Rich�Ulrich�<[EMAIL PROTECTED]>�wrote�on�6/13/03�4:14:55�PM: > >>�The�F-test�can�answer�the�question�"Are�these�two�models significantly >>�different�at�the�X%�level?". > >This�bothers�me�-�that's�not�the�way�that�I�would�describe >the�question.��Down�below:��Clearly�the�two�models�are�*different*, >anyway,�by�a�factor�of�1/T.����Does�one�have�a�lower�residual? > They�will�almost�certainly�have�different�residuals�for�a�particular set�of�data.�The�F-test�quantifies�the�confidence�that�two�residuals are�indeed�different. >You�can�solve�by�least-squares,�comparing�predicted�to� >observed,�if�the�errors�of�prediction�are�of�similar�size� >across�the�range.��But�that's�the�non-assumption,�for� >non-linear�regression,��isn't�it? I�purposely�avoided�the�issue�of�what�to�do�if�the�errors�vary�across the�range�of�variables. >>� >>�One�model�can�be�constant;�i.e.�assume�that�data�scatter�is entirely >>�random�about�the�dependent�variable�mean�&�not�dependent�on�the >>�independent�variable(s)�at�all. >>� >>�Or�one�can�see�if�adding�another�parameter�to�a�fitting�function makes >>�a�significant�difference. >>� > >Nested�models;�assessed�by�reduction�of�residuals >of�least�squares,�say,�or�by�increase�of�Likelihood. > >>�Or�one�can�find�out�which�of�two�arbitrary�models�best�fits�the data; >>�this�is�very�useful�for�comparing�two�theories,� nb.�even�if�the�two�models�are�related�such�that�one�has�one�extra parameter. > >AIC��and�BIC��are�keywords�for�looking�up�comparisons >of�non-nested�models. > I�will�follow�up�on�AIC�and�BIC.�Thanks. A�simple�decrease�in�residuals�can�be�purely�by�chance.�
It�is�not�necessary�to�use�nested�model�logic for the F-test to be applicable. >>� >>�As�an�example,�alternate�theories�for�chemical�diffusivity�D�are: >>� >>�(1)�D=(K)*exp(-Q/T) >>�(2)�D=(K/T)*exp(-Q/T)� >>�where�K�and�Q�are�experimentally�determined�constants�and�T�is >>�temperature. >>� >>�Given�a�set�of�(D,T)�data�the�F-test�can�be�used�to�see�if�there�is a >>�significant�difference�between�these�two�models. >>� > >The�F-test�is�used,�by�theory�and�by�custom,�to�test >models�that�are�*nested*,�using�the�difference�in�d.f.��as� >the�numerator�degrees�of�freedom.��Here,�that�d.f.��seems >to�be�zero....�� >I'll�try�to�find�something�in�this�library�book�I�have�on�the�topic. > It�is�true�that�the�F-test�is�often�used�to�tell�if�the�change�in residual�due�to�an�additional�parameter�is�significant.� However,�by�its�defiinition,�the�F-test�can�be�applied�to�test�the ratio�of�residuals�with�the�same�or�different� degrees�of�freedom.�There�is�no�reason�these�residuals�must�come�from functionally�nested�or�even�related� models. For�example�one�could�use�the�F-test�with�some�velocity/distance measurements�to�see�if�velocity�varies� exponentially�or�linearly�with�distance�for�a�falling�body. >This�does�seem�to�be�a�curious�example.�� >I�think�I�would�show�folks��the�F-test,�assuming�one� >d.f.,��as�a�'demonstration'�of�the�size�of�the�difference.�� > >But�these�models�are�surely��*different*��in�a�way >that�seems�pretty�strong.���I�guess,�I�am�accustomed� >to�worrying�more�about�whether�a�variable�is�*in*��a >model�at�all,�instead�of�worrying�about�what�form�it�takes. This�is�a�real�example�drawn�from�materials�science/physical chemistry.�The�two�models�have�the�same�degrees� of�freedom�(N-2)�that's�why�I�chose�to�use�it�as�the�example. I've�come�across�cases�in�my�research�where�one�really�couldn't distinguish�between�these�two�models�with�a� particular�set�of�experimental�data. >Biomedical�data�with�subjective�reports�is�usually�not� >so�definitive,�not�so�well-measured�as�to�select�between >models�like�that.� That's�why�it�is�important�to�apply�such�a�test.�Many�times�the impression�is�given�that�an�experiment�is�consistent� with�model�"A"��to�the�actual�or�implied�exclusion�of�model�"B" without�an�appropriate�confidence�being�stated. I'd�say�that�one�should�use�an�F-test�to�be�sure�the�models�differ. PS.�The�models�I�presented�can�be�linearized�by�taking�logs.�Whether one�should�do�so�depends�on�how�the� measurement�errors�depend�on�variable�magnitude.�If�errors�vary�in proportion�to�magnitude�one�is�best�off�fitting� data�that's�been�linearized�by�taking�logs.��However,�numeric experiments�I've�done�imply�this�isn't�as�big�a�factor� as�one�might�think. . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
