hoeffding, W. G. Madow, and H. B. Mann, 278–292. Menlo Park, CA: Stanford University Press. Markowski, C. A., and E. P. Markowski. 1990. Conditions for the effectiveness of a preliminary test of variance. American Statistician 44: 322–326. Seed, P. T. 2000. sbe33: Comparing several methods of measuring the same quantity. Stata Technical Bulletin 55: 2–9. Reprinted in Stata Technical Bulletin Reprints, vol. 10, pp. 73–82. College Station, TX: Stata Press. Tobı́as, A. 1998. gr28: A graphical procedure to test equality of variances. Stata Technical Bulletin 42: 4–6. Reprinted in Stata Technical Bulletin Reprints, vol. 7, pp. 68–70. College Station, TX: Stata Press. Also see [R] ttest — t tests (mean-comparison tests)
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