Fitting Vector Generalized Linear Models

vglm fits vector generalized linear models (VGLMs). This very large class of models includes generalized linear models (GLMs) as a special case.

Usage

vglm(formula,
     family = stop("argument 'family' needs to be assigned"),
     data = list(), weights = NULL, subset = NULL,
     na.action, etastart = NULL, mustart = NULL,
     coefstart = NULL, control = vglm.control(...), offset = NULL,
     method = "vglm.fit", model = FALSE, x.arg = TRUE, y.arg = TRUE,
     contrasts = NULL, constraints = NULL, extra = list(),
     form2 = NULL, qr.arg = TRUE, smart = TRUE, ...)

Arguments

formula

a symbolic description of the model to be fit. The RHS of the formula is applied to each linear predictor. The effect of different variables in each linear predictor can be controlled by specifying constraint matrices—see constraints below.

family

a function of class "vglmff" (see vglmff-class) describing what statistical model is to be fitted. This is called a “VGAM family function”. See CommonVGAMffArguments for general information about many types of arguments found in this type of function. The argument name "family" is used loosely and for the ease of existing glm users; there is no concept of a formal “error distribution” for VGLMs. Possibly the argument name should be better "model" but unfortunately that name has already been taken.

data

an optional data frame containing the variables in the model. By default the variables are taken from environment(formula), typically the environment from which vglm is called.

weights

an optional vector or matrix of (prior fixed and known) weights to be used in the fitting process. If the VGAM family function handles multiple responses ($Q > 1$ of them, say) then weights can be a matrix with $Q$ columns. Each column matches the respective response. If it is a vector (the usually case) then it is recycled into a matrix with $Q$ columns. The values of weights must be positive; try setting a very small value such as 1.0e-8 to effectively delete an observation.

Currently the weights argument supports sampling weights from complex sampling designs via svyVGAM. Some details can be found at https://CRAN.R-project.org/package=svyVGAM.

subset

an optional logical vector specifying a subset of observations to be used in the fitting process.

na.action

a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The “factory-fresh” default is na.omit which is known as complete case analysis and applied to both sides of the formula.

etastart

optional starting values for the linear predictors. It is a $M$-column matrix with the same number of rows as the response. If $M = 1$ then it may be a vector. Note that etastart and the output of predict(fit) should be comparable. Here, fit is the fitted object. Almost all VGAM family functions are self-starting.

mustart

optional starting values for the fitted values. It can be a vector or a matrix; if a matrix, then it has the same number of rows as the response. Usually mustart and the output of fitted(fit) should be comparable. Most family functions do not make use of this argument because it is not possible to compute all $M$ columns of eta from mu.

coefstart

optional starting values for the coefficient vector. The length and order must match that of coef(fit).

control

a list of parameters for controlling the fitting process. See vglm.control for details.

offset

a vector or $M$-column matrix of offset values. These are a priori known and are added to the linear/additive predictors during fitting.

method

the method to be used in fitting the model. The default (and presently only) method vglm.fit() uses iteratively reweighted least squares (IRLS).

model

a logical value indicating whether the model frame should be assigned in the model slot.

x.arg, y.arg

logical values indicating whether the LM matrix and response vector/matrix used in the fitting process should be assigned in the x and y slots. Note that the model matrix is the LM matrix; to get the VGLM matrix type model.matrix(vglmfit) where vglmfit is a vglm object.

contrasts

an optional list. See the contrasts.arg of model.matrix.default.

constraints

an optional list of constraint matrices. The components of the list must be named (labelled) with the term it corresponds to (and it must match in character format exactly—see below). There are two types of input: "lm"-type and "vlm"-type. The former is a subset of the latter. The former has a matrix for each term of the LM matrix. The latter has a matrix for each column of the big VLM matrix. After fitting, the constraints extractor function may be applied; it returns the "vlm"-type list of constraint matrices by default. If "lm"-type are returned by constraints then these can be fed into this argument and it should give the same model as before.

If the constraints argument is used then the family function's zero argument (if it exists) needs to be set to NULL. This avoids what could be a probable contradiction. Sometimes setting other arguments related to constraint matrices to FALSE is also a good idea, e.g., parallel = FALSE, exchangeable = FALSE.

Properties: each constraint matrix must have $M$ rows, and be of full-column rank. By default, constraint matrices are the $M$ by $M$ identity matrix unless arguments in the family function itself override these values, e.g., parallel (see CommonVGAMffArguments). If constraints is used then it must contain all the terms; an incomplete list is not accepted.

As mentioned above, the labelling of each constraint matrix must match exactly, e.g., list("s(x2,df=3)"=diag(2)) will fail as as.character(~ s(x2,df=3)) produces white spaces: "s(x2, df = 3)". Thus list("s(x2, df = 3)" = diag(2)) is needed. See Example 6 below. More details are given in Yee (2015; Section 3.3.1.3) which is on p.101. Note that the label for the intercept is "(Intercept)".

extra

an optional list with any extra information that might be needed by the VGAM family function.

form2

the second (optional) formula. If argument xij is used (see vglm.control) then form2 needs to have all terms in the model. Also, some VGAM family functions such as micmen use this argument to input the regressor variable. If given, the slots @Xm2 and @Ym2 may be assigned. Note that smart prediction applies to terms in form2 too.

qr.arg

logical value indicating whether the slot qr, which returns the QR decomposition of the VLM model matrix, is returned on the object.

smart

logical value indicating whether smart prediction (smartpred) will be used.

...

further arguments passed into vglm.control.

Details

A vector generalized linear model (VGLM) is loosely defined as a statistical model that is a function of $M$ linear predictors and can be estimated by Fisher scoring. The central formula is given by $$\eta_j = \beta_j^T x$$ where $x$ is a vector of explanatory variables (sometimes just a 1 for an intercept), and $\beta_j$ is a vector of regression coefficients to be estimated. Here, $j=1,\ldots,M$, where $M$ is finite. Then one can write $\eta=(\eta_1,\ldots,\eta_M)^T$ as a vector of linear predictors.

Most users will find vglm similar in flavour to glm. The function vglm.fit actually does the work.

Value

An object of class "vglm", which has the following slots. Some of these may not be assigned to save space, and will be recreated if necessary later.

extra

the list extra at the end of fitting.

family

the family function (of class "vglmff").

iter

the number of IRLS iterations used.

predictors

a $M$-column matrix of linear predictors.

assign

a named list which matches the columns and the (LM) model matrix terms.

call

the matched call.

coefficients

a named vector of coefficients.

constraints

a named list of constraint matrices used in the fitting.

contrasts

the contrasts used (if any).

control

list of control parameter used in the fitting.

criterion

list of convergence criterion evaluated at the final IRLS iteration.

df.residual

the residual degrees of freedom.

df.total

the total degrees of freedom.

dispersion

the scaling parameter.

effects

the effects.

fitted.values

the fitted values, as a matrix. This is often the mean but may be quantiles, or the location parameter, e.g., in the Cauchy model.

misc

a list to hold miscellaneous parameters.

model

the model frame.

na.action

a list holding information about missing values.

offset

if non-zero, a $M$-column matrix of offsets.

post

a list where post-analysis results may be put.

preplot

used by plotvgam, the plotting parameters may be put here.

prior.weights

initially supplied weights (the weights argument). Also see weightsvglm.

qr

the QR decomposition used in the fitting.

R

the R matrix in the QR decomposition used in the fitting.

rank

numerical rank of the fitted model.

residuals

the working residuals at the final IRLS iteration.

ResSS

residual sum of squares at the final IRLS iteration with the adjusted dependent vectors and weight matrices.

smart.prediction

a list of data-dependent parameters (if any) that are used by smart prediction.

terms

the terms object used.

weights

the working weight matrices at the final IRLS iteration. This is in matrix-band form.

x

the model matrix (linear model LM, not VGLM).

xlevels

the levels of the factors, if any, used in fitting.

y

the response, in matrix form.

This slot information is repeated at vglm-class.

References

Yee, T. W. (2015). Vector Generalized Linear and Additive Models: With an Implementation in R. New York, USA: Springer.

Yee, T. W. and Hastie, T. J. (2003). Reduced-rank vector generalized linear models. Statistical Modelling, 3, 15–41.

Yee, T. W. and Wild, C. J. (1996). Vector generalized additive models. Journal of the Royal Statistical Society, Series B, Methodological, 58, 481–493.

Yee, T. W. (2014). Reduced-rank vector generalized linear models with two linear predictors. Computational Statistics and Data Analysis, 71, 889–902.

Yee, T. W. (2008). The VGAM Package. R News, 8, 28–39.

Author

Thomas W. Yee

Note

This function can fit a wide variety of statistical models. Some of these are harder to fit than others because of inherent numerical difficulties associated with some of them. Successful model fitting benefits from cumulative experience. Varying the values of arguments in the VGAM family function itself is a good first step if difficulties arise, especially if initial values can be inputted. A second, more general step, is to vary the values of arguments in vglm.control. A third step is to make use of arguments such as etastart, coefstart and mustart.

Some VGAM family functions end in "ff" to avoid interference with other functions, e.g., binomialff, poissonff. This is because VGAM family functions are incompatible with glm (and also gam() in gam and gam in the mgcv library).

The smart prediction (smartpred) library is incorporated within the VGAM library.

The theory behind the scaling parameter is currently being made more rigorous, but it it should give the same value as the scale parameter for GLMs.

In Example 5 below, the xij argument to illustrate covariates that are specific to a linear predictor. Here, lop/rop are the ocular pressures of the left/right eye (artificial data). Variables leye and reye might be the presence/absence of a particular disease on the LHS/RHS eye respectively. See vglm.control and fill1 for more details and examples.

WARNING

See warnings in vglm.control. Also, see warnings under weights above regarding sampling weights from complex sampling designs.

Examples

# Example 1. See help(glm)
(d.AD <- data.frame(treatment = gl(3, 3),
                    outcome = gl(3, 1, 9),
                    counts = c(18,17,15,20,10,20,25,13,12)))
#>   treatment outcome counts
#> 1         1       1     18
#> 2         1       2     17
#> 3         1       3     15
#> 4         2       1     20
#> 5         2       2     10
#> 6         2       3     20
#> 7         3       1     25
#> 8         3       2     13
#> 9         3       3     12
vglm.D93 <- vglm(counts ~ outcome + treatment, poissonff,
                 data = d.AD, trace = TRUE)
#> Iteration 1: deviance = 5.181115
#> Iteration 2: deviance = 5.129147
#> Iteration 3: deviance = 5.129141
#> Iteration 4: deviance = 5.129141
summary(vglm.D93)
#> 
#> Call:
#> vglm(formula = counts ~ outcome + treatment, family = poissonff, 
#>     data = d.AD, trace = TRUE)
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  3.045e+00  1.709e-01  17.815   <2e-16 ***
#> outcome2    -4.543e-01  2.022e-01  -2.247   0.0246 *  
#> outcome3    -2.930e-01  1.927e-01  -1.520   0.1285    
#> treatment2   5.532e-16  2.000e-01   0.000   1.0000    
#> treatment3   3.790e-16  2.000e-01   0.000   1.0000    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Name of linear predictor: loglink(lambda) 
#> 
#> Residual deviance: 5.1291 on 4 degrees of freedom
#> 
#> Log-likelihood: -23.3807 on 4 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 4 
#> 


# Example 2. Multinomial logit model
pneumo <- transform(pneumo, let = log(exposure.time))
vglm(cbind(normal, mild, severe) ~ let, multinomial, pneumo)
#> 
#> Call:
#> vglm(formula = cbind(normal, mild, severe) ~ let, family = multinomial, 
#>     data = pneumo)
#> 
#> 
#> Coefficients:
#> (Intercept):1 (Intercept):2         let:1         let:2 
#>    11.9750920     3.0390622    -3.0674665    -0.9020936 
#> 
#> Degrees of Freedom: 16 Total; 12 Residual
#> Residual deviance: 5.347382 
#> Log-likelihood: -25.25054 
#> 
#> This is a multinomial logit model with 3 levels


# Example 3. Proportional odds model
fit3 <- vglm(cbind(normal, mild, severe) ~ let, propodds, pneumo)
coef(fit3, matrix = TRUE)
#>             logitlink(P[Y>=2]) logitlink(P[Y>=3])
#> (Intercept)          -9.676093         -10.581725
#> let                   2.596807           2.596807
constraints(fit3)
#> $`(Intercept)`
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> $let
#>      [,1]
#> [1,]    1
#> [2,]    1
#> 
model.matrix(fit3, type = "lm")  # LM model matrix
#>   (Intercept)      let
#> 1           1 1.757858
#> 2           1 2.708050
#> 3           1 3.068053
#> 4           1 3.314186
#> 5           1 3.511545
#> 6           1 3.676301
#> 7           1 3.828641
#> 8           1 3.941582
#> attr(,"assign")
#> attr(,"assign")$`(Intercept)`
#> [1] 1
#> 
#> attr(,"assign")$let
#> [1] 2
#> 
#> attr(,"orig.assign.lm")
#> [1] 0 1
model.matrix(fit3)               # Larger VGLM (or VLM) matrix
#>     (Intercept):1 (Intercept):2      let
#> 1:1             1             0 1.757858
#> 1:2             0             1 1.757858
#> 2:1             1             0 2.708050
#> 2:2             0             1 2.708050
#> 3:1             1             0 3.068053
#> 3:2             0             1 3.068053
#> 4:1             1             0 3.314186
#> 4:2             0             1 3.314186
#> 5:1             1             0 3.511545
#> 5:2             0             1 3.511545
#> 6:1             1             0 3.676301
#> 6:2             0             1 3.676301
#> 7:1             1             0 3.828641
#> 7:2             0             1 3.828641
#> 8:1             1             0 3.941582
#> 8:2             0             1 3.941582
#> attr(,"assign")
#> attr(,"assign")$`(Intercept)`
#> [1] 1 2
#> 
#> attr(,"assign")$let
#> [1] 3
#> 
#> attr(,"vassign")
#> attr(,"vassign")$`(Intercept):1`
#> [1] 1
#> 
#> attr(,"vassign")$`(Intercept):2`
#> [1] 2
#> 
#> attr(,"vassign")$let
#> [1] 3
#> 
#> attr(,"constraints")
#> attr(,"constraints")$`(Intercept)`
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 
#> attr(,"constraints")$let
#>      [,1]
#> [1,]    1
#> [2,]    1
#> 
#> attr(,"orig.assign.lm")
#> [1] 0 1


# Example 4. Bivariate logistic model
fit4 <- vglm(cbind(nBnW, nBW, BnW, BW) ~ age, binom2.or, coalminers)
coef(fit4, matrix = TRUE)
#>             logitlink(mu1) logitlink(mu2) loglink(oratio)
#> (Intercept)     -6.5858766    -4.23347003        2.832535
#> age              0.1029391     0.06534392        0.000000
depvar(fit4)  # Response are proportions
#>        nBnW        nBW         BnW          BW
#> 1 0.9431352 0.04866803 0.003586066 0.004610656
#> 2 0.9235064 0.05862647 0.005025126 0.012841988
#> 3 0.8816848 0.08376716 0.008991955 0.025556081
#> 4 0.8469278 0.09234639 0.017247575 0.043478261
#> 5 0.7818821 0.12005277 0.023746702 0.074318382
#> 6 0.7154200 0.13539490 0.036773924 0.112411199
#> 7 0.6334928 0.11722488 0.055980861 0.193301435
#> 8 0.5525714 0.12857143 0.086857143 0.232000000
#> 9 0.4630282 0.11619718 0.093309859 0.327464789
weights(fit4, type = "prior")
#>   [,1]
#> 1 1952
#> 2 1791
#> 3 2113
#> 4 2783
#> 5 2274
#> 6 2393
#> 7 2090
#> 8 1750
#> 9 1136


# Example 5. The use of the xij argument (simple case).
# The constraint matrix for 'op' has one column.
nn <- 1000
eyesdat <- round(data.frame(lop = runif(nn),
                            rop = runif(nn),
                             op = runif(nn)), digits = 2)
eyesdat <- transform(eyesdat, eta1 = -1 + 2 * lop,
                              eta2 = -1 + 2 * lop)
eyesdat <- transform(eyesdat,
           leye = rbinom(nn, 1, prob = logitlink(eta1, inv = TRUE)),
           reye = rbinom(nn, 1, prob = logitlink(eta2, inv = TRUE)))
head(eyesdat)
#>    lop  rop   op  eta1  eta2 leye reye
#> 1 0.75 0.91 0.33  0.50  0.50    1    0
#> 2 0.70 0.02 0.47  0.40  0.40    1    1
#> 3 0.10 0.19 0.73 -0.80 -0.80    0    0
#> 4 0.61 0.38 0.74  0.22  0.22    1    0
#> 5 0.02 0.31 0.89 -0.96 -0.96    0    0
#> 6 0.65 0.53 0.30  0.30  0.30    0    1
fit5 <- vglm(cbind(leye, reye) ~ op,
             binom2.or(exchangeable = TRUE, zero = 3),
             data = eyesdat, trace = TRUE,
             xij = list(op ~ lop + rop + fill1(lop)),
             form2 = ~  op + lop + rop + fill1(lop))
#> Iteration 1: deviance = 2727.2353
#> Iteration 2: deviance = 2726.0386
#> Iteration 3: deviance = 2726.0015
#> Iteration 4: deviance = 2726.0004
#> Iteration 5: deviance = 2726.0004
#> Iteration 6: deviance = 2726.0004
coef(fit5)
#> (Intercept):1 (Intercept):2            op 
#>   -0.52717612    0.08909758    1.01465047 
coef(fit5, matrix = TRUE)
#>             logitlink(mu1) logitlink(mu2) loglink(oratio)
#> (Intercept)     -0.5271761     -0.5271761      0.08909758
#> op               1.0146505      1.0146505      0.00000000
constraints(fit5)
#> $`(Intercept)`
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    1    0
#> [3,]    0    1
#> 
#> $op
#>      [,1]
#> [1,]    1
#> [2,]    1
#> [3,]    0
#> 
fit5@control$xij
#> [[1]]
#> op ~ lop + rop + fill1(lop)
#> <environment: 0x6141fb8805e8>
#> 
head(model.matrix(fit5))
#>     (Intercept):1 (Intercept):2   op
#> 1:1             1             0 0.75
#> 1:2             1             0 0.91
#> 1:3             0             1 0.00
#> 2:1             1             0 0.70
#> 2:2             1             0 0.02
#> 2:3             0             1 0.00


# Example 6. The use of the 'constraints' argument.
as.character(~ bs(year,df=3))  # Get the white spaces right
#> [1] "~"                "bs(year, df = 3)"
clist <- list("(Intercept)"      = diag(3),
              "bs(year, df = 3)" = rbind(1, 0, 0))
fit1 <- vglm(r1 ~ bs(year,df=3), gev(zero = NULL),
             data = venice, constraints = clist, trace = TRUE)
#> Iteration 1: loglikelihood = -217.53311
#> Iteration 2: loglikelihood = -215.92358
#> Iteration 3: loglikelihood = -215.79841
#> Iteration 4: loglikelihood = -215.79279
#> Iteration 5: loglikelihood = -215.79236
#> Iteration 6: loglikelihood = -215.79233
#> Iteration 7: loglikelihood = -215.79232
#> Iteration 8: loglikelihood = -215.79232
coef(fit1, matrix = TRUE)  # Check
#>                   location loglink(scale) logofflink(shape, offset = 0.5)
#> (Intercept)       93.45545       2.664918                      -0.7177728
#> bs(year, df = 3)1 25.60150       0.000000                       0.0000000
#> bs(year, df = 3)2 10.42492       0.000000                       0.0000000
#> bs(year, df = 3)3 36.32781       0.000000                       0.0000000