## Stata 15 help for sdtest

```
[R] sdtest -- Variance-comparison tests

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 #obs1 {#mean1 | . } #sd1 #obs2 {#mean2 | . } #sd2 [,
level(#)]

Robust tests for equality of variances

robvar  varname [if] [in] , by(groupvar)

by is allowed with sdtest and robvar; see [D] by.

sdtest

Statistics > Summaries, tables, and tests > Classical tests of
hypotheses > Variance-comparison test

sdtesti

Statistics > Summaries, tables, and tests > Classical tests of
hypotheses > Variance-comparison test calculator

robvar

Statistics > Summaries, tables, and tests > Classical tests of
hypotheses > Robust equal-variance test

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 immed.

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 (W_0) 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 (W_50)
replaces the mean with the median.  The second alternative replaces the
mean with the 10% trimmed mean (W_10).

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.

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.

Examples

---------------------------------------------------------------------------
Setup
. sysuse auto

Test standard deviation of mpg against 5
. sdtest mpg == 5

---------------------------------------------------------------------------
Setup
. webuse fuel

Test that mpg1 and mpg2 have same standard deviation
. sdtest mpg1 == mpg2

---------------------------------------------------------------------------
Setup
. webuse fuel2

Test that the two groups of treat have same standard deviation
. sdtest mpg, by(treat)

Test sd=6 when observed sd=6.5 and n=75
. sdtesti 75 . 6.5 6

Test sd1=sd2 when observed n1=75, sd1=6.5, n2=65, and sd2= 7.5
. sdtesti 75 . 6.5 65 . 7.5

---------------------------------------------------------------------------
Setup
. webuse stay

Test whether length of stay differs by gender
. robvar lengthstay, by(sex)
---------------------------------------------------------------------------

Stored results

sdtest and sdtesti store the following in r():

Scalars
r(N)           number of observations
r(p_l)         lower one-sided p-value
r(p_u)         upper one-sided p-value
r(p)           two-sided p-value
r(F)           F statistic
r(sd)          standard deviation
r(sd_1)        standard deviation for first variable
r(sd_2)        standard deviation for second variable
r(df)          degrees of freedom
r(df_1)        numerator degrees of freedom
r(df_2)        denominator degrees of freedom
r(chi2)        chi-squared

robvar stores the following in r():

Scalars
r(N)           number of observations
r(w50)         Brown and Forsythe's F statistic (median)
r(p_w50)       Brown and Forsythe's p-value
r(w0)          Levene's F statistic
r(p_w0)        Levene's p-value
r(w10)         Brown and Forsythe's F statistic (trimmed mean)
r(p_w10)       Brown and Forsythe's p-value (trimmed mean)
r(df_1)        numerator degrees of freedom
r(df_2)        denominator degrees of freedom

References

Brown, M. B., and A. B. Forsythe. 1974. Robust test for the equality of
variances. Journal of the American Statistical Association 69:
364-367.

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.

```