Version: 0.2.2
Title: Likelihood Based Inference
Description: Maximum likelihood estimation and likelihood ratio test are essential for modern statistics. This package supports in calculating likelihood based inference. Reference: Pawitan Y. (2001, ISBN:0-19-850765-8).
Depends: R (≥ 3.0.0)
Author: Kyun-Seop Bae [aut, cre, cph]
Maintainer: Kyun-Seop Bae <k@acr.kr>
Copyright: 2023-, Kyun-Seop Bae
License: GPL-3
Repository: CRAN
URL: https://cran.r-project.org/package=LBI
Packaged: 2025-01-22 08:03:13 UTC; Kyun-SeopBae
NeedsCompilation: no
Date/Publication: 2025-01-22 08:20:05 UTC

Likelihood Based Inference

Description

It conducts likelihood based inference.

Details

Modern likelihood concept and maximum likelihood estimation are established by Fisher RA, while Likelihood Ratio Test (LRT) is established by Neyman J. Post-Fisher methods - generalized linear model, survival analysis, and mixed effects model - are all likelihood based. Inferences from the perspective of Fisherian and pure likelihoodist are suggested here.

Author(s)

Kyun-Seop Bae <k@acr.kr>

References

  1. Wilks SS. The Large-sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses. Ann Math Stat. 1938;9(1):60-62.

  2. Edwards AWF. Likelihood. 1972.

  3. Fisher RA. Statistical Methods and Scientific Inference. 3e. 1973.

  4. Bates DM, Watts DG. Nonlinear Regression Analysis and its Application. 1988.

  5. Ruppert D, Cressie N, Carroll RJ. A Transformation/Weighting Model for Estimating Michaelis-Menten Parameters. Cornell University Technical Report 796. 1988.

  6. Royall R. Statistical Evidence. 1997.

  7. Pinheiro JC, Bates DM. Mixed Effects Models in S and S-PLUS. 2000.

  8. Pawitan Y. In All Likelihood: Statistical Modelling and Inference Using Likelihood. 2001.

  9. Lehmann EL. Fisher, Nayman, and the Creation of Classical Statistics. 2011.

  10. Rohde CA. Introductory Statistical Inference with the Likelihood Function. 2014.

  11. Held L, Bové DS. Likelihood and Bayesian Inference. 2020.

Likelihood Based Confidence Interval of sd and variance assuming Normal Distribution

Description

Likelihood based confidence interval of sd and variance assuming normal distribution. It usually shows narrower interval than convenrtional chi-square interval. This uses estimated likelihood, not profile likelihood.

Usage

  LBCIvar(x, conf.level=0.95)

Arguments

x

a vector of observation

conf.level

confidence level

Details

It calculates (same height) likelihood based confidence interval of sd and variance assuming normal distribution in one group. The likelihood interval is asymmetric and there is no standard error in the output.

Value

PE

maximum likelihood estimate

LL

lower limit of likelihood interval

UL

upper limit of likelihood interval

Author(s)

Kyun-Seop Bae k@acr.kr

Examples

  LBCIvar(lh)
  (length(lh) - 1)*var(lh)/qchisq(c(0.975, 0.025), length(lh) - 1)

Likelihood Interval for a Proportion or a Binomial Distribution

Description

Likelihood interval of a proportion in one group

Usage

  LIbin(y, n, k, conf.level=0.95, eps=1e-8)

Arguments

y

positive event count of a group

n

total count of a group

k

1/k likelihood interval will be calculated

conf.level

approximately corresponding confidence level. If k is specified, this is ignored.

eps

Values less than eps are considered as 0.

Details

It calculates likelihood interval of a proportion in one group. The likelihood interval is asymmetric and there is no standard error in the output. If you need percent scale, multiply the output by 100.

Value

y

positive (concerning) event count

n

total trial count

PE

maximum likelihood estimate on the proportion

LL

lower limit of likelihood interval

UL

upper limit of likelihood interval

Author(s)

Kyun-Seop Bae k@acr.kr

References

Fisher RA. Statistical methods and scientific inference. 3e. 1973. pp68-76.

See Also

binom.test, prop.test

Examples

  LIbin(3, 14, k=2)
  LIbin(3, 14, k=5)
  LIbin(3, 14, k=15)
  LIbin(3, 14)
# binom.test(3, 14)
# prop.test(3, 14)

Likelihood Interval of mean, sd and variance assuming Normal Distribution

Description

Likelihood interval of mean and sd assuming normal distribution. This is estimated likelihood interval, not profile likelihood interval.

Usage

  LInorm(x, k, conf.level=0.95, PLOT="", LOCATE=FALSE, Resol=201)

Arguments

x

a vector of observation

k

1/k likelihood interval will be calculated

conf.level

approximately corresponding confidence level. If k is specified, this is ignored.

PLOT

"1d" for profile plot or "2d" for contour plot.

LOCATE

use locater. This works only with PLOT="2D" option.

Resol

resolution for plot. This works only with PLOT=TRUE option.

Details

It calculates likelihood interval of mean and sd assuming normal distribution in one group. There is no standard error in the output.

Value

PE

maximum likelihood estimate

LL

lower limit of likelihood interval

UL

upper limit of likelihood interval

Author(s)

Kyun-Seop Bae k@acr.kr

Examples

  x = c(-5.3, -4.5, -1.0, -0.7, 3.7, 3.9, 4.2, 5.5, 6.8, 7.4, 9.3)
  LInorm(x, k=1/0.15) # Pawitan Ex10-9 p289
  LInorm(x)
  LInorm(x, PLOT="1d")
  LInorm(x, PLOT="2d", LOCATE=TRUE)

Likelihood Interval of sd and variance assuming Normal Distribution

Description

Likelihood interval of sd and variance assuming normal distribution. This is estimated likelihood interval, not profile likelihood interval.

Usage

  LInormVar(x, k, conf.level=0.95)

Arguments

x

a vector of observation

k

1/k likelihood interval will be calculated

conf.level

approximately corresponding confidence level. If k is specified, this is ignored.

Details

It calculates likelihood interval of sd and variance assuming normal distribution in one group. The likelihood interval is asymmetric and there is no standard error in the output.

Value

PE

maximum likelihood estimate

LL

lower limit of likelihood interval

UL

upper limit of likelihood interval

Author(s)

Kyun-Seop Bae k@acr.kr

Examples

  x = c(-5.3, -4.5, -1.0, -0.7, 3.7, 3.9, 4.2, 5.5, 6.8, 7.4, 9.3)
  LInormVar(x, k=1/0.15) # Pawitan Ex10-9 p289
  LInormVar(x)

Likelihood Interval of the Mean assuming Poisson Distribution

Description

Likelihood interval of lambda assuming Poisson distribution.

Usage

  LIpois(x, k, n = 1, conf.level = 0.95, eps = 1e-8)

Arguments

x

raw data vector or a mean value. If the length of x is 1, x is considered as a mean.

k

1/k likelihood interval will be calculated.

n

number of observations. If the length of x is 1, x is considered as the mean.

conf.level

approximately corresponding confidence level. If k is specified, this is ignored.

eps

estimated values less than this eps are considered as 0.

Details

It calculates likelihood interval of mean(lambda) assuming Poisson distribution. The likelihood interval is asymmetric and there is no standard error in the output.

Value

PE

maximum likelihood estimate on the lambda

LL

lower limit of likelihood interval

UL

upper limit of likelihood interval

n

number of observations

k

1/k likelihood interval provided

logk

log(k) of k value

maxLL without factorial

maximum log likelihood without factorial part

Author(s)

Kyun-Seop Bae k@acr.kr

Examples

  LIpois(4, k=15)     # Fisher
  LIpois(4, k=exp(2)) # Edwards
  LIpois(4, k=1/0.15) # Pawitan
  LIpois(4, k=8)      # Rhode
  LIpois(4, n=4)      # Bae
  LIpois(4)           # Bae
#  poisson.test(4)
  LIpois(4, k=32)     # 0.7454614 11.7893612
  LIpois(2.1, n=60)   # 1.750222   2.493533

Likelihood Interval of the ratio of two variances from two groups

Description

Likelihood interval of the ratio of two variances from two groups assuming normal distribution. Likelihood interval usually gives a narrower interval when the likelihood function is asymmetric.

Usage

  LIvRatio(x, y, k, conf.level=0.95)

Arguments

x

observations from the first group, the test group, used for the numerator

y

observations from the second group, the control group, used for the denominator

k

1/k likelihood interval will be provided

conf.level

approximate confidence level

Details

It calculates likelihood interval of the ratio of two variances from two groups. Likelihood interval usually gives a narrower interval when the likelihood function is asymmetric.

Value

PE

maximum likelihood estimate on the ratio

LL

lower limit of likelihood interval

UL

upper limit of likelihood interval

logk

log(k) value used for LI

maxLL

maximum log likelihood

conf.level

approximate confidence level

Author(s)

Kyun-Seop Bae k@acr.kr

Examples

  LIvRatio(mtcars$drat, mtcars$wt)
  var.test(mtcars$drat, mtcars$wt)
  LIvRatio(mtcars$qsec, mtcars$wt)
  var.test(mtcars$qsec, mtcars$wt)
  LIvtest(sd(mtcars$qsec), nrow(mtcars), sd(mtcars$wt), nrow(mtcars))

Likelihood Interval of variance and sd assuming Normal Distribution with sample mean and sample size

Description

Likelihood interval of sd and variance assuming normal distribution. This is estimated likelihood interval, not profile likelihood interval.

Usage

  LIvar(s1, n1, k, conf.level=0.95)

Arguments

s1

standard deviation of the sample

n1

sample size

k

1/k likelihood interval will be calculated

conf.level

approximately corresponding confidence level. If k is specified, this is ignored.

Details

It calculates likelihood interval of sd and variance assuming normal distribution in one group. The likelihood interval is asymmetric and there is no standard error in the output.

Value

PE

maximum likelihood estimate on the population variance

LL

lower limit of likelihood interval

UL

upper limit of likelihood interval

Author(s)

Kyun-Seop Bae k@acr.kr

Examples

  x = c(-5.3, -4.5, -1.0, -0.7, 3.7, 3.9, 4.2, 5.5, 6.8, 7.4, 9.3)
  LInormVar(x)
  LIvar(sd(x), length(x))

Likelihood Interval of variance and sd assuming Normal Distribution using means and SDs

Description

Likelihood interval of variance and sd assuming normal distribution. This is estimated likelihood interval, not profile likelihood interval.

Usage

  LIvtest(s1, n1, s2, n2, k, conf.level=0.95)

Arguments

s1

sample standard deviation of the first group

n1

sample size of the first group

s2

sample standard deviation of the second group

n2

sample size of the second group

k

1/k likelihood interval will be calculated

conf.level

approximate confidence level. If k is specified, this is ignored.

Details

It calculates likelihood interval of variance and sd using sufficient statistics. There is no standard error in the output.

Value

PE

maximum likelihood estimate on the ratio

LL

lower limit of likelihood interval

UL

upper limit of likelihood interval

logk

log(k) value used for LI

maxLL

maximum log likelihood

conf.level

approximate confidence level

Author(s)

Kyun-Seop Bae k@acr.kr

Examples

  LIvtest(10.5, 3529, 8.9, 5190)
  LIvtest(3, 10, 2, 10)
  LIvtest(3, 10, 2, 10, k=15)

Likelihood Ratio Test

Description

Likelihood ratio test with given fitting results, sample size, number of parameters, log-likelihoods, and alpha

Usage

  LRT(n, pFull, pReduced, logLikFull, logLikReduced, alpha=0.05, Wilks=FALSE)

Arguments

n

number of observations

pFull

number of parameters of full model

pReduced

number of parameters of reduced model

logLikFull

log likelihood of full model

logLikReduced

log likelihood of reduced model

alpha

alpha value for type I error, significance level

Wilks

if TRUE, Wilks theorem (chi-square distribution) will be used, otherwise F distribution will be used.

Details

It performs likelihood ratio test with given fitting results. The default test is using F distribution. For small n (i.e. less than 100), you need to use F distribution. If the residuals are normally distributed, the delta -2 log likelihood (the difference between -2LL, the objective function value of each model) follows exactly an F-distribution, independent of sample size. When the distribution of the residuals is not normal (no matter what the distribution of the residuals is), it approaches a chi-square distribution as sample size increases (Wilks' theorem). The extreme distribution of the F-distribution (when the degrees of freedom in the denominator go to infinity) is chi-square distribution. The p-value from the F-distribution is slightly larger than the p-value from the chi-square distribution, meaning the F-distribution is more conservative. The difference decreases as sample size increases.

Value

n

number of observations

paraFull

number of parameters of full model

paraReduced

number of parameters of reduced model

deltaPara

difference of parameter counts

cutoff

cutoff, threshold, critical value of log-likelihood for the test

deltaLogLik

difference of log likelihood, if negative 0 is used.

Chisq or Fval

statistics according to the used distribution Chi-square of F

pval

p-value of null hypothesis. i.e. the reduced model is better.

Verdict

the model preferred.

Author(s)

Kyun-Seop Bae k@acr.kr

References

  1. Ruppert D, Cressie N, Carroll RJ. A Transformation/Weighting Model For Estimating Michaelis-Menten Parameters. School of Operations Research and Industrial Engineering, College of Engineering, Cornell University. Technical Report No. 796. May 1988.

  2. Scheffé H. The Analysis of Variance. Wiley. 1959.

  3. Wilks SS. The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses. Annals Math. Statist. 1938;9:60-62

Examples

  LRT(20, 4, 2, -58.085, -60.087)
  LRT(20, 4, 2, -58.085, -60.087, Wilks=TRUE)
  LRT(20, 4, 2, -57.315, -66.159)
  LRT(20, 4, 2, -57.315, -66.159, Wilks=TRUE)

  r1 = lm(mpg ~ disp + drat + wt, mtcars)
  r2 = lm(mpg ~ disp + drat, mtcars)
  anova(r2, r1)
  LRT(nrow(mtcars), r1$rank, r2$rank, logLik(r1), logLik(r2))

Odds Ratio and its Likelihood Interval between two groups without strata

Description

Odds ratio and its likelihood interval between two groups without stratification

Usage

  ORLI(y1, n1, y2, n2, conf.level=0.95, k, eps=1e-8)

Arguments

y1

positive event count of test (the first) group

n1

total count of the test (the first) group. Maximum allowable value is 1e8.

y2

positive event count of control (the second) group

n2

total count of control (the second) group. Maximum allowable value is 1e8.

conf.level

approximate confidence level to calculate k when k is missing.

k

1/k likelihood interval will be provided

eps

absolute value less than eps is regarded as negligible

Details

It calculates risk (proportion) difference and its likelihood interval between the two groups. The likelihood interval is asymmetric, and there is no standard error in the output. This does not support stratification.

Value

There is no standard error.

odd1

odd from the first group, y1/(n1 - y1)

odd2

odd from the second group, y2/(n2 - y2)

OR

odds ratio, odd1/odd2

lower

lower likelihood limit of OR

upper

upper likelihood limit of OR

Author(s)

Kyun-Seop Bae k@acr.kr

Examples

  ORLI(7, 10, 3, 10)
  ORLI(3, 10, 7, 10)

Likelihood Ratio Test for One group vs Two group gaussian mixture model

Description

With a given vector, it performs likelihood ratio test which model - one or two group - is better.

Usage

  OneTwo(x, alpha=0.05)

Arguments

x

a vector of numbers

alpha

alpha value for type I error, significance level

Details

It performs likelihood ratio test using both F distribution and Chi-square distribution (by Wilks' theorem).

Value

Estimate

n, Mean, SD for each group assumption and prior probability of each group in two group model

Delta

delta number of parameters and log-likelihoods

Statistic

Statistics from both the F distribution and Chi-square distribtuion. Cutoff is in terms of log-likelihood not the statistic.

Author(s)

Kyun-Seop Bae k@acr.kr

Examples

  OneTwo(c(7, 5, 17, 13, 16, 5, 7, 3, 8, 10, 8, 14, 14, 11, 14, 17, 2, 12, 15, 19))
  OneTwo(c(5, 3, 0, 6, 5, 2, 6, 6, 4, 4, 15, 13, 18, 18, 19, 14, 19, 13, 19, 18))

Risk (Proportion) Difference and its Likelihood Interval between two groups without strata

Description

Risk difference and its likelihood interval between two groups without stratification

Usage

  RDLI(y1, n1, y2, n2, conf.level=0.95, k, eps=1e-8)

Arguments

y1

positive event count of test (the first) group

n1

total count of the test (the first) group. Maximum allowable value is 1e8.

y2

positive event count of control (the second) group

n2

total count of control (the second) group. Maximum allowable value is 1e8.

conf.level

approximate confidence level to calculate k when k is missing.

k

1/k likelihood interval will be provided

eps

absolute value less than eps is regarded as negligible

Details

It calculates risk (proportion) difference and its likelihood interval between the two groups. The likelihood interval is asymmetric, and there is no standard error in the output. This does not support stratification.

Value

There is no standard error.

p1

proportion from the first group, y1/n1

p2

proportion from the second group, y2/n2

RD

risk difference, p1 - p2

lower

lower likelihood limit of RD

upper

upper likelihood limit of RD

Author(s)

Kyun-Seop Bae k@acr.kr

Examples

  RDLI(7, 10, 3, 10)
  RDLI(3, 10, 7, 10)

Relative Risk and its Likelihood Interval between two groups without strata

Description

Relative risk and its likelihood interval between two groups without stratification

Usage

  RRLI(y1, n1, y2, n2, conf.level=0.95, k, eps=1e-8)

Arguments

y1

positive event count of test (the first) group

n1

total count of the test (the first) group. Maximum allowable value is 1e8.

y2

positive event count of control (the second) group

n2

total count of control (the second) group. Maximum allowable value is 1e8.

conf.level

approximate confidence level to calculate k when k is missing.

k

1/k likelihood interval will be provided

eps

absolute value less than eps is regarded as negligible

Details

It calculates relative risk and its likelihood interval between the two groups. The likelihood interval is asymmetric, and there is no standard error in the output. This does not support stratification.

Value

There is no standard error.

p1

proportion from the first group, y1/n1

p2

proportion from the second group, y2/n2

RR

relative risk, p1/p2

lower

lower likelihood limit of RR

upper

upper likelihood limit of RR

Author(s)

Kyun-Seop Bae k@acr.kr

Examples

  RRLI(7, 10, 3, 10)
  RRLI(3, 10, 7, 10)