help tostrrp
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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 Description
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Main
delta0(#) the value defining an equivalence interval (the lower value, if deltaupper is used)
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tostrrp and tostrrpi options Description
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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
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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
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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: https://www.alexisdinno.com/stata/tost.html
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