*! version 3.1.5 12oct2024
help tostmcc

Title

tostmcc -- Paired z test for equivalence of marginal probabilities in binary data

Syntax

tostmcc var_exposed_case var_exposed_control [if] [in]
[weight] [, eqvtype(type)
eqvlevel(#)
uppereqvlevel(#)
yates
edwards
alpha(#)
relevance]

tostmcci #a #b #c #d
[, eqvtype(type)
eqvlevel(#)
uppereqvlevel(#)
yates
edwards
alpha(#)
relevance]

tostmcc options

Main
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

yates include a Yates continuity correction

edwards include an Edwards continuity correction

alpha(#) set nominal type I level; default is alpha(0.05)

relevance perform & report combined tests for difference and equivalence

fweights are allowed; see weight.

Description

tostmcc tests for equivalence of the marginal probabilities of exposure in matched
case-control data. It calculates a Wald-type asymptotic z test (Liu, et al., 2002)
in a two one-sided tests approach (Schuirmann, 1987).
tostmcci is the immediate form of tostmcc; see immed. Typically the
null hypotheses of the corresponding McNemar's chi-square test
(McNemar, 1947) for difference in marginal probabilities
are framed from an assumption of equality of marginal probability of exposure
between cases and controls (e.g. Ho: P(exposure|case) = P(exposure|controls),
rejecting this assumption only with sufficient evidence. When performing tests
for equivalence of marginal probabilities, the null hypothesis is framed as the
difference in marginal probabilities is at least as much as the equivalence
interval as defined by some chosen level of tolerance (as specified by eqvtype
and eqvlevel).

With respect to a z test, a negativist null hypothesis takes one of the
following two forms depending on whether tolerance is defined in terms of
Delta (equivalence expressed in the units of the probability of counts of
discordant pairs) or in terms of epsilon (equivalence expressed in the units of
the z distribution):

Ho: |b - c| >= Delta,

where the equivalence interval ranges from |b - c|-Delta to
|b - c|+Delta, and where b is the count of pairs with cases exposed, but
controls unexposed, and and c is the count of pairs with cases unexposed and
controls exposed. This null hypothesis translates directly into two one-sided
null hypotheses:

Ho1: Delta - (b - c) <= 0; and

Ho2: (b - c) + 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: (b - c) <= Delta_lower, or (b - c) >= Delta_upper,

where the equivalence interval ranges from (b - c) + Delta_lower to
(b - c) + Delta_upper. This also translates directly into two one-sided null
hypotheses:

Ho1: Delta_upper - (b - c) <= 0; and

Ho2: (b - c) - Delta_lower <= 0

-OR-

Ho: Z <= epsilon_lower, or Z >= epsilon_upper,

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 McNemar's test for difference of maginal probabilities, 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 for mcc and mcci

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 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 = n*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 over n for a given alpha. tostmcc reports when either
of these conditions obtain. Tolerances should be specified using delta by
considering the difference in P(b) and P(c).

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.

yates specifies that the test statistics incorporate a Yates continuity
correction (Yates, 1934) using the term [(b - c)-0.5]
for z1, and the term [(b - c)+0.5] for z2. yates is exclusive of edwards. This
continuity correction is only applied to the two one-sided z test statistics,
and not to any of the confidence intervals.

edwards specifies that the test statistics incorporate an Edwards continuity
correction (Edwards, 1947) using the term [(b - c)-1]
for z1, and the term [(b - c)+1] for z2. edwards is exclusive of yates. This
continuity correction is only applied to the two one-sided z test statistics,
and not to any of the 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 of marginal probabilities (of exposure) for a specific
alpha, eqvtype, and eqvlevel. See the end of the
Discussion section in tost for more details on inference from combined tests.

Examples

Setup

. webuse mccxmpl

Relevance test in paired binary data

. tostmcc case control [fw=pop], eqvt(delta) eqvlevel(.2) rel

Same as above command, but using immediate form

. tostmcci 8 8 3 8, eqvt(delta) eqvlevel(.2) rel

Different example using asymetric intervals specified with epsilon.

Note: 2.1448536 = invnormal(.95) + .5, and 2.6448536 = invnormal(.95) + 1

. tostmcci 4 9 8 5, eqvt(epsilon) eqvlevel(2.1448536) upper(2.6448536) rel

Saved results

tostmcc and tostmcci save the following in r():

Scalars

r(z1) z test statistic for Ho1 (upper)

r(z2) z test statistic for Ho2 (lower)

r(p1) P(Z >= z1)

r(p2) P(Z >= z2)

r(D_f) difference in proportion with exposure

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.

Suggested citation

Dinno, A. 2024. tostmcc: Paired z test for equivalence of marginal probabilities
in paired binary data. In: tost Stata software package. URL:

Reference

Edwards, A. 1948. Note on the "correction for continuity" in testing the significance of the difference between correlated proportions.
Psychometrika 13: 185-187

Liu, J., et al., 2002. Tests for equivalence or non-inferiority for paired binary data.
Statistics In Medicine 21: 231-245.

McNemar, Q. 1947. Note on the sampling error of the difference between correlated proportions or percentages.
Psychometrika 12: 153-157

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

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, tostrrp, pkequiv, mcc