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 `VKEEP`

statement 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