R: Summary for a GAM fit

summary.gam {mgcv} | R Documentation

## Summary for a GAM fit

### Description

Takes a fitted `gam` object produced by `gam()` and produces various useful summaries from it. (See `sink` to divert output to a file.)

### Usage

    ## S3 method for class 'gam':
    summary(object, dispersion=NULL, freq=FALSE, ...)

    ## S3 method for class 'summary.gam':
    print(x,digits = max(3, getOption("digits") - 3),
                      signif.stars = getOption("show.signif.stars"),...)

### Arguments

`object` | a fitted `gam` object as produced by `gam()`.
`x` | a `summary.gam` object produced by `summary.gam()`.
`dispersion` | A known dispersion parameter. `NULL` to use estimate or default (e.g. 1 for Poisson).
`freq` | By default p-values for individual terms are calculated using the frequentist estimated covariance matrix of the parameter estimators. If this is set to FALSE then the Bayesian covariance matrix of the parameters is used instead. See details.
`digits` | controls number of digits printed in output.
`signif.stars` | Should significance stars be printed alongside output.
`...` | other arguments.

### Details

Model degrees of freedom are taken as the trace of the influence (or hat) matrix A for the model fit. Residual degrees of freedom are taken as number of data minus model degrees of freedom. Let P_i be the matrix giving the parameters of the ith smooth when applied to the data (or pseudodata in the generalized case) and let X be the design matrix of the model. Then tr(XP_i) is the edf for the ith term. Clearly this definition causes the edf's to add up properly!

`print.summary.gam` tries to print various bits of summary information useful for term selection in a pretty way.

If `freq=TRUE` then the frequentist approximation for p-values of smooth terms described in section 4.8.5 of Wood (2006) is used. The approximation is not great. If p_i is the parameter vector for the ith smooth term, and this term has estimated covariance matrix V_i then the statistic is p_i'V_i^{k-}p_i, where V_i^{k-} is the rank k pseudo-inverse of V_i, and k is estimated rank of V_i. p-values are obtained as follows. In the case of known dispersion parameter, they are obtained by comparing the chi.sq statistic to the chi-squared distribution with k degrees of freedom, where k is the estimated rank of V_i. If the dispersion parameter is unknown (in which case it will have been estimated) the statistic is compared to an F distribution with k upper d.f. and lower d.f. given by the residual degrees of freedom for the model. Typically the p-values will be somewhat too low.

If `freq=FALSE` then `Bayesian p-values' are returned for the smooth terms, based on a test statistic motivated by Nychka's (1988) analysis of the frequentist properties of Bayesian confidence intervals for smooths. These appear to have better frequentist performance (in terms of power and distribution under the null) than the alternative strictly frequentist approximation. Let f denote the vector of values of a smooth term evaluated at the original covariate values and let V_f denote the corresponding Bayesian covariance matrix. Let V*_f denote the rank r pseudoinverse of V_f, where r is the EDF for the term, rounded up, (or the numerical rank of V_f if this is smaller). The statistic used is then

T = f'V*_f f

(this can be calculated efficiently without forming the pseudoinverse explicitly). T is compared to a chi-squared distribution with degrees of freedom given by the EDF for the term + 0.5, or T is used as a component in an F ratio statistic if the scale parameter has been estimated. The heuristic justification for the reference DoF is that, away from the lower bound on the EDF, the theoretically correct reference level appears to be r, which if on average close to EDF + 0.5. However EDF + 0.5 behaves better than r when the EDF is near its lower limit.

The definition of Bayesian p-value used is: the probability of observing an f less probable than bf 0, under the approximation for the posterior for f implied by the truncation used in the test statistic.

### Value

`summary.gam` produces a list of summary information for a fitted `gam` object.

`p.coeff` | is an array of estimates of the strictly parametric model coefficients.
`p.t` | is an array of the `p.coeff`'s divided by their standard errors.
`p.pv` | is an array of p-values for the null hypothesis that the corresponding parameter is zero. Calculated with reference to the t distribution with the estimated residual degrees of freedom for the model fit if the dispersion parameter has been estimated, and the standard normal if not.
`m` | The number of smooth terms in the model.
`chi.sq` | An array of test statistics for assessing the significance of model smooth terms. See details.
`s.pv` | An array of approximate p-values for the null hypotheses that each smooth term is zero. Be warned, these are only approximate.
`se` | array of standard error estimates for all parameter estimates.
`r.sq` | The adjusted r-squared for the model. Defined as the proportion of variance explained, where original variance and residual variance are both estimated using unbiased estimators. This quantity can be negative if your model is worse than a one parameter constant model, and can be higher for the smaller of two nested models! Note that proportion null deviance explained is probably more appropriate for non-normal errors.
`dev.expl` | The proportion of the null deviance explained by the model.
`edf` | array of estimated degrees of freedom for the model terms.
`residual.df` | estimated residual degrees of freedom.
`n` | number of data.
`gcv` | minimized GCV score for the model, if GCV used.
`ubre` | minimized UBRE score for the model, if UBRE used.
`scale` | estimated (or given) scale parameter.
`family` | the family used.
`formula` | the original GAM formula.
`dispersion` | the scale parameter.
`pTerms.df` | the degrees of freedom associated with each parameteric term (excluding the constant).
`pTerms.chi.sq` | a Wald statistic for testing the null hypothesis that the each parametric term is zero.
`pTerms.pv` | p-values associated with the tests that each term is zero. For penalized fits these are approximate. The reference distribution is an appropriate chi-squared when the scale parameter is known, and is based on an F when it is not.
`cov.unscaled` | The estimated covariance matrix of the parameters (or estimators if `freq=TRUE`), divided by scale parameter.
`cov.scaled` | The estimated covariance matrix of the parameters (estimators if `freq=TRUE`).
`p.table` | significance table for parameters
`s.table` | significance table for smooths
`p.Terms` | significance table for parametric model terms

### WARNING

The p-values are approximate.

### Author(s)

Simon N. Wood simon.wood@r-project.org with substantial improvements by Henric Nilsson.

### References

Nychka (1988) Bayesian Confidence Intervals for Smoothing Splines. Journal of the American Statistical Association 83:1134-1143.

Wood S.N. (2006) Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC Press.

### See Also

`gam`, `predict.gam`, `gam.check`, `anova.gam`

### Examples

    library(mgcv)
    set.seed(0)
    dat <- gamSim(1,n=200,scale=2) ## simulate data

    b <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),data=dat)
    plot(b,pages=1)
    summary(b)

    ## now check the p-values by using a pure regression spline.....
    b.d <- round(summary(b)$edf)+1 ## get edf per smooth
    b.d <- pmax(b.d,3) # can't have basis dimension less than 3!
    bc<-gam(y~s(x0,k=b.d[1],fx=TRUE)+s(x1,k=b.d[2],fx=TRUE)+
            s(x2,k=b.d[3],fx=TRUE)+s(x3,k=b.d[4],fx=TRUE),data=dat)
    plot(bc,pages=1)
    summary(bc)

    ## p-value check - increase k to make this useful!
    n<-200;p<-0;k<-20
    for (i in 1:k)
    { b<-gam(y~s(x,z),data=data.frame(y=rnorm(n),x=runif(n),z=runif(n)))
      p[i]<-summary(b)$s.p[1]
    }
    plot(((1:k)-0.5)/k,sort(p))
    abline(0,1,col=2)

[Package mgcv version 1.4-1.1 Index]