summary.gam               package:mgcv               R Documentation

_S_u_m_m_a_r_y _f_o_r _a _G_A_M _f_i_t

_D_e_s_c_r_i_p_t_i_o_n:

     Takes a fitted 'gam' object produced by 'gam()' and produces
     various useful summaries from it.

_U_s_a_g_e:

     summary.gam(object,freq=TRUE,...)
     print.summary.gam(x,...)

_A_r_g_u_m_e_n_t_s:

  object: a fitted 'gam' object as produced by 'gam()'.

       x: a 'summary.gam' object produced by 'summary.gam()'.

    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. 

     ...: other arguments.

_D_e_t_a_i_l_s:

     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=FALSE' then the Bayesian parameter covariance matrix,
     'object$Vp', is used to calculate test statistics for terms, and
     the degrees of freedom for reference distributions is taken as the
     estimated degrees of freedom for the term concerned. This is not
     easy to justify theoretically, and the resulting `Bayesian
     p-values' are difficult to interpret and often have much worse
     frequentist performance than the default p-values.

_V_a_l_u_e:

     '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  pseudo-inverse of V_i, and k is
          estimated rank of  V_i.

    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 k degrees of freedom,
          where k is the estimated rank of  V_i. In the GCV case (in 
          which the scale parameter 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,
          because they are conditional on the smoothing parameters,
          which are usually uncertain.  

      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, being conditional
          on the smoothing parameters. The reference distribution is an
          appropriate chi-squared when the scale parameter is known,
          and is based on an F when it is not.

_W_A_R_N_I_N_G:

     The supplied p-values will often be underestimates if smoothing
     parameters have been estimated as part of model fitting.

_A_u_t_h_o_r(_s):

     Simon N. Wood simon.wood@r-project.org

_R_e_f_e_r_e_n_c_e_s:

     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

     Wood, S.N. (2004a) Stable and efficient multiple smoothing
     parameter estimation for generalized additive models. J. Amer.
     Statist. Ass.

     Wood, S.N. (2004b) On confidence intervals for GAMs based on
     penalized regression splines. Technical Report 04-12 Department of
     Statistics, University of Glasgow.

     <URL: http://www.stats.gla.ac.uk/~simon/>

_S_e_e _A_l_s_o:

     'gam', 'predict.gam', 'gam.check', 'anova.gam'

_E_x_a_m_p_l_e_s:

     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))

