Performs equivalence, non-inferiority and non-superiority tests (R.W. Payne).
Options
PRINT = string tokens |
Controls printed output (confidence, description, test); default desc, test |
PLOT = string token |
Controls plotting of the confidence intervals (confidence); default * |
CLASSIFICATION = pointer |
Specifies the factors classifying the table of means; must be supplied for a multi-way table |
METHOD = string token |
Type of test required (equivalence, noninferiority, nonsuperiority); default equi |
CIPROBABILITY = scalar |
The probability level for the confidence interval; default 0.95 |
EQLIMITS = scalar or variate |
Limits for equivalence, non-inferiority or non-superiority |
TITLE = text |
Title for the graph of confidence intervals; default 'Confidence plot' |
WINDOW = scalar |
Window for the graph of confidence intervals; default uses a window defined to fill the screen |
SCREEN = string token |
Whether to clear the screen before plotting the confidence intervals (clear, keep); default clea |
Parameters
MEANS = tables or variates |
Means to be compared |
CONTROL = scalars, texts or pointers |
Specifies the control treatment |
SED = symmetric matrix or scalar |
Standard errors of differences of the means |
DF = symmetric matrix or scalar |
Degrees of freedom for the standard errors of differences |
TSTATISTICS = tables or variates |
Saves the t-statistics for the tests |
PROBABILITIES = tables or variates |
Saves the probabilities from the tests |
DIFFERENCES = tables or variates |
Saves the differences from the control |
SEDCONTROL = tables or variates |
Saves the standard errors for the differences from the control |
DFCONTROL = tables or variates |
Saves the degrees of freedom for the differences from the control |
LOWER = tables or variates |
Saves the lower limits of the confidence intervals |
UPPER = tables or variates |
Saves the upper limits of the confidence intervals |
Description
TEQUIVALENCE performs tests that can be used to assess whether a treatment is acceptably similar to a control (or standard) treatment.
For an equivalence test, you specify a lower and an upper limit for the difference between the mean of the treatment and the mean of the control. These define the zone within which the treatment can be regarded as equivalent to the control. The null hypothesis is that the treatment is not equivalent to the control i.e. that the difference in means lies outside that zone. The test calculates t-statistics for the distance of the difference above the lower limit, and its distance below the upper limit. Their probabilities provide the evidence to assess whether the difference lies within the equivalence zone, at the lower and upper end respectively. The procedure reports the larger (i.e. the less significant) of the two probabilities together with its t-statistic. You can also check the tests by printing or plotting the confidence limits. Both tests need to be significant, and thus both ends of the confidence interval be within the zone, to conclude that the treatments are equivalent.
For non-inferiority, the difference between the mean of the treatment and the mean of the control must not be less than a (negative) limit. Any positive difference is acceptable, and a negative difference must be greater than the limit. The null hypothesis is that the treatment is inferior to the control i.e. that the difference is less than the limit. There is just one t-statistic, assessing whether the difference is greater than the limit, and the confidence interval is unbounded at the positive end.
Similarly, for non-superiority, the difference between the mean of the treatment and the mean of the control must not be greater than a (positive) limit. Any negative difference is acceptable, and a positive difference must be less than the limit. The null hypothesis is that the treatment is superior to the control i.e. that the difference greater than the limit. There is just one t-statistic, assessing whether the difference is less than the limit, and the confidence interval is unbounded at the negative end.
The MEANS parameter specifies a table or a variate containing the means that are to be assessed. For a variate, the CONTROL parameter specifies a scalar containing the number of the unit containing the control mean. For a table classified by a single factor, it specifies a scalar or a single-valued text to indicate the level or label of the factor for the control treatment. For a multi-way table, the CLASSIFICATION option must specify a pointer containing its classifying factors. The CONTROL parameter then specifies a pointer, containing scalars or single-valued texts, indicating the levels or labels of the classifying factors for the control (specified in the same order as in the CLASSIFICATION pointer). All the other means are tested against the control.
The SED and DF parameters specify standard errors for differences for the means and their numbers of degrees of freedom, respectively. These can supply scalars if they are the same for every pair of means, or otherwise symmetric matrices. The order of the rows in a symmetric matrix must be compatible with the order of the means in the table. To ensure compatibility, you should save the standard errors of differences and degrees of freedom from the same AKEEP, PREDICT or VKEEPstatement as the means.
The METHOD option specifies the type of test. It can be set to either equivalence (default), noninferiority or nonsuperiority. The EQLIMITS option supplies a variate with the two limits for an equivalence test. The first value must be negative and the second must be positive. For a non-inferiority test, it supplies a scalar containing the (negative) limit. For a non-superiority test it supplies a scalar containing the (positive) limit.
Printed output is controlled by the PRINT option with settings:
description control mean and limit(s),
test t-statistic and probability level, and
confidence confidence interval for the difference between means of treatments and control.
The default is PRINT=description,test. Usually a 95% confidence interval is calculated, but this can be changed by setting the CIPROBABILITY option to the corresponding probability. For the equivalence tests, the confidence interval is an amalgamation of two one-sided intervals, as you are making two one-sided tests. Each limit is therefore calculated for twice the distance from 100% (e.g. 90% instead of 95%, corresponding to a significance level of 5% for the test of equivalence).
You can plot the confidence intervals by setting option PLOT=confidence. The TITLE option specifies the title for the plot; default 'Confidence plot'. The WINDOW option specifies the window to use. If this is not set, TEQUIVALENCE uses a window defined to fill the whole (0,1) × (0,1) square. The SCREEN option allows you to add the plot to an existing graphics screen; by default the screen is cleared.
The TSTATISTICS parameter can save the t-statistics for the tests, in a variate or a table according to the setting of the MEANS parameter. The PROBABILITIES parameter can similarly save the probabilities of the tests. The DIFFERENCES parameter can save the differences of the means from the control mean, again in either a variate or a table. The SEDCONTROL and DFCONTROL parameters can similarly save the standard errors of their differences and their degrees of freedom, respectively. The LOWER and UPPER parameters can save the lower and upper confidence limits, again in either a variate or a table.
Options: PRINT, PLOT, CLASSIFICATION, METHOD, CIPROBABILITY, EQLIMITS, TITLE, WINDOW, SCREEN.
Parameters: MEANS, CONTROL, SED, DF, TSTATISTICS, PROBABILITIES, DIFFERENCES, SEDCONTROL, DFCONTROL, LOWER, UPPER.
See also
Procedure: TTEST.
Commands for: Basic and nonparametric statistics, Analysis of variance.
Example
CAPTION 'TEQUIVALENCE example',\
!t('Effect of 6 different diets on gain in weight of rats',\
'(datafrom Snedecor & Cochran, Statistical Methods p.305)');\
STYLE=meta,plain
SPLOAD '%data%/Ratfactorial.gsh'
TREATMENTS Source*Amount
ANOVA Gain
AKEEP Source; MEANS=MS; SED=SEDS; DFMEANS=DFMS
" Test the non-superiority of the meat diets to the cereal diet,
with a limit for non-superiority of 12. (Conclusion: the Pork
diet is non-superior, but the Beef diet is on the borderline.) "
TEQUIVALENCE [PRINT=description,test,confidence; METHOD=nonsuperiority;\
EQLIMIT=12] MEANS=MS; CONTROL='Cereal'; SED=SEDS; DF=DFMS