Smple Lear Regresso Regresso equato a equato that descrbes the average relatoshp betwee a respose (depedet) ad a eplaator (depedet) varable. 6 8 Slope-tercept equato for a le m b (,6) slope. (,) 6 6 8 Determstc Model A model that defes a eact relatoshp betwee varables. Eample:. There s o allowace for error the predcto of for a gve. Fahrehet 8 96 9 F C - Celsus
Probablstc Model A model that accouts for radom error. Icludes both a determstc compoet ad a radom error compoet.. radom error Ths model hpotheszes a probablstc relatoshp betwee ad.. 6 8 6 Probablstc Model Geeral Form Frst-Order (Straght Le) Probablstc Model = Determstc compoet + Radom compoet where = Depedet varable = Idepedet varable where s the varable of terest. Assume that the mea value of the radom error s zero the mea value of, E(), equals the determstc compoet of the model = populato -tercept of the le the pot at whch the le tersects or cuts through the -as = populato slope of the le the amout of crease (or decrease) the determstc compoet of for ever -ut crease (or decrease). = radom error compoet 8
Frst-Order (Straght Le) Probablstc Model ad are populato parameters. The wll ol be kow f the populato of all (, ) measuremets are avalable. ad, alog wth a specfc value of the depedet varable determe the mea value of the depedet varable. 9 Model Developmet ad wll geerall be ukow. The process of developg a model, estmatg model parameters, ad usg the model ca be summarzed these -steps:. Hpothesze the determstc compoet of the model that relates the mea, E() to the depedet varable E. Use sample data to estmate ukow model parameters fd estmates: ˆ or b, ˆ or b Model Developmet (cotued). Specf the probablt dstrbuto of the radom error term ad estmate the SD of ths dstrbuto ~ N, wll revst ths later. Statstcall evaluate the usefuless of the model. Use model for predcto, estmato or other purposes Eample: Reacto tme versus drug percetage Subject Amout of Drug (%) Reacto Tme (secods)
Eample: Reacto tme versus drug percetage Eample: Reacto tme versus drug percetage Reacto Tme (sec.) -. -...... Reacto Tme (sec.) -. -...... Eample: Reacto tme versus drug percetage Errors of predcto---vertcal dffereces betwee the observed ad the predcted values of (-) = (-) = (-) = (-) = - (-) = Sum of errors = Sum of squared errors (SSE) = Least Squares Le Also called regresso le, or the least squares predcto equato Method to fd ths le s called the method of least squares For our eample, we have a sample of = pars of (, ) values. The ftted le that we wll calculate s wrtte as ŷ b b ŷ s a estmator of the mea value of, E ; ad are estmators of ad b b 6
Least Squares Le (cotued) Defe the sum of squares of the devatos of the values about ther predcted values for all data pots as: ˆ b b SSE We wat to fd b ad b to make the SSE a mmum---termed least squares estmates ŷ b b s called the least squares le Formulas for the Least Squares Estmates SS SD Slope: b or b r SS SD -tercept: s = SS b b b = sample sze s = SS 8 LS Calculatos for Drug/Reacto Eample 9 6 SS 6 8 b. b.. SS.. 9 LS Le for Drug/Reacto Eample Reacto Tme (sec.) -. -...... ŷ..
LS Calculatos for Drug/Reacto Eample ŷ.. ˆ ˆ.6.... (-.6) =. (-.) = -. (-.) = (-.) = -. (-.) =.6 Sum of errors =.6.9..9.6 Sum of squared errors (SSE) =. The LS le has a sum of errors =, but SSE =. <. for vsual model Least Squares Le Iterpretato of ŷ.. Estmated tercept s egatve that the estmated mea reacto tme s equal to -. secods whe the amout of drug s %. What does ths mea sce egatve reacto tmes are ot possble? Model parameters should be terpreted ol wth the sampled rage of the depedet varable. Least Squares Le Iterpretato of ŷ.. The slope of. mples that for ever ut crease of, the mea value of s estmated to crease b. uts. I the cotet of the problem: For ever % crease the amout of drug the bloodstream, the mea reacto tme s estmated to crease b. secods over the sampled rage of drug amouts from % to %. Coeffcet of Determato A measure of the cotrbuto of predctg Assumg that provdes o formato for the predcto of, the best predcto for the value of s Reacto Tme (sec.) -. -...... ˆ SS 6
Coeffcet of Determato (cotued) Reacto Tme (sec.) -. -...... ˆ SSE Coeffcet of Determato (cotued) SS ˆ SSE SS SSE SS SSE SS --total sample varato aroud mea --ueplaed sample varablt after fttg --eplaed sample varablt attrbutable to lear relatoshp eplaed total proporto of total sample varablt eplaed b the lear relatoshp 6 Coeffcet of Determato (cotued) } SS SSE SSE r Ueplaed SS SS varablt I smple lear regresso r s computed as the square of the correlato coeffcet, r r Iterpretato r. meas that the sum of squared devatos of the values about ther predcted values has bee reduced b % b the use ŷ, stead of, to predct of the least squares equato.