(Semi)Parametric Models vs Nonparametric Models

Transcription

1 buay, 2003 Pobablty Models (Sem)Paametc Models vs Nonpaametc Models I defne paametc, sempaametc, and nonpaametc models n the two sample settng My defnton of sempaametc models s a lttle stonge than some statstcans The dstncton s to solate models wth assumptons that I thnk too stong Notaton fo two sample pobablty model Teatment : Contol :, K,, K, 1 1 d n d m ~ F ~ G Nonpaametc Models: 1 Nonpaametc Models: 2 Pobablty Models Pobablty Models Paametc models F, G ae known up to some fnte dmensonal paamete vectos F () t = Ψ( t, ) () = t, ) whee : (, ) Ψ has known fom s fnte dmensonal and unknown Paametc models: Examples Nomal : Benoull : Exponental : ~ N ~ B ~ E 2 2 ( µ, σ ) ~ N ( ν, τ ) ( 1, µ ) ~ B( 1, ν ) ( µ ) ~ E( ν ) Nonpaametc Models: 3 Nonpaametc Models: 4

2 buay, 2003 Pobablty Models Sempaametc models Foms of F, G ae unknown, but elated to each othe by some fnte dmensonal paamete vecto G can be detemned fom F and a fnte dmensonal paamete (Most often: unde the null hypothess, F = G) Pobablty Models Sempaametc models Foms of F, G ae unknown, but elated to each othe by some fnte dmensonal paamete vecto G can be detemned fom F and a fnte dmensonal paamete F() t = Ψ( t, ) = Ψ t, () ( ) whee : (, ) Ψ has unknown fom (n t) s fnte dmensonal and known (dentfablty) s fnte dmensonal and unknown Nonpaametc Models: 5 Nonpaametc Models: 6 Pobablty Models Pobablty Models Sempaametc models: Examples Shft : Shft - scale : Accel falue : Pop hzd : () = F( t µ ) () t µ = F σ () = F( tγ ) G() t = [ 1 F() t ] γ 1- Nonpaametc models Foms of F, G ae completely abtay and unknown An nfnte dmensonal paamete s needed to deve the fom of G fom F (I demand that the above hold unde all hypotheses, unless the test s consstent when F G) Examples of tuly nonpaametc analyses: Kolmogoov-Smnov test t-test wth unequal vaances (lage samples) Nonpaametc Models: 7 Nonpaametc Models: 8

3 buay, 2003 The Poblem: A Logcal Dsconnect Because the lght s so much bette hee unde the steetlamp - a dunk lookng fo the keys he lost half a block away Nonpaametc Models: 9 Nonpaametc Models: 10 The Poblem The Poblem In the development of statstcal models, and even moeso n the teachng of statstcs, paametc pobablty models have eceved undue emphass Examples: t test s typcally pesented n the context of the nomal pobablty model theoy of lnea models stesses small sample popetes andom effects specfed paametcally Bayesan (and especally heachcal Bayes) models ae eplete wth paametc dstbutons ASSERTION: Such emphass s not typcally n keepng wth the state of knowledge as an expement s beng conducted The paametc assumptons ae moe detaled than the hypothess beng tested, e.g.,: Queston: How does the nteventon affect the fst moment of the pobablty dstbuton? Assumpton: We know how the nteventon affects the 2nd, 3d,, cental moments of the pobablty dstbuton. Nonpaametc Models: 11 Nonpaametc Models: 12

4 buay, 2003 The Poblem Condtons unde whch an nteventon mght be expected to affect many aspects of a pobablty dstbuton Example 1: Cell polfeaton n cance peventon Wthn subect dstbuton of outcome s skewed (cance s a focal dsease) Such skewed measuements ae only obseved n a subset of the subects The nteventon affects only hypepolfeaton (ou deal) The Poblem Condtons unde whch an nteventon mght be expected to affect many aspects of a pobablty dstbuton (cont.) Example 2: Teatment of hypetenson Hypetenson has multple causes Any gven nteventon mght teat only subgoups of subects (and subgoup membeshp s a latent vaable) The teated populaton has a mxtue dstbuton (and note that we mght expect geate vaance n the goup wth the lowe mean) Nonpaametc Models: 13 Nonpaametc Models: 14 The Poblem The Poblem Condtons unde whch an nteventon mght be expected to affect many aspects of a pobablty dstbuton (cont.) Example 3: Effects on ates The nteventon affects ates The outcome measues a cumulatve state Abtaly complex mean-vaance elatonshps can esult These and othe mechansms would seem to make t lkely that the poblems n whch a fully paametc model o even a sempaametc model s coect consttute a set of measue zeo Excepton: ndependent bnay data must be bnomally dstbuted n the populaton fom whch they wee sampled andomly (exchangeably?) Nonpaametc Models: 15 Nonpaametc Models: 16

5 The Poblem Impact on what we teach about optmalty of statstcal models Clealy, paametc theoy may be elevant n an exact sense (though as gudelnes t s stll useful) Much of what we teach about the optmalty of nonpaametc tests s based on sempaametc models e.g., Lehmann, 1975: locaton-shft models The Poblem Example: the Wlcoxon ank sum test Common teachng: Not too bad aganst nomal data Bette than t test when data have heavy tals Moe accuate gudelnes: Above holds when a shft model holds fo some monotonc tansfomaton of the data If popensty to outles (mxtue dstbutons) s dffeent between goups, the t test may be bette even n pesence of heavy tals In the geneal case, the t test and the Wlcoxon ae not testng the same summay measue Nonpaametc Models: 17 Nonpaametc Models: 18 buay, 2003