help tostt -------------------------------------------------------------------------------------------------------------------------------------- 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 -------------------------------------------------------------------------------------------------------------------------------- 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 -------------------------------------------------------------------------------------------------------------------------------- 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 +------+ ----+ 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 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 Tryon’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