Generalized additive models II
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1 Generalized additive models II Patrick Breheny October 13 Patrick Breheny BST 764: Applied Statistical Modeling 1/23
2 Coronary heart disease study Today s lecture will feature several case studies involving the application of smoothing splines and GAMs to real data sets Our first study looks at many potential risk factors for coronary heart disease (CHD): sbp: Systolic blood pressure tobacco: Cumulative lifetime tobacco consumption (kg) ldl: Low-density lipoprotein adiposity: % of weight from fat obesity: Body mass index famhist: Family history of CHD typea: A measure of stress alcohol: Current alcohol consumption age Patrick Breheny BST 764: Applied Statistical Modeling 2/23
3 CHD study: Plan of analysis There are many ways one could analyze this data, but since we are interested in all the potential risk factors, one reasonable way to proceed is to model each of the continuous factors using smoothing splines, with the degrees of freedom for each smooth estimated by GCV Here there is one categorical factor, family history, which is highly significant (p < ) with an odds ratio of 2.6 (1.6, 4.1) Patrick Breheny BST 764: Applied Statistical Modeling 3/23
4 CHD study: Smoothing term tests The remaining 8 predictors are continuous; cumulatively they add degrees of freedom to the model We are interested in carrying out likelihood ratio tests for these terms Because there are so many smooth terms, it is desirable to fix the smoothing parameters {λ j } at their full-model values; otherwise the hypothesis tests will become muddled by the fact that the reduced model differs not only in terms of its components, but also how flexible each term is allowed to be Patrick Breheny BST 764: Applied Statistical Modeling 4/23
5 CHD GAM Case study #1 Overall effect: p = 0.21 Linearity: p = 0.12 Overall effect: p = < Linearity: p = 0.02 s(sbp,1.24) s(tobacco,5.87) sbp tobacco Overall effect: p = < 0.01 Overall effect: p = 0.02 Linearity: p = 0.03 s(ldl,1) s(obesity,2.2) ldl obesity Patrick Breheny BST 764: Applied Statistical Modeling 5/23
6 CHD GAM (cont d) Overall effect: p = 0.20 Overall effect: p = < 0.01 Linearity: p = 0.05 s(adiposity,1) s(typea,3.33) adiposity typea Overall effect: p = 0.90 Overall effect: p = < 0.01 Linearity: p = 0.15 s(alcohol,1) s(age,3.39) alcohol age Patrick Breheny BST 764: Applied Statistical Modeling 6/23
7 Asthma CAFO study As another example, I collaborated with Dr. Sanderson on a project involving exposure to air pollution from concentrated animal feeding operations (CAFOs) and its effect on developing asthma in children The exposure metric was calculated based on the proximity of the home to a CAFO (or CAFOs), the size of those CAFOs, and adjusted for wind patterns to develop an overall CAFO exposure score Based on previous studies of asthma, we also adjusted for a number of risk factors: age, gender, socioeconomic status, allergies, premature birth, early respiratory infection, and whether the individual worked with pigs Patrick Breheny BST 764: Applied Statistical Modeling 7/23
8 CAFO study: Plan of analysis We could once again model each continuous term using smoothing splines, although since the focus of the study was on a single risk factor, I modeled CAFO exposure nonparametrically and included the others as simple linear terms The estimation of how CAFO exposure affects asthma risk was the purpose of the study; an interesting additional question is whether or not any interactions are present involving this exposure In the course of the study, we found some evidence for an interaction with early respiratory illness (the test for interaction had a p-value of 0.06) Patrick Breheny BST 764: Applied Statistical Modeling 8/23
9 CAFO study: Results log(cafo) Log odds (Asthma) Patrick Breheny BST 764: Applied Statistical Modeling 9/23
10 CAFO study: Results (cont d) Deviance Model explained Base 28.8% Linear 31.1% Spline 33.0% p-value for overall effect of CAFO:.001 p-value for nonlinear component of CAFO:.03 Patrick Breheny BST 764: Applied Statistical Modeling 10/23
11 CAFO study: Possible interaction No early respiratory infection log(cafo) Effect on log(odds) Early respiratory infection log(cafo) Effect on log(odds) Patrick Breheny BST 764: Applied Statistical Modeling 11/23
12 GAMMs Case study #1 I am glossing over a complication present in the CAFO study: the observations were not all independent Specifically, sampling was done at the household/family level and one would expect some correlation among subjects within a household Possibilities for dealing with this are to include a random intercept (a Generalized Additive Mixed Model, or GAMM), or to use generalized estimating equations Patrick Breheny BST 764: Applied Statistical Modeling 12/23
13 Case study #3 GAMs are wonderful tools, but they do rely on that assumption of additivity; what if effects are not additive? To put it another way, suppose we have x 1 and x 2, both continuous variables; can we estimate E(y x) = f(x 1, x 2 ) in a completely nonparametric way, without making any assumptions on f? Patrick Breheny BST 764: Applied Statistical Modeling 13/23
14 Tensor product splines Case study #3 One approach to constructing multidimensional splines is using the tensor product basis Suppose we specify a set of basis functions {h 1k } for x 1 and {h 2k } for x 2, with M 1 and M 2 elements, respectively The tensor product basis for the two-dimensional smooth function of x 1 and x 2 is given by and has M 1 M 2 elements g jk (x 1, x 2 ) = h 1j (x 1 )h 2k (x 2 ) Patrick Breheny BST 764: Applied Statistical Modeling 14/23
15 Thin plate splines Case study #1 Case study #3 The multidimensional analog of smoothing splines are called thin plate regression splines For two dimensions, we find f(x 1, x 2 ) that minimizes n l{y i, f(x 1i, x 2i )}+λ i=1 [ 2 ] 2 [ f 2 ] 2 f u u v [ 2 f v 2 Thin plate regression splines have fairly complicated basis functions ] 2 dudv Patrick Breheny BST 764: Applied Statistical Modeling 15/23
16 2D smoothing Case study #1 Case study #3 Thin plate spline ldl age Patrick Breheny BST 764: Applied Statistical Modeling 16/23
17 Case study #3 Restrictions imposed by various models GLM GLM w/ interaction ldl ldl age age GAM Thin plate ldl ldl age age Patrick Breheny BST 764: Applied Statistical Modeling 17/23
18 The curse of dimensionality Case study #3 Thin plate regression splines can be extended further into higher dimensions, but they become rather computationally intensive as the dimension exceeds 2 Also, the curse of dimensionality implies that we need an exponentially increasing amount of data to maintain accuracy as p increases Furthermore, multidimensional smooth functions are harder to visualize and interpret In other words, it is nice to have the option of including interactions, but one generally cannot stray too far from the structure provided by the additive model Patrick Breheny BST 764: Applied Statistical Modeling 18/23
19 WISEWOMAN Case study #1 Case study #3 Our final case study involves WISEWOMAN, a project initiated by the CDC offering early screenings of risk factors for heart disease, diabetes, and other chronic diseases in financially disadvantaged women In Iowa, the project contained an intervention component designed to effect lifestyle changes in at-risk individuals Patrick Breheny BST 764: Applied Statistical Modeling 19/23
20 Attendance Case study #1 Case study #3 However, no intervention can be successful if the individuals targeted do not attend In order to design better interventions, it is of interest to study factors associated with attendance vs. nonattendance In particular, investigators are interested in the role that travel distance plays in attendance Patrick Breheny BST 764: Applied Statistical Modeling 20/23
21 WISEWOMAN study: Analysis Case study #3 As with the CAFO study, the emphasis was on a particular term, so distance was modeled using a smoothing spline while other factors affecting attendance were controlled for using linear terms In this study, we found significant evidence for an interaction between travel distance and age Furthermore, we found evidence of a three-way interaction, in that the above two-dimensional smooth function differed in urban and rural settings Patrick Breheny BST 764: Applied Statistical Modeling 21/23
22 WISEWOMAN: Rural results Case study #3 Probability(Attend) 0.9 Probability(Attend) Distance Age Age Distance Patrick Breheny BST 764: Applied Statistical Modeling 22/23
23 Case study #3 WISEWOMAN: Urban vs. rural (cross-sections) Patrick Breheny BST 764: Applied Statistical Modeling 23/23
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