| Title: | Provides Tests and Graphics for Assessing Tests of Equivalence |
|---|---|
| Description: | Provides statistical tests and graphics for assessing tests of equivalence. Such tests have similarity as the alternative hypothesis instead of the null. Sample data sets are included. |
| Authors: | Andrew Robinson <[email protected]> |
| Maintainer: | Andrew Robinson <[email protected]> |
| License: | GPL-2 |
| Version: | 0.7.2 |
| Built: | 2024-10-31 20:38:15 UTC |
| Source: | CRAN |
This function wraps the regression-based TOST equivalence test inside a bootstrap, extracts and reports the useful quantities, and reports the outcome of the test. The function was written for validating models, and requires paired data points. To use it for this purpose, pass the model predictions as the predictor variable, and the observations (which the predictions are intended to match) as the response variable.
equiv.boot(x, y, alpha = 0.05, b0.ii = 0.25, b1.ii = 0.25, reps = 100, b0.ii.absolute = FALSE)equiv.boot(x, y, alpha = 0.05, b0.ii = 0.25, b1.ii = 0.25, reps = 100, b0.ii.absolute = FALSE)
x |
the predictor variable (commonly predictions) |
y |
the response variable (commonly observations) |
alpha |
the size of the test |
b0.ii |
the half-length of the region of similarity for the intercept, expressed as a proportion of the estimate or in the same units as the estimate (see b0.ii.absolute). |
b1.ii |
the half-length of the region of similarity for the slope, expressed as a proportion of the estimate. |
reps |
the number of bootstrap replicates required |
b0.ii.absolute |
option to express b0.ii in the same units as the estimate of the intercept. |
In each case, if the two one-sided confidence interval is inside the region of similarity then the null hypothesis of dissimilarity is rejected.
A list of length 10, comprising
n |
The effective (non-missing) sample size |
ci.b0 |
The intercept TOST confidence interval |
rs.b0 |
The intercept region of similarity |
q.b0 |
The proportions of simulations below, within, and above, the intercept region of similarity |
Test.b0 |
The outcome of the test of the null hypothesis of dissimilarity for the intercept (Reject/Not Reject) |
ci.b1 |
The slope TOST confidence interval |
rs.b1 |
The slope region of similarity |
q.b1 |
The proportions of simulations below, within, and above, the slope region of similarity |
Test.b1 |
The outcome of the test of the null hypothesis of dissimilarity for the slope (Reject/Not Reject) |
eff.alpha |
The corrected alpha for each of the two independent tests. |
Feedback from Mohammad Al-Ahmadi has been very useful for this function.
Andrew Robinson [email protected]
Robinson, A.P., R.A. Duursma, and J.D. Marshall. 2005. A regression-based equivalence test for model validation: shifting the burden of proof. Tree Physiology 25, 903-913.
# Approximately reproduces the first row from Table 2 of Robinson et al. (2005) data(pref.4PG) equiv.boot(pref.4PG$volinc4PG, pref.4PG$stemvolinc)# Approximately reproduces the first row from Table 2 of Robinson et al. (2005) data(pref.4PG) equiv.boot(pref.4PG$volinc4PG, pref.4PG$stemvolinc)
This function generates the TOST intervals for the intercept and the slope of the regression of y on x, and determines the smallest region of indifference in each case that would reject the null hypothesis of dissimilarity.
equiv.p(x, y, alpha = 0.05)equiv.p(x, y, alpha = 0.05)
x |
The predictor variable - perhaps the model predictions |
y |
The response variable - perhaps the observations |
alpha |
The size of the test |
The generated confidence intervals are corrected for experiment-level size of alpha using Bonferroni.
A list of two items:
Intercept |
The smallest half-length of the interval that leads to rejection of the null hypothesis of dissimilarity for the intercept, in the units of y. |
Slope |
The smallest half-length of the interval that leads to rejection of the null hypothesis of dissimilarity for the slope, in the units of the slope. |
The accuracy of the output of this function is contingent on the usual regression assumptions, which are not checked here. Caveat emptor!
Andrew Robinson [email protected]
Robinson, A.P., and R.E. Froese. 2004. Model validation using equivalence tests. Ecological Modelling 176, 349–358.
Robinson, A.P., R.A. Duursma, and J.D. Marshall. 2005. A regression-based equivalence test for model validation: shifting the burden of proof. Tree Physiology 25, 903-913.
data(ufc) equiv.p(ufc$Height.m.p, ufc$Height.m)data(ufc) equiv.p(ufc$Height.m.p, ufc$Height.m)
This package provides tools to perform several equivalence tests.
Recent changes: the tost.data function is deprecated as of version 0.5.0; please use tost, which provides more functionality.
Andrew Robinson [email protected]
Schuirmann, D.L. 1981. On hypothesis testing to determine if the mean of a normal distribution is contained in a known interval. Biometrics 37 617.
Robinson, A.P., and R.E. Froese. 2004. Model validation using equivalence tests. Ecological Modelling 176, 349–358.
Wellek, S. 2003. Testing statistical hypotheses of equivalence. Chapman and Hall/CRC. 284 pp.
Westlake, W.J. 1981. Response to T.B.L. Kirkwood: bioequivalence testing - a need to rethink. Biometrics 37, 589-594.
data(ufc) ### Tost tost(ufc$Height.m.p, ufc$Height.m, epsilon = 1) ### equivalence plot ufc.ht <- ufc[!is.na(ufc$Height),] equivalence.xyplot(ufc.ht$Height.m ~ ufc.ht$Height.m.p, alpha=0.05, b0.ii=0.1, b1.ii=0.2, xlab="Predicted height (m)", ylab="Measured height (m)")data(ufc) ### Tost tost(ufc$Height.m.p, ufc$Height.m, epsilon = 1) ### equivalence plot ufc.ht <- ufc[!is.na(ufc$Height),] equivalence.xyplot(ufc.ht$Height.m ~ ufc.ht$Height.m.p, alpha=0.05, b0.ii=0.1, b1.ii=0.2, xlab="Predicted height (m)", ylab="Measured height (m)")
These functions are provided for compatibility with older versions of R only, and may be defunct as soon as the next release.
tost.data(x, null = 0, alpha = 0.05, Epsilon = 0.36, absolute = FALSE)tost.data(x, null = 0, alpha = 0.05, Epsilon = 0.36, absolute = FALSE)
x |
the sample of paired differences |
null |
the value of the parameter in the null hypothesis |
alpha |
test size |
Epsilon |
magnitude of region of similarity |
absolute |
choose units: absolute (TRUE) or relative to the standard deviation (FALSE). |
The original help page for these functions is
available at help("oldName-deprecated") (note the quotes).
Functions in packages other than the base package are listed in
help("pkg-deprecated").
tost.data is superceded by tost.
Implements regression-based tests of equivalence within lattice graphics.
equivalence.xyplot(formula, alpha, b0.ii, b1.ii, add.smooth=FALSE, b0.absolute=FALSE, ...)equivalence.xyplot(formula, alpha, b0.ii, b1.ii, add.smooth=FALSE, b0.absolute=FALSE, ...)
formula |
a formula describing the form of conditioning plot. See
the manual entry for |
alpha |
the size of the test |
b0.ii |
the half-length of the region of similarity for the intercept, can be relative or absolute (see below). |
b1.ii |
the half-length of the region of similarity for the slope. |
add.smooth |
adds a loess smooth to the graph. |
b0.absolute |
is b0.ii in absolute or relative units? |
... |
extra arguments passed on to xyplot |
The graphic created by this function was proposed by Robinson et al. (2005) as a visual summary of the regression-based TOST. At first glance the graph will look messy; interpretation eases with practice. The following points should be noted.
LS line:A black, solid line of best fit is added.
Mean:A grey vertical bar indicates the mean of x and the TOST confidence interval for the intercept.
b0 R.S.:A shaded polygon is the region of similarity of the intercept, to test the model bias.
Test b0:If the grey vertical bar is within the shaded polygon then reject the null hypothesis of dissimilarity. This is a test of bias.
...1 -If the region is too low then the predictions are too low.
...2 -If the region is too high then the predictions are too high.
...3 -If the region is too narrow then the predictions are too variable.
b1 C.I.:A black vertical bar undermeath the grey bar represents a confidence interval for the slope of the line of best fit.
b1 R.S.:Two black dashed lines are added representing the region of similarity.
Test b1:If the black bar is within the angle described by the dashed black lines then the slope of the observed/predicted regression is significantly similar to 1. This is a test of proportionality.
...1 -If the bar is too high then the slope is too high; the model over-predicts the higher observations and under-predicts the lower observations.
...2 -If the bar is too low then the slope is too low; the model underpredicts the higher observations and overpredicts the lower observations (analogous to regression to the mean).
...3 -If the bar is too narrow then the predictions are too variable.
The implementation in Robinson et al. (2005) required shifting so that the predictor has 0 mean. This hack has been removed here so that the basic graph object is a plot of the two variables being compared.
Run for its side effect of producing a lattice plot object.
The accuracy of the output of this function is contingent on the usual regression assumptions, which are not checked here. Caveat emptor! Consider using equiv.boot() for a bootstrap-based solution. Transforming either variable will probably complicate the analysis considerably.
Feedback from Mohammad Al-Ahmadi has been very useful for this function.
This version produces a regression-based TOST for each level of the conditioning factor. There may be an argument for pooling the test across these levels, in which case some prepanel computations will be helpful.
The TOST requires only estimates and standard errors from the data. Therefore the linear model used in the panel function can be replaced by any model that will produce suitable estimates. For example, in applying this function to hierarchical data we have had success using lme() instead.
I'm not entirely convinced that all these lines on one image are a good idea. It's straightforward to remove some, or change the colours. Recommendations for graphics that are visually cleaner are welcome.
Andrew Robinson [email protected]
Robinson, A.P., R.A. Duursma, and J.D. Marshall. 2005. A regression-based equivalence test for model validation: shifting the burden of proof. Tree Physiology 25, 903-913.
data(pref.4PG) equivalence.xyplot(pref.4PG$stemvolinc ~ pref.4PG$volinc4PG, alpha=0.05, b0.ii=0.25, b1.ii=0.25, add.smooth=TRUE, xlab=expression(paste("4PG decadal volume growth (", m^3, ha^-1, decade^-1, ")", sep="")), ylab=expression(paste("Measured decadal volume growth (", m^3, ha^-1, decade^-1, ")", sep=""))) data(pref.LAI) equivalence.xyplot(pref.LAI$lai.pa ~ pref.LAI$lai.bl, alpha=0.05, b0.ii=0.25, b1.ii=0.25, xlab=expression(paste("LAI Beer-Lambert (", m^2, m^-2, ")", sep="")), ylab=expression(paste("LAI Ceptometer (", m^2, m^-2, ")", sep=""))) data(ufc) ufc.ht <- ufc[!is.na(ufc$Height),] equivalence.xyplot(ufc.ht$Height.m ~ ufc.ht$Height.m.p, alpha=0.05, b0.ii=0.1, b1.ii=0.2, xlab="Predicted height (m)", ylab="Measured height (m)") equivalence.xyplot(ufc.ht$Height.m ~ ufc.ht$Height.m.p | ufc.ht$Species, alpha=0.05, b0.ii=0.1, b1.ii=0.2, xlab="Predicted height (m)", ylab="Measured height (m)", subset=ufc.ht$Species %in% levels(ufc.ht$Species)[table(ufc.ht$Species)>5])data(pref.4PG) equivalence.xyplot(pref.4PG$stemvolinc ~ pref.4PG$volinc4PG, alpha=0.05, b0.ii=0.25, b1.ii=0.25, add.smooth=TRUE, xlab=expression(paste("4PG decadal volume growth (", m^3, ha^-1, decade^-1, ")", sep="")), ylab=expression(paste("Measured decadal volume growth (", m^3, ha^-1, decade^-1, ")", sep=""))) data(pref.LAI) equivalence.xyplot(pref.LAI$lai.pa ~ pref.LAI$lai.bl, alpha=0.05, b0.ii=0.25, b1.ii=0.25, xlab=expression(paste("LAI Beer-Lambert (", m^2, m^-2, ")", sep="")), ylab=expression(paste("LAI Ceptometer (", m^2, m^-2, ")", sep=""))) data(ufc) ufc.ht <- ufc[!is.na(ufc$Height),] equivalence.xyplot(ufc.ht$Height.m ~ ufc.ht$Height.m.p, alpha=0.05, b0.ii=0.1, b1.ii=0.2, xlab="Predicted height (m)", ylab="Measured height (m)") equivalence.xyplot(ufc.ht$Height.m ~ ufc.ht$Height.m.p | ufc.ht$Species, alpha=0.05, b0.ii=0.1, b1.ii=0.2, xlab="Predicted height (m)", ylab="Measured height (m)", subset=ufc.ht$Species %in% levels(ufc.ht$Species)[table(ufc.ht$Species)>5])
These data are the juxtaposition of model output and field measurements for the Priest River Experimental Forest in northern Idaho, USA. The model was a process-based aspatial forest stand model called 4-PG, developed from 3-PG by Duursma (2004). The data were used to demonstrate a regression-based TOST in Robinson et al (2005).
data(pref.4PG)data(pref.4PG)
A dataset with 35 observations on 33 variables, of which the following two were used for the model validation exercise.
measured decadal stem growth in $m^3/ha$
predicted decadal stem growth in $m^3/ha$
Nothing in particular. Devil's club (Oplopanax horridus) is very painful.
The data are documented in Duursma (2004) and Robinson et al (2005).
Duursma, R.A. 2004. A simple process-based model of forest growth, and a test for the Priest River Experimental Forest. Ph.D. Thesis, University of Idaho, 169 p.
Robinson, A.P., R.A. Duursma, and J.D. Marshall. 2005. A regression-based equivalence test for model validation: shifting the burden of proof. Tree Physiology 25, 903-913
data(pref.4PG)data(pref.4PG)
These data are the juxtaposition of model output and field measurements for 36 forest plots in the Priest River Experimental Forest in northern Idaho, USA. The model was based on the Beer-Lambert model (Duursma et al. 2003). The data were used to demonstrate a regression-based TOST in Robinson et al (2005).
data(pref.LAI)data(pref.LAI)
A dataset with 36 observations on 8 variables, the following two of which were used for the test of equivalence:
LAI as estimated using allometric functions applied to measurements of trees within the sample plot.
LAI as estimated using the Beer-Lambert model from measurements of interception using a hand-held ceptometer.
Nothing in particular. PREF, and northern Idaho, are very attractive.
The data are documented in Duursma et al. (2003) and Robinson et al (2005).
Duursma, R.A., J.D. Marshall, and A.P. Robinson. 2003. Leaf area index inferred from solar beam transmission in mixed conifer forests on complex terrain. Agricultural and Forest Meteorology 118, 221-236.
Robinson, A.P., R.A. Duursma, and J.D. Marshall. 2005. A regression-based equivalence test for model validation: shifting the burden of proof. Tree Physiology 25, 903-913.
data(pref.LAI)data(pref.LAI)
Printing objects of class '"tost"' by simple 'print' methods.
## S3 method for class 'tost' print(x, ...)## S3 method for class 'tost' print(x, ...)
x |
object of class '"tost"' |
... |
arguments to be passed to other functions. |
The function inherits infrastructure from R's print.htest, so a number of elements have been copied from the help file of that function.
the argument 'x', invisibly, as for all 'print' methods.
Andrew Robinson [email protected]
data(ufc) tost(ufc$Height.m.p, ufc$Height.m, epsilon = 1, paired = TRUE)data(ufc) tost(ufc$Height.m.p, ufc$Height.m, epsilon = 1, paired = TRUE)
This function computes the test and key test quantities for the paired t-test for equivalence, as documented in Wellek (2003, pp 77-80). This function computes the test from a sample of a normally-distributed population.
ptte.data(x, alpha = 0.05, Epsilon = 0.25)ptte.data(x, alpha = 0.05, Epsilon = 0.25)
x |
paired differences |
alpha |
test size |
Epsilon |
magnitude of region of similarity |
This test requires the assumption of normality of the population. Under that assumption the test is the uniformly most powerful invariant test (Wellek, 2003, pp. 78-79).
The function as documented by Wellek (2003) uses units relative to the standard deviation, noting (p. 12) that 0.25 corresponds to a strict test and 0.5 to a liberal test.
A list with the following components
Dissimilarity |
the outcome of the test of the null hypothesis of dissimilarity |
Mean |
the mean of the sample |
StdDev |
the standard deviation of the sample |
n |
the sample size |
alpha |
the size of the test |
missing |
the number of observations missing |
Epsilon |
the magnitude of the region of similarity |
cutoff |
the critical value |
Tstat |
the test statistic; if Tstat < cutoff then the null hypothesis is rejected. |
Power |
the power of the test evaluated at the observed value |
This test requires the assumption of normality of the population. Under that assumption the test is the uniformly most powerful invariant test (Wellek, 2003, pp. 78-79). The exposition in Robinson and Froese (2004) mistakenly omits the square root of the F-quantile.
Andrew Robinson[email protected]
Robinson, A.P., and R.E. Froese. 2004. Model validation using equivalence tests. Ecological Modelling 176, 349–358.
Wellek, S. 2003. Testing statistical hypotheses of equivalence. Chapman and Hall/CRC. 284 pp.
data(ufc) ptte.data(ufc$Height.m.p - ufc$Height.m)data(ufc) ptte.data(ufc$Height.m.p - ufc$Height.m)
This function computes the test and key test quantities for the paired t-test for equivalence, as documented in Wellek (2003, pp 77-80). This function computes the test from the mean and standard deviation of a sample of paired differences from a normally-distributed population.
ptte.stat(mean, std, n, alpha = 0.05, Epsilon = 0.25)ptte.stat(mean, std, n, alpha = 0.05, Epsilon = 0.25)
mean |
the sample mean |
std |
the sample standard deviation |
n |
sample size |
alpha |
test size |
Epsilon |
magnitude of region of similarity |
This test requires the assumption of normality of the population. Under that assumption the test is the uniformly most powerful invariant test (Wellek, 2003, pp. 78-79). This version of the test can be applied post-hoc to any testing situation in which you have the mean, standard deviation, and sample size, and are confident that the sample is drawn from a normally-distributed population.
The function as documented by Wellek (2003) uses units relative to the standard deviation, noting (p. 12) that 0.25 corresponds to a strict test and 0.5 to a liberal test.
A list with the following components
Dissimilarity |
the outcome of the test of the null hypothesis of dissimilarity |
Mean |
the mean of the sample |
StdDev |
the standard deviation of the sample |
n |
the sample size |
alpha |
the size of the test |
Epsilon |
the magnitude of the region of similarity |
cutoff |
the critical value |
Tstat |
the test statistic; if Tstat < cutoff then the null hypothesis is rejected. |
Power |
the power of the test evaluated at the observed value |
The exposition in Robinson and Froese (2004) mistakenly omits the square root of the F-quantile.
Andrew Robinson [email protected]
Robinson, A.P., and R.E. Froese. 2004. Model validation using equivalence tests. Ecological Modelling 176, 349–358.
Wellek, S. 2003. Testing statistical hypotheses of equivalence. Chapman and Hall/CRC. 284 pp.
data(ufc) ptte.stat(mean(ufc$Height.m.p - ufc$Height.m, na.rm=TRUE), sd(ufc$Height.m.p - ufc$Height.m, na.rm=TRUE), sum(!is.na(ufc$Height.m.p - ufc$Height.m)))data(ufc) ptte.stat(mean(ufc$Height.m.p - ufc$Height.m, na.rm=TRUE), sd(ufc$Height.m.p - ufc$Height.m, na.rm=TRUE), sum(!is.na(ufc$Height.m.p - ufc$Height.m)))
This function computes the TOST and key TOST quantities for the two one-sided test for equivalence [Schuirmann (1981) and Westlake (1981)], using the robust t-test of Yuen [Yuen and Dixon (1973), Yuen (1974)] in place of the standard Welch t test (t.test stats). The yuen t test makes no assumption of normality. The function computes the robust TOST for a sample of paired differences or for two samples. The function performs almost as well as the Welch t test when the population distribution is normal and is more robust than the Welch t test in the face of non-normality (e.g., distributions that are long-tailed, heteroscedastic, or contaminated by outliers)[Yuen and Dixon (1973), Yuen (1974)].
rtost(x, y = NULL, alpha = 0.05, epsilon = 0.31, tr = 0.2, ...)rtost(x, y = NULL, alpha = 0.05, epsilon = 0.31, tr = 0.2, ...)
x |
the first (or only) sample |
y |
the second sample |
alpha |
test size |
tr |
the proportion (percent/100) of the data set to be trimmed |
epsilon |
magnitude of region of similarity |
... |
arguments to be passed to yuen.t.test |
The rtost function is wrapped around the yuen PairedData t test, a robust variant of the t test using trimmed means and winsorized variances. It provides tosts for the same range of designs, accepts the same arguments, and handles missing values the same way as tost equivalence. For the tost, the user must set epsilon, which is the magnitude of region similarity. Warning: with tr > 0.25 type I error control might be poor.
A list with the following components
mean.diff |
the mean of the difference |
se.diff |
the standard error of the difference |
alpha |
the size of the test |
ci.diff |
the 1-alpha confidence interval for the difference |
df |
the degrees of freedom used for the confidence interval |
epsilon |
the magnitude of the region of similarity |
result |
the outcome of the test of the null hypothesis of dissimilarity |
p.value |
the p-value of the significance test |
check.me |
the confidence interval corresponding to the p-value |
This test is the tost for equivalence (tost equivalence) wrapped around the robust, trimmed mean, winsorized variance yuen.t.test (yuen PairedData).
Gregory Belenky [email protected]
Schuirmann, D.L. 1981. On hypothesis testing to determine if the mean of a normal distribution is contained in a known interval. Biometrics 37, 617.
Robinson, A.P., and R.E. Froese. 2004. Model validation using equivalence tests. Ecological Modelling 176, 349–358.
Wellek, S. 2003. Testing statistical hypotheses of equivalence. Chapman and Hall/CRC. 284 pp.
Westlake, W.J. 1981. Response to T.B.L. Kirkwood: bioequivalence testing - a need to rethink. Biometrics 37, 589-594.
Yuen, K.K. (1974) The two-sample trimmed t for unequal population variances. Biometrika, 61, 165-170.
Yuen, K.K., and Dixon, W.J. (1973) The approximate behavior and performance of the two-sample trimmed t. Biometrika, 60, 369-374.
data(ufc) rtost(ufc$Height.m.p, ufc$Height.m, epsilon = 1, tr = 0.2)data(ufc) rtost(ufc$Height.m.p, ufc$Height.m, epsilon = 1, tr = 0.2)
This function computes the test and key test quantities for the two one-sided test for equivalence, as documented in Schuirmann (1981) and Westlake (1981). The function computes the test for a sample of paired differences or two samples, assumed to be from a normally-distributed population.
Much code in the function has been copied and adapted from R's t.test.default function.
tost(x, y = NULL, epsilon = 1, paired = FALSE, var.equal = FALSE, conf.level = 0.95, alpha = NULL, ...)tost(x, y = NULL, epsilon = 1, paired = FALSE, var.equal = FALSE, conf.level = 0.95, alpha = NULL, ...)
x |
the first (or only) sample |
y |
the second sample |
epsilon |
magnitude of region of similarity |
paired |
a logical indicating whether you want a paired tost |
var.equal |
a logical variable indicating whether to treat the two variances as being equal. If 'TRUE' then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used. |
conf.level |
confidence level of the interval |
alpha |
test size (for backwards-compatibility, overrides conf.level) |
... |
arguments to be passed to other functions. |
The function inherits infrastructure from R's t.test.default, so a number of elements have been copied from the help file of that function.
This test requires the assumption of normality of the population, or an invocation of large-sample theory. The function wraps around the t.test function, so it provides tosts for the same range of designs, accepts the same arguments, and handles missing values the same way.
If 'paired' is 'TRUE' then both 'x' and 'y' must be specified and they must be the same length. Missing values are silently removed (in pairs if 'paired' is 'TRUE'). If 'var.equal' is 'TRUE' then the pooled estimate of the variance is used. By default, if 'var.equal' is 'FALSE' then the variance is estimated separately for both groups and the Welch modification to the degrees of freedom is used.
A list with the following components
estimate |
the mean of the difference |
se.diff |
the standard error of the difference |
alpha |
the size of the test |
data.name |
a character string giving the name(s) of the data |
ci.diff |
the 1-alpha confidence interval for the difference |
parameter |
the degrees of freedom used for the confidence interval |
epsilon |
the magnitude of the region of similarity |
result |
the outcome of the test of the null hypothesis of dissimilarity |
method |
a character string indicating what type of t-test was performed |
null.value |
the specified hypothesized value of the mean or mean difference depending on whether it was a one-sample tost or a two-sample tost. |
tost.p.value |
the p-value of the tost significance test |
tost.interval |
the two one-sided confidence interval corresponding to the test. |
This test requires the assumption of normality of the population. The components of the test are t-based confidence intervals, so the Central Limit Theorem and Slutsky's Theorem may be relevant to its application in large samples.
Andrew Robinson [email protected]
Schuirmann, D.L. 1981. On hypothesis testing to determine if the mean of a normal distribution is contained in a known interval. Biometrics 37 617.
Robinson, A.P., and R.E. Froese. 2004. Model validation using equivalence tests. Ecological Modelling 176, 349–358.
Wellek, S. 2003. Testing statistical hypotheses of equivalence. Chapman and Hall/CRC. 284 pp.
Westlake, W.J. 1981. Response to T.B.L. Kirkwood: bioequivalence testing - a need to rethink. Biometrics 37, 589-594.
data(ufc) tost(ufc$Height.m.p, ufc$Height.m, epsilon = 1, paired = TRUE)data(ufc) tost(ufc$Height.m.p, ufc$Height.m, epsilon = 1, paired = TRUE)
This function computes the test and key test quantities for the two one-sided test for equivalence, as documented in Schuirmann (1981) and Westlake (1981). This function computes the test from the statistics of a sample of paired differences of a normally-distributed population.
tost.stat(mean, std, n, null = 0, alpha = 0.05, Epsilon = 0.36)tost.stat(mean, std, n, null = 0, alpha = 0.05, Epsilon = 0.36)
mean |
sample mean |
std |
sample standard deviation |
n |
sample size |
null |
the value of the parameter in the null hypothesis |
alpha |
test size |
Epsilon |
magnitude of region of similarity |
This test requires the assumption of normality of the population.
A list with the following components
Dissimilarity |
the outcome of the test of the null hypothesis of dissimilarity |
Mean |
the mean of the sample |
StdDev |
the standard deviation of the sample |
n |
the non-missing sample size |
alpha |
the size of the test |
Epsilon |
the magnitude of the region of similarity |
Interval |
the half-length of the two one-sided interval |
This test requires the assumption of normality of the population. The components of the test are t-based confidence intervals, so the Central Limit Theorem and Slutsky's Theorem may be relevant to its application in large samples.
Andrew Robinson [email protected]
Schuirmann, D.L. 1981. On hypothesis testing to determine if the mean of a normal distribution is contained in a known interval. Biometrics 37 617.
Wellek, S. 2003. Testing statistical hypotheses of equivalence. Chapman and Hall/CRC. 284 pp.
Westlake, W.J. 1981. Response to T.B.L. Kirkwood: bioequivalence testing - a need to rethink. Biometrics 37, 589-594.
data(ufc) tost.stat(mean(ufc$Height.m.p - ufc$Height.m, na.rm=TRUE), sd(ufc$Height.m.p - ufc$Height.m, na.rm=TRUE), sum(!is.na(ufc$Height.m.p - ufc$Height.m)))data(ufc) tost.stat(mean(ufc$Height.m.p - ufc$Height.m, na.rm=TRUE), sd(ufc$Height.m.p - ufc$Height.m, na.rm=TRUE), sum(!is.na(ufc$Height.m.p - ufc$Height.m)))
These are forest measurement data from the Upper Flat Creek unit of the University of Idaho Experimental Forest, measured in 1991. The inventory was based on variable radius plots with 6.43 sq. m. per ha. BAF (Basal Area Factor). The forest stand was 121.5 ha.
data(ufc)data(ufc)
A data frame with 633 observations on the following 12 variables.
plot label
tree label
species kbd
tree diameter at 1.37 m. from the ground, measured in millimetres.
tree height measured in decimetres
tree height converted to feet
tree diameter converted to inches
predicted tree height in feet
tree height in metres
predicted tree height in metres
Plots that were measured with no trees are signified in the dataset by lines that have blank species codes and missing DBH.
The data are provided courtesy of Harold Osborne and Ross Appelgren of the University of Idaho Experimental Forest. The predicted height comes from the height/diameter model documented in Wykoff et al. (1982). The data and model were used in Robinson et al. (2005).
Robinson, A.P., R.A. Duursma, and J.D. Marshall. 2005. A regression-based equivalence test for model validation: shifting the burden of proof. Tree Physiology 25, 903-913.
Wykoff,W.R., N.L. Crookston, and A.R. Stage. 1982. User's guide to the stand Prognosis model. GTR-INT 133, USDA Forest Service, Intermountain Research Station, Ogden, UT, 113 p.
data(ufc)data(ufc)