Title: | Statistical Modeling |
---|---|
Description: | A collection of algorithms and functions to aid statistical modeling. Includes limiting dilution analysis (aka ELDA), growth curve comparisons, mixed linear models, heteroscedastic regression, inverse-Gaussian probability calculations, Gauss quadrature and a secure convergence algorithm for nonlinear models. Also includes advanced generalized linear model functions including Tweedie and Digamma distributional families, secure convergence and exact distributional calculations for unit deviances. |
Authors: | Gordon Smyth [cre, aut], Lizhong Chen [aut], Yifang Hu [ctb], Peter Dunn [ctb], Belinda Phipson [ctb], Yunshun Chen [ctb] |
Maintainer: | Gordon Smyth <[email protected]> |
License: | GPL-2 | GPL-3 |
Version: | 1.5.0 |
Built: | 2024-11-07 03:10:38 UTC |
Source: | https://github.com/cran/statmod |
This package includes a variety of functions for numerical analysis and statistical modelling. The functions are briefly summarized by type of application below.
The function tweedie
defines a large class of generalized linear model families with power variance functions.
It used in conjunction with the glm function, and widens the class of families that can be fitted.
qresiduals
implements randomized quantile residuals for generalized linear models.
The functions
canonic.digamma
,
unitdeviance.digamma
,
varfun.digamma
,
cumulant.digamma
,
d2cumulant.digamma
,
meanval.digamma
and logmdigamma
are used to fit double-generalized models, in which a link-linear model is fitted to the dispersion as well as to the mean.
Spefically they are used to fit the dispersion submodel associated with a gamma glm.
compareGrowthCurves
,
compareTwoGrowthCurves
and
meanT
are functions to test for differences between growth curves with repeated measurements on subjects.
The limdil
function is used in the analysis of stem cell frequencies.
It implements limiting dilution analysis using complemenary log-log binomial generalized linear model regression, with some improvements on previous programs.
The functions
qinvgauss
,
dinvgauss
,
pinvgauss
and
rinvgauss
provide probability calculations for the inverse Gaussian distribution.
gauss.quad
and
gauss.quad.prob
compute Gaussian Quadrature with probability distributions.
hommel.test
performs Hommel's multiple comparison tests.
power.fisher.test
computes the power of Fisher's Exact Test for comparing proportions.
sage.test
is a fast approximation to Fisher's exact test for each tag for comparing two Serial Analysis of Gene Expression (SAGE) libraries.
permp
computes p-values for permutation tests when the permutations are randomly drawn.
mixedModel2
,
mixedModel2Fit
and
glmgam.fit
fit mixed linear models.
remlscore
and remlscoregamma
fit heteroscedastic and varying dispersion models by REML.
welding
is an example data set.
matvec
and vecmat
facilitate multiplying matrices by vectors.
Gordon Smyth
Produces a Digamma generalized linear model family object. The Digamma distribution is the distribution of the unit deviance for a gamma response.
Digamma(link = "log") unitdeviance.digamma(y, mu) cumulant.digamma(theta) meanval.digamma(theta) d2cumulant.digamma(theta) varfun.digamma(mu) canonic.digamma(mu)
Digamma(link = "log") unitdeviance.digamma(y, mu) cumulant.digamma(theta) meanval.digamma(theta) d2cumulant.digamma(theta) varfun.digamma(mu) canonic.digamma(mu)
link |
character string, number or expressing specifying the link function. See |
y |
numeric vector of (positive) response values |
mu |
numeric vector of (positive) fitted values |
theta |
numeric vector of values of the canonical variable, equal to |
This family is useful for dispersion modelling with gamma generalized linear models.
The Digamma distribution describes the distribution of the unit deviances for a gamma family, in the same way that the gamma distribution itself describes the distribution of the unit deviances for Gaussian or inverse Gaussian families.
The Digamma distribution is so named because it is dual to the gamma distribution in the above sense, and because the digamma function
appears in its mean function.
Suppose that follows a gamma distribution with mean
and dispersion parameter
, so the variance of
is
.
Write
for the gamma distribution unit deviance.
Then
meanval.digamma(-1/phi)
gives the mean of and
2*d2cumulant.digamma(-1/phi)
gives the variance.
Digamma
produces a glm family object, which is a list of functions and expressions used by glm
in its iteratively reweighted least-squares algorithm. See family
for details.
The other functions take vector arguments and produce vector values of the same length and called by Digamma
.
unitdeviance.digamma
gives the unit deviances of the family, equal to the squared deviance residuals.
cumulant.digamma
is the cumulant function. If the dispersion is unity, then successive derivatives of the cumulant function give successive cumulants of the Digamma distribution. meanvalue.digamma
gives the first derivative, which is the expected value.
d2cumulant.digamma
gives the second derivative, which is the variance.
canonic.digamma
is the inverse of meanvalue.digamma
and gives the canonical parameter as a function of the mean parameter.
varfun.digamma
is the variance function of the Digamma family, the variance as a function of the mean.
Gordon Smyth
Smyth, G. K. (1989). Generalized linear models with varying dispersion. J. R. Statist. Soc. B, 51, 47-61. doi:10.1111/j.2517-6161.1989.tb01747.x
# Test for log-linear dispersion trend in gamma regression y <- rchisq(20,df=1) x <- 1:20 out.gam <- glm(y~x,family=Gamma(link="log")) d <- residuals(out.gam)^2 out.dig <- glm(d~x,family=Digamma(link="log")) summary(out.dig,dispersion=2)
# Test for log-linear dispersion trend in gamma regression y <- rchisq(20,df=1) x <- 1:20 out.gam <- glm(y~x,family=Gamma(link="log")) d <- residuals(out.gam)^2 out.dig <- glm(d~x,family=Digamma(link="log")) summary(out.dig,dispersion=2)
Fit single-hit model to a dilution series using complementary log-log binomial regression.
elda(response, dose, tested=rep(1,length(response)), group=rep(1,length(response)), observed=FALSE, confidence=0.95, test.unit.slope=FALSE) limdil(response, dose, tested=rep(1,length(response)), group=rep(1,length(response)), observed=FALSE, confidence=0.95, test.unit.slope=FALSE) eldaOneGroup(response, dose, tested, observed=FALSE, confidence=0.95, tol=1e-8, maxit=100, trace=FALSE)
elda(response, dose, tested=rep(1,length(response)), group=rep(1,length(response)), observed=FALSE, confidence=0.95, test.unit.slope=FALSE) limdil(response, dose, tested=rep(1,length(response)), group=rep(1,length(response)), observed=FALSE, confidence=0.95, test.unit.slope=FALSE) eldaOneGroup(response, dose, tested, observed=FALSE, confidence=0.95, tol=1e-8, maxit=100, trace=FALSE)
response |
numeric vector giving number of positive cases out of |
dose |
numeric vector of expected number of cells in assay. Values must be positive. |
tested |
numeric vector giving number of trials at each dose. Should take integer values. |
group |
vector or factor giving group to which the response belongs. |
observed |
logical, is the actual number of cells observed? |
confidence |
numeric level for confidence interval. Should be strictly between 0 and 1. |
test.unit.slope |
logical, should the adequacy of the single-hit model be tested? |
tol |
convergence tolerance. |
maxit |
maximum number of Newton iterations to perform. |
trace |
logical, if |
elda
and limdil
are alternative names for the same function.
(limdil
was the older name before the 2009 paper by Hu and Smyth.)
eldaOneGroup
is a lower-level function that does the computations when there is just one group, using a globally convergent Newton iteration.
It is called by the other functions.
These functions implement maximum likelihood analysis of limiting dilution data using methods proposed by Hu and Smyth (2009). The functions gracefully accommodate situations where 0% or 100% of the assays give positive results, which is why we call it "extreme" limiting dilution analysis. The functions provide the ability to test for differences in stem cell frequencies between groups, and to test goodness of fit in a number of ways. The methodology has been applied to the analysis of stem cell assays (Shackleton et al, 2006).
The statistical method is explained by Hu and Smyth (2009).
A binomial generalized linear model is fitted for each group with cloglog link and offset log(dose)
.
If observed=FALSE
, a classic Poisson single-hit model is assumed, and the Poisson frequency of the stem cells is the exp
of the intercept.
If observed=TRUE
, the values of dose
are treated as actual cell numbers rather than expected values.
This doesn't change the generalized linear model fit, but it does change how the frequencies are extracted from the estimated model coefficient (Hu and Smyth, 2009).
The confidence interval is a Wald confidence interval, unless the responses are all negative or all positive, in which case Clopper-Pearson intervals are computed.
If group
takes several values, then separate confidence intervals are computed for each group.
In this case a likelihood ratio test is conducted for differences in active cell frequencies between the groups.
These functions compute a number of different tests of goodness of fit.
One test is based on the coefficient for log(dose)
in the generalized linear model.
The nominal slope is 1.
A slope greater than one suggests a multi-hit model in which two or more cells are synergistically required to produce a positive response.
A slope less than 1 suggests some sort of cell interference.
Slopes less than 1 can also be due to heterogeneity of the stem cell frequency between assays.
elda
conducts likelihood ratio and score tests that the slope is equal to one.
Another test is based on the coefficient for dose
.
This idea is motivated by a suggestion of Gart and Weiss (1967), who suggest that heterogeneity effects are more likely to be linear in dose
than log(dose)
.
These functions conducts score tests that the coefficient for dose
is non-zero.
Negative values for this test suggest heterogeneity.
These functions produce objects of class "limdil"
.
There are print
and plot
methods for "limdil"
objects.
elda
and limdil
produce an object of class "limdil"
. This is a list with the following components:
CI |
numeric matrix giving estimated stem cell frequency and lower and upper limits of Wald confidence interval for each group |
test.difference |
numeric vector giving chisquare likelihood ratio test statistic and p-value for testing the difference between groups |
test.slope.wald |
numeric vector giving wald test statistics and p-value for testing the slope of the offset equal to one |
test.slope.lr |
numeric vector giving chisquare likelihood ratio test statistics and p-value for testing the slope of the offset equal to one |
test.slope.score.logdose |
numeric vector giving score test statistics and p-value for testing multi-hit alternatives |
test.slope.score.dose |
numeric vector giving score test statistics and p-value for testing heterogeneity |
response |
numeric of integer counts of positive cases, out of |
tested |
numeric vector giving number of trials at each dose |
dose |
numeric vector of expected number of cells in assay |
group |
vector or factor giving group to which the response belongs |
num.group |
number of groups |
Yifang Hu and Gordon Smyth
Hu, Y, and Smyth, GK (2009). ELDA: Extreme limiting dilution analysis for comparing depleted and enriched populations in stem cell and other assays. Journal of Immunological Methods 347, 70-78. doi:10.1016/j.jim.2009.06.008 http://www.statsci.org/smyth/pubs/ELDAPreprint.pdf
Shackleton, M., Vaillant, F., Simpson, K. J., Stingl, J., Smyth, G. K., Asselin-Labat, M.-L., Wu, L., Lindeman, G. J., and Visvader, J. E. (2006). Generation of a functional mammary gland from a single stem cell. Nature 439, 84-88. doi:10.1038/nature04372
Gart, JJ, and Weiss, GH (1967). Graphically oriented tests for host variability in dilution experiments. Biometrics 23, 269-284.
plot.limdil
and print.limdil
are methods for limdil
class objects.
A web interface to this function is available at https://bioinf.wehi.edu.au/software/elda/.
# When there is one group Dose <- c(50,100,200,400,800) Responses <- c(2,6,9,15,21) Tested <- c(24,24,24,24,24) out <- elda(Responses,Dose,Tested,test.unit.slope=TRUE) out plot(out) # When there are four groups Dose <- c(30000,20000,4000,500,30000,20000,4000,500,30000,20000,4000,500,30000,20000,4000,500) Responses <- c(2,3,2,1,6,5,6,1,2,3,4,2,6,6,6,1) Tested <- c(6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6) Group <- c(1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4) elda(Responses,Dose,Tested,Group,test.unit.slope=TRUE)
# When there is one group Dose <- c(50,100,200,400,800) Responses <- c(2,6,9,15,21) Tested <- c(24,24,24,24,24) out <- elda(Responses,Dose,Tested,test.unit.slope=TRUE) out plot(out) # When there are four groups Dose <- c(30000,20000,4000,500,30000,20000,4000,500,30000,20000,4000,500,30000,20000,4000,500) Responses <- c(2,3,2,1,6,5,6,1,2,3,4,2,6,6,6,1) Tested <- c(6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6) Group <- c(1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4) elda(Responses,Dose,Tested,Group,test.unit.slope=TRUE)
Expected value and variance of the scaled unit deviance for common generalized linear model families.
expectedDeviance(mu, family="binomial", binom.size, nbinom.size, gamma.shape)
expectedDeviance(mu, family="binomial", binom.size, nbinom.size, gamma.shape)
mu |
numeric vector or matrix giving mean of response variable. |
family |
character string indicating the linear exponential family. Possible values are |
binom.size |
integer vector giving the number of binomial trials when |
nbinom.size |
numeric vector giving the negative binomial size parameter when |
gamma.shape |
numeric vector giving the gamma shape parameter when |
For a generalized linear model (GLM), the scaled unit deviances can be computed using d <- f$dev.resids(y, mu, wt=1/phi)
where f
is the GLM family object, y
is the response variable, mu
is the vector of means and phi
is the vector of GLM dispersions (incorporating any prior weights).
The scaled unit deviances are often treated as being chiquare distributed on 1 df, so the mean should be 1 and the variance should be 2.
This distribution result only holds however when the saddlepoint approximation is accurate for the response variable distribution (Dunn and Smyth, 2018).
In other cases, the expected value and variance of the unit deviances can be far from the nominal values.
The expectedDeviance
function returns the exact mean and variance of the unit deviance for the usual GLM familes assuming that mu
is the true mean and phi
is the true dispersion.
When family
is "poisson"
, "binomial"
or "negative.binomial"
, the expected values and variances are computed using Chebyshev polynomial approximations.
When family = "Gamma"
, the function uses exact formulas derived by Smyth (1989).
A list with the components
mean |
expected values |
variance |
variances |
both of which have the same length and dimensions as the input mu
.
Lizong Chen and Gordon Smyth
Dunn PK, Smyth GK (2018). Generalized linear models with examples in R. Springer, New York, NY. doi:10.1007/978-1-4419-0118-7
Smyth, G. K. (1989). Generalized linear models with varying dispersion. J. R. Statist. Soc. B, 51, 47-61. doi:10.1111/j.2517-6161.1989.tb01747.x
family
, meanval.digamma
, d2cumulant.digamma
.
# Poisson example lambda <- 3 nsim <- 1e4 y <- rpois(nsim, lambda=lambda) d <- poisson()$dev.resids(y=y, mu=rep(lambda,nsim), wt=1) c(mean=mean(d), variance=var(d)) unlist(expectedDeviance(mu=lambda, family="poisson")) # binomial example n <- 10 p <- 0.01 y <- rbinom(nsim, prob=p, size=n) d <- binomial()$dev.resids(y=y/n, mu=rep(p,nsim), wt=n) c(mean=mean(d), variance=var(d)) unlist(expectedDeviance(mu=p, family="binomial", binom.size=n)) # gamma example alpha <- 5 beta <- 2 y <- beta * rgamma(1e4, shape=alpha) d <- Gamma()$dev.resids(y=y, mu=rep(alpha*beta,n), wt=alpha) c(mean=mean(d), variance=var(d)) unlist(expectedDeviance(mu=alpha*beta, family="Gamma", gamma.shape=alpha)) # negative binomial example library(MASS) mu <- 10 phi <- 0.2 y <- rnbinom(nsim, mu=mu, size=1/phi) f <- MASS::negative.binomial(theta=1/phi) d <- f$dev.resids(y=y, mu=rep(mu,nsim), wt=1) c(mean=mean(d), variance=var(d)) unlist(expectedDeviance(mu=mu, family="negative.binomial", nbinom.size=1/phi)) # binomial expected deviance tends to zero for p small: p <- seq(from=0.001,to=0.11,len=200) ed <- expectedDeviance(mu=p,family="binomial",binom.size=10) plot(p,ed$mean,type="l")
# Poisson example lambda <- 3 nsim <- 1e4 y <- rpois(nsim, lambda=lambda) d <- poisson()$dev.resids(y=y, mu=rep(lambda,nsim), wt=1) c(mean=mean(d), variance=var(d)) unlist(expectedDeviance(mu=lambda, family="poisson")) # binomial example n <- 10 p <- 0.01 y <- rbinom(nsim, prob=p, size=n) d <- binomial()$dev.resids(y=y/n, mu=rep(p,nsim), wt=n) c(mean=mean(d), variance=var(d)) unlist(expectedDeviance(mu=p, family="binomial", binom.size=n)) # gamma example alpha <- 5 beta <- 2 y <- beta * rgamma(1e4, shape=alpha) d <- Gamma()$dev.resids(y=y, mu=rep(alpha*beta,n), wt=alpha) c(mean=mean(d), variance=var(d)) unlist(expectedDeviance(mu=alpha*beta, family="Gamma", gamma.shape=alpha)) # negative binomial example library(MASS) mu <- 10 phi <- 0.2 y <- rnbinom(nsim, mu=mu, size=1/phi) f <- MASS::negative.binomial(theta=1/phi) d <- f$dev.resids(y=y, mu=rep(mu,nsim), wt=1) c(mean=mean(d), variance=var(d)) unlist(expectedDeviance(mu=mu, family="negative.binomial", nbinom.size=1/phi)) # binomial expected deviance tends to zero for p small: p <- seq(from=0.001,to=0.11,len=200) ed <- expectedDeviance(mu=p,family="binomial",binom.size=10) plot(p,ed$mean,type="l")
Fit a multi-group negative-binomial model to SAGE data, with Pearson estimation of the common overdispersion parameter.
fitNBP(y, group=NULL, lib.size=colSums(y), tol=1e-5, maxit=40, verbose=FALSE)
fitNBP(y, group=NULL, lib.size=colSums(y), tol=1e-5, maxit=40, verbose=FALSE)
y |
numeric matrix giving counts. Rows correspond to tags (genes) and columns to SAGE libraries. |
group |
factor indicating which library belongs to each group. If |
lib.size |
vector giving total number of tags in each library. |
tol |
small positive numeric tolerance to judge convergence |
maxit |
maximum number of iterations permitted |
verbose |
logical, if |
The overdispersion parameter is estimated equating the Pearson goodness of fit to its expectation. The variance is assumed to be of the form Var(y)=mu*(1+phi*mu) where E(y)=mu and phi is the dispersion parameter. All tags are assumed to share the same dispersion.
For given dispersion, the model for each tag is a negative-binomial generalized linear model with log-link and log(lib.size)
as offset.
The coefficient parametrization used is that corresponding to the formula ~0+group+offset(log(lib.size)
.
Except for the dispersion being common rather than genewise, the model fitted by this function is equivalent to that proposed by Lu et al (2005). The numeric algorithm used is that of alternating iterations (Smyth, 1996) using Newton's method as the outer iteration for the dispersion parameter starting at phi=0. This iteration is monotonically convergent for the dispersion.
List with components
coefficients |
numeric matrix of rates for each tag (gene) and each group |
fitted.values |
numeric matrix of fitted values |
dispersion |
estimated dispersion parameter |
This function has been made obsolete by the edgeR package on Bioconductor.
Gordon Smyth
Lu, J, Tomfohr, JK, Kepler, TB (2005). Identifying differential expression in multiple SAGE libraries: an overdispersed log-linear model approach. BMC Bioinformatics 6,165.
Smyth, G. K. (1996). Partitioned algorithms for maximum likelihood and other nonlinear estimation. Statistics and Computing, 6, 201-216. doi:10.1007/BF00140865
The edgeR package on Biconductor provides new and better functions to fit negative-binomial glms to SAGE or RNA-seq data.
See particularly the glmFit
and mglmOneWay
functions.
# True value for dispersion is 1/size=2/3 # Note the Pearson method tends to under-estimate the dispersion y <- matrix(rnbinom(10*4,mu=4,size=1.5),10,4) lib.size <- rep(50000,4) group <- c(1,1,2,2) fit <- fitNBP(y,group=group,lib.size=lib.size) logratio <- fit$coef %*% c(-1,1)
# True value for dispersion is 1/size=2/3 # Note the Pearson method tends to under-estimate the dispersion y <- matrix(rnbinom(10*4,mu=4,size=1.5),10,4) lib.size <- rep(50000,4) group <- c(1,1,2,2) fit <- fitNBP(y,group=group,lib.size=lib.size) logratio <- fit$coef %*% c(-1,1)
Fit a multi-group negative-binomial model to SAGE data, with Pearson estimation of the common overdispersion parameter.
forward(y, x, xkept=NULL, intercept=TRUE, nvar=ncol(x))
forward(y, x, xkept=NULL, intercept=TRUE, nvar=ncol(x))
y |
numeric response vector. |
x |
numeric matrix of covariates, candidates to be added to the regression. |
xkept |
numeric matrix of covariates to be included in the starting regression. |
intercept |
logical, should an intercept be added to |
nvar |
integer, number of covariates from |
This function has the advantage that x
can have many more columns than the length of y
.
Integer vector of length nvar
, giving the order in which columns of x
are added to the regression.
Gordon Smyth
y <- rnorm(10) x <- matrix(rnorm(10*5),10,5) forward(y,x)
y <- rnorm(10) x <- matrix(rnorm(10*5),10,5) forward(y,x)
Calculate nodes and weights for Gaussian quadrature.
gauss.quad(n, kind = "legendre", alpha = 0, beta = 0)
gauss.quad(n, kind = "legendre", alpha = 0, beta = 0)
n |
number of nodes and weights |
kind |
kind of Gaussian quadrature, one of |
alpha |
parameter for Jacobi or Laguerre quadrature, must be greater than -1 |
beta |
parameter for Jacobi quadrature, must be greater than -1 |
The integral from a
to b
of w(x)*f(x)
is approximated by sum(w*f(x))
where x
is the vector of nodes and w
is the vector of weights. The approximation is exact if f(x)
is a polynomial of order no more than 2n-1
.
The possible choices for w(x)
, a
and b
are as follows:
Legendre quadrature: w(x)=1
on (-1,1)
.
Chebyshev quadrature of the 1st kind: w(x)=1/sqrt(1-x^2)
on (-1,1)
.
Chebyshev quadrature of the 2nd kind: w(x)=sqrt(1-x^2)
on (-1,1)
.
Hermite quadrature: w(x)=exp(-x^2)
on (-Inf,Inf)
.
Jacobi quadrature: w(x)=(1-x)^alpha*(1+x)^beta
on (-1,1)
. Note that Chebyshev quadrature is a special case of this.
Laguerre quadrature: w(x)=x^alpha*exp(-x)
on (0,Inf)
.
The algorithm used to generated the nodes and weights is explained in Golub and Welsch (1969).
A list containing the components
nodes |
vector of values at which to evaluate the function |
weights |
vector of weights to give the function values |
Gordon Smyth, using Netlib Fortran code https://netlib.org/go/gaussq.f, and including a suggestion from Stephane Laurent
Golub, G. H., and Welsch, J. H. (1969). Calculation of Gaussian quadrature rules. Mathematics of Computation 23, 221-230.
Golub, G. H. (1973). Some modified matrix eigenvalue problems. Siam Review 15, 318-334.
Smyth, G. K. (1998). Numerical integration. In: Encyclopedia of Biostatistics, P. Armitage and T. Colton (eds.), Wiley, London, pages 3088-3095. http://www.statsci.org/smyth/pubs/NumericalIntegration-Preprint.pdf
Smyth, G. K. (1998). Polynomial approximation. In: Encyclopedia of Biostatistics, P. Armitage and T. Colton (eds.), Wiley, London, pages 3425-3429. http://www.statsci.org/smyth/pubs/PolyApprox-Preprint.pdf
Stroud, AH, and Secrest, D (1966). Gaussian Quadrature Formulas. Prentice-Hall, Englewood Cliffs, N.J.
# mean of gamma distribution with alpha=6 out <- gauss.quad(10,"laguerre",alpha=5) sum(out$weights * out$nodes) / gamma(6)
# mean of gamma distribution with alpha=6 out <- gauss.quad(10,"laguerre",alpha=5) sum(out$weights * out$nodes) / gamma(6)
Calculate nodes and weights for Gaussian quadrature in terms of probability distributions.
gauss.quad.prob(n, dist = "uniform", l = 0, u = 1, mu = 0, sigma = 1, alpha = 1, beta = 1)
gauss.quad.prob(n, dist = "uniform", l = 0, u = 1, mu = 0, sigma = 1, alpha = 1, beta = 1)
n |
number of nodes and weights |
dist |
distribution that Gaussian quadrature is based on, one of |
l |
lower limit of uniform distribution |
u |
upper limit of uniform distribution |
mu |
mean of normal distribution |
sigma |
standard deviation of normal distribution |
alpha |
positive shape parameter for gamma distribution or first shape parameter for beta distribution |
beta |
positive scale parameter for gamma distribution or second shape parameter for beta distribution |
This is a rewriting and simplification of gauss.quad
in terms of probability distributions.
The probability interpretation is explained by Smyth (1998).
For details on the underlying quadrature rules, see gauss.quad
.
The expected value of f(X)
is approximated by sum(w*f(x))
where x
is the vector of nodes and w
is the vector of weights. The approximation is exact if f(x)
is a polynomial of order no more than 2n-1
.
The possible choices for the distribution of X
are as follows:
Uniform on (l,u)
.
Normal with mean mu
and standard deviation sigma
.
Beta with density x^(alpha-1)*(1-x)^(beta-1)/B(alpha,beta)
on (0,1)
.
Gamma with density x^(alpha-1)*exp(-x/beta)/beta^alpha/gamma(alpha)
.
A list containing the components
nodes |
vector of values at which to evaluate the function |
weights |
vector of weights to give the function values |
Gordon Smyth, using Netlib Fortran code https://netlib.org/go/gaussq.f, and including corrections suggested by Spencer Graves
Smyth, G. K. (1998). Polynomial approximation. In: Encyclopedia of Biostatistics, P. Armitage and T. Colton (eds.), Wiley, London, pages 3425-3429. http://www.statsci.org/smyth/pubs/PolyApprox-Preprint.pdf
# the 4th moment of the standard normal is 3 out <- gauss.quad.prob(10,"normal") sum(out$weights * out$nodes^4) # the expected value of log(X) where X is gamma is digamma(alpha) out <- gauss.quad.prob(32,"gamma",alpha=5) sum(out$weights * log(out$nodes))
# the 4th moment of the standard normal is 3 out <- gauss.quad.prob(10,"normal") sum(out$weights * out$nodes^4) # the expected value of log(X) where X is gamma is digamma(alpha) out <- gauss.quad.prob(32,"gamma",alpha=5) sum(out$weights * log(out$nodes))
Computes score test statistics (z-statistics) for adding covariates to a generalized linear model.
glm.scoretest(fit, x2, dispersion=NULL)
glm.scoretest(fit, x2, dispersion=NULL)
fit |
generalized linear model fit object, of class |
x2 |
vector or matrix with each column a covariate to be added. |
dispersion |
the dispersion for the generalized linear model family. |
Rao's score test is a type of asymptotic test that is an alternative to Wald tests or likelihood ratio tests (LRTs) (Dunn and Smyth, 2018). Wald tests are computed by dividing parameter estimates by their standard errors. LRTs are computed from differences in the log-likihoods between the null and alternative hypotheses. Score tests are computed from log-likelihood derivatives. All three types of tests (Wald, score and LRT) are asymptotically equivalent under ideal circumstances, but the score and LRT tests are invariant under-reparametrization whereas Wald tests are not.
One of the main differences between the tests is the need for estimation of parameters under the null and alternative hypotheses. Wald tests require maximum likelihood estimates (MLEs) to be computed only under the alternative hypothesis, LRT tests require both the null and alternative models to be fitted, while score tests require only the null hypothesis to be fitted.
When a generalized linear model (GLM) is fitted in R using the glm
function, the summary()
function presents Wald tests for all the coefficients in the linear model while anova()
is able to compute likelihood ratio tests.
GLM output in R has historically not included score tests, although score tests can be a very computationally coefficient choice when one wants to test for many potential additional covariates being added to a relatively simple null model.
A number of well known Pearson chisquare statistics, including goodness of fit statistics and the Pearson test for independence in a contingency table can be derived as score tests (Smyth, 2003; Dunn and Smyth, 2018).
This function computes score test statistics for adding a single numerical covariate to a GLM, given the glm
output for the null model.
It makes very efficient use of the quantities already stored in the GLM fit object.
A computational formula for the score test statistics is given in Section 7.2.6 of Dunn and Smyth (2018).
The dispersion parameter is treated as for summary.glm
.
If NULL
, the Pearson estimator is used, except for the binomial, Poisson and negative binomial
families, for which the dispersion is one.
Note that the anova.glm
function in the stats package has offered a Rao score test option since 2011, but it requires model fits under the alternative as well as the null hypothesis, which does not take advantage of the computational advantage of score test.
The glm.scoretest
is more efficient as it does not require a full model fit.
On the other hand, anova.glm
can compute score tests for factors and multiple covariates, which glm.scoretest
does not currently do.
numeric vector containing the z-statistics, one for each covariate. The z-statistics can be treated as standard normal under the null hypothesis.
Gordon Smyth
Dunn, PK, and Smyth, GK (2018). Generalized linear models with examples in R. Springer, New York, NY. doi:10.1007/978-1-4419-0118-7
Lovison, G (2005). On Rao score and Pearson X^2 statistics in generalized linear models. Statistical Papers, 46, 555-574.
Pregibon, D (1982). Score tests in GLIM with applications. In GLIM82: Proceedings of the International Conference on Generalized Linear Models, R Gilchrist (ed.), Lecture Notes in Statistics, Volume 14, Springer, New York, pages 87-97.
Smyth, G. K. (2003). Pearson's goodness of fit statistic as a score test statistic. In: Science and Statistics: A Festschrift for Terry Speed, D. R. Goldstein (ed.), IMS Lecture Notes - Monograph Series, Volume 40, Institute of Mathematical Statistics, Beachwood, Ohio, pages 115-126. http://www.statsci.org/smyth/pubs/goodness.pdf
# Pearson's chisquare test for independence # in a contingency table is a score test. # First the usual test y <- c(20,40,40,30) chisq.test(matrix(y,2,2), correct=FALSE) # Now same test using glm.scoretest a <- gl(2,1,4) b <- gl(2,2,4) fit <- glm(y~a+b, family=poisson) x2 <- c(0,0,0,1) z <- glm.scoretest(fit, x2) z^2
# Pearson's chisquare test for independence # in a contingency table is a score test. # First the usual test y <- c(20,40,40,30) chisq.test(matrix(y,2,2), correct=FALSE) # Now same test using glm.scoretest a <- gl(2,1,4) b <- gl(2,2,4) fit <- glm(y~a+b, family=poisson) x2 <- c(0,0,0,1) z <- glm.scoretest(fit, x2) z^2
Fit a generalized linear model with secure convergence.
glmgam.fit(X, y, coef.start = NULL, tol = 1e-6, maxit = 50, trace = FALSE)
glmgam.fit(X, y, coef.start = NULL, tol = 1e-6, maxit = 50, trace = FALSE)
X |
design matrix, assumed to be of full column rank. Missing values not allowed. |
y |
numeric vector of responses. Negative or missing values not allowed. |
coef.start |
numeric vector of starting values for the regression coefficients |
tol |
small positive numeric value giving convergence tolerance |
maxit |
maximum number of iterations allowed |
trace |
logical value. If |
This function implements a modified Fisher scoring algorithm for generalized linear models, similar to the Levenberg-Marquardt algorithm for nonlinear least squares. The Levenberg-Marquardt modification checks for a reduction in the deviance at each step, and avoids the possibility of divergence. The result is a very secure algorithm that converges for almost all datasets.
glmgam.fit
is in principle equivalent to glm.fit(X,y,family=Gamma(link="identity"))
but with much more secure convergence.
List with the following components:
coefficients |
numeric vector of regression coefficients |
fitted |
numeric vector of fitted values |
deviance |
residual deviance |
iter |
number of iterations used to convergence. If convergence was not achieved then |
Gordon Smyth and Yunshun Chen
Dunn, PK, and Smyth, GK (2018). Generalized linear models with examples in R. Springer, New York, NY. doi:10.1007/978-1-4419-0118-7
glmgam.fit
is called by mixedModel2Fit
.
glm
is the standard glm fitting function in the stats package.
y <- rgamma(10, shape=5) X <- cbind(1, 1:10) fit <- glmgam.fit(X, y, trace=TRUE)
y <- rgamma(10, shape=5) X <- cbind(1, 1:10) fit <- glmgam.fit(X, y, trace=TRUE)
Fit a generalized linear model with secure convergence.
glmnb.fit(X, y, dispersion, weights = NULL, offset = 0, coef.start = NULL, start.method = "mean", tol = 1e-6, maxit = 50L, trace = FALSE)
glmnb.fit(X, y, dispersion, weights = NULL, offset = 0, coef.start = NULL, start.method = "mean", tol = 1e-6, maxit = 50L, trace = FALSE)
X |
design matrix, assumed to be of full column rank. Missing values not allowed. |
y |
numeric vector of responses. Negative or missing values not allowed. |
dispersion |
numeric vector of dispersion parameters for the negative binomial distribution. If of length 1, then the same dispersion is assumed for all observations. |
weights |
numeric vector of positive weights, defaults to all one. |
offset |
offset vector for linear model |
coef.start |
numeric vector of starting values for the regression coefficients |
start.method |
method used to find starting values, possible values are |
tol |
small positive numeric value giving convergence tolerance |
maxit |
maximum number of iterations allowed |
trace |
logical value. If |
This function implements a modified Fisher scoring algorithm for generalized linear models, analogous to the Levenberg-Marquardt algorithm for nonlinear least squares. The Levenberg-Marquardt modification checks for a reduction in the deviance at each step, and avoids the possibility of divergence. The result is a very secure algorithm that converges for almost all datasets.
glmnb.fit
is in principle equivalent to glm.fit(X,y,family=negative.binomial(link="log",theta=1/dispersion))
but with more secure convergence.
Here negative.binomial
is a function in the MASS package.
The dispersion
parameter is the same as 1/theta
for the MASS::negative.binomial
function or 1/size
for the stats::rnbinom
function.
dispersion=0
corresponds to the Poisson distribution.
List with the following components:
coefficients |
numeric vector of regression coefficients |
fitted |
numeric vector of fitted values |
deviance |
residual deviance |
iter |
number of iterations used to convergence. If convergence was not achieved then |
Gordon Smyth and Yunshun Chen
Dunn, PK, and Smyth, GK (2018). Generalized linear models with examples in R. Springer, New York, NY. doi:10.1007/978-1-4419-0118-7
The glmFit
function in the edgeR package on Bioconductor is a high-performance version of glmnb.fit
for many y
vectors at once.
glm
is the standard glm fitting function in the stats package.
negative.binomial
in the MASS package defines a negative binomial family for use with glm
.
y <- rnbinom(10, mu=1:10, size=5) X <- cbind(1, 1:10) fit <- glmnb.fit(X, y, dispersion=0.2, trace=TRUE)
y <- rnbinom(10, mu=1:10, size=5) X <- cbind(1, 1:10) fit <- glmnb.fit(X, y, dispersion=0.2, trace=TRUE)
Do all pairwise comparisons between groups of growth curves using a permutation test.
compareGrowthCurves(group, y, levels=NULL, nsim=100, fun=meanT, times=NULL, verbose=TRUE, adjust="holm", n0=0.5) compareTwoGrowthCurves(group, y, nsim=100, fun=meanT, n0=0.5) plotGrowthCurves(group, y, levels=sort(unique(group)), times=NULL, col=NULL,...)
compareGrowthCurves(group, y, levels=NULL, nsim=100, fun=meanT, times=NULL, verbose=TRUE, adjust="holm", n0=0.5) compareTwoGrowthCurves(group, y, nsim=100, fun=meanT, n0=0.5) plotGrowthCurves(group, y, levels=sort(unique(group)), times=NULL, col=NULL,...)
group |
vector or factor indicating group membership. Missing values are allowed in |
y |
matrix of response values with rows for individuals and columns for times. The number of rows must agree with the length of |
levels |
a character vector containing the identifiers of the groups to be compared. By default all groups with two more more members will be compared. |
nsim |
number of permutations to estimated p-values. |
fun |
a function defining the statistic used to measure the distance between two groups of growth curves.
Defaults to |
times |
a numeric vector containing the column numbers on which the groups should be compared. By default all the columns are used. |
verbose |
should progress results be printed? |
adjust |
method used to adjust for multiple testing, see |
n0 |
offset used for numerator and denominator of p-value calculation. |
col |
vector of colors corresponding to distinct groups |
... |
other arguments passed to |
compareTwoGrowthCurves
performs a permutation test of the difference between two groups of growth curves.
compareGrowthCurves
does all pairwise comparisons between two or more groups of growth curves.
The permutation p-values are computed as p = (ngt + neq/2 + n0) / (nsim + n0) where ngt is the number of permutations with test statistics greater than observed, neq is the number of permuttation with test statistics equal to that observed, and n0 is an offset to avoid p-values of zero (Phipson & Smyth 2010).
The offset n0 improves the type I error rate control and can be interpreted as allowing for the observed data as one of the permutations.
High resolution p-values can be obtained by setting nsim
to some large value, nsim=10000
say.
compareTwoGrowthCurves
returns a list with two components, stat
and p.value
, containing the observed statistics and the estimated p-value. compareGrowthCurves
returns a data frame with components
Group1 |
name of first group in a comparison |
Group2 |
name of second group in a comparison |
Stat |
observed value of the statistic |
P.Value |
permutation p-value |
adj.P.Value |
p-value adjusted for multiple testing |
Gordon Smyth
Elso, C. M., Roberts, L. J., Smyth, G. K., Thomson, R. J., Baldwin, T. M., Foote, S. J., and Handman, E. (2004). Leishmaniasis host response loci (lmr13) modify disease severity through a Th1/Th2-independent pathway. Genes and Immunity 5, 93-100.
Baldwin, T., Sakthianandeswaren, A., Curtis, J., Kumar, B., Smyth, G. K., Foote, S., and Handman, E. (2007). Wound healing response is a major contributor to the severity of cutaneous leishmaniasis in the ear model of infection. Parasite Immunology 29, 501-513.
Phipson B, Smyth GK (2010). Permutation P-values should never be zero: calculating exact P-values when permutations are randomly drawn. Statistical Applications in Genetics and Molecular Biology, Volume 9, Issue 1, Article 39. doi:10.2202/1544-6115.1585, doi:10.48550/arXiv.1603.05766.
meanT
, compareGrowthCurves
, compareTwoGrowthCurves
# A example with only one time data(PlantGrowth) compareGrowthCurves(PlantGrowth$group,as.matrix(PlantGrowth$weight)) # Can make p-values more accurate by nsim=10000
# A example with only one time data(PlantGrowth) compareGrowthCurves(PlantGrowth$group,as.matrix(PlantGrowth$weight)) # Can make p-values more accurate by nsim=10000
Given a set of p-values and a test level, returns vector of test results for each hypothesis.
hommel.test(p, alpha=0.05)
hommel.test(p, alpha=0.05)
p |
numeric vector of p-values |
alpha |
numeric value, desired significance level |
This function implements the multiple testing procedure of Hommel (1988).
Hommel's method is also implemented as an adjusted p-value method in the function p.adjust
but the accept/reject approach used here is faster.
logical vector indicating whether each hypothesis is accepted
Gordon Smyth
Hommel, G. (1988). A stagewise rejective multiple test procedure based on a modified Bonferroni test. Biometrika, 75, 383-386.
Shaffer, J. P. (1995). Multiple hypothesis testing. Annual Review of Psychology 46, 561-576. (An excellent review of the area.)
p <- sort(runif(100))[1:10] cbind(p,p.adjust(p,"hommel"),hommel.test(p))
p <- sort(runif(100))[1:10] cbind(p,p.adjust(p,"hommel"),hommel.test(p))
Density, cumulative probability, quantiles and random generation for the inverse Gaussian distribution.
dinvgauss(x, mean=1, shape=NULL, dispersion=1, log=FALSE) pinvgauss(q, mean=1, shape=NULL, dispersion=1, lower.tail=TRUE, log.p=FALSE) qinvgauss(p, mean=1, shape=NULL, dispersion=1, lower.tail=TRUE, log.p=FALSE, maxit=200L, tol=1e-14, trace=FALSE) rinvgauss(n, mean=1, shape=NULL, dispersion=1)
dinvgauss(x, mean=1, shape=NULL, dispersion=1, log=FALSE) pinvgauss(q, mean=1, shape=NULL, dispersion=1, lower.tail=TRUE, log.p=FALSE) qinvgauss(p, mean=1, shape=NULL, dispersion=1, lower.tail=TRUE, log.p=FALSE, maxit=200L, tol=1e-14, trace=FALSE) rinvgauss(n, mean=1, shape=NULL, dispersion=1)
x , q
|
vector of quantiles. |
p |
vector of probabilities. |
n |
sample size. If |
mean |
vector of (positive) means. |
shape |
vector of (positive) shape parameters. |
dispersion |
vector of (positive) dispersion parameters. Ignored if |
lower.tail |
logical; if |
log |
logical; if |
log.p |
logical; if |
maxit |
maximum number of Newton iterations used to find |
tol |
small positive numeric value giving the convergence tolerance for the quantile. |
trace |
logical, if |
The inverse Gaussian distribution takes values on the positive real line.
It is somewhat more right skew than the gamma distribution, with variance given by dispersion*mean^3
.
The distribution has applications in reliability and survival analysis and is one of the response distributions used in generalized linear models.
Giner and Smyth (2016) show that the inverse Gaussian distribution converges to an inverse chi-squared distribution as the mean becomes large.
The functions provided here implement numeric algorithms developed by Giner and Smyth (2016) that achieve close to full machine accuracy for all possible parameter values. Giner and Smyth (2016) show that the probability calculations provided by these functions are considerably more accurate, and in most cases faster, than previous implementations of inverse Gaussian functions. The improvement in accuracy is most noticeable for extreme probability values and for large parameter values.
The shape and dispersion parameters are alternative parametrizations for the variability, with dispersion=1/shape
.
Only one of these two arguments needs to be specified.
If both are set, then shape
takes precedence.
Output values give density (dinvgauss
), probability (pinvgauss
), quantile (qinvgauss
) or random sample (rinvgauss
) for the inverse Gaussian distribution with mean mean
and dispersion dispersion
.
Output is a vector of length equal to the maximum length of any of the arguments x
, q
, mean
, shape
or dispersion
.
If the first argument is the longest, then all the attributes of the input argument are preserved on output, for example, a matrix x
will give a matrix on output.
Elements of input vectors that are missing will cause the corresponding elements of the result to be missing, as will non-positive values for mean
or dispersion
.
Gordon Smyth.
Very early S-Plus versions of these functions, using simpler algorithms, were published 1998 at http://www.statsci.org/s/invgauss.html.
Paul Bagshaw (Centre National d'Etudes des Telecommunications, France) contributed the original version of qinvgauss
in December 1998.
Trevor Park (Department of Statistics, University of Florida) suggested improvements to a version of rinvguass
in 2005.
Giner, G., and Smyth, G. K. (2016). statmod: Probability calculations for the inverse Gaussian distribution. R Journal 8(1), 339-351. https://journal.r-project.org/archive/2016-1/giner-smyth.pdf
q <- rinvgauss(10, mean=1, disp=0.5) # generate vector of 10 random numbers p <- pinvgauss(q, mean=1, disp=0.5) # p should be uniformly distributed # Quantile for small right tail probability: qinvgauss(1e-20, mean=1.5, disp=0.7, lower.tail=FALSE) # Same quantile, but represented in terms of left tail probability on log-scale qinvgauss(-1e-20, mean=1.5, disp=0.7, lower.tail=TRUE, log.p=TRUE)
q <- rinvgauss(10, mean=1, disp=0.5) # generate vector of 10 random numbers p <- pinvgauss(q, mean=1, disp=0.5) # p should be uniformly distributed # Quantile for small right tail probability: qinvgauss(1e-20, mean=1.5, disp=0.7, lower.tail=FALSE) # Same quantile, but represented in terms of left tail probability on log-scale qinvgauss(-1e-20, mean=1.5, disp=0.7, lower.tail=TRUE, log.p=TRUE)
The difference between the log
and digamma
functions.
logmdigamma(x)
logmdigamma(x)
x |
numeric vector or array of positive values. Negative or zero values will return |
digamma(x)
is asymptotically equivalent to log(x)
. logmdigamma(x)
computes log(x) - digamma(x)
without subtractive cancellation for large x
.
Gordon Smyth
Abramowitz, M., and Stegun, I. A. (1970). Handbook of mathematical functions. Dover, New York.
log(10^15) - digamma(10^15) # returns 0 logmdigamma(10^15) # returns value correct to 15 figures
log(10^15) - digamma(10^15) # returns 0 logmdigamma(10^15) # returns value correct to 15 figures
Multiply the rows or columns of a matrix by the elements of a vector.
matvec(M, v) vecmat(v, M)
matvec(M, v) vecmat(v, M)
M |
numeric matrix, or object which can be coerced to a matrix. |
v |
numeric vector, or object which can be coerced to a vector. Length should match the number of columns of |
matvec(M,v)
is equivalent to M %*% diag(v)
but is faster to execute.
Similarly vecmat(v,M)
is equivalent to diag(v) %*% M
but is faster to execute.
A matrix of the same dimensions as M
.
Gordon Smyth
A <- matrix(1:12,3,4) A matvec(A,c(1,2,3,4)) vecmat(c(1,2,3),A)
A <- matrix(1:12,3,4) A matvec(A,c(1,2,3,4)) vecmat(c(1,2,3),A)
The mean-t statistic of the distance between two groups of growth curves.
meanT(y1, y2)
meanT(y1, y2)
y1 |
matrix of response values for the first group, with a row for each individual and a column for each time. Missing values are allowed. |
y2 |
matrix of response values for the second group. Must have the same number of columns as |
This function computes the pooled two-sample t-statistic between the response values at each time, and returns the mean of these values weighted by the degrees of freedom.
This function is used by compareGrowthCurves
.
numeric vector of length one containing the mean t-statistic.
Gordon Smyth
compareGrowthCurves
, compareTwoGrowthCurves
y1 <- matrix(rnorm(4*3),4,3) y2 <- matrix(rnorm(4*3),4,3) meanT(y1,y2) data(PlantGrowth) compareGrowthCurves(PlantGrowth$group,as.matrix(PlantGrowth$weight)) # Can make p-values more accurate by nsim=10000
y1 <- matrix(rnorm(4*3),4,3) y2 <- matrix(rnorm(4*3),4,3) meanT(y1,y2) data(PlantGrowth) compareGrowthCurves(PlantGrowth$group,as.matrix(PlantGrowth$weight)) # Can make p-values more accurate by nsim=10000
Fits a mixed linear model by REML. The linear model contains one random factor apart from the unit errors.
mixedModel2(formula, random, weights=NULL, only.varcomp=FALSE, data=list(), subset=NULL, contrasts=NULL, tol=1e-6, maxit=50, trace=FALSE) mixedModel2Fit(y, X, Z, w=NULL, only.varcomp=FALSE, tol=1e-6, maxit=50, trace=FALSE) randomizedBlock(formula, random, weights=NULL, only.varcomp=FALSE, data=list(), subset=NULL, contrasts=NULL, tol=1e-6, maxit=50, trace=FALSE) randomizedBlockFit(y, X, Z, w=NULL, only.varcomp=FALSE, tol=1e-6, maxit=50, trace=FALSE)
mixedModel2(formula, random, weights=NULL, only.varcomp=FALSE, data=list(), subset=NULL, contrasts=NULL, tol=1e-6, maxit=50, trace=FALSE) mixedModel2Fit(y, X, Z, w=NULL, only.varcomp=FALSE, tol=1e-6, maxit=50, trace=FALSE) randomizedBlock(formula, random, weights=NULL, only.varcomp=FALSE, data=list(), subset=NULL, contrasts=NULL, tol=1e-6, maxit=50, trace=FALSE) randomizedBlockFit(y, X, Z, w=NULL, only.varcomp=FALSE, tol=1e-6, maxit=50, trace=FALSE)
The arguments formula
, weights
, data
, subset
and contrasts
have the same meaning as in lm
.
The arguments y
, X
and w
have the same meaning as in lm.wfit
.
formula |
formula specifying the fixed model. |
random |
vector or factor specifying the blocks corresponding to random effects. |
weights |
optional vector of prior weights. |
only.varcomp |
logical value, if |
data |
an optional data frame containing the variables in the model. |
subset |
an optional vector specifying a subset of observations to be used in the fitting process. |
contrasts |
an optional list. See the |
tol |
small positive numeric tolerance, passed to |
maxit |
maximum number of iterations permitted, passed to |
trace |
logical value, passed to |
y |
numeric response vector |
X |
numeric design matrix for fixed model |
Z |
numeric design matrix for random effects |
w |
optional vector of prior weights |
Note that randomizedBlock
and mixedModel2
are alternative names for the same function.
This function fits the model where
is a vector of fixed coefficients and
is a vector of random effects.
Write
for the length of
and
for the length of
.
The random effect vector
is assumed to be normal, mean zero, with covariance matrix
while
is normal, mean zero, with covariance matrix
.
If
is an indicator matrix, then this model corresponds to a randomized block experiment.
The model is fitted using an eigenvalue decomposition that transforms the problem into a Gamma generalized linear model.
To the knowledge of the author, this is an original and unpublished approach to the problem of fitting mixed models.
Note that the block variance component varcomp[2]
is not constrained to be non-negative.
It may take negative values corresponding to negative intra-block correlations.
However the correlation varcomp[2]/sum(varcomp)
must lie between -1
and 1
.
Missing values in the data are not allowed.
This function is in principle equivalent to lme(fixed=formula,random=~1|random)
except that the block variance component is not constrained to be non-negative.
If the block variance component is non-negative, then this function is faster and more accurate than lme
for small to moderate size data sets but slower than lme
when the number of observations is large.
This function tends to be fast and reliable, compared to competitor functions that fit randomized block models, when then number of observations is small, say no more than 200.
However it becomes quadratically slow as the number of observations increases because of the need to compute two singular value decompositions of order nearly equal to the number of observations, although this can be limited to only one decomposition if only.varcomp = TRUE
).
For these reasons, this function is a good choice when fitting large numbers of mixed models to small datasets but is not optimal as currently implemented for fitting mixed models to very large data sets.
A list with the components:
varcomp |
numeric vector of length two containing the residual and block components of variance. |
se.varcomp |
standard errors for the variance components (if |
coefficients |
numeric vector of fixed coefficients (if |
se.coefficients |
standard errors for the fixed coefficients (if |
Gordon Smyth
Venables, W., and Ripley, B. (2002). Modern Applied Statistics with S-Plus, Springer.
glmgam.fit
, lme
, lm
, lm.fit
# Compare with first data example from Venable and Ripley (2002), # Chapter 10, "Linear Models" library(MASS) data(petrol) out <- mixedModel2(Y~SG+VP+V10+EP, random=No, data=petrol) cbind(varcomp=out$varcomp,se=out$se.varcomp)
# Compare with first data example from Venable and Ripley (2002), # Chapter 10, "Linear Models" library(MASS) data(petrol) out <- mixedModel2(Y~SG+VP+V10+EP, random=No, data=petrol) cbind(varcomp=out$varcomp,se=out$se.varcomp)
Robust estimation of a scale parameter using Hampel's redescending psi function.
mscale(u, na.rm=FALSE)
mscale(u, na.rm=FALSE)
u |
numeric vector of residuals. |
na.rm |
logical. Should missing values be removed? |
Estimates a scale parameter or standard deviation using an M-estimator with 50% breakdown.
This means the estimator is highly robust to outliers.
If the input residuals u
are a normal sample, then mscale(u)
should be equal to the standard deviation.
numeric constant giving the estimated scale.
Gordon Smyth
Yohai, V. J. (1987). High breakdown point and high efficiency robust estimates for regression. Ann. Statist. 15, 642-656.
Stromberg, A. J. (1993). Computation of high breakdown nonlinear regression parameters. J. Amer. Statist. Assoc. 88, 237-244.
Smyth, G. K., and Hawkins, D. M. (2000). Robust frequency estimation using elemental sets. Journal of Computational and Graphical Statistics 9, 196-214.
u <- rnorm(100) sd(u) mscale(u)
u <- rnorm(100) sd(u) mscale(u)
Calculates exact p-values for permutation tests when permutations are randomly drawn with replacement.
permp(x, nperm, n1, n2, total.nperm=NULL, method="auto", twosided=TRUE)
permp(x, nperm, n1, n2, total.nperm=NULL, method="auto", twosided=TRUE)
x |
number of permutations that yielded test statistics at least as extreme as the observed data. May be a vector or an array of values. |
nperm |
total number of permutations performed. |
n1 |
sample size of group 1. Not required if |
n2 |
sample size of group 2. Not required if |
total.nperm |
total number of permutations allowable from the design of the experiment. |
method |
character string indicating computation method. Possible values are |
twosided |
logical, is the test two-sided and symmetric between the two groups? |
This function can be used for calculating exact p-values for permutation tests where permutations are sampled with replacement, using theory and methods developed by Phipson and Smyth (2010).
The input values are the total number of permutations done (nperm
) and the number of these that were considered at least as extreme as the observed data (x
).
total.nperm
is the total number of distinct values of the test statistic that are possible.
This is generally equal to the number of possible permutations, unless a two-sided test is conducted with equal sample sizes, in which case total.nperm
is half the number of permutations, because the test statistic must then be symmetric in the two groups.
Usually total.nperm
is computed automatically from n1
and n2
, but can also be supplied directly by the user.
When method="exact"
, the p-values are computed to full machine precision by summing a series terms.
When method="approximate"
, an approximation is used that is faster and uses less memory.
If method="auto"
, the exact calculation is used when total.nperm
is less than or equal to 10,000 and the approximation is used otherwise.
vector or array of p-values, of same dimensions as x
Belinda Phipson and Gordon Smyth
Phipson B, Smyth GK (2010). Permutation P-values should never be zero: calculating exact P-values when permutations are randomly drawn. Statistical Applications in Genetics and Molecular Biology, Volume 9, Issue 1, Article 39. doi:10.2202/1544-6115.1585, doi:10.48550/arXiv.1603.05766.
x <- 0:5 # Both calls give same results permp(x=x, nperm=99, n1=6, n2=6) permp(x=x, nperm=99, total.nperm=462)
x <- 0:5 # Both calls give same results permp(x=x, nperm=99, n1=6, n2=6) permp(x=x, nperm=99, total.nperm=462)
Plot or print the results of a limiting dilution analysis.
## S3 method for class 'limdil' print(x, ...) ## S3 method for class 'limdil' plot(x, col.group=NULL, cex=1, lwd=1, legend.pos="bottomleft", ...)
## S3 method for class 'limdil' print(x, ...) ## S3 method for class 'limdil' plot(x, col.group=NULL, cex=1, lwd=1, legend.pos="bottomleft", ...)
x |
object of class |
col.group |
vector of colors for the groups of the same length as |
cex |
relative symbol size |
lwd |
relative line width |
legend.pos |
positioning on plot of legend when there are multiple groups |
... |
other arguments to be passed to |
The print method formats results similarly to a regression or anova summary in R.
The plot method produces a plot of a limiting dilution experiment similar to that in Bonnefoix et al (2001). The basic design of the plot was made popular by Lefkovits and Waldmann (1979).
The plot shows frequencies and confidence intervals for the multiple groups. A novel feature is that assays with 100% successes are included in the plot and are represented by down-pointing triangles.
Yifang Hu and Gordon Smyth
Bonnefoix, T, Bonnefoix, P, Callanan, M, Verdiel, P, and Sotto, JJ (2001). Graphical representation of a generalized linear model-based statistical test estimating the fit of the single-hit poisson model to limiting dilution assays. The Journal of Immunology 167, 5725-5730.
Lefkovits, I, and Waldmann, H (1979). Limiting dilution analysis of cells in the immune system. Cambridge University Press, Cambridge.
limdil describes the limdil
class.
Calculate by simulation the power of Fisher's exact test for comparing two proportions given two margin counts.
power.fisher.test(p1, p2, n1, n2, alpha=0.05, nsim=100, alternative="two.sided")
power.fisher.test(p1, p2, n1, n2, alpha=0.05, nsim=100, alternative="two.sided")
p1 |
first proportion to be compared. |
p2 |
second proportion to be compared. |
n1 |
first sample size. |
n2 |
second sample size. |
alpha |
significance level. |
nsim |
number of data sets to simulate. |
alternative |
indicates the alternative hypothesis and must be one of "two.sided", "greater" or "less". |
Estimates the power of Fisher's exact test for testing the null hypothesis that p1
equals p2
against the alternative that they are not equal.
The power is estimated by simulation.
The function generates nsim
pairs of binomial deviates and calls fisher.test
to obtain nsim
p-values.
The required power is tnen the proportion of the simulated p-values that are less than alpha
.
Estimated power of the test.
Gordon Smyth
power.fisher.test(0.5,0.9,20,20) # 70% chance of detecting difference
power.fisher.test(0.5,0.9,20,20) # 70% chance of detecting difference
Compute randomized quantile residuals for generalized linear models.
qresiduals(glm.obj,dispersion=NULL) qresid(glm.obj,dispersion=NULL) qres.binom(glm.obj) qres.pois(glm.obj) qres.nbinom(glm.obj) qres.gamma(glm.obj,dispersion=NULL) qres.invgauss(glm.obj,dispersion=NULL) qres.tweedie(glm.obj,dispersion=NULL) qres.default(glm.obj,dispersion=NULL)
qresiduals(glm.obj,dispersion=NULL) qresid(glm.obj,dispersion=NULL) qres.binom(glm.obj) qres.pois(glm.obj) qres.nbinom(glm.obj) qres.gamma(glm.obj,dispersion=NULL) qres.invgauss(glm.obj,dispersion=NULL) qres.tweedie(glm.obj,dispersion=NULL) qres.default(glm.obj,dispersion=NULL)
glm.obj |
Object of class |
dispersion |
a positive real number. Specifies the value of the
dispersion parameter for a Gamma or inverse Gaussian generalized linear
model if known. If |
Quantile residuals are based on the idea of inverting the estimated distribution function for each observation to obtain exactly standard normal residuals. In the case of discrete distributions, such as the binomial and Poisson, some randomization is introduced to produce continuous normal residuals. Quantile residuals are the residuals of choice for generalized linear models in large dispersion situations when the deviance and Pearson residuals can be grossly non-normal. Quantile residuals are the only useful residuals for binomial or Poisson data when the response takes on only a small number of distinct values.
Numeric vector of standard normal quantile residuals.
Gordon Smyth
Dunn, K. P., and Smyth, G. K. (1996). Randomized quantile residuals. Journal of Computational and Graphical Statistics 5, 1-10. http://www.statsci.org/smyth/pubs/residual.html
Dunn, PK, and Smyth, GK (2018). Generalized linear models with examples in R. Springer, New York, NY. doi:10.1007/978-1-4419-0118-7
# Poisson example: quantile residuals show no granularity y <- rpois(20,lambda=4) x <- 1:20 fit <- glm(y~x, family=poisson) qr <- qresiduals(fit) qqnorm(qr) abline(0,1) # Gamma example: # Quantile residuals are nearly normal while usual resids are not y <- rchisq(20, df=1) fit <- glm(y~1, family=Gamma) qr <- qresiduals(fit, dispersion=2) qqnorm(qr) abline(0,1) # Negative binomial example: if(require("MASS")) { fit <- glm(Days~Age,family=negative.binomial(2),data=quine) summary(qresiduals(fit)) fit <- glm.nb(Days~Age,link=log,data = quine) summary(qresiduals(fit)) }
# Poisson example: quantile residuals show no granularity y <- rpois(20,lambda=4) x <- 1:20 fit <- glm(y~x, family=poisson) qr <- qresiduals(fit) qqnorm(qr) abline(0,1) # Gamma example: # Quantile residuals are nearly normal while usual resids are not y <- rchisq(20, df=1) fit <- glm(y~1, family=Gamma) qr <- qresiduals(fit, dispersion=2) qqnorm(qr) abline(0,1) # Negative binomial example: if(require("MASS")) { fit <- glm(Days~Age,family=negative.binomial(2),data=quine) summary(qresiduals(fit)) fit <- glm.nb(Days~Age,link=log,data = quine) summary(qresiduals(fit)) }
Fits a heteroscedastic regression model using residual maximum likelihood (REML).
remlscore(y, X, Z, trace=FALSE, tol=1e-5, maxit=40)
remlscore(y, X, Z, trace=FALSE, tol=1e-5, maxit=40)
y |
numeric vector of responses |
X |
design matrix for predicting the mean |
Z |
design matrix for predicting the variance |
trace |
Logical variable. If true then output diagnostic information at each iteration. |
tol |
Convergence tolerance |
maxit |
Maximum number of iterations allowed |
Write and
for the expectation and variance of the
th response.
We assume the heteroscedastic regression model
where and
are vectors of covariates, and
and
are vectors of regression coefficients affecting the mean and variance respectively.
Parameters are estimated by maximizing the REML likelihood using REML scoring as described in Smyth (2002).
List with the following components:
beta |
vector of regression coefficients for predicting the mean |
se.beta |
vector of standard errors for beta |
gamma |
vector of regression coefficients for predicting the variance |
se.gam |
vector of standard errors for gamma |
mu |
estimated means |
phi |
estimated variances |
deviance |
minus twice the REML log-likelihood |
h |
numeric vector of leverages |
cov.beta |
estimated covariance matrix for beta |
cov.gam |
estimated covarate matrix for gamma |
iter |
number of iterations used |
Gordon Smyth
Smyth, G. K. (2002). An efficient algorithm for REML in heteroscedastic regression. Journal of Computational and Graphical Statistics 11, 836-847. doi:10.1198/106186002871
data(welding) attach(welding) y <- Strength # Reproduce results from Table 1 of Smyth (2002) X <- cbind(1,(Drying+1)/2,(Material+1)/2) colnames(X) <- c("1","B","C") Z <- cbind(1,(Material+1)/2,(Method+1)/2,(Preheating+1)/2) colnames(Z) <- c("1","C","H","I") out <- remlscore(y,X,Z) cbind(Estimate=out$gamma,SE=out$se.gam)
data(welding) attach(welding) y <- Strength # Reproduce results from Table 1 of Smyth (2002) X <- cbind(1,(Drying+1)/2,(Material+1)/2) colnames(X) <- c("1","B","C") Z <- cbind(1,(Material+1)/2,(Method+1)/2,(Preheating+1)/2) colnames(Z) <- c("1","C","H","I") out <- remlscore(y,X,Z) cbind(Estimate=out$gamma,SE=out$se.gam)
Estimates structured dispersion effects using approximate REML with gamma responses.
remlscoregamma(y, X, Z, mlink="log", dlink="log", trace=FALSE, tol=1e-5, maxit=40)
remlscoregamma(y, X, Z, mlink="log", dlink="log", trace=FALSE, tol=1e-5, maxit=40)
y |
numeric vector of responses. |
X |
design matrix for predicting the mean. |
Z |
design matrix for predicting the variance. |
mlink |
character string or numeric value specifying link for mean model. |
dlink |
character string or numeric value specifying link for dispersion model. |
trace |
logical value. If |
tol |
convergence tolerance. |
maxit |
maximum number of iterations allowed. |
This function fits a double generalized linear model (glm) with gamma responses.
As for ordinary gamma glms, a link-linear model is assumed for the expected values.
The double glm assumes a separate link-linear model for the dispersions as well.
The responses y
are assumed to follow a gamma generalized linear model with link mlink
and design matrix
X
.
The dispersions follow a link-linear model with link dlink
and design matrix Z
.
Write for the
th response.
The
are assumed to be independent and gamma distributed with
and var
.
The link-linear model for the means can be written as
where is the mean-link function defined by
mlink
and is the vector of means.
The dispersion link-linear model can be written as
where is the dispersion-link function defined by
dlink
and is the vector of dispersions.
The parameters are estimated by approximate REML likelihood using an adaption of the algorithm described by Smyth (2002).
See also Smyth and Verbyla (1999a,b) and Smyth and Verbyla (2009).
Having estimated
and
, the
are estimated as usual for a gamma glm.
The estimated values for ,
,
and
are return as
beta
, mu
, gamma
and phi
respectively.
List with the following components:
beta |
numeric vector of regression coefficients for predicting the mean. |
se.beta |
numeric vector of standard errors for beta. |
gamma |
numeric vector of regression coefficients for predicting the variance. |
se.gam |
numeric vector of standard errors for gamma. |
mu |
numeric vector of estimated means. |
phi |
numeric vector of estimated dispersions. |
deviance |
minus twice the REML log-likelihood. |
h |
numeric vector of leverages. |
Smyth, G. K., and Verbyla, A. P. (1999a). Adjusted likelihood methods for modelling dispersion in generalized linear models. Environmetrics 10, 695-709. http://www.statsci.org/smyth/pubs/ties98tr.html
Smyth, G. K., and Verbyla, A. P. (1999b). Double generalized linear models: approximate REML and diagnostics. In Statistical Modelling: Proceedings of the 14th International Workshop on Statistical Modelling, Graz, Austria, July 19-23, 1999, H. Friedl, A. Berghold, G. Kauermann (eds.), Technical University, Graz, Austria, pages 66-80. http://www.statsci.org/smyth/pubs/iwsm99-Preprint.pdf
Smyth, G. K. (2002). An efficient algorithm for REML in heteroscedastic regression. Journal of Computational and Graphical Statistics 11, 836-847.
Smyth, GK, and Verbyla, AP (2009). Leverage adjustments for dispersion modelling in generalized nonlinear models. Australian and New Zealand Journal of Statistics 51, 433-448.
data(welding) attach(welding) y <- Strength X <- cbind(1,(Drying+1)/2,(Material+1)/2) colnames(X) <- c("1","B","C") Z <- cbind(1,(Material+1)/2,(Method+1)/2,(Preheating+1)/2) colnames(Z) <- c("1","C","H","I") out <- remlscoregamma(y,X,Z)
data(welding) attach(welding) y <- Strength X <- cbind(1,(Drying+1)/2,(Material+1)/2) colnames(X) <- c("1","B","C") Z <- cbind(1,(Material+1)/2,(Method+1)/2,(Preheating+1)/2) colnames(Z) <- c("1","C","H","I") out <- remlscoregamma(y,X,Z)
Computes p-values for differential abundance for each tag between two digital libraries, conditioning on the total count for each tag. The counts in each group as a proportion of the whole are assumed to follow a binomial distribution.
sage.test(x, y, n1=sum(x), n2=sum(y))
sage.test(x, y, n1=sum(x), n2=sum(y))
x |
integer vector giving counts in first library. Non-integer values are rounded to the nearest integer. |
y |
integer vector giving counts in second library. Non-integer values are rounded to the nearest integer. |
n1 |
total number of tags in first library. Non-integer values are rounded to the nearest integer. |
n2 |
total number of tags in second library. Non-integer values are rounded to the nearest integer. |
This function was originally written for comparing SAGE libraries (a method for counting the frequency of sequence tags in samples of RNA). It can however be used for comparing any two digital libraries from RNA-Seq, ChIP-Seq or other technologies with respect to technical variation.
An exact two-sided binomial test is computed for each tag.
This test is closely related to Fisher's exact test for 2x2 contingency tables but, unlike Fisher's test, it conditions on the total number of counts for each tag.
The null hypothesis is that the expected counts are in the same proportions as the library sizes, i.e., that the binomial probability for the first library is n1/(n1+n2)
.
The two-sided rejection region is chosen analogously to Fisher's test. Specifically, the rejection region consists of those values with smallest probabilities under the null hypothesis.
When the counts are reasonably large, the binomial test, Fisher's test and Pearson's chisquare all give the same results. When the counts are smaller, the binomial test is usually to be preferred in this context.
This function is a later version of the earlier sage.test
function in the sagenhaft Bioconductor package.
This function has been made obsolete by binomTest
in the edgeR package.
Numeric vector of p-values.
This function is kept in the statmod package so as not to break code that depends on it
but it has been replaced by binomTest
in the edgeR Bioconductor package and
is no longer updated.
It may be removed in a later release of this package.
Gordon Smyth
https://en.wikipedia.org/wiki/Binomial_test
https://en.wikipedia.org/wiki/Fisher's_exact_test
https://en.wikipedia.org/wiki/Serial_analysis_of_gene_expression
https://en.wikipedia.org/wiki/RNA-Seq
The binomTest
function in the edgeR package on Bioconductor is a newer and better version of this function.
binom.test
in the stats package performs univariate binomial tests.
sage.test(c(0,5,10),c(0,30,50),n1=10000,n2=15000) # Univariate equivalents: binom.test(5,5+30,p=10000/(10000+15000))$p.value binom.test(10,10+50,p=10000/(10000+15000))$p.value
sage.test(c(0,5,10),c(0,30,50),n1=10000,n2=15000) # Univariate equivalents: binom.test(5,5+30,p=10000/(10000+15000))$p.value binom.test(10,10+50,p=10000/(10000+15000))$p.value
Produces a generalized linear model family object with any power variance function and any power link. Includes the Gaussian, Poisson, gamma and inverse-Gaussian families as special cases.
tweedie(var.power = 0, link.power = 1 - var.power)
tweedie(var.power = 0, link.power = 1 - var.power)
var.power |
index of power variance function |
link.power |
index of power link function. |
This function provides access to a range of generalized linear model (GLM) response distributions that are not otherwise provided by R.
It is also useful for accessing distribution/link combinations that are disallowed by the R glm
function.
The variance function for the GLM is assumed to be V(mu) = mu^var.power, where mu is the expected value of the distribution.
The link function of the GLM is assumed to be mu^link.power for non-zero values of link.power or log(mu) for var.power=0.
For example, var.power=1
produces the identity link.
The canonical link for each Tweedie family is link.power = 1 - var.power
.
The Tweedie family of GLMs is discussed in detail by Dunn and Smyth (2018).
Each value of var.power
corresponds to a particular type of response distribution.
The values 0, 1, 2 and 3 correspond to the normal distribution, the Poisson distribution, the gamma distribution and the inverse-Gaussian distribution respectively.
For these choices of var.power
, the Tweedie family is exactly equivalent to the usual GLM famly except with a greater choice of link powers.
For example, tweedie(var.power = 1, link.power = 0)
is exactly equivalent to poisson(link = "log")
.
The most interesting Tweedie families occur for var.power
between 1 and 2.
For these GLMs, the response distribution has mass at zero (i.e., it has exact zeros) but is otherwise continuous on the positive real numbers (Smyth, 1996; Hasan et al, 2012).
These GLMs have been used to model rainfall for example.
Many days there is no rain at all (exact zero) but, if there is any rain, then the actual amount of rain is continuous and positive.
Generally speaking, var.power
should be chosen so that the theoretical response distribution matches the type of response data being modeled.
Hence var.power
should be chosen between 1 and 2 only if the response observations are continuous and positive except for exact zeros and var.power
should be chosen greater than or equal to 2 only if the response observations are continuous and strictly positive.
There are no theoretical Tweedie GLMs with var.power between 0 and 1 (Jorgensen 1987).
The tweedie
function will work for those values but the family should be interpreted in a quasi-likelihood sense.
Theoretical Tweedie GLMs do exist for negative values of var.power, but they are of little practical application.
These distributions assume
The tweedie
function will work for those values but the family should be interpreted in a quasi-likelihood sense.
The name Tweedie has been associated with this family by Joergensen (1987) in honour of M. C. K. Tweedie. Joergensen (1987) gives a mathematical derivation of the Tweedie distributions proving that no distributions exist for var.power between 0 and 1.
Mathematically, a Tweedie GLM assumes the following.
Let be the expectation of the
th response. We assume that
where is a vector of covariates and b is a vector of regression cofficients, for some
,
and
.
This family is specified by
var.power = p
and link.power = q
.
A value of zero for is interpreted as
.
The following table summarizes the possible Tweedie response distributions:
var.power | Response distribution |
0 | Normal |
1 | Poisson |
(1, 2) | Compound Poisson, non-negative with mass at zero |
2 | Gamma |
3 | Inverse-Gaussian |
> 2 | Stable, with support on the positive reals |
A family object, which is a list of functions and expressions used by glm
and gam
in their iteratively reweighted least-squares algorithms.
See family
and glm
in the R base help for details.
Gordon Smyth
Dunn, P. K., and Smyth, G. K, (2018). Generalized linear models with examples in R. Springer, New York, NY. doi:10.1007/978-1-4419-0118-7 (Chapter 12 gives an overall discussion of Tweedie GLMs with R code and case studies.)
Hasan, M.M. and Dunn, P.K. (2012). Understanding the effect of climatology on monthly rainfall amounts in Australia using Tweedie GLMs. International Journal of Climatology, 32(7) 1006-1017. (An example with var.power between 1 and 2)
Joergensen, B. (1987). Exponential dispersion models. J. R. Statist. Soc. B 49, 127-162. (Mathematical derivation of Tweedie response distributions)
Tweedie, M. C. K. (1984). An index which distinguishes between some important exponential families. In Statistics: Applications and New Directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference. (Eds. J. K. Ghosh and J. Roy), pp. 579-604. Calcutta: Indian Statistical Institute. (The original mathematical paper from which the family is named)
Smyth, G. K. (1996). Regression modelling of quantity data with exact zeroes. Proceedings of the Second Australia-Japan Workshop on Stochastic Models in Engineering, Technology and Management. Technology Management Centre, University of Queensland, pp. 572-580. http://www.statsci.org/smyth/pubs/RegressionWithExactZerosPreprint.pdf (Derivation and examples of Tweedie GLMS with var.power between 0 and 1)
Smyth, G. K., and Verbyla, A. P., (1999). Adjusted likelihood methods for modelling dispersion in generalized linear models. Environmetrics 10, 695-709.
http://www.statsci.org/smyth/pubs/Ties98-Preprint.pdf
(Includes examples of Tweedie GLMs with var.power=2
and var.power=4
)
y <- rgamma(20,shape=5) x <- 1:20 # Fit a poisson generalized linear model with identity link glm(y~x,family=tweedie(var.power=1,link.power=1)) # Fit an inverse-Gaussion glm with log-link glm(y~x,family=tweedie(var.power=3,link.power=0))
y <- rgamma(20,shape=5) x <- 1:20 # Fit a poisson generalized linear model with identity link glm(y~x,family=tweedie(var.power=1,link.power=1)) # Fit an inverse-Gaussion glm with log-link glm(y~x,family=tweedie(var.power=3,link.power=0))
This is a highly fractionated two-level factorial design employed as a screening design in an off-line welding experiment performed by the National Railway Corporation of Japan. There were 16 runs and 9 experimental factors. The response variable is the observed tensile strength of the weld, one of several quality characteristics measured. All other variables are at plus and minus levels.
data(welding)
data(welding)
A data frame containing the following variables.
All the explanatory variables are numeric with two levels, -1
and 1
.
Variable | Description | |
Rods | Kind of welding rods | |
Drying | Period of drying | |
Material | Welded material | |
Thickness | Thickness | |
Angle | Angle | |
Opening | Opening | |
Current | Current | |
Method | Welding method | |
Preheating | Preheating | |
Strength | Tensile strength of the weld in kg/mm. The response variable. | |
http://www.statsci.org/data/general/welding.html
Smyth, G. K., Huele, F., and Verbyla, A. P. (2001). Exact and approximate REML for heteroscedastic regression. Statistical Modelling 1, 161-175.
Smyth, G. K. (2002). An efficient algorithm for REML in heteroscedastic regression. Journal of Computational and Graphical Statistics 11, 1-12.