| Type: | Package |
| Title: | Optimal Level of Significance for Regression and Other Statistical Tests |
| Version: | 2.2 |
| Imports: | pwr |
| Date: | 2022-06-29 |
| Author: | Jae H. Kim <jaekim8080@gmail.com> |
| Maintainer: | Jae H. Kim <jaekim8080@gmail.com> |
| Description: | The optimal level of significance is calculated based on a decision-theoretic approach. The optimal level is chosen so that the expected loss from hypothesis testing is minimized. A range of statistical tests are covered, including the test for the population mean, population proportion, and a linear restriction in a multiple regression model. The details are covered in Kim and Choi (2020) <doi:10.1111/abac.12172>, and Kim (2021) <doi:10.1080/00031305.2020.1750484>. |
| License: | GPL-2 |
| NeedsCompilation: | no |
| Packaged: | 2022-07-03 03:23:48 UTC; jh808 |
| Repository: | CRAN |
| Date/Publication: | 2022-07-03 12:30:14 UTC |
Optimal Level of Significance for Regression and Other Statistical Tests
Description
The optimal level of significance is calculated based on a decision-theoretic approach. The optimal level is chosen so that the expected loss from hypothesis testing is minimized. A range of statistical tests are covered, including the test for the population mean, population proportion, and a linear restriction in a multiple regression model. The details are covered in Kim and Choi (2020) <doi:10.1111/abac.12172>, and Kim (2021) <doi:10.1080/00031305.2020.1750484>.
Details
The DESCRIPTION file:
| Package: | OptSig |
| Type: | Package |
| Title: | Optimal Level of Significance for Regression and Other Statistical Tests |
| Version: | 2.2 |
| Imports: | pwr |
| Date: | 2022-06-29 |
| Author: | Jae H. Kim <jaekim8080@gmail.com> |
| Maintainer: | Jae H. Kim <jaekim8080@gmail.com> |
| Description: | The optimal level of significance is calculated based on a decision-theoretic approach. The optimal level is chosen so that the expected loss from hypothesis testing is minimized. A range of statistical tests are covered, including the test for the population mean, population proportion, and a linear restriction in a multiple regression model. The details are covered in Kim and Choi (2020) <doi:10.1111/abac.12172>, and Kim (2021) <doi:10.1080/00031305.2020.1750484>. |
| License: | GPL-2 |
Index of help topics:
Opt.sig.norm.test Optimal significance level calculation for the
mean of a normal distribution (known variance)
Opt.sig.t.test Optimal significance level calculation for
t-tests of means (one sample, two samples and
paired samples)
OptSig-package Optimal Level of Significance for Regression
and Other Statistical Tests
OptSig.2p Optimal significance level calculation for the
test for two proportions (same sample sizes)
OptSig.2p2n Optimal significance level calculation for the
test for two proportions (different sample
sizes)
OptSig.Boot Optimal Significance Level for the F-test using
the bootstrap
OptSig.BootWeight Weighted Optimal Significance Level for the
F-test based on the bootstrap
OptSig.Chisq Optimal Significance Level for a Chi-square
test
OptSig.F Optimal Significance Level for an F-test
OptSig.Weight Weighted Optimal Significance Level for the
F-test based on the assumption of normality in
the error term
OptSig.anova Optimal significance level calculation for
balanced one-way analysis of variance tests
OptSig.p Optimal significance level calculation for
proportion tests (one sample)
OptSig.r Optimal significance level calculation for
correlation test
OptSig.t2n Optimal significance level calculation for two
samples (different sizes) t-tests of means
Power.Chisq Function to calculate the power of a Chi-square
test
Power.F Function to calculate the power of an F-test
R.OLS Restricted OLS estimation and F-test
data1 Data for the U.S. production function
estimation
The package accompanies the paper: Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach. Abacus. Wiley.
It oprovides functions for the optimal level of significance for the test for linear restiction in a regeression model.
Other basic statistical tests, including those for population mean and proportion, are also covered using the functions from the pwr package.
Author(s)
Jae H. Kim <jaekim8080@gmail.com>
Maintainer: Jae H. Kim <jaekim8080@gmail.com>
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach: Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.
Stephane Champely (2017). pwr: Basic Functions for Power Analysis. R package version 1.2-1. https://CRAN.R-project.org/package=pwr
See Also
Leamer, E. 1978, Specification Searches: Ad Hoc Inference with Nonexperimental Data, Wiley, New York.
Kim, JH and Ji, P. 2015, Significance Testing in Empirical Finance: A Critical Review and Assessment, Journal of Empirical Finance 34, 1-14. <DOI:http://dx.doi.org/10.1016/j.jempfin.2015.08.006>
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
data(data1)
y=data1$lnoutput; x=cbind(data1$lncapital,data1$lnlabor)
# Restriction matrices to test for constant returns to scale
Rmat=matrix(c(0,1,1),nrow=1); rvec=matrix(0.94,nrow=1)
# Model Estimation and F-test
M=R.OLS(y,x,Rmat,rvec)
# Degrees of Freedom and estimate of non-centrality parameter
K=ncol(x)+1; T=length(y)
df1=nrow(Rmat);df2=T-K; NCP=M$ncp
# Optimal level of Significance: Under Normality
OptSig.F(df1,df2,ncp=NCP,p=0.5,k=1, Figure=TRUE)
Optimal significance level calculation for the mean of a normal distribution (known variance)
Description
Computes the optimal significance level for the mean of a normal distribution (known variance)
Usage
Opt.sig.norm.test(ncp=NULL,d=NULL,n=NULL,p=0.5,k=1,alternative="two.sided",Figure=TRUE)
Arguments
ncp |
Non-centrality parameter |
d |
Effect size, Cohen's d |
n |
Sample size |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
alternative |
a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less" |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
Refer to Kim and Choi (2020) for the details of k and p
Either ncp or d value should be given.
In a general term, if X ~ N(mu,sigma^2); let H0:mu = mu0; and H1:mu = mu1;
ncp = sqrt(n)(mu1-mu0)/sigma
d = (mu1-mu0)/sigma: Cohen's d
Value
alpha.opt |
Optimal level of significance |
beta.opt |
Type II error probability at the optimal level |
Note
Also refer to the manual for the pwr package
The black curve in the figure is the line of enlightened judgement: see Kim and Choi (2019). The red dot inticates the optimal significance level that minimizes the expected loss: (alpha.opt,beta.opt). The blue horizontal line indicates the case of alpha = 0.05 as a reference point.
Author(s)
Jae H. Kim (using a function from the pwr package)
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach: Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.
Stephane Champely (2017). pwr: Basic Functions for Power Analysis. R package version 1.2-1. https://CRAN.R-project.org/package=pwr
See Also
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
Opt.sig.norm.test(d=0.2,n=60,alternative="two.sided")
Optimal significance level calculation for t-tests of means (one sample, two samples and paired samples)
Description
Computes the optimal significance level for the test for t-tests of means
Usage
Opt.sig.t.test(ncp=NULL,d=NULL,n=NULL,p=0.5,k=1,
type="one.sample",alternative="two.sided",Figure=TRUE)
Arguments
ncp |
Non-centrality parameter |
d |
Effect size |
n |
Sample size |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
type |
Type of t test : one- two- or paired-sample |
alternative |
a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less" |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
Refer to Kim and Choi (2020) for the details of k and p
Either ncp or d value should be given, with the value of n.
In a general term, if X ~ N(mu,sigma^2); let H0:mu = mu0; and H1:mu = mu1;
ncp = sqrt(n)(mu1-mu0)/sigma
d = (mu1-mu0)/sigma: Cohen's d
Value
alpha.opt |
Optimal level of significance |
beta.opt |
Type II error probability at the optimal level |
Note
Also refer to the manual for the pwr package
The black curve in the figure is the line of enlightened judgement: see Kim and Choi (2020). The red dot inticates the optimal significance level that minimizes the expected loss: (alpha.opt,beta.opt). The blue horizontal line indicates the case of alpha = 0.05 as a reference point.
Author(s)
Jae H. Kim (using a function from the pwr package)
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach: Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.
Stephane Champely (2017). pwr: Basic Functions for Power Analysis. R package version 1.2-1. https://CRAN.R-project.org/package=pwr
See Also
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
Opt.sig.t.test(d=0.2,n=60,type="one.sample",alternative="two.sided")
Optimal significance level calculation for the test for two proportions (same sample sizes)
Description
Computes the optimal significance level for the test for two proportions
Usage
OptSig.2p(ncp=NULL,h=NULL,n=NULL,p=0.5,k=1,alternative="two.sided",Figure=TRUE)
Arguments
ncp |
Non-centrality parameter |
h |
Effect size, Cohen's h |
n |
Number of observations (per sample) |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
alternative |
a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less" |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
Refer to Kim and Choi (2020) for the details of k and p
Either ncp or h value should be specified.
For h, refer to Cohen (1988) or Champely (2017)
In a general term, if X ~ N(mu,sigma^2); let H0:mu = mu0; and H1:mu = mu1;
ncp = sqrt(n)(mu1-mu0)/sigma
Value
alpha.opt |
Optimal level of significance |
beta.opt |
Type II error probability at the optimal level |
Note
Also refer to the manual for the pwr package,
The black curve in the figure is the line of enlightened judgement: see Kim and Choi (2020). The red dot inticates the optimal significance level that minimizes the expected loss: (alpha.opt,beta.opt). The blue horizontal line indicates the case of alpha = 0.05 as a reference point.
Author(s)
Jae H. Kim (using a function from the pwr package)
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach: Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.
Stephane Champely (2017). pwr: Basic Functions for Power Analysis. R package version 1.2-1. https://CRAN.R-project.org/package=pwr
See Also
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
OptSig.2p(h=0.2,n=60,alternative="two.sided")
Optimal significance level calculation for the test for two proportions (different sample sizes)
Description
Computes the optimal significance level for the test for two proportions
Usage
OptSig.2p2n(ncp=NULL,h=NULL,n1=NULL,n2=NULL,p=0.5,k=1,alternative="two.sided",Figure=TRUE)
Arguments
ncp |
Non-centrality parameter |
h |
Effect size, Cohen's h |
n1 |
Number of observations (1st sample) |
n2 |
Number of observations (2nd sample) |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
alternative |
a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less" |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
Refer to Kim and Choi (2020) for the details of k and p
Either ncp or h value should be specified.
For h, refer to Cohen (1988) or Chapmely (2017)
Assume X ~ N(mu,sigma^2); and let H0:mu = mu0; and H1:mu = mu1;
ncp = sqrt(n)(mu1-mu0)/sigma
Value
alpha.opt |
Optimal level of significance |
beta.opt |
Type II error probability at the optimal level |
Note
Also refer to the manual for the pwr package
The black curve in the figure is the line of enlightened judgement: see Kim and Choi (2020). The red dot inticates the optimal significance level that minimizes the expected loss: (alpha.opt,beta.opt). The blue horizontal line indicates the case of alpha = 0.05 as a reference point.
Author(s)
Jae H. Kim (using a function from the pwr package)
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach: Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.
Stephane Champely (2017). pwr: Basic Functions for Power Analysis. R package version 1.2-1. https://CRAN.R-project.org/package=pwr
See Also
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
OptSig.2p2n(h=0.30,n1=80,n2=245,alternative="greater")
Optimal Significance Level for the F-test using the bootstrap
Description
The function calculates the optimal level of significance for the F-test
The bootstrap can be conducted using either iid resampling or wild bootstrap.
Usage
OptSig.Boot(y,x,Rmat,rvec,p=0.5,k=1,nboot=3000,wild=FALSE,Figure=TRUE)
Arguments
y |
a matrix of dependent variable, T by 1 |
x |
a matrix of K independent variable, T by K |
Rmat |
a matrix for J restrictions, J by (K+1) |
rvec |
a vector for restrictions, J by 1 |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
nboot |
the number of bootstrap iterations, the default is 3000 |
wild |
if TRUE, wild bootsrap is conducted; if FALSE (default), bootstrap is based on iid residual resampling |
Figure |
show graph if TRUE (default). No graph otherwise |
Details
See Kim and Choi (2020)
Value
alpha.opt |
Optimal level of significance |
crit.opt |
Critical value at the optimal level |
beta.opt |
Type II error probability at the optimal level |
Note
Applicable to a linear regression model
The black curve in the figure plots the denity under H0; The blue curve in the figure plots the denity under H1.
Author(s)
Jae H. Kim
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach, Abacus, Wiley. <https://doi.org/10.1111/abac.12172>
See Also
Leamer, E. 1978, Specification Searches: Ad Hoc Inference with Nonexperimental Data, Wiley, New York.
Kim, JH and Ji, P. 2015, Significance Testing in Empirical Finance: A Critical Review and Assessment, Journal of Empirical Finance 34, 1-14. <DOI:http://dx.doi.org/10.1016/j.jempfin.2015.08.006>
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
data(data1)
# Define Y and X
y=data1$lnoutput; x=cbind(data1$lncapital,data1$lnlabor)
# Restriction matrices to test for constant returns to scale
Rmat=matrix(c(0,1,1),nrow=1); rvec=matrix(0.94,nrow=1)
OptSig.Boot(y,x,Rmat,rvec,p=0.5,k=1,nboot=1000,Figure=TRUE)
Weighted Optimal Significance Level for the F-test based on the bootstrap
Description
The function calculates the weighted optimal level of significance for the F-test
The weights are obtained from the bootstrap distribution of the non-centrality parameter estimates
Usage
OptSig.BootWeight(y,x,Rmat,rvec,p=0.5,k=1,nboot=3000,wild=FALSE,Figure=TRUE)
Arguments
y |
a matrix of dependent variable, T by 1 |
x |
a matrix of K independent variable, T by K |
Rmat |
a matrix for J restrictions, J by (K+1) |
rvec |
a vector for restrictions, J by 1 |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
nboot |
the number of bootstrap iterations, the default is 3000 |
wild |
if TRUE, wild bootsrap is conducted (default); if FALSE, bootstrap is based on iid resampling |
Figure |
show graph if TRUE . No graph if FALSE (default) |
Details
The bootstrap can be conducted using either iid resampling or wild bootstrap.
Value
alpha.opt |
Optimal level of significance |
crit.opt |
Critical value at the optimal level |
Note
Applicable to a linear regression model
Author(s)
Jae H. Kim
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach. Abacus, Wiley. <https://doi.org/10.1111/abac.12172>
See Also
Leamer, E. 1978, Specification Searches: Ad Hoc Inference with Nonexperimental Data, Wiley, New York.
Kim, JH and Ji, P. 2015, Significance Testing in Empirical Finance: A Critical Review and Assessment, Journal of Empirical Finance 34, 1-14. <DOI:http://dx.doi.org/10.1016/j.jempfin.2015.08.006>
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
data(data1)
# Define Y and X
y=data1$lnoutput; x=cbind(data1$lncapital,data1$lnlabor)
# Restriction matrices to test for constant returns to scale
Rmat=matrix(c(0,1,1),nrow=1); rvec=matrix(0.94,nrow=1)
OptSig.Boot(y,x,Rmat,rvec,p=0.5,k=1,nboot=1000,Figure=TRUE)
Optimal Significance Level for a Chi-square test
Description
The function calculates the optimal level of significance for a Ch-square test
Usage
OptSig.Chisq(w=NULL, N=NULL, ncp=NULL, df, p = 0.5, k = 1, Figure = TRUE)
Arguments
w |
Effect size, Cohen's w |
N |
Total number of observations |
ncp |
a value of the non-centality paramter |
df |
the degrees of freedom |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
See Kim and Choi (2020)
Value
alpha.opt |
Optimal level of significance |
crit.opt |
Critical value at the optimal level |
beta.opt |
Type II error probability at the optimal level |
Note
Applicable to any Chi-square test Either ncp or w (with N) should be given.
The black curve in the figure is the line of enlightened judgement: see Kim and Choi (2020). The red dot inticates the optimal significance level that minimizes the expected loss: (alpha.opt,beta.opt). The blue horizontal line indicates the case of alpha = 0.05 as a reference point.
Author(s)
Jae. H Kim
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach: Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
See Also
Leamer, E. 1978, Specification Searches: Ad Hoc Inference with Nonexperimental Data, Wiley, New York.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.
Kim, JH and Ji, P. 2015, Significance Testing in Empirical Finance: A Critical Review and Assessment, Journal of Empirical Finance 34, 1-14. <DOI:http://dx.doi.org/10.1016/j.jempfin.2015.08.006>
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
# Optimal level of Significance for the Breusch-Pagan test: Chi-square version
data(data1) # call the data: Table 2.1 of Gujarati (2015)
# Extract Y and X
y=data1$lnoutput; x=cbind(data1$lncapital,data1$lnlabor)
# Restriction matrices for the slope coefficents sum to 1
Rmat=matrix(c(0,1,1),nrow=1); rvec=matrix(1,nrow=1)
# Model Estimation
M=R.OLS(y,x,Rmat,rvec); print(M$coef)
# Breusch-Pagan test for heteroskedasticity
e = M$resid[,1] # residuals from unrestricted model estimation
# Restriction matrices for the slope coefficients being 0
Rmat=matrix(c(0,0,1,0,0,1),nrow=2); rvec=matrix(0,nrow=2)
# Model Estimation for the auxilliary regression
M1=R.OLS(e^2,x,Rmat,rvec);
# Degrees of Freedom and estimate of non-centrality parameter
df1=nrow(Rmat); NCP=M1$ncp
# LM stat and p-value
LM=nrow(data1)*M1$Rsq[1,1]
pval=pchisq(LM,df=df1,lower.tail = FALSE)
OptSig.Chisq(df=df1,ncp=NCP,p=0.5,k=1, Figure=TRUE)
Optimal Significance Level for an F-test
Description
The function calculates the optimal level of significance for an F-test
Usage
OptSig.F(df1, df2, ncp, p = 0.5, k = 1, Figure = TRUE)
Arguments
df1 |
the first degrees of freedom for the F-distribution |
df2 |
the second degrees of freedom for the F-distribution |
ncp |
a value of of the non-centality paramter |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
See Kim and Choi (2020)
Value
alpha.opt |
Optimal level of significance |
crit.opt |
Critical value at the optimal level |
beta.opt |
Type II error probability at the optimal level |
Note
Applicable to any F-test, following F-distribution
The black curve in the figure is the line of enlightened judgement: see Kim and Choi (2020). The red dot inticates the optimal significance level that minimizes the expected loss: (alpha.opt,beta.opt). The blue horizontal line indicates the case of alpha = 0.05 as a reference point.
Author(s)
Jae. H Kim
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach: Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
See Also
Leamer, E. 1978, Specification Searches: Ad Hoc Inference with Nonexperimental Data, Wiley, New York.
Kim, JH and Ji, P. 2015, Significance Testing in Empirical Finance: A Critical Review and Assessment, Journal of Empirical Finance 34, 1-14. <DOI:http://dx.doi.org/10.1016/j.jempfin.2015.08.006>
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
data(data1)
# Define Y and X
y=data1$lnoutput; x=cbind(data1$lncapital,data1$lnlabor)
# Restriction matrices to test for constant returns to scale
Rmat=matrix(c(0,1,1),nrow=1); rvec=matrix(0.94,nrow=1)
# Model Estimation and F-test
M=R.OLS(y,x,Rmat,rvec)
# Degrees of Freedom and estimate of non-centrality parameter
K=ncol(x)+1; T=length(y)
df1=nrow(Rmat);df2=T-K; NCP=M$ncp
# Optimal level of Significance: Under Normality
OptSig.F(df1,df2,ncp=NCP,p=0.5,k=1, Figure=TRUE)
Weighted Optimal Significance Level for the F-test based on the assumption of normality in the error term
Description
The function calculates the weighted optimal level of significance for the F-test
The weights are obtained from a folded-normal distribution with mean m and staradrd deviation delta
Usage
OptSig.Weight(df1, df2, m, delta = 2, p = 0.5, k = 1, Figure = TRUE)
Arguments
df1 |
the first degrees of freedom for the F-distribution |
df2 |
the second degrees of freedom for the F-distribution |
m |
a value of of the non-centality paramter, the mean of the folded-normal distribution |
delta |
standard deviation of the folded-normal distribution |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
See Kim and Choi (2020)
Value
alpha.opt |
Optimal level of significance |
crit.opt |
Critical value at the optimal level |
Note
The figure shows the folded-normal distribution
Author(s)
Jae H. Kim
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach, Abacus, Wiley. <https://doi.org/10.1111/abac.12172>
See Also
Leamer, E. 1978, Specification Searches: Ad Hoc Inference with Nonexperimental Data, Wiley, New York.
Kim, JH and Ji, P. 2015, Significance Testing in Empirical Finance: A Critical Review and Assessment, Journal of Empirical Finance 34, 1-14. <DOI:http://dx.doi.org/10.1016/j.jempfin.2015.08.006>
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
data(data1)
# Define Y and X
y=data1$lnoutput; x=cbind(data1$lncapital,data1$lnlabor)
# Restriction matrices to test for constant returns to scale
Rmat=matrix(c(0,1,1),nrow=1); rvec=matrix(0.94,nrow=1)
# Model Estimation and F-test
M=R.OLS(y,x,Rmat,rvec)
# Degrees of Freedom and estimate of non-centrality parameter
K=ncol(x)+1; T=length(y)
df1=nrow(Rmat);df2=T-K; NCP=M$ncp
OptSig.Weight(df1,df2,m=NCP,delta=3,p=0.5,k=1,Figure=TRUE)
Optimal significance level calculation for balanced one-way analysis of variance tests
Description
Computes the optimal significance level for the test for balanced one-way analysis of variance tests
Usage
OptSig.anova(K = NULL, n = NULL, f = NULL, p = 0.5, k = 1, Figure = TRUE)
Arguments
K |
Number of groups |
n |
Number of observations (per group) |
f |
Effect size |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
Refer to Kim and Choi (2020) for the details of k and p
For the value of f, refer to Cohen (1988) or Champely (2017)
Value
alpha.opt |
Optimal level of significance |
beta.opt |
Type II error probability at the optimal level |
Note
Also refer to the manual for the pwr package
The black curve in the figure is the line of enlightened judgement: see Kim and Choi (2020). The red dot inticates the optimal significance level that minimizes the expected loss: (alpha.opt,beta.opt). The blue horizontal line indicates the case of alpha = 0.05 as a reference point.
Author(s)
Jae H. Kim (using a function from the pwr package)
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach: Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.
Stephane Champely (2017). pwr: Basic Functions for Power Analysis. R package version 1.2-1. https://CRAN.R-project.org/package=pwr
See Also
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
OptSig.anova(f=0.28,K=4,n=20)
Optimal significance level calculation for proportion tests (one sample)
Description
Computes the optimal significance level for proportion tests (one sample)
Usage
OptSig.p(ncp=NULL,h=NULL,n=NULL,p=0.5,k=1,alternative="two.sided",Figure=TRUE)
Arguments
ncp |
Non-centraity parameter |
h |
Effect size, Cohen's h |
n |
Number of observations (per sample) |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
alternative |
a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less" |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
Refer to Kim and Choi (2020) for the details of k and p
Either ncp or h value should be given
For h, refer to Cohen (1988) or Chapmely (2017)
In a general term, if X ~ N(mu,sigma^2); let H0:mu = mu0; and H1:mu = mu1;
ncp = sqrt(n)(mu1-mu0)/sigma
Value
alpha.opt |
Optimal level of significance |
beta.opt |
Type II error probability at the optimal level |
Note
Also refer to the manual for the pwr package
The black curve in the figure is the line of enlightened judgement: see Kim and Choi (2020). The red dot inticates the optimal significance level that minimizes the expected loss: (alpha.opt,beta.opt). The blue horizontal line indicates the case of alpha = 0.05 as a reference point.
Author(s)
Jae H. Kim (using a function from the pwr package)
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach: Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.
Stephane Champely (2017). pwr: Basic Functions for Power Analysis. R package version 1.2-1. https://CRAN.R-project.org/package=pwr
See Also
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
OptSig.p(h=0.2,n=60,alternative="two.sided")
Optimal significance level calculation for correlation test
Description
Computes the optimal significance level for the correlation test
Usage
OptSig.r(r=NULL,n=NULL,p=0.5,k=1,alternative="two.sided",Figure=TRUE)
Arguments
r |
Linear correlation coefficient |
n |
sample size |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II error, k = L2/L1, default is k = 1 |
alternative |
a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less" |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
Refer to Kim and Choi (2020) for the details of k and p
In a general term, if X ~ N(mu,sigma^2); let H0:mu = mu0; and H1:mu = mu1;
ncp = sqrt(n)(mu1-mu0)/sigma
Value
alpha.opt |
Optimal level of significance |
beta.opt |
Type II error probability at the optimal level |
Note
Also refer to the manual for the pwr package
The black curve in the figure is the line of enlightened judgement: see Kim and Choi (2020). The red dot inticates the optimal significance level that minimizes the expected loss: (alpha.opt,beta.opt). The blue horizontal line indicates the case of alpha = 0.05 as a reference point.
Author(s)
Jae H. Kim (using a function from the pwr package)
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach: Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.
Stephane Champely (2017). pwr: Basic Functions for Power Analysis. R package version 1.2-1. https://CRAN.R-project.org/package=pwr
See Also
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
OptSig.r(r=0.2,n=60,alternative="two.sided")
Optimal significance level calculation for two samples (different sizes) t-tests of means
Description
Computes the optimal significance level for two samples (different sizes) t-tests of means
Usage
OptSig.t2n(ncp=NULL,d=NULL,n1=NULL,n2=NULL,p=0.5,k=1,alternative="two.sided",Figure=TRUE)
Arguments
ncp |
Non-centrality parameter |
d |
Effect size |
n1 |
umber of observations in the first sample |
n2 |
umber of observations in the second sample |
p |
prior probability for H0, default is p = 0.5 |
k |
relative loss from Type I and II errors, k = L2/L1, default is k = 1 |
alternative |
a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less" |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
Refer to Kim and Choi (2020) for the details of k and p
Either ncp or d value should be specified.
In a general term, if X ~ N(mu,sigma^2); let H0:mu = mu0; and H1:mu = mu1;
ncp = sqrt(n)(mu1-mu0)/sigma
d = (mu1-mu0)/sigma: Cohen's d
Value
alpha.opt |
Optimal level of significance |
beta.opt |
Type II error probability at the optimal level |
Note
Also refer to the manual for the pwr package
The black curve in the figure is the line of enlightened judgement: see Kim and Choi (2020). The red dot inticates the optimal significance level that minimizes the expected loss: (alpha.opt,beta.opt). The blue horizontal line indicates the case of alpha = 0.05 as a reference point.
Author(s)
Jae H. Kim (using a function from the pwr package)
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach: Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale,NJ: Lawrence Erlbaum.
Stephane Champely (2017). pwr: Basic Functions for Power Analysis. R package version 1.2-1. https://CRAN.R-project.org/package=pwr
See Also
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
OptSig.t2n(d=0.6,n1=90,n2=60,alternative="greater")
Function to calculate the power of a Chi-square test
Description
This function calculates the power of a Chi-square test, given the value of non-centrality parameter
Usage
Power.Chisq(df, ncp, alpha, Figure = TRUE)
Arguments
df |
degree of freedom |
ncp |
a value of of the non-centality paramter |
alpha |
the level of significance |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
See Kim and Choi (2020)
Value
Power |
Power of the test |
Crit.val |
Critical value at alpha level of signifcance |
Note
See Application Section and Appendix of Kim and Choi (2017)
Author(s)
Jae H. Kim
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach, Abacus, Wiley. <https://doi.org/10.1111/abac.12172>
See Also
Leamer, E. 1978, Specification Searches: Ad Hoc Inference with Nonexperimental Data, Wiley, New York.
Kim, JH and Ji, P. 2015, Significance Testing in Empirical Finance: A Critical Review and Assessment, Journal of Empirical Finance 34, 1-14. <DOI:http://dx.doi.org/10.1016/j.jempfin.2015.08.006>
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
Power.Chisq(df=5,ncp=5,alpha=0.05,Figure=TRUE)
Function to calculate the power of an F-test
Description
This function calculates the power of an F-test, given the value of non-centrality parameter
Usage
Power.F(df1, df2, ncp, alpha, Figure = TRUE)
Arguments
df1 |
the first degrees of freedom for the F-distribution |
df2 |
the second degrees of freedom for the F-distribution |
ncp |
a value of of the non-centality paramter |
alpha |
the level of significance |
Figure |
show graph if TRUE (default); No graph if FALSE |
Details
See Kim and Choi (2020)
Value
Power |
Power of the test |
Crit.val |
Critical value at alpha level of signifcance |
Note
See Application Section and Appendix of Kim and Choi (2020)
Author(s)
Jae H. Kim
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach, Abacus, Wiley. <https://doi.org/10.1111/abac.12172>
See Also
Leamer, E. 1978, Specification Searches: Ad Hoc Inference with Nonexperimental Data, Wiley, New York.
Kim, JH and Ji, P. 2015, Significance Testing in Empirical Finance: A Critical Review and Assessment, Journal of Empirical Finance 34, 1-14. <DOI:http://dx.doi.org/10.1016/j.jempfin.2015.08.006>
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
data(data1)
# Define Y and X
y=data1$lnoutput; x=cbind(data1$lncapital,data1$lnlabor)
# Restriction matrices to test for constant returns to scale
Rmat=matrix(c(0,1,1),nrow=1); rvec=matrix(0.94,nrow=1)
# Model Estimation and F-test
M=R.OLS(y,x,Rmat,rvec)
# Degrees of Freedom and estimate of non-centrality parameter
K=ncol(x)+1; T=length(y)
df1=nrow(Rmat);df2=T-K; NCP=M$ncp
Power.F(df1,df2,ncp=NCP,alpha=0.20747,Figure=TRUE)
Restricted OLS estimation and F-test
Description
Function to calcuate the Restricted (under H0) OLS Estimators and F-test statistic
Usage
R.OLS(y, x, Rmat, rvec)
Arguments
y |
a matrix of dependent variable, T by 1 |
x |
a matrix of K independent variable, T by K |
Rmat |
a matrix for J restrictions, J by (K+1) |
rvec |
a vector for restrictions, J by 1 |
Details
Rmat and rvec are the matrices for the linear restrictions, which a user should supply.
Refer to an econometrics textbook for details.
Value
coef |
matrix of estimated coefficients, (K+1) by 2, under H1 and H0 |
RSq |
R-square values under H1 and H0, 2 by 1 |
resid |
residual vector under H1 and H0, T by 2 |
F.stat |
F-statistic and p-value |
ncp |
non-centrality parameter, estimated by replaicing unknowns using OLS estimates |
Note
The function automatically adds an intercept, so the user need not include a vector of ones in x matrix.
Author(s)
Jae H. Kim
References
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach, Abacus, Wiley. <https://doi.org/10.1111/abac.12172>
See Also
Leamer, E. 1978, Specification Searches: Ad Hoc Inference with Nonexperimental Data, Wiley, New York.
Kim, JH and Ji, P. 2015, Significance Testing in Empirical Finance: A Critical Review and Assessment, Journal of Empirical Finance 34, 1-14. <DOI:http://dx.doi.org/10.1016/j.jempfin.2015.08.006>
Kim, Jae H., 2020, Decision-theoretic hypothesis testing: A primer with R package OptSig, The American Statistician. <https://doi.org/10.1080/00031305.2020.1750484.>
Examples
data(data1)
# Define Y and X
y=data1$lnoutput; x=cbind(data1$lncapital,data1$lnlabor)
# Restriction matrices to test for constant returns to scale
Rmat=matrix(c(0,1,1),nrow=1); rvec=matrix(1,nrow=1)
# Model Estimation and F-test
M=R.OLS(y,x,Rmat,rvec)
Data for the U.S. production function estimation
Description
US production, captal, labour in natrual logs for the year 2005
Usage
data("data1")Format
A data frame with 51 observations on the following 3 variables.
lnoutputnatrual log of output
lnlabornatrual log of labor
lncapitalnatrual log of capital
Details
The data contains 51 observations for 50 US states and Washington DC
Source
Gujarati, D. 2015, Econometrics by Example, Second edition, Palgrave.
References
See Section 2.2 of Gujarari (2015)
Kim and Choi, 2020, Choosing the Level of Significance: A Decision-theoretic Approach, Abacus: a Journal of Accounting, Finance and Business Studies. Wiley. <https://doi.org/10.1111/abac.12172>
Examples
data(data1)