help tostpr
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title
tostpr -- One- and two-sample z tests of proportion-equivalence
Syntax
One-sample proportion-equivalence z test
tostpr varname == # [if] [in] [, eqvtype(type) eqvlevel(#) uppereqvlevel(#) alpha(#) relevance]
Two-sample proportion-equivalence z test
tostpr varname1 == varname2 [if] [in] [, eqvtype(type) eqvlevel(#) yates ha uppereqvlevel(#) alpha(#) relevance]
Two-group proportion-equivalence z test
tostpr varname [if] [in] , by(groupvar) [eqvtype(type) eqvlevel(#) uppereqvlevel(#) yates ha alpha(#) relevance]
Immediate form of one-sample z test of proportion-equivalence
tostpri #obs1 #p1 #p2 [, eqvtype(type) eqvlevel(#) uppereqvlevel(#) count alpha(#) relevance]
Immediate form of two-sample z test of proportion-equivalence
tostpri #obs1 #p1 #obs2 #p2 [, eqvtype(type) eqvlevel(#) uppereqvlevel(#) count yates ha alpha(#) relevance]
by is allowed with the non-immediate form of tostpri; see [D] by.
tostpr & tostpri options Description
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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
by(groupvar) variable defining the two groups
count integers, not proportions are used with tostpri
yates use the Yates continuity correction
ha use the Hauck-Anderson continuity correction
alpha(#) set nominal type I level; default is alpha(0.05)
relevance perform & report combined tests for difference and equivalence
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Description
tostpr tests for the equivalence of proportions within a symmetric equivalence interval defined by eqvtype and eqvlevel using a two one-sided z tests approach (Hauck and Anderson, 1984; 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 for 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 eqvlevel).
With respect to a z test of proportions, 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 prop(x) and prop(y) or in terms of epsilon (equivalence expressed in the units of the z distribution):
Ho: |prop(x) - prop(y)| >= Delta,
where the equivalence interval ranges from diff-Delta to diff+Delta, and where diff is the difference in proportions. This translates directly into two one-sided null hypotheses:
Ho1: Delta - [prop(x) - prop(y)] <= 0; and
Ho2: [prop(x) - prop(y)] + Delta <= 0
-OR-
Ho: |z| >= epsilon,
where the equivalence interval ranges from -epsilon to epsilon. This also translates directly into two one-sided null hypotheses:
Ho1: epsilon - Z <= 0; and
Ho2: Z + epsilon <= 0
When an asymmetric equivalence interval is defined using the uppereqvlevel option the general negativist null hypothesis becomes:
Ho: [prop(x) - prop(y)] <= Delta_lower, or [prop(x) - prop(y)] >= Delta_upper,
where the equivalence interval ranges from [prop(x) - prop(y)]+Delta_lower to [prop(x) - prop(y)]+Delta_upper. This also translates directly into two one-sided null hypotheses:
Ho1: Delta_upper - [prop(x) - prop(y)] <= 0; and
Ho2: [prop(x) - prop(y)] - Delta_lower <= 0
-OR-
Ho: Z <= epsilon_lower, or Z >= epsilon_upper, and
Ho1: epsilon_upper - Z <= 0; and
Ho2: Z - epsilon_lower <= 0
NOTE: the appropriate level of alpha is precisely the same as in the corresponding two-sided test of proportion 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 (Wellek, 2010).
Options
+------+
----+ Main +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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 proportion of the variable being tested, and when epsilon is specified, the evqlevel is measured in multiples of the standard deviation of the Z
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 z for a given alpha. 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 z for a given alpha. tostpr and tostpri 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 0.1 when delta is the type, and is 2 when epsilon is the type.
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 are analogous 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 tostpr will use to test the hypothesis that their proportions are different. Specifying by(groupvar) implies a two sample z test. Do
not confuse the by() option with the by prefix; you can specify both.
count specifies that integer counts instead of proportions be used in tostpri. In the first syntax, tostpri expects that #obs1 and #p1 are counts -- #p1 < #obs1 -- and #p2 is a proportion. In the second
syntax, tostpri expects that all four numbers are integer counts, that #obs1 > #p1, and that #obs2 > #p2.
yates specifies that the test statistics incorporate the Yates continuity correction (Yates 1934). This option is included for convenience although the Hauck-Anderson correction (see ha) is preferred (Tu
1997). This continuity correction is only applied to the two one-sided z test statistics, and not to any of the Wald confidence intervals.
ha specifies that the test statistics incorporate the Hauck-Anderson continuity correction (Hauck and Anderson 1986). This is the preferred continuity correction option (Tu 1997). This continuity
correction is only applied to the two one-sided z test statistics, and not to any of the Wald confidence intervals.
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 (Tryon and Lewis 2008), when both tests for 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 proportions 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 proportions 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 proportions 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 prtest:
. sysuse auto (setup)
. tostpr foreign==.4, eqvt(delta) eqvl(.15) upper(.2) (one-sample proportion-equivalence test)
. webuse cure (setup)
. tostpr cure1==cure2, eqvt(epsilon) eqvl(2.645) rel (two-sample proportion-relevance test)
(epsilon = invnormal(.95)+1 = 2.645)
. webuse cure2 (setup)
. tostpr cure, by(sex) ha delta(.25) alpha(.1) (two-group proportion-equivalence test)
(cure has same proportion for males and females)
(inlcudes Hauck-Anderson continuity correction)
. tostpri 50 .52 .70, eqvt(delta) eqvl(.1) (immediate form of one-sample test of proportion-equivalence)
(note warning about value of Delta!)
. tostpri 30 4 .7, count eqvt(epsilon) eqvl(3.145) rel (first two numbers are counts)
(epsilon = invnormal(.95)+1.5 = 3.145)
. tostpri 30 .4 45 .67, eqvt(delta) eqvl(.2) (immediate form of two-sample test of proportion-equivalence)
. tostpri 30 4 45 17, count eqvt(delta) eqvl(.2) (all numbers are counts)
Saved results
The one-sample forms of tostpr and tostpri saves the following in r():
Scalars
r(z2) z statistic under Ho2
r(z1) z statistic under Ho1
r(P_1) proportion for variable 1
r(N_1) number of observations for variable 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 tostpr and tostpri save the following in r():
Scalars
r(z2) z statistic under Ho2
r(z1) z statistic under Ho1
r(P_2) proportion for variable 2
r(N_2) number of observations for variable 2
r(P_1) proportion for variable 1
r(N_1) number of observations for variable 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
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. 2024. tostpr: One- and two-sample z tests of proportion-equivalence. In tost Stata software package. URL: https://www.alexisdinno.com/stata/tost.html
References
Hauck, W. W., and Anderson, S. 1984. A new statistical procedure for testing equivalence in two-group comparative bioavailability trials. Journal of Pharmacokinetics and Pharmacodynamics. 12: 83-91
Hauck, W. W., and Anderson, S. 1986. A comparison of large-sample confidence interval methods for the difference of two binomial probabilities. The American Statistician, 40: 318-322
Schuirmann, D. A. 1987. A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. Journal of Pharmacokinetics and
Biopharmaceutics. 15: 657-680
Tryon, W. W., and Lewis, C. 2008. An inferential confidence interval method of establishing statistical equivalence that corrects Tryon's (2001) reduction factor. Psychological Methods. 13: 272-277
Tu, D. 1997. Two one-sided tests procedures in establishing therapeutic equivalence with binary clinical endpoints: Fixed sample performances and sample size determination. Journal of Statistical
Computing and Simmulation. 59: 271-290
Yates, F. 1934. Contingency tables involving small numbers and the chi-squared test. Supplement to the Journal of the Royal Statistical Society. 1: 217-235
Wellek, S. 2010. Testing Statistical Hypotheses of Equivalence and Noninferiority, second edition. Chapman and Hall/CRC Press. p. 31
Also See
Help: tost, pkequiv, prtest