help tostt
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Title
tostt -- Mean-equivalence t tests
Syntax
One-sample mean-equivalence t test
tostt varname == # [if] [in] [, eqvtype(type) eqvlevel(#) uppereqvlevel(#) alpha(#) relevance]
Two-sample unpaired mean-equivalence t test
tostt varname1 == varname2 [if] [in] , unpaired [eqvtype(type) eqvlevel(#) uppereqvlevel(#) unequal welch alpha(#)
relevance]
Two-sample paired mean-equivalence t test
tostt varname1 == varname2 [if] [in] [, eqvtype(type) eqvlevel(#) uppereqvlevel(#) alpha(#) relevance]
Two-group unpaired mean-equivalence t test
tostt varname [if] [in] , by(groupvar) [eqvtype(type) eqvlevel(#) uppereqvlevel(#) unequal welch alpha(#) relevance]
Immediate form of one-sample mean-equivalence t test
tostti #obs #mean #sd #val [, eqvtype(type) eqvlevel(#) uppereqvlevel(#) xname(string) alpha(#) relevance]
Immediate form of two-sample mean-equivalence t test
tostti #obs1 #mean1 #sd1 #obs2 #mean2 #sd2 [, eqvtype(type) eqvlevel(#) uppereqvlevel(#) unequal welch xname(string)
yname(string) alpha(#) relevance]
options Description
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Miscellaneous
eqvtype(string) specify equivalence threshold with Delta or epsilon
eqvlevel(#) the level of tolerance defining the equivalence interval
uppereqvlevel(#) the upper value of an asymmetric equivalence interval
unpaired the data are unpaired
by(groupvar) variable defining the two groups (implies unpaired)
unequal unpaired data have unequal variances
welch use Welch's approximation (implies unequal)
xname(string) the name of the first variable
yname(string) the name of the second variable
alpha(#) set nominal type I level; default is alpha(0.05)
relevance perform & report combined tests for difference and equivalence
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Description
tostt tests for the equivalence of means within a symmetric equivalence interval defined by eqvtype and eqvlevel using a two
one-sided t tests approach (Schuirmann, 1987). Typically null hypotheses are framed from an assumption of a lack of difference
between two quantities, and reject this assumption only with sufficient evidence. When performing tests of equivalence, one
frames a null hypothesis with the assumption that two quantities are different within an equivalence interval defined by some
chosen level of tolerance (as specified by eqvtype and eqvlevel).
With respect to an unpaired t test, an equivalence null hypothesis takes one of the following two forms depending on whether
equivalence is defined in terms of Delta (equivalence expressed in the same units as the x and y) or in terms of epsilon
(equivalence expressed in the units of the t distribution with the given degrees of freedom):
Ho: |mean(x) - mean(y)| >= Delta,
where the equivalence interval ranges from diff-Delta to diff+Delta, and where diff is either the mean difference or the
difference in means depending on whether the test is paired or unpaired. This translates directly into two one-sided null
hypotheses:
Ho1: Delta - [mean(x) - mean(y)] <= 0; and
Ho2: [mean(x) - mean(y)] + Delta <= 0
-OR-
Ho: |t| >= epsilon,
where the equivalence interval ranges from -epsilon to epsilon. This also translates directly into two one-sided null
hypotheses:
Ho1: epsilon - t <= 0; and
Ho2: t + epsilon <= 0
When an asymmetric equivalence interval is defined using the uppereqvlevel option the general negativist null hypothesis
becomes:
Ho: [mean(x) - mean(y)] <= Delta_lower, or [mean(x) - mean(y)] >= Delta_upper,
where the equivalence interval ranges from [mean(x) - mean(y)] + Delta_lower to [mean(x) - mean(y)] + Delta_upper. This also
translates directly into two one-sided null hypotheses:
Ho1: Delta_upper - [mean(x) - mean(y)] <= 0; and
Ho2: [mean(x) - mean(y)] - Delta_lower <= 0
-OR-
Ho: t <= epsilon_lower, or t >= epsilon_upper,
Ho1: epsilon_upper - t <= 0; and
Ho2: t - epsilon_lower <= 0
NOTE: the appropriate level of alpha is precisely the same as in the corresponding two-sided test of mean difference, so that,
for example, if one wishes to make a type I error %5 of the time, one simply conducts both of the one-sided tests of Ho1 and Ho2
by comparing the resulting p-value to 0.05 (Tryon and Lewis, 2008).
Options
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eqvtype(string) defines whether the equivalence interval will be defined in terms of Delta or epsilon (delta, or epsilon).
These options change the way that evqlevel is interpreted: when delta is specified, the evqlevel is measured in the units of
the variable being tested, and when epsilon is specified, the evqlevel is measured in multiples of the standard deviation of
the t distribution; put another way epsilon = Delta/standard error. The default is delta.
Defining tolerance in terms of epsilon means that it is not possible to reject any test of mean equivalence Ho if epsilon <=
the critical value of t for a given alpha and degrees of freedom. Because epsilon = Delta/standard error, we can see that it
is not possible to reject any Ho if Delta <= the product of the standard error and critical value of t for a given alpha and
degrees of freedom. tostt and tostti now report when either of these conditions obtain.
eqvlevel(#) defines the equivalence threshold for the tests depending on whether eqvtype is delta or epsilon (see above).
Researchers are responsible for choosing meaningful values of Delta or epsilon. The default value is 1 when delta is the
eqvtype and 2 when epsilon is the eqvtype.
uppereqvlevel(#) defines the upper equivalence threshold for the test, and transforms the meaning of eqvlevel to mean the lower
equivalence threshold for the test. Also, eqvlevel is assumed to be a negative value. Taken together, these correspond to
Schuirmann's (1987) asymmetric equivalence intervals. If uppereqvlevel==|eqvlevel|, then uppereqvlevel will be ignored.
by(groupvar) specifies the groupvar that defines the two groups that tostt will use to test the hypothesis that their means are
different. Specifying by(groupvar) implies an unpaired (two sample) t test. Do not confuse the by() option with the by
prefix; you can specify both.
unpaired specifies that the data be treated as unpaired. The unpaired option is used when the two set of values to be compared
are in different variables.
unequal specifies that the unpaired data not be assumed to have equal variances.
welch specifies that the approximate degrees of freedom for the test be obtained from Welch's formula (1947) rather than
Satterthwaite's approximation formula (1946), which is the default when unequal is specified. Specifying welch implies
unequal.
xname(string) specifies how the first variable will be labeled in the output. The default value of xname is x.
yname(string) specifies how the second variable will be labeled in the output. The default value of yname is y.
alpha(#) specifies the nominal type I error rate. The default is alpha(0.05).
relevance reports results and inference for combined tests for difference and equivalence for a specific alpha, eqvtype, and
eqvlevel. See the Remarks section more details on inference from combined tests.
Remarks
As described by Tryon and Lewis (2008), when both tests of difference and equivalence are taken together, there are four
possible interpretations:
1. One may reject the null hypothesis of no difference, but fail to reject the null hypothesis of difference, and conclude that
there is a relevant difference in means at least as large as Delta or epsilon.
2. One may fail to reject the null hypothesis of no difference, but reject the null hypothesis of difference, and conclude that
the means are equivalent within the equivalence range (i.e. defined by Delta or epsilon).
3. One may reject both the null hypothesis of no difference and the null hypothesis of difference, and conclude that the means
are trivially different, within the equivalence range (i.e. defined by Delta or epsilon).
4. One may fail to reject both the null hypothesis of no difference and the null hypothesis of difference, and draw an
indeterminate conclusion, because the data are underpowered to detect difference or equivalence.
Examples
These examples correspond to those written in the help file for ttest:
. sysuse auto (setup)
. tostt mpg==20, eqvt(delta) eqvl(2.5) upper(3) (one-sample mean-equivalence test)
. webuse fuel (setup)
. tostt mpg1==mpg2, eqvt(epsilon) eqvl(3) rel (two-sample paired mean relevance test)
. webuse fuel3 (setup)
. tostt mpg, by(treated) (two-group unpaired mean-comparison test)
(note warning about value of Delta!)
(no setup required)
. tostti 24 62.6 15.8 75, rel (immediate form; n=24, m=62.6, sd=15.8;
test m=75)
Author
Alexis Dinno
Portland State University
alexis.dinno@pdx.edu
Development of tost is ongoing, please contact me with any questions, bug reports or suggestions for improvement. Fixing bugs
will be facilitated by sending along (1) a copy of the data (de-labeled or anonymized is fine), (2) a copy of the command used
and (3) a copy of the exact output of the command.
I am endebted to my winter 2013 students for their inspiration.
Suggested citation
Dinno A. 2017. tostt: Mean-equivalence t tests. Stata software package. URL: https://www.alexisdinno.com/stata/tost.html
Saved results
The one-sample form of tostt saves the following in r():
Scalars
r(sd_1) standard deviation for the variable
r(se) estimate of standard error
r(p2) P(T >= t2); upper one-sided p-value under Ho2
r(p1) P(T >= t1); upper one-sided p-value under Ho1
r(t2) t statistic under Ho2
r(t1) t statistic under Ho1
r(df_t) degrees of freedom
r(mu_1) x_1 bar, mean for the population
r(N_1) sample size n_1
r(Delta) Delta, tolerance level defining the equivalence interval; OR
r(Du) Delta_upper, tolerance level defining the equivalence interval's upper side; AND
r(Dl) Delta_lower, tolerance level defining the equivalence interval's lower side; OR
r(epsilon) epsilon, tolerance level defining the equivalence interval
r(eu) epsilon_upper, tolerance level defining the equivalence interval's upper side; AND
r(el) epsilon_lower, tolerance level defining the equivalence interval's lower side
r(relevance) Relevance test conclusion for given alpha and Delta/epsilon
The two-sample and two-group forms of tostt save the following in r():
Scalars
r(sd_2) standard deviation for second variable
r(sd_1) standard deviation for first variable
r(se) estimate of standard error
r(p2) P(T >= t2); upper one-sided p-value under Ho2
r(p1) P(T >= t1); upper one-sided p-value under Ho1
r(t2) t statistic under Ho2
r(t1) t statistic under Ho1
r(df_t) degrees of freedom
r(mu_2) x_2 bar, mean for population 2
r(N_2) sample size n_2
r(mu_1) x_1 bar, mean for population 1
r(N_1) sample size n_1
r(Delta) Delta, tolerance level defining the equivalence interval; OR
r(Du) Delta_upper, tolerance level defining the equivalence interval's upper side; AND
r(Dl) Delta_lower, tolerance level defining the equivalence interval's lower side; OR
r(epsilon) epsilon, tolerance level defining the equivalence interval
r(eu) epsilon_upper, tolerance level defining the equivalence interval's upper side; AND
r(el) epsilon_lower, tolerance level defining the equivalence interval's lower side
r(relevance) Relevance test conclusion for given alpha and Delta/epsilon
References
Satterthwaite, F. E. 1946. An approximate distribution of estimates of variance components. Biometrics Bulletin 2: 110-114.
Schuirmann, D. A. 1987. A comparison of the two one-sided tests procedure and the power approach for assessing the
equivalence of average bioavailability. Pharmacometrics. 15: 657-680
Tryon, W. W., and C. Lewis. 2008. An inferential confidence interval method of establishing statistical equivalence that
corrects Tryonâ€™s (2001) reduction factor. Psychological Methods. 13: 272-277
Welch, B. L. 1947. The generalization of `Student's' problem when several different population variances are involved.
Biometrika 34: 28-35.
Also See
Help: tost, pkequiv, ttest