Package 'relevance' reference manual

Title:	Calculate Relevance and Significance Measures
Description:	Calculates relevance and significance values for simple models and for many types of regression models. These are introduced in 'Stahel, Werner A.' (2021) "Measuring Significance and Relevance instead of p-values." <https://stat.ethz.ch/~stahel/relevance/stahel-relevance2103.pdf>. These notions are also applied to replication studies, as described in the manuscript 'Stahel, Werner A.' (2022) "'Replicability': Terminology, Measuring Success, and Strategy" available in the documentation.
Authors:	Werner A. Stahel
Maintainer:	Werner A. Stahel <[email protected]>
License:	GPL-2
Version:	2.1
Built:	2024-03-26 07:41:08 UTC
Source:	CRAN

Calculate Relevance and Significance Measures

Description

Calculates relevance and significance values for simple models and for many types of regression models. These are introduced in 'Stahel, Werner A.' (2021) "Measuring Significance and Relevance instead of p-values." <https://stat.ethz.ch/~stahel/relevance/stahel-relevance2103.pdf>. These notions are also applied to replication studies, as described in the manuscript 'Stahel, Werner A.' (2022) "'Replicability': Terminology, Measuring Success, and Strategy" available in the documentation.

Details

The DESCRIPTION file:

Package:	relevance
Type:	Package
Title:	Calculate Relevance and Significance Measures
Version:	2.1
Date:	2024-01-24
Author:	Werner A. Stahel
Maintainer:	Werner A. Stahel <[email protected]>
Depends:	R (>= 3.5.0)
Imports:	stats, utils, graphics
Suggests:	MASS, survival, knitr
VignetteBuilder:	knitr
Description:	Calculates relevance and significance values for simple models and for many types of regression models. These are introduced in 'Stahel, Werner A.' (2021) "Measuring Significance and Relevance instead of p-values." <https://stat.ethz.ch/~stahel/relevance/stahel-relevance2103.pdf>. These notions are also applied to replication studies, as described in the manuscript 'Stahel, Werner A.' (2022) "'Replicability': Terminology, Measuring Success, and Strategy" available in the documentation.
License:	GPL-2
NeedsCompilation:	no
Packaged:	2024-01-25 16:36:07 UTC; stahel
Repository:	CRAN
Date/Publication:	2024-01-25 17:00:02 UTC

Index of help topics:

asinp                   arc sine Transformation
confintF                Confidence Interval for the Non-Central F and
                        Chisquare Distribution
correlation             Correlation with Relevance and Significance
                        Measures
d.blast                 Blasting for a tunnel
d.everest               Data of an 'anchoring' experiment in psychology
d.negposChoice          Data of an 'anchoring' experiment in psychology
d.osc15                 Data from the OSC15 replication study
d.osc15Onesample        Data from the OSC15 replication study, one
                        sample tests
drop1Wald               Drop Single Terms of a Model and Calculate
                        Respective Wald Tests
dropNA                  drop or replace NA values
dropdata                Drop Observations from a Data.frame
formatNA                Print NA values by a Desired Code
getcoeftable            Extract Components of a Fit
inference               Calculate Confidence Intervals and Relevance
                        and Significance Values
last                    Last Elements of a Vector or of a Matrix
logst                   Started Logarithmic Transformation
ovarian                 ovarian
plconfint               Plot Confidence Intervals
plot.inference          Plot Inference Results
print.inference         Print Tables with Inference Measures
relevance-package       Calculate Relevance and Significance Measures
relevance.options       Options for the relevnance Package
replication             Inference for Replication Studies
rlvClass                Relevance Class
rplClass                Reproducibility Class
shortenstring           Shorten Strings
showd                   Show a Part of a Data.frame
sumNA                   Count NAs
termeffects             All Coefficients of a Model Fit
termtable               Statistics for Linear Models, Including
                        Relevance Statistics
twosamples              Relevance and Significance for One or Two
                        Samples

Further information is available in the following vignettes:

`relevance-descr`	'Calculate Relevance and Significance Measures' (source, pdf)

Relevance is a measure that expresses the (scientific) relevance of an effect. The simplest case is a single sample of supposedly normally distributed observations, where interest lies in the expectation, estimated by the mean of the observations. There is a threshold for the expectation, below which an effect is judged too small to be of interest.

The estimated relevance ‘ $Rle$ ’ is then simply the estimated effect divided by the threshold. If it is larger than 1, the effect is thus judged relevant. The two other values that characterize the relevance are the limits of the confidence interval for the true value of the relevance, called the secured relevance ‘ $Rls$ ’ and the potential relevance ‘ $Rlp$ ’.

If $Rle > 1$ , then one might say that the effect is “significantly relevant”.

Another useful measure, meant to replace the p-value, is the “significance” ‘Sg0’. In the simple case, it divides the estimated effect by the critical value of the (t-) test statistic. Thus, the statistical test of the null hypothesis of zero expectation is significant if ‘Sg0’ is larger than one, $Sg0 > 1$ .

These measures are also calculated for the comparison of two groups, for proportions, and most importantly for regression models. For models with linear predictors, relevances are obtained for standardized coefficients as well as for the effect of dropping terms and the effect on prediction.

The most important functions are

twosamples():: calculate the measures for two paired or unpaired sampless or a simple mean. This function calls
inference():: calculates the confidence interval and siginificance based on an estimate and a standard error, and adds relevance for a standardized effect.
termtable():: deals with fits of regression models with a linear predictor. It calculates confidence intervals and significances for the coefficients of terms with a single degree of freedom. It includes the effect of dropping each term (based on the drop1 function) and the respective significance and relevance measures.
termeffects():: calculates the relevances for the coefficients related to each term. These differ from the enties of termtable only for terms with more than one degree of freedom.

Author(s)

Werner A. Stahel

Maintainer: Werner A. Stahel <[email protected]>

References

Stahel, Werner A. (2021). New relevance and significance measures to replace p-values. To appear in PLoS ONE

Examples

data(swiss)
  rr <- lm(Fertility ~ . , data = swiss)
  termtable(rr)

arc sine Transformation

Description

Calculates the sqrt arc sine of x/100, rescaled to be in the unit interval.
This transformation is useful for analyzing percentages or proportions of any kind.

Usage

asinp(x)

Arguments

`x`	vector of data values

Value

vector of transformed values

Note

This very simple function is provided in order to simplify formulas. It has an attribute "inverse" that contains the inverse function, see example.

Author(s)

Werner A. Stahel, ETH Zurich

Examples

asinp(seq(0,100,10))
( y <- asinp(c(1,50,90,95,99)) )
attr(asinp, "inverse")(y)

Confidence Interval for the Non-Central F and Chisquare Distribution

Description

Confidence Interval for the Non-Central F and Chisquare Distribution

Usage

confintF(f, df1, df2, testlevel = 0.05)

Arguments

`f`	observed F value(s)
`df1`	degrees of freedom for the numerator of the F distribution
`df2`	degrees of freedom for the denominator of the F distribution
`testlevel`	level of the (two-sided) test that determines the confidence interval, 1 - confidence level

Details

The confidence interval is calculated by solving the two implicit equations qf(f, df1, df2, x) = testlevel/2 and ... = 1 - testlevel/2. For f>100, the usual f +- standard error interval is used as a rather crude approximation.

A confidence interval for the non-centrality of the Chisquare distribution is obtained by setting df2 to Inf (the default) and f=x2/df1 if x2 is the observed Chisquare value.

Value

vector of lower and upper limit of the confidence interval, or, if any of the arguments has length >1, matrix containing the intervals as rows.

Author(s)

Werner A. Stahel

Examples

confintF(5, 3, 200)
## [1] 2.107 31.95
confintF(1:5, 5, 20)   ## lower limit is 0 for the first 3 f values

Correlation with Relevance and Significance Measures

Description

Inference for a correlation coefficient: Collect quantities, including Relevance and Significance measures

Usage

correlation(x, y = NULL, method = c("pearson", "spearman"),
  hypothesis = 0, testlevel=getOption("testlevel"),
  rlv.threshold=getOption("rlv.threshold"), ...)

Arguments

`x`	data for the first variable, or matrix or data.frame containing both variables
`y`	data for the second variable
`hypothesis`	the null effect to be tested, and anchor for the relevance
`method`	type of correlation, either `"pearson"` for the ordinary Pearson product moment correlation, or `"spearman"` for the nonparametric measures
`testlevel`	level for the test, also determining the confidence level
`rlv.threshold`	Relevance threshold, or a vector of thresholds from which the element `corr` is taken
`...`	further arguments, ignored

Value

an object of class 'inference', a vector with components

effect:: correlation, transformed with Fisher's z transformation
ciLow, ciUp:: confidence interval for the effect
Rle, Rls, Rlp:: relevance measures: estimated, secured, potential
Sig0:: significance measure for test or 0 effect
Sigth:: significance measure for test of effect == relevance threshold
p.value:: p value for test against 0

In addition, it has attributes

method:: type of correlation
effectname:: label for the effect
hypothesis:: the null effect
n:: number(s) of observations
estimate:: estimated correlation
conf.int:: confidence interval on correlation scale
statistic:: test statistic
data:: data.frame containing the two variables
rlv.threshold:: relevance threshold

Author(s)

Werner A. Stahel

References

see those in relevance-package.

Examples

correlation(iris[1:50,1:2])

Blasting for a tunnel

Description

Blasting causes tremor in buildings, which can lead to damages. This dataset shows the relation between tremor and distance and charge of blasting.

Usage

data("d.blast")

Format

A data frame with 388 observations on the following 7 variables.

date: date in Date format
location: Code for location of the building, loc1 to loc8
device: Number of measuring device, 1 to 4
distance: Distance between blasting and location of measurement
charge: Charge of blast
tremor: Tremor energy (target variable)

Details

The charge of the blasting should be controled in order to avoid tremors that exceed a threshold. This dataset can be used to establish the suitable rule: For a given distance, how large can charge be in order to avoid exceedance of the threshold?

Source

Basler and Hoffmann AG, Zurich

Examples

data(d.blast)

summary(lm(log10(tremor)~location+log10(distance)+log10(charge),
           data=d.blast))

Data of an 'anchoring' experiment in psychology

Description

Are answers to questions influenced by providing partial information?

Students were asked to guesstimate the height of Mount Everest. One group was 'anchored' by telling them that it was more than 2000 feet, the other group was told that it was less than 45,500 feet. The hypothesis was that respondents would be influenced by their 'anchor,' such that the first group would produce smaller numbers than the second. The true height is 29,029 feet.

The data is taken from the 'many labs' replication study (see 'source'). The first 20 values from PSU university are used here.

Usage

data("d.everest")

Format

A data frame with 20 observations on the following 2 variables.

y: numeric: guesstimates of the height
g: factor with levels low high: anchoring group

Source

Klein RA, Ratliff KA, Vianello M et al. (2014). Investigating variation in replicability: A "many labs" replication project. Social Psychology. 2014; 45(3):142-152. https://doi.org/10.1027/1864-9335/a000178

Examples

data(d.everest)

(rr <- twosamples(log(y)~g, data=d.everest, var.equal=TRUE))
print(rr, show="classical")

pltwosamples(log(y)~g, data=d.everest)

Data of an 'anchoring' experiment in psychology

Description

Is a choice influenced by the formulation of the options?

Here is the question: Confronted with a new contagious disease, the government has a choice between action A that would save 200 out of 600 people or action B which would save all 600 with probability 1/3. This was the 'positive' description. The negative one was that either (A) 400 would die or (B) all 600 would die with probability 2/3.

The dataset encompasses the results for Penn State (US) and Tilburg (NL) universities.

Usage

data("d.negposChoice")

Format

A data frame with 4 observations on the following 4 variables.

uni: character: university
negpos: character: formulation of the options
A: number of students choosing option A
B: number of students choosing option B

Source

Examples

data(d.negposChoice)

d1 <- d.negposChoice[d.negposChoice$uni=="PSU",-1]
(r1 <- twosamples(table=d1[,-1]))
d2 <- d.negposChoice[d.negposChoice$uni=="Tilburg",-1]
r2 <- twosamples(table=d2[,-1])

Data from the OSC15 replication study

Description

The data of the famous replication study of the Open Science Collaboration published in 2015

Usage

data("d.osc15")

Format

d.osc15: The data frame of OSC15, with 100 observations on 149 variables, of which only the most important are described here. For a description of all variables, see the repository https://osf.io/jrxtm/

Study.Num: Identification number of the study
EffSize.O, EffSize.R: effect size as defined by OSC15, original paper and replication, respectively
Tst.O, Tst.R: test statistic, original and replication
N.O, N.R: number of observations, original and replication

Source

Data repository https://osf.io/jrxtm/

References

Open Science Collaboration (2015). Estimating the reproducibility of psychological science. Science 349, 943-952

Examples

data(d.osc15)

## plot effect sizes of replication against original
## row 9 has an erroneous EffSize.R, and there are 4 missing effect sizes
dd <- na.omit(d.osc15[-9,c("EffSize.O","EffSize.R")]) 
## change sign for negative original effects
dd[dd$EffSize.O<0,] <- -dd[dd$EffSize.O<0,] 
plot(dd)
abline(h=0)

Data from the OSC15 replication study, one sample tests

Description

A small subset of the data of the famous replication study of the Open Science Collaboration published in 2015, comprising the one sample and paired sample tests, used for illustration of the determination of succcess of the replications as defined by Stahel (2022)

Usage

data("d.osc15Onesample")

Format

d.osc15:

row.names: identification number of the study
teststatistico, teststatisticr: test statistic, original paper and replication, respectively
no, nr: number of observations, original and replication
effecto, effectr: effect size as defined by OSC15, original and replication

Source

Data repository https://osf.io/jrxtm/

References

Open Science Collaboration (2015). Estimating the reproducibility of psychological science. Science 349, 943-952

Examples

data(d.osc15Onesample)

plot(effectr~effecto, data=d.osc15Onesample, xlim=c(0,3.5),ylim=c(0,2.5),
     xaxs="i", yaxs="i")
abline(0,1)

## Compare confidence intervals between original paper and replication
to <- structure(d.osc15Onesample[,c("effecto","teststatistico","no")],
      names=c("effect","teststatistic","n"))
tr <- structure(d.osc15Onesample[,c("effectr","teststatisticr","nr")],
      names=c("effect","teststatistic","n"))
( rr <- replication(to, tr, rlv.threshold=0.1) )
plconfint(rr, refline=c(0,0.1))
plconfint(attr(rr, "estimate"), refline=c(0,0.1))

Drop Single Terms of a Model and Calculate Respective Wald Tests

Description

drop1Wald calculates tests for single term deletions based on the covariance matrix of estimated coefficients instead of re-fitting a reduced model. This helps in cases where re-fitting is not feasible, inappropriate or costly.

Usage

drop1Wald(object, scope=NULL, scale = NULL, test = NULL, k = 2, ...)

Arguments

`object`	a fitted model.
`scope`	a formula giving the terms to be considered for dropping. If 'NULL', 'drop.scope(object)' is obtained
`scale`	an estimate of the residual mean square to be used in computing Cp. Ignored if '0' or 'NULL'.
`test`	see `drop1`
`k`	the penalty constant in AIC / Cp.
`...`	further arguments, ignored

Details

The test statistics and Cp and AIC values are calculated on the basis of the estimated coefficients and their (unscaled) covariance matrix as provided by the fit object. The function may be used for all model fitting objects that contain these two components as $coefficients and $cov.unscaled.

Value

An object of class 'anova' summarizing the differences in fit between the models.

Note

drop1Wald is used for models of class 'lm' or 'lmrob' for preparing a termtable.

Author(s)

Werner A. Stahel

Examples

data(d.blast)
r.blast <- lm(log10(tremor)~location+log10(distance)+log10(charge),
              data=d.blast)
drop1(r.blast)
drop1Wald(r.blast)

## Example from example(glm)
dd <- data.frame(treatment = gl(3,3), outcome = gl(3,1,9),
           counts = c(18,17,15,20,10,20,25,13,12)) 
r.glm <- glm(counts ~ outcome + treatment, data = dd, family = poisson())
drop1(r.glm, test="Chisq")
drop1Wald(r.glm)

Drop Observations from a Data.frame

Description

Allows for dropping observations (rows) determined by row names or factor levels from a data.frame or matrix.

Usage

dropdata(data, rowid = NULL, incol = "row.names", colid = NULL)

Arguments

`data`	a data.frame of matrix
`rowid`	vector of character strings identifying the rows to be dropped
`incol`	name or index of the column used to identify the observations (rows)
`colid`	vector of character strings identifying the columns to be dropped

Value

The data.frame or matrix without the dropped observations and/or variables. Attributes are passed on.

Note

Ordinary subsetting by [...,...] drops attributes. Furthermore, the convenient way to drop rows or columns by giving negative indices to [...,...] cannot be used with names of rows or columns.

Author(s)

Werner A. Stahel, ETH Zurich

Examples

dd <- data.frame(rbind(a=1:3,b=4:6,c=7:9,d=10:12))
dropdata(dd,"b")
dropdata(dd, col="X3")

d1 <- dropdata(dd,"d")
d2 <- dropdata(d1,"b")
naresid(attr(d2,"na.action"),as.matrix(d2))

dropdata(letters, 3:5)

drop or replace NA values

Description

dropNA returns the vector 'x', without elements that are NA or NaN or, if 'inf' is TRUE, equal to Inf or -Inf. replaceNA replaces these values by values from the second argument

Usage

dropNA(x, inf = TRUE)
replaceNA(x, na, inf = TRUE)

Arguments

`x`	vector from which the non-real values should be dropped or replaced
`na`	replacement or vector from which the replacing values are taken.
`inf`	logical: should 'Inf' and '-Inf' be considered "non-real"?

Value

For dropNA: Vector containing the 'real' values of 'x' only
For replaceNA: Vector with 'non-real' values replaced by the respective elements of na.

Note

The differences to 'na.omit(x)' are: 'Inf' and '-Inf' are also dropped, unless 'inf==FALSE'.\ no attribute 'na.action' is appended.

Author(s)

Werner A. Stahel

Examples

dd <- c(1, NA, 0/0, 4, -1/0, 6)
dropNA(dd)
na.omit(dd)

replaceNA(dd, 99)
replaceNA(dd, 100+1:6)

Print NA values by a Desired Code

Description

Recodes the NA entries in output by a desired code like " ."

Usage

formatNA(x, na.print = " .", digits = getOption("digits"), ...)

Arguments

`x`	object to be printed, usually a numeric vector or data.frame
`na.print`	code to be used for `NA` values
`digits`	number of digits for formatting numeric values
`...`	other arguments to `format`

Details

The na.encode argument of print only applies to character objects. formatNA does the same for numeric arguments.

Value

Should mimik the value of format

Author(s)

Werner A. Stahel

Examples

formatNA(c(1,NA,3))

dd <- data.frame(X=c(1,NA,3), Y=c(4,5, NA), g=factor(c("a",NA,"b")))
(rr <- formatNA(dd, na.print="???"))
str(rr)

Extract Components of a Fit

Description

Retrieve the table of coefficients and standard errors, or the scale parameter, or the factors needed for standardizing coefficients from diverse model fitting results

Usage

getcoeftable(object)
getscalepar(object)
getcoeffactor(object, standardize = TRUE)

Arguments

`object`	an R object resulting from a model fitting function
`standardize`	ligical: should a scaling factor for the response variable be determined (calling `getscalepar`) and used?

Details

Object regrModelClasses contains the names of the classes for which the result should work. For other model classes, the function is not tested and may fail.

Value

For getcoeftable: Matrix containing at least the two columns containing the estimated coefficients (first column) and the standard errors (second column).

For getscalepar: scale parameter.

For getcoeffactor: vector of multiplicative factors, with attributes scale, fitclass and family or dist according to object.

Author(s)

Werner A. Stahel

Examples

rr <- lm(Fertility ~ . , data = swiss)
  getcoeftable(rr) # identical to  coef(summary(rr))  or also summary(rr)$coefficients
  getscalepar(rr)

 if(requireNamespace("survival", quietly=TRUE)) {
  data(ovarian) ## , package="survival"
  rs <- survival::survreg(survival::Surv(futime, fustat) ~ ecog.ps + rx,
                data = ovarian, dist = "weibull")
  getcoeftable(rs)
  getcoeffactor(rs)
 }

Calculate Confidence Intervals and Relevance and Significance Values

Description

Calculates confidence intervals and relevance and significance values given estimates, standard errors and, for relevance, additional quantities.

Usage

inference(object = NULL, estimate = NULL, teststatistic = NULL,
  se = NA, n = NULL, df = NULL,
  stcoef = TRUE, rlv = TRUE, rlv.threshold = getOption("rlv.threshold"),
  testlevel = getOption("testlevel"), ...)

Arguments

`object`	A data.frame containing, as its variables, the arguments `estimate` to `df`, as far as needed, or a vector to be used as `estimate` if `estimate` is not specified... ... or a model fit object
`estimate`	estimate(s) of the parameter(s)
`teststatistic`	test statistic(s)
`se`	standard error(s) of the estimate(s)
`n`	number(s) of observations
`df`	degrees of freedom of the residuals
`stcoef`	standardized coefficients. If `NULL`, these will be calculated from `object`, if the latter is a model fit.
`rlv`	logical: Should relevances be calculated?
`rlv.threshold`	Relevance threshold(s). May be a simple number for simple inference, or a vector containing the elements `stand`: threshold for (simple) standardized effects `rel`: for relative effects, `coef`: for standardized coefficients, `drop`: for drop effects, `pred`: for prediction intervals.
`testlevel`	1 - confidence level
`...`	furter arguments, passed to `termtable` and `termeffects`

Details

The estimates divided by standard errors are assumed to be t-distributed with df degrees of freedom. For df==Inf, this is the standard normal distribution.

Value

A data.frame of class "inference", with the variables

`effect`, `se`	estimated effect(s), often coefficients, and their standard errors
`ciLow`, `ciUp`	lower and upper limit of the confidence interval
`teststatistic`	t-test statistic
`p.value`	p value
`Sig0`	significance value, i.e., test statistic divided by critical value, which in turn is the `1-testlevel/2`-quantile of the t-distribution.
`ciLow`, `ciUp`	confidence interval for `effect`

If rlv is TRUE,

`stcoef`	standardized coefficient
`st.Low`, `st.Up`	confidence interval for `stcoef`
`Rle`	estimated relevance of `coef`
`Rls`	secured relevance, lower end of confidence interval for the relevance of `coef`
`Rlp`	potential relevance, upper end of confidence interval ...
`Rls.symbol`	symbols for the secured relevance
`Rlvclass`	relevance class

Author(s)

Werner A. Stahel

References

Werner A. Stahel (2020). New relevance and significance measures to replace p-values. PLOS ONE 16, e0252991, doi: 10.1371/journal.pone.0252991

Examples

data(d.blast)
rr <-
  lm(log10(tremor)~location+log10(distance)+log10(charge),
    data=d.blast) 
inference(rr)

Last Elements of a Vector or of a Matrix

Description

Selects or drops the last element or the last n elements of a vector or the last n rows or ncol columns of a matrix

Usage

last(data, n = NULL, ncol=NULL, drop=is.matrix(data))

Arguments

`data`	vector or matrix or data.frame from which to select or drop
`n`	if >0, `last` selects the last `n` elements (rows) form the result. if <0, the last `abs(n)` elements (rows) are dropped, and the first `length(data)-abs(n)` ones from the result
`ncol`	if `data` is a matrix or data.frame, the last `ncol` columns are selected (if `ncol` is positive) or dropped (if negative).
`drop`	if only one row or column of a matrix (or one column of a data.frame) is selected or left over, should the result be a vector or a row or column matrix (or one variable data.frame)

Value

The selected elements of the vector or matrix or data.frame

Note

This is a very simple function. It is defined mainly for selecting from the results of other functions without storing them.

Author(s)

Werner Stahel

Examples

x <- runif(rpois(1,10))
  last(sort(x), 3)
  last(sort(x), -5)
##
  df <- data.frame(X=c(2,5,3,8), F=LETTERS[1:4], G=c(TRUE,FALSE,FALSE,TRUE))
  last(df,3,-2)

Started Logarithmic Transformation

Description

Transforms the data by a log10 transformation, modifying small and zero observations such that the transformation yields finite values.

Usage

logst(data, calib=data, threshold=NULL, mult = 1)

Arguments

`data`	a vector or matrix of data, which is to be transformed
`calib`	a vector or matrix of data used to calibrate the transformation(s), i.e., to determine the constant `c` needed
`threshold`	constant c that determines the transformation, possibly a vector with a value for each variable.
`mult`	a tuning constant affecting the transformation of small values, see Details

Details

Small values are determined by the threshold c. If not given by the argument threshold, then it is determined by the quartiles $q_1$ and $q_3$ of the non-zero data as those smaller than $c=q_1 / (q_3/q_1)^{mult}$ . The rationale is that for lognormal data, this constant identifies 2 percent of the data as small. Beyond this limit, the transformation continues linear with the derivative of the log curve at this point. See code for the formula.

The function chooses log10 rather than natural logs because they can be backtransformed relatively easily in the mind.

Value

the transformed data. The value c needed for the transformation is returned as attr(.,"threshold").

Note

The names of the function alludes to Tudey's idea of "started logs".

Author(s)

Werner A. Stahel, ETH Zurich

Examples

dd <- c(seq(0,1,0.1),5*10^rnorm(100,0,0.2))
dd <- sort(dd)
r.dl <- logst(dd)
plot(dd, r.dl, type="l")
abline(v=attr(r.dl,"threshold"),lty=2)

ovarian

Description

copy of ovarian from package 'survival'. Will disappear

Usage

data("ovarian")

Format

A data frame with 26 observations on the following 6 variables.

futime: a numeric vector
fustat: a numeric vector
age: a numeric vector
resid.ds: a numeric vector
rx: a numeric vector
ecog.ps: a numeric vector

Details

This copy is here since the package was rejected because the checking procedure did not find it in the package

Examples

data(ovarian)
summary(ovarian)

Plot Confidence Intervals

Description

Plot confidence or relevance interval(s) for several samples and for the comparison of two samples, also useful for replications and original studies

Usage

plconfint(x, y = NULL, select=NULL, overlap = NULL, pos = NULL,
          xlim = NULL, refline = 0, add = FALSE, bty = "L", col = NULL,
          plpars = list(lwd=c(2,3,1,4,2), posdiff=0.35,
                        markheight=c(1, 0.6, 0.6), extend=NA, reflinecol="gray70"),
          label = TRUE, label2 = NULL, xlab="", ...)

pltwosamples(x, ...)
## Default S3 method:
pltwosamples(x, y = NULL, overlap = TRUE, ...)
## S3 method for class 'formula'
pltwosamples(formula, data = NULL, ...)

Arguments

`x`	For `plconfint`: A vector of length >=2 or a matrix with this number of columns, containing `[,1]`: the estimate `[,2]`: if `x` is of length 2: width of (symmetric) confidence interval `[,2:3]`: if of length >2: the interval end points `[,4:5]`: if of length >=5: values for additional ticks on the intervals, typically indicating the end points of a shortened interal, see Details For `pltwosamples`: A formula or the data for the first sample – or a list or matrix or data.frame with two components/columns corresponding to the two samples
`y`	data for a second confidence interval (for `plconfint` or the second sample (for `pltwosamples`)
`select`	selects samples, effects, or studies
`overlap`	logical: should shortened intervals be shown to show significance of differences? see Details
`pos`	positions of the bars in vertical direction
`xlim`	limits for the horizontal axis. `NA`s will be replaced by the respective element of the range of the x values.
`refline`	`x` values for which vertical reference lines are drawn
`add`	logical: should the plotted elements be added to an existing plot?
`bty`	type of 'box' around the plot, see `par`
`col`	color to be used for the confidence intervals, usually a vector of colors if used.
`plpars`	graphical options, see Details
`label`, `label2`	labels for intervals (or intervall pairs) to be dislayed on the left and right hand margin, respectivly. If `label` is `TRUE`, row.names of `x` are used.
`xlab`	label for horizontal axis
`formula`, `data`	formula and data for the `formula` method
`...`	further arguments to the call of `plconfint`

Details

Columns 4 and 5 of x are typically used to indicate an "overlap interval", which allows for a graphical assessment of the significance of the test for zero difference(s), akin the "notches" in box plots: The difference between a pair of groups is siginificant if their overlap intervals do not overlap. For equal standard errors of the groups, the standard error of the difference between two of them is larger by the factor sqrt(2). Therefore, the intervals should be shortened by this factor, or multiplied by 1/sqrt(2), which is the default for overlapfactor. If only two groups are to be shown, the factor is adjusted to unequal standard errors, and accurate quantiles of a t distribution are used.

The graphical options are:

lwd:: line widths for: [1] the interval, [2] middle mark, [3] end marks, [4] overlap interval marks, [5] vertical line marking the relevance threshold
markheight:: determines the length of the middle mark, the end marks and the marks for the overlap interval as a multiplier of the default length
extend:: extension of the vertical axis beyond the range
reflinecol:: color to be used for the vertical lines at relevances 0 and 1

Value

none

Author(s)

Werner A. Stahel

Examples

## --- regression
data(swiss)
rr <- lm(Fertility ~ . , data = swiss)
rt <- termtable(rr)
plot(rt)

## --- termeffects
data(d.blast)
rlm <- lm(log10(tremor)~location+log10(distance)+log10(charge), data=d.blast)
rte <- termeffects(rlm)
plot(rte, single=TRUE)

## --- replication
data(d.osc15Onesample)
td <- d.osc15Onesample
tdo <- structure(td[,c(1,2,6)], names=c("effect", "n", "teststatistic"))
tdr <- structure(td[,c(3,4,7)], names=c("effect", "n", "teststatistic"))
rr <- replication(tdo,tdr)

plconfint(attr(rr, "estimate"), refline=c(0,1))

Plot Inference Results

Description

Plot confidence or relevance interval(s) for one or several items

Usage

## S3 method for class 'inference'
plot(x, pos = NULL, overlap = FALSE, 
  refline = c(0,1,-1), xlab = "relevance", ...)
## S3 method for class 'termeffects'
plot(x, pos = NULL, single=FALSE,
  overlap = TRUE, termeffects.gap = 0.2, refline = c(0, 1, -1),
  xlim=NULL, ylim=NULL, xlab = "relevance", mar=NA,
  labellength=getOption("labellength"), ...)

Arguments

`x`	a vector or matrix of class `inference`.
`pos`	positions of the bars in vertical direction
`overlap`	logical: should shortened intervals be shown to show significance of differences? see Details
`refline`	values for vertical reference lines
`single`	logical: should terms with a single degree of freedom be plotted?
`termeffects.gap`	gap between blocks corresponding to terms
`xlim`, `ylim`	limits of plotting area, as usual
`xlab`	label for horizontal axis
`mar`	plot margins. If `NULL` (default), the left side margin will be adjusted to accomodate the labels of effects of factor levels
`labellength`	maximum number of characters for label strings
`...`	further arguments to the call of `plot.inference` (for`plot.termeffects`) and `plot`

Details

The overlap interval allows for a graphical assessment of the significance of the test for zero difference(s), akin the notches in the box plots: The difference between a pair of groups is siginificant if their overlap intervals do not overlap. For equal standard errors of the groups, the standard error of the difference between two of them is larger by the factor sqrt(2). Therefore, the intervals should be shortened by this factor, or multiplied by 1/sqrt(2), which is the default for overlapfactor. If only two groups are to be shown, the factor is adjusted to unequal standard errors.

The graphical options are:

lwd:: line widths for: [1] the interval, [2] middle mark, [3] end marks, [4] overlap interval marks, [5] vertical line marking the relevance threshold
markheight:: determines the length of the middle mark, the end marks and the marks for the overlap interval as a multiplier of the default length
extend:: extension of the vertical axis beyond the range
framecol:: color to be used for the framing lines: axis and vertical lines at relevances 0 and 1

Value

none

Note

plot.inference displays termtable objects, too, since they inherit from class inference.

Author(s)

Werner A. Stahel

Examples

## --- regression
data(swiss)
rr <- lm(Fertility ~ . , data = swiss)
rt <- termtable(rr)
plot(rt)

## --- termeffects
data(d.blast)
rlm <- lm(log10(tremor)~location+log10(distance)+log10(charge), data=d.blast)
rte <- termeffects(rlm)
plot(rte, single=TRUE)

Print Tables with Inference Measures

Description

Print methods for objects of class "inference", "termtable", "termeffects", or "printInference".

Usage

## S3 method for class 'inference'
print(x, show = getOption("show.inference"), print=TRUE,
  digits = getOption("digits.reduced"), transpose.ok = TRUE,
  legend = NULL, na.print = getOption("na.print"), ...)

## S3 method for class 'termtable'
print(x, show = getOption("show.inference"), ...)

## S3 method for class 'termeffects'
print(x, show = getOption("show.inference"),
  transpose.ok = TRUE, single = FALSE, print = TRUE, warn = TRUE, ...)

## S3 method for class 'printInference'
print(x, ...)

Arguments

`x`	object to be printed
`show`	determines items (columns) to be shown
`digits`	number of significant digits to be printed
`transpose.ok`	logical: May a single column be shown as a row?
`single`	logical: Should components with a single coefficient be printed?
`legend`	logical: should the legend(s) for the symbols characterizing p-values and relevances be printed? Defaults to `regroptions("show.symbolLegend")`.
`na.print`	string by which `NA`s are shown
`print`	logical: if `FALSE`, no printing will occur, used to edit the result before printing it.
`warn`	logical: Should the warning be issued if `termeffects` has nothing to print since there are no terms with more than one degree of freedom
`...`	further arguments, passed to `print.data.frame()`.

Details

The value, if assigned to rr, say, can be printed by using print.printInference, writing print(rr), which is just what happens internally unless print=FALSE is used. This allows for editing the result before printing it, see Examples.

printInference objects can be a vector, a data.frame or a matrix, or a list of such items. Each item can have an attribute head of mode character that is printed by cat before the item, and analogous with a tail attribute.

Value

A kind of formatted version of x, with class printInference. For print.inference, it will be a character vector or a data.frame with attributes head and tail if applicable. For print.termeffects, it will be a list of such elements, with its own head and tail. It is invisibly returned.

Author(s)

Werner A. Stahel

Examples

data(d.blast)
r.blast <-
  lm(log10(tremor)~location+log10(distance)+log10(charge), data=d.blast)
rt <- termtable(r.blast)
## print() : first default, then "classical" :
rt
print(rt, show="classical")

class(te <- termeffects(r.blast)) #  "termeffects"
rr <- print(te, print=FALSE)
attr(rr, "head") <- sub("lm", "Linear Regression", attr(rr, "head"))
class(rr) # "printInference"
rr # <==>  print(rr)

str(rr)

Options for the relevnance Package

Description

List of options used in the relevnance package to select items and formats for printing inference elements

Usage

relevance.options
rlv.symbols
p.symbols

Format

The format is: List of 22 $ digits.reduced : 3 $ testlevel : 0.05 $ rlv.threshold : stand rel prop corr coef drop pred 0.10 0.10 0.10 0.10 0.10 0.10 0.05 $ termtable : TRUE $ show.confint : TRUE $ show.doc : TRUE $ show.inference : "relevance" $ show.simple.relevance : "Rle" "Rlp" "Rls" "Rls.symbol" $ show.simple.test : "Sig0" "p.symbol" $ show.simple.classical : "statistic" "p.value" "p.symbol" $ show.term.relevance : "df" "R2.x" "coefRlp" "coefRls" ... $ show.term.test : "df" "ciLow" "ciUp" "R2.x" ... $ show.term.classical : "statistic" "df" "ciLow" "ciUp" ... $ show.termeff.relevance: "coef" "coefRls.symbol" $ show.termeff.test : "coef" "p.symbol" $ show.termeff.classical: "coef" "p.symbol" $ show.symbollegend : TRUE $ na.print : "." $ p.symbols : List, see below $ rlv.symbols : List, see below

rlv.symbols List $ symbol : " " "." "+" "++" "+++" $ cutpoint: -Inf 0 1 2 5 Inf

p.symbols List $ symbol : "***" "**" "*" "." " " $ cutpoint: 0 0.001 0.01 0.05 0.1 1

Examples

relevance.options
options(relevance.options) ## restores the package's default options

Inference for Replication Studies

Description

Calculate inference for a replication study and for its comparison with the original

Usage

replication(original, replication, testlevel=getOption("testlevel"),
           rlv.threshold=getOption("rlv.threshold") )

Arguments

`original`	list of class `inference`, providing the effect estimate (`["effect"]`), its standard error (`["se"]`), the number of observations (`["n"]`), and the scatter (`["scatter"]`) for the 'original' study, or a matrix or data.frame containing this information as the first row.
`replication`	the same, for the replication study; if empty or `NULL`, the second row of argument `original` is assumed to contain the information about the replication.
`testlevel`	level of statistical tests
`rlv.threshold`	threshold of relevance; if this is a vector, the first element will be used.

Value

A list of class inference and replication containing the results of the comparison between the studies and, as an attribute, the results for the replication.

Author(s)

Werner A. Stahel

References

Werner A. Stahel (2020). Measuring Significance and Relevance instead of p-values. Submitted; available in the documentation.

Examples

data(d.osc15Onesample)
tx <- structure(d.osc15Onesample[,c("effecto","teststatistico","no")],
      names=c("effect","teststatistic","n"))
ty <- structure(d.osc15Onesample[,c("effectr","teststatisticr","nr")],
      names=c("effect","teststatistic","n"))
replication(tx, ty, rlv.threshold=0.1)

Relevance Class

Description

Find the class of relevance on the basis of the confidence interval and the relevance threshold

Usage

rlvClass(effect, ci=NULL, relevance=NA)

Arguments

`effect`	either a list of class `"inference"` (in which case the remaining arguments will be ignored) or the estimated effect
`ci`	confidence interval for `estimate` or width of confidence interval (if of equal length as `estimate`)
`relevance`	relevance threshold

Value

Character string: the relevance class, either "Rlv" if the effect is statistically proven to be larger than the threshold, "Amb" if the confidence interval contains the threshold, "Ngl" if the interval only covers values lower than the threshold, but contains 0, and "Ctr" if the interval only contains negative values.

Author(s)

Werner A. Stahel

References

Werner A. Stahel (2020). New relevance and significance measures to replace p-values. PLOS ONE 16, e0252991, doi: 10.1371/journal.pone.0252991

Examples

rlvClass(2.3, 1.6, 0.4)  ##  "Rlv"
  rlvClass(2.3, 1.6, 1)  ##  "Sig"

Reproducibility Class

Description

Find the classes of relevance and of reprodicibility.

Usage

rplClass(rlvclassd, rlvclassr, rler=NULL)

Arguments

`rlvclassd`	relevance class of the difference between rplication and original study
`rlvclassr`	relevance class of the replication's effect estimate
`rler`	estimated relevance of the replication

Value

Character string: the replication outcome class

Author(s)

Werner A. Stahel

References

Werner A. Stahel (2020). Measuring Significance and Relevance instead of p-values. Submitted

Examples

data(d.osc15Onesample)
tx <- structure(d.osc15Onesample[,c("effecto","teststatistico","no")],
      names=c("effect","teststatistic","n"))
ty <- structure(d.osc15Onesample[,c("effectr","teststatisticr","nr")],
      names=c("effect","teststatistic","n"))
rplClass(tx, ty)

Shorten Strings

Description

Strings are shortened if they are longer than n

Usage

shortenstring(x, n = 50, endstring = "..", endchars = NULL)

Arguments

`x`	a string or a vector of strings
`n`	maximal character length
`endstring`	string(s) to be appended to the shortened strings
`endchars`	number of last characters to be shown at the end of the abbreviated string. By default, it adjusts to `n`.

Value

Abbreviated string(s)

Author(s)

Werner A. Stahel

Examples

shortenstring("abcdefghiklmnop", 8)

shortenstring(c("aaaaaaaaaaaaaaaaaaaaaa","bbbbc",
  "This text is certainly too long, don't you think?"),c(8,3,20))

Show a Part of a Data.frame

Description

Shows a part of the data.frame which allows for grasping the nature of the data. The function is typically used to make sure that the data is what was desired and to grasp the nature of the variables in the phase of getting acquainted with the data.

Usage

showd(data, first = 3, nrow. = 4, ncol. = NULL, digits=getOption("digits"))

Arguments

`data`	a data.frame, a matrix, or a vector
`first`	the first `first` rows will be shown and ...
`nrow.`	a selection of `nrow.` rows will be shown in addition. They will be selected with equal row number differences. The last row is always included.
`ncol.`	number of columns (variables) to be shown. The first and last columns will also be included. If `ncol.` has more than one element, it is used to identify the columns directly.
`digits`	number of significant digits used in formatting numbers

Value

returns invisibly the character vector containing the formatted data

Author(s)

Werner A. Stahel, ETH Zurich

Examples

showd(iris)

data(d.blast)
names(d.blast)
## only show 3 columns, including the first and last
showd(d.blast, ncol=3)  

showd(cbind(1:100))

Count NAs

Description

Count the missing or non-finite values for each column of a matrix or data.frame

Usage

sumNA(object, inf = TRUE)

Arguments

`object`	a vector, matrix, or data.frame
`inf`	if TRUE, Inf and NaN values are counted along with NAs

Value

numerical vector containing the missing value counts for each column

Note

This is a simple shortcut for apply(is.na(object),2,sum) or apply(!is.finite(object),2,sum)

Author(s)

Werner A. Stahel, ETH Zurich

Examples

t.d <- data.frame(V1=c(1,2,NA,4), V2=c(11,12,13,Inf), V3=c(21,NA,23,Inf))
sumNA(t.d)

All Coefficients of a Model Fit

Description

A list of all coefficients of a model fit, possibly with respective statistics

Usage

termeffects(object, se = 2, df = df.residual(object), rlv = TRUE,
  rlv.threshold = getOption("rlv.threshold"), ...)

Arguments

`object`	a model fit, produced, e.g., by a call to `lm` or `regr`.
`se`	logical: Should inference statistics be generated?
`df`	degrees of freedom for t-test
`rlv`	logical: Should relevances be calculated?
`rlv.threshold`	Relevance thresholds, see `inference`
`...`	further arguments, passed to `inference`

Value

a list with a component for each term in the model formula. Each component is a termtable for the coefficients corresponding to the term.

Author(s)

Werner A. Stahel

Examples

data(d.blast)
  r.blast <-
    lm(log10(tremor)~location+log10(distance)+log10(charge), data=d.blast)
  termeffects(r.blast)

Statistics for Linear Models, Including Relevance Statistics

Description

Calculate a table of statistics for (multiple) regression mdels with a linear predictor

Usage

termtable(object, summary = summary(object), testtype = NULL,
  r2x = TRUE, rlv = TRUE, rlv.threshold = getOption("rlv.threshold"), 
  testlevel = getOption("testlevel"), ...)

relevance.modelclasses

Arguments

`object`	result of a model fitting function like `lm`
`summary`	result of `summary(object)`. If `NULL`, the `summary` will be called.
`testtype`	type of test to be applied for dropping each term in turn. If `NULL`, it is selected according to the class of the object, see Details.
`r2x`	logical: should the collinearity measures “`R2.x`” (see below) for the terms be calculated?
`rlv`	logical: Should relevances be calculated?
`rlv.threshold`	Relevance thresholds, vector containing the elements `rel`: threshold for relative effects, `coef`: for standardized coefficients, `drop`: for drop effects, `pred`: for prediction intervals.
`testlevel`	1 - confidence level
`...`	further arguments, ignored

Details

relevance.modelclasses collects the names of classes of model fitting results that can be handled by termtable.

If testtype is not specified, it is determined by the class of object and its attribute family as follows:

"F":: or t for objects of class lm, lmrob and glm with families quasibinomial and quasipoisson,
"Chi-squared":: for other glms and survreg

Value

data.frame with columns

coef:: coefficients for terms with a single degree of freedom
df:: degrees of freedom
se:: standard error of coef
statistic:: value of the test statistic
p.value, p.symbol:: p value and symbol for it
Sig0:: significance value for the test of coef==0
ciLow, ciUp:: confidence interval for coef
stcoef:: standardized coefficient (standardized using the standard deviation of the 'error' term, sigma, instead of the response's standard deviation)
st.Low, st.Up:: confidence interval for stcoef
R2.x:: collinearity measure ( $= 1 - 1 / vif$ , where $vif$ is the variance inflation factor)
coefRle:: estimated relevance of coef
coefRls:: secured relevance, lower end of confidence interval for the relevance of coef
coefRlp:: potential relevance, the upper end of the confidence interval.
dropRle, dropRls, dropRlp:: analogous values for drop effect
predRle, predRls, predRlp:: analogous values for prediction effect

In addition, it has attributes

testtype:: as determined by the argument testtype or the class and attributes of object.
fitclass:: class and attributes of object.
family, dist:: more specifications if applicable

Author(s)

Werner A. Stahel

References

Werner A. Stahel (2020). Measuring Significance and Relevance instead of p-values. Submitted

Examples

data(swiss)
  rr <- lm(Fertility ~ . , data = swiss)
  rt <- termtable(rr)
  rt

Relevance and Significance for One or Two Samples

Description

Inference for a difference between two independent samples or for a single sample: Collect quantities for inference, including Relevance and Significance measures

Usage

twosamples(x, ...)
onesample(x, ...)

## Default S3 method:
twosamples(x, y = NULL, paired = FALSE, table = NULL, 
  hypothesis = 0,var.equal = TRUE,
  testlevel=getOption("testlevel"), log = NULL, standardize = NULL, 
  rlv.threshold=getOption("rlv.threshold"), ...)
## S3 method for class 'formula'
twosamples(x, data = NULL, subset, na.action, log = NULL, ...)
## S3 method for class 'table'
twosamples(x, ...)

Arguments

`x`	a formula or the data for the first or the single sample
`y`	data for the second sample
`table`	A `table` summarizing the data in case of binary (binomial) data. If given, `x` and `y` are ignored.
`paired`	logical: In case `x` and `y` are given. are their values paired?
`hypothesis`	the null effect to be tested, and anchor for the relevance
`var.equal`	logical: In case of two samples, should the variances be assumed equal? Only applies for quantitative data.
`testlevel`	level for the test, also determining the confidence level
`log`	logical...: Is the target variable on log scale? – or character: either "log" or "log10" (or "logst"). If so, no standardization is applied to it. By default, the function examines the formula to check whether the left hand side of the formula contains a log transformation.
`standardize`	logical: Should the effect be standardized (for quantiative data)?
`rlv.threshold`	Relevance threshold, or a vector of thresholds from which the element `stand` is taken for quantitative data and the element `prop`, for binary data.

For the formula method:

`formula`	formula of the form y~x giving the target y and condition x variables. For a one-sample situation, use y~1.
`data`	data from which the variables are obtained
`subset`, `na.action`	subset and na.action to be applied to `data`
`...`	further arguments, ignored

Details

Argument log: If log10 (or logst from package plgraphics) is used, rescaling is done (by log(10)) to obtain the correct relevance. Therefore, log needs to be set appropriately in this case.

Value

an object of class 'inference', a vector with elements

effect:

for quantitative data: estimated difference between expectations of the two samples, or mean in case of a single sample.

For binary data: log odds (for one sample or paired samples) or log odds ratio (for two samples)

se:

standard error of effect

teststatistic:

test statistic

p.value:

p value for test against 0

Sig0:

significance measure for test or 0 effect

ciLow, ciUp:

confidence interval for the effect

Rle, Rls, Rlp:

relevance measures: estimated, secured, potential

Sigth:

significance measure for test of effect == relevance threshold

In addition to the columns/components, it has attributes

type:: type of relevance: simple
method:: problem and inference method
effectname:: label for the effect
hypothesis:: the null effect
n:: number(s) of observations
estimate:: estimated parameter, with standard error or confidence interval, if applicable; in the case of 2 independent samples: their means
teststatistic:: test statistic
V:: single observation variance
df:: degrees of freedom for the t distribution
data:: if paired, vector of differences; if single sample, vector of data; if two independent samples, list containing the two samples
rlv.threshold:: relevance threshold

Note

onesample and twosamples are identical. twosamples.table(x,...) just calls twosamples.default(table=x, ...).

Author(s)

Werner A. Stahel

References

see those in relevance-package.

Examples

data(sleep)
t.test(sleep[sleep$group == 1, "extra"], sleep[sleep$group == 2, "extra"])
twosamples(sleep[sleep$group == 1, "extra"], sleep[sleep$group == 2, "extra"])

## Two-sample test, wilcox.test example,  Hollander & Wolfe (1973), 69f.
## Permeability constants of the human chorioamnion (a placental membrane)
## at term and between 12 to 26 weeks gestational age
d.permeabililty <-
  data.frame(perm = c(0.80, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46,
                      1.15, 0.88, 0.90, 0.74, 1.21), atterm = rep(1:0, c(10,5))
             )
t.test(perm~atterm, data=d.permeabililty)
twosamples(perm~atterm, data=d.permeabililty)

## one sample
onesample(sleep[sleep$group == 2, "extra"])

## plot two samples
pltwosamples(extra ~ group, data=sleep)

Package 'relevance'

Help Index

Calculate Relevance and Significance Measures

Description

Details

Author(s)

References

See Also

Examples

arc sine Transformation

Description

Usage

Arguments

Value

Note

Author(s)

Examples

Confidence Interval for the Non-Central F and Chisquare Distribution

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Correlation with Relevance and Significance Measures

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Blasting for a tunnel

Description

Usage

Format

Details

Source

Examples

Data of an 'anchoring' experiment in psychology

Description

Usage

Format

Source

Examples

Data of an 'anchoring' experiment in psychology

Description

Usage

Format

Source

Examples

Data from the OSC15 replication study

Description

Usage

Format

Source

References

See Also

Examples

Data from the OSC15 replication study, one sample tests

Description

Usage

Format

Source

References

See Also

Examples

Drop Single Terms of a Model and Calculate Respective Wald Tests

Description

Usage

Arguments

Details

Value

Note

Author(s)

See Also

Examples