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.

### Usage

    summary.gam(object,...)
    print.summary.gam(x,...)

### Arguments

`object` | a fitted `gam` object as produced by `gam()`.
`x` | a `summary.gam` object produced by `summary.gam()`.
`...` | 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.

### 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.
`m` | The number of smooth terms in the model.
`chi.sq` | An array of test statistics for assessing the significance of model smooth terms. 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-1 pseudo-inverse of V_i, and k is the basis dimension.

`s.pv`

An array of approximate p-values for the null hypotheses that each smooth term is zero. Be warned, these are only approximate. In the case in which UBRE has been used, they are obtained by comparing the chi.sq statistic given above to the chi-squared distribution with degrees of freedom given by the estimated degrees of freedom for the term. In the GCV case (in which the scale parameter will have been estimated) the statistic is compared to an F distribution with upper d.f. given by the estimate degrees of freedom for the term, and lower d.f. given by the residual degrees of freedom for the model . Use at your own risk! Typically the p-values will be somewhat inaccurate because they are conditional on the smoothing parameters and the distributional assumption doesn't have a firm theoretical basis. A pragmatic approach to the latter issue is to check p-values by refitting the model using regression splines with each basis dimension set to one more than the rounded edf for the term (see example, below). In this latter case the distributional assumption would be fine if the smoothing parameters were known, but of course they are not, and conditioning on the smoothing parameters will always be problematic. The difficulty is that GCV has used the data to select the most plausible smoothing parameters for each term - not surprisingly this tends to mean that the terms appear a little more `significant' than they should. Of course this problem is no different to the standard difficulty in interpreting p-values for model terms when the model has been selected by hypothesis testing methods such as backwards elimination.

`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.

### WARNING

The supplied p-values are only approximate and should be treated with scepticism.

### Author(s)

Simon N. Wood simon@stats.gla.ac.uk

### References

Gu and Wahba (1991) Minimizing GCV/GML scores with multiple smoothing parameters via the Newton method. SIAM J. Sci. Statist. Comput. 12:383-398

Wood, S.N. (2000) Modelling and Smoothing Parameter Estimation with Multiple Quadratic Penalties. J.R.Statist.Soc.B 62(2):413-428

Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114

http://www.stats.gla.ac.uk/~simon/

### See Also

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

### Examples

    library(mgcv)
    set.seed(0)
    n<-200
    sig2<-4
    x0 <- runif(n, 0, 1)
    x1 <- runif(n, 0, 1)
    x2 <- runif(n, 0, 1)
    x3 <- runif(n, 0, 1)
    pi <- asin(1) * 2
    y <- 2 * sin(pi * x0)
    y <- y + exp(2 * x1) - 3.75887
    y <- y + 0.2 * x2^11 * (10 * (1 - x2))^6 + 10 * (10 * x2)^3 * (1 - x2)^10 - 1.396
    e <- rnorm(n, 0, sqrt(abs(sig2)))
    y <- y + e
    b<-gam(y~s(x0)+s(x1)+s(x2)+s(x3))
    plot(b,pages=1)
    summary(b)
    # now check the p-values by using a pure regression spline.....
    b.d<-round(b$edf)+1
    b.d<-pmax(b.d,3) # can't have basis dimension less than this!
    bc<-gam(y~s(x0,k=b.d[1],fx=TRUE)+s(x1,k=b.d[2],fx=TRUE)+
            s(x2,k=b.d[3],fx=TRUE)+s(x3,k=b.d[4],fx=TRUE))
    plot(bc,pages=1)
    summary(bc)
    # p-value check - increase k to make this useful!
    n<-200;p<-0;k<-20
    for (i in 1:k)
    { b<-gam(y~s(x,z),data=data.frame(y=rnorm(n),x=runif(n),z=runif(n)))
      p[i]<-summary(b)$s.p[1]
    }
    plot(((1:k)-0.5)/k,sort(p))

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