Lecture 16 Statistical Analysis in Biomaterials Research (Part II)
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1 3.051J/0.340J 1 Lecture 16 Statstcal Analyss n Bomaterals Research (Part II) C. F Dstrbuton Allows comparson of varablty of behavor between populatons usng test of hypothess: σ x = σ x amed for Brtsh statstcan Sr Ronald A. Fsher. Defne a statstc: χ =ν 1 S /σ ν 1 where ν 1 s degrees of freedom. χ v 1 /ν then F = 1 χ /ν v For σ x = σ x S x F = S x ' F Procedure to test varablty hypothess: 1. Calculate S x and S x (wth ν 1 =Ν 1 and ν = -1, respectvely). Compute F 3. Look n F-dstrbuton tables for crtcal F for ν 1, ν, and desred confdence level P 4. For F 1 P < F < F 1+ P σx = σ x
2 3.051J/0.340J Case Example: Measurements of C5a producton for blood exposure to an extracorpeal fltraton devce and tubng gave same means, but dfferent varabltes. Are the standard devatons dfferent wthn 95% confdence? Control (tubng only): S x = 6 (µg/ml), ν =9 Fltraton devce: S x = 3 (µg/ml), ν 1 = 7 1. Calculate S x and S x (provded). Compute F S F = x S = 3/6 = 1.31 x ' 3. Determne crtcal F values from F-dstrbuton chart ν 1 =7 and ν =9 (m, n for use wth tables) 1 P = 0.05 F 0.05 = P = F = 4.0 For 0.07 F 4.0 σ x = σ x F=1.31 falls wthn ths nterval. Conclude σ values for two systems are the same!
3 3.051J/0.340J 3 D. Other dstrbutons of nterest Radoactve decay Posson dstrbuton Only possble outcomes Bnomal dstrbuton 3. Standard devatons of computed values If quantty z of nterest s a functon of measured parameters z = f(x,y,...) What s S z? We assume: z = f ( x, y,...) Devatons (δz) of z from ts unversal value can be wrtten as: z z z = x + y +... x y The standard devaton for z s calculated: S z = z z x S x + S y +... y Case Example: We measure motlty (µ) and persstence (P) of a cell and want to know the standard devaton of the speed: S P µ = rearranges to S = µ P
4 3.051J/0.340J 4 S = 0.5 µ P 3/ = S S S = 0.5 = P P µ P µ µ S S = S S S S S µ == P S P + µ 4 P S P + S µ 4 µ 4. Least Squares Analyss of Data (Lnear Regresson) Computng the straght lne that best fts data. Suppose we have some measured data for bndng of a lgand to ts receptor: L + R = C Lgand - Receptor Bndng Data 1/ν /[L] Ths equlbrum s descrbed by: K = [C]/[L][R] ν = fracton of occuped receptors = [C]/([C] + [R]) = K[L]/(1 + K[L]) 1/ν = 1 + 1/K[L]
5 3.051J/0.340J 5 Queston: How can we numercally obtan the lnear equaton that best represents the data? Answer: Mnmze the squared devaton of the lne from each pont. OTE: Ths s a generc tool n data regresson, ndependent of the fttng functon. 1/ν 3.5 Statstcally Best Ft Lne Mnmzes: (A-B) + (C-D) + (E-F) + (G-H) E G H C F A D B /[L] The devaton of any gven measured pont (x, y ) from the lne s: y y lne = y (mx + b) Where m and b are the slope and ntercept of the lne. Our mnmzaton crteron can thus be wrtten: M= [y (mx +b) = mnmum ] =1 Mathematcally we requre: M m M = 0, =0 b
6 3.051J/0.340J 6 Solvng these two equatons for the two unknowns (best ft m and b for the lne), we get: (x y ) y =1 =1 =1 m = b = x y m x x ( x ) =1 =1 =1 =1 Quantfyng Error of the Straght-Lne Ft If the error on each y s unknown (e.g., a sngle measurement was made): The standard devaton for the regresson lne s gven by: Ths assumes: σ = M - n denomnator snce degrees of freedom are taken n calculatng m and b (two ponts make a lne, so σ for = s meanngless.) - a normal dstrbuton of data ponts about the lne - spread of ponts s of smlar magntude for full data range Goodness of ft can be further characterzed by the correlaton coeffcent, r (or coeffcent of determnaton, r ), calculated as: ( y y ) M r = =1 For a perfect ft ( y y ) M=0 r =1 =1 For r =1, <y> represents data as well as a lne
7 3.051J/0.340J 7 Many calculators, spreadsheets & other math tools are programmed to perform lnear least-squares fttng, as well as fts to more complex equatons followng a smlar premse. Many nonlnear equatons can be lnearzed by takng the log of both sdes e.g., m y = bx becomes ln y = mln x + ln b or y ' = mx '+ b ' Multple regresson In some cases, we wsh to ft data dependent on more than one ndependent varable. The procedure wll be exactly analogous to that used above, and solutons can be obtaned through matrx algebra. Here we wll consder the smple case of a lnear dependence on ndependent varables. Our mnmzaton crteron can thus be wrtten: M = [ y (a+ bx + cz )] = mnmum =1 Mathematcally we requre: M M M = 0, = 0, = 0 a b c whch yelds the 3 equatons:
8 3.051J/0.340J 8 a + x b + z c = y =1 =1 =1 x a + x b + (x z ) c = (x y ) =1 =1 =1 =1 z a + x z b + ( ) c = z (z y ) =1 =1 =1 =1 These equatons can be solved to obtan a, b and c. References 1) D.C. Bard, Expermentaton: An Introducton to Measurement Theory and Experment Desgn, nd Ed., Prentce Hall, Englewood Clffs, J (1988). ) D.C. Montgomery, Desgn and Analyss of Experments, 3 rd Ed., John Wley and Sons, ew York, Y (1991). 3) A. Goldsten, Bostatstcs: An Introductory Text, MacMllan Co., ew York, Y (1964). 4) C.I. Blss, Statstcs n Bology, Volume, McGraw-Hll, Inc. ew York, Y (1970). 5) R.J. Larson and M.L. Marx, An Intro. to Mathematcal Statstcs and t Applcatons, nd ed., Prentce-Hall, Englewood, J (1986). 6) A.C. Bajpa, I.M. Calus and J.A. Farley, Statstcal Methods for Engneers and Scentsts: A Students Course Book, John Wley and Sons, Chchester, Great Brtan (1978).
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