*! version 3.1.5 12oct2024
help tostrrp

Title

tostrrp -- Test for equivalence of relative risk and unity in paired binary data

Syntax

tostrrp treatment_1_outcome_variable treatment_2_outcome_variable [if] [in]
,
delta0(#)
[, deltaupper(#)
alpha(#)
exactchisq
treatment1(treatment 1 name)
treatment2(treatment 2 name)
outcome(positive outcome label)
nooutcome(negative outcome label)
relevance]

tostrrpi #a #b #c #n #delta0
[, deltaupper(#)
alpha(#)
exactchisq
treatment1(treatment 1 name)
treatment2(treatment 2 name)
outcome(positive outcome label)
nooutcome(negative outcome label)
relevance]

tostrrp options

Main
delta0(#) the value defining an equivalence interval (the lower value, if deltaupper is used)

tostrrp and tostrrpi options

Main
deltaupper(#) the upper value of a geometrically asymmetric equivalence interval

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

exactchisq use the exact chi-square p-value

treatment1(string) The name of the first treatment group

treatment2(string) The name of the second treatment group

outcome(string) The label for having a positive outcome

nooutcome(string) The label for having a negative outcome

relevance perform & report combined tests for difference and equivalence

Description

tostrrp test for equivalence of the relative risk of a positive outcome
and unity in paired (or matched) randomized control trial or paired (or matched)
cohort design data. It calculates an asymptotic z test statistic based on a
reparameterized multinomial model (Tang, et al., 2003)
in a two one-sided tests approach (Schuirmann, 1987).
tostrrpi is the immediate form of tostrrp; see immed. The equivalence
interval for the test is defined by a chosen level of tolerance, as specified by
delta0, or by #delta0 in the immediate form of the command.

The two one-sided null hypotheses take on the following form based on the
relative risk (RR), and the threshold delta0:

Ho1: RR <= delta0, or

Ho2: RR >= 1/delta0,

where the equivalence interval ranges from delta0 to 1/delta0.

When a geometrically asymmetric equivalence interval is defined using the deltaupper option
the two one-sided null hypotheses become:

Ho1: RR <= delta0, or

Ho2: RR >= deltaupper,

where the equivalence interval ranges from delta0 to
deltaupper.

The two z test statistics, z1 and z2, are both constructed with rejection
probabilities in the upper tails. So p1 = P(Z >= z1), and p2 = P(Z >= z2).

NOTES: When delta0 = 1, the Tang, Tang and Chan test statistic reduces to
McNemar's test statistic (McNemar, 1947). When a = b = c = 0,
there are no positve outcomes in either treatment group, and the RR and test
statistics become undefined. If a > 0, and b = c = 0, then there
is complete concordance, and z1 = z2, so p1 = p2. As is standard with two
one-sided tests for equivalence, 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 tostrrp and tostrrpi

Main

delta0(#) defines the equivalence threshold for the tests, with the lower
boundary of equivalence equal to delta0, and the upper boundary equal to
1/delta0. For example, delta0 = 0.8 gives an equivalence interval
of 0.8, 1.25 (because 1.25 = 1/0.8). Researchers are responsible for choosing
meaningful values of delta0.

deltaupper(#) defines the upper equivalence threshold for the test,
and restricts the meaning of delta0 to the lower equivalence threshold
for the test. Also, delta0 must be a positive value less than or equal to 1,
and deltaupper must be a positive value that is greater than or equal to 1.
Taken together, these correspond to geometrically asymmetric equivalence intervals.

alpha(#) specifies the nominal type I error rate. The default is alpha(0.05).

exactchisq calculate and use the exact chi-square p-value for making the
rejection decision for the positivist test. The default is the asymptotic p-value
because the Tang-Tang-Chan test statistics are also asymptotic.

treatment1(string) labels the name of the first treatment group in the output table.
For tostppr this defaults to the variable label for treatment_1_outcome_variable.

treatment2(string) labels the name of the second treatment group in the output table.
For tostppr this defaults to the variable label for treatment_2_outcome_variable.

outcome(string) labels the value corresponding to positive for the outcome.
For tostppr this defaults to the label for the value = 1 in treatment_1_outcome_variable
(or in treatment_2_outcome_variable if treatment_1_outcome_variable is unlabeled).

nooutcome(string) labels the value corresponding to negative for the outcome.
For tostppr this defaults to the label for the value = 0 in treatment_1_outcome_variable
(or in treatment_2_outcome_variable if treatment_1_outcome_variable is unlabeled).

relevance reports results and inference for combined tests for difference
and equivalence of marginal probabilities (of exposure) for a specific
alpha, and delta0. See the end of the Discussion section in tost
for more details on inference from combined tests.

Examples

Setup

. use hivfluid (requires that you install hivfluid.dta)

Relevance test example from Tang, et al., 2003, Table II, based on data from Lachenbruch and Lynch, 1998

. tostrrp plasma alternate, delta0(0.95) rel

Same as above command, but using immediate form

. tostrrpi 446 5 16 1157 0.95, rel treatment1("Plasma sample") treatment2("Alternative fluid") outcome("HIV Positive") nooutcome("HIV Negative")

Example from Tang, et al., 2003, Table V, based on data from Tango, 1998

. tostrrpi 43 0 1 44 0.9, treatment1("Thermal") treatment2("Chemical") outcome("Effective") nooutcome("Ineffective")

Saved results

tostrrp and tostrrpi save the following in r():

Scalars

r(RR) relative risk (aka incidence rate ratio) of positive outcome for treatment 2 vs. treatment 1

r(sdRR) standard deviation of relative risk based on the score statistic per (Tang, et al., 2003)

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(delta0) delta0, tolerance level defining the equivalence interval; OR

r(deltalower) delta_lower, tolerance level defining the equivalence interval's lower side; AND

r(deltaupper) delta_upper, tolerance level defining the equivalence interval's upper side

r(relevance) Relevance test conclusion for given alpha and delta0

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. tostrrp: Test for equivalence of relative risk and unity in
paired binary data. In: tost Stata software package. URL:

Reference

Lachenbruch, P. A., and Lynch, C. J. 1998. Assessing screening tests: Extensions of McNemar's test.
Statistics In Medicine 17: 2207-2217

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

Tang, N.-S., Tang, M.-L., and Chan, I. S. F. 2003. On tests of equivalence via non-unity relative risk for matched-pair design. Statistics In Medicine 22: 1217-1233.

Tango, T. 1998. Equivalence test and confidence interval for the difference in proportions for the paired-sample design.
Statistics In Medicine, 17: 891-908

Wellek, S. 2010. Testing Statistical Hypotheses of Equivalence and Noninferiority,
second edition. Chapman and Hall/CRC Press. p. 31

Also See

Help: tost, mcc, tostmcc