Lecture 15 Statistical Analysis in Biomaterials Research

Transcription

1 3.05/BE.340 Lecture 5 tatstcal Analyss n Bomaterals Research. Ratonale: Why s statstcal analyss needed n bomaterals research? Many error sources n measurements on lvng systems! Eamples of Measured Values and Error n Bomaterals Data Image removed due to copyrght consderatons. Image removed due to copyrght consderatons.

2 3.05/BE.340 Case Eample: cell sedmentaton assay of % of cells adherng to a bomateral vs. control (the smplest cell assay) ources of data varaton: Contamnaton of surface cytotocty or modfed surface chemstry Varatons n the number of cells seeded on each surface Varatons n bomateral synthess (reagent amounts, T, tme, etc.) Varatons n cells (dfferent cell passages) Varatons n meda (e.g., dfferent concentratons, protens agglomerated) Varatons n sterlzaton procedure Researcher error (e.g., sneezng on samples) Behavor s best characterzed by a populaton or dstbuton of values Case Eample: Ttraton of aoh soluton wth equmolar HCl soluton. Image removed due to copyrght consderatons.

3 3.05/BE Data converge to a populaton as : Image removed due to copyrght consderatons. Our Goal: to make enough measurements to accurately characterze the dstrbuton of behavors, va The mean of the data dstrbuton <> from measurements The wdth of the dstrbuton, or standard devaton, : ( < > ) ote that the standard devaton,, s dvded by - to avod basng, snce we do not know the true mean, µ.. Τhe unversal standard devaton, σ, s dvded by.

4 3.05/BE Equvalently the varance,, s defned by: ( ) Ellpsometer Measurements σ - +σ + mean Flm Thckness (Angstroms). Important Dstrbuton Functons A. Gaussan (or normal) dstrbuton descrbes processes domnated by dffusve forces Eamples: cell mgraton processes subject to random error or fluctuaton: (+ or equally lkely) Eample: cell number ppetted onto surface

5 3.05/BE For a Gaussan dstrbuton, the probablty P of measurng a value s gven by: P ( ) e σ ( µ ) σ π where µ s the true (unversal) mean of the dstrbuton and σ s the unversal standard devaton. 68% of values fall wthn µ±σ 95% of values fall wthn µ±σ P Real Datasets vs. Dstrbuton σ σ µ σ σ nvolve a fnte set of ponts approach the theoretcal dstrbuton only as must account for standard devaton of measured mean (<>) (<> µ as ) standard devaton of the mean, m (also known as the standard error): m 68% of measured <> values fall wthn µ± m 95% of measured <> values fall wthn µ± m Use as confdence nterval for datasets wth large

6 3.05/BE B. tudent s t Dstrbuton Approprate for small datasets ( < 30) (can be appled to any sze) amed for W.. Gossett, who publshed work on statstcs under pseudonym tudent n early 900s. P P0 t ( + ) / where P 0 s a constant chosen so that the area under the probablty curve (obtaned by ntegratng P) and t s the statstc: < > µ t Y ormal n n t Uses of the t Dstrbuton ) Calculate confdence nterval for a mean derved from small datasets: t < µ < + t + P m + P m Interval n over whch you are P% confdent of fndng the unversal mean µ.

7 3.05/BE t + P s the crtcal t value for a gven confdence level P (e.g., 90%, 95%, 99%) and a dataset wth degrees of freedom ν - t + P s obtaned from t p Dstrbuton Percentle chart (see handout) Case Eample: measurements of mgraton speed of cells are made. What s the 95% confdence nterval for the mean mgraton speed? measurement # mgraton speed (µm/mn) ν0 t.5(+.95) t (from table). Calculate <> 48.4

8 3.05/BE Calculate m ( ) (9, 000) (53) (0) 8. m 8./() The true mean (µ) mgraton speed s 95% lkely to be found n the nterval: t < µ < + t + P m + P m (5.46) < µ < (5.46) 36. < µ < 60.6 ) Evaluate whether sample populatons are statstcally dfferent For a two-sample t-test: t ' ( µ µ ) σ p + ' ' test that µ µ to some confdence level

9 3.05/BE where σ p standard devaton of pooled populaton ' ( < > ) + ( < ' > j) ( ) + ( ' ) ' σ p ( ) + ( ' ) + ' Case Eample: Does a surface modfcaton change the % cells adhered? fracton of cells adhered to control surface fracton of cells adhered to modfed surface Calculate <>, < >,, ' <> 0.3 < > 0.0 ( ) 8(0.436) (.855) 8(7) 0.045

10 3.05/BE ' ' ' '( ' ) 0(0.4406) (.097) 0(9) Calculate σ p σ p ' ( ) + ( ' ) + ' 7(0.045) + 9( ) Calculate two-sample t value for µ µ t ' ( µ µ ) σ p + ' ' t < > < ' > σ p + ' Test µ µ at 99% confdence level (P<0.0): t < t < t P for + P + µ µ

11 3.05/BE.340 ν + 6 t.5(+.99) t (from table) -.9 t.9 t3.86 does not fall wthn ths nterval. The means are statstcally dfferent wth P<0.0, so a change n % cells adhered was observed. Reject µ µ Area Reject µ µ t as tatstcal gnfcance In eperments nvolvng bologcal systems, P < 0.05 s generally accepted to ndcate measured change s not attrbutable to random error (stll may not be practcally meanngful!) "sgnfcant at the 5% level"

12 3.05/BE.340 C. F Dstrbuton Allows comparson of varablty of behavor between populatons usng test of hypothess: σ σ amed for Brtsh statstcan r Ronald A. Fsher. Defne a statstc: ν / where ν s degrees of freedom. χ ν σ then F χ χ v v / ν / ν For σ σ F ' F Procedure to test varablty hypothess:. Calculate and (wth ν Ν and ν -, respectvely). Compute F 3. Look n F-dstrbuton tables for crtcal F for ν, ν, and desred confdence level P F F F 4. For P < < + P σ σ

13 3.05/BE Case Eample: Measurements of C5a producton for blood eposure to an etracorpeal fltraton devce and tubng gave same means, but dfferent varabltes. Are the standard devatons dfferent wthn 95% confdence? Control (tubng only): 6 (µg/ml), ν 9 Fltraton devce: 3 (µg/ml), ν 7. Calculate and (provded). Compute F F 3/6.3 ' 3. Determne crtcal F values from F-dstrbuton chart ν 7 and ν 9 (m, n for use wth tables) P + P 0.05 F F For 0.07 F 4.0 σ σ F.3 falls wthn ths nterval. Conclude σ values for two systems are the same!

14 3.05/BE D. Other dstrbutons of nterest Radoactve decay Posson dstrbuton Only possble outcomes Bnomal dstrbuton 3. tandard devatons of computed values If quantty z of nterest s a functon of measured parameters z f(,y,...) What s z? We assume: z f (, y,...) Devatons (δz) of z from ts unversal value can be wrtten as: z z z y... + y + The standard devaton for z s calculated: z z z + y y +... Case Eample: We measure motlty (µ) and persstence (P) of a cell and want to know the standard devaton of the speed: P µ rearranges to µ P

15 3.05/BE P 3/ 0.5 µ P P 0.5 µ P µ µ P + µ P + µ P µ 4 P 4 µ 4. Least quares Analyss of Data (Lnear Regresson) Computng the straght lne that best fts data. uppose we have some measured data for bndng of a lgand to ts receptor: L + R C Lgand - Receptor Bndng Data /ν /[L] Ths equlbrum s descrbed by: K [C]/[L][R] ν fracton of occuped receptors [C]/([C] + [R]) K[L]/( + K[L]) /ν + /K[L]

16 3.05/BE 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. /ν 3.5 tatstcally 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 (, y ) from the lne s: y y lne y (m + b) Where m and b are the slope and ntercept of the lne. Our mnmzaton crteron can thus be wrtten: M [ y ( m + b)] mnmum Mathematcally we requre: M m M 0, 0 b

17 3.05/BE olvng these two equatons for the two unknowns (best ft m and b for the lne), we get: m ( y ) y ( ) b y m Quantfyng Error of the traght-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: r ( ) ( y y ) y y M For a perfect ft M0 r For r, <y> represents data as well as a lne

18 3.05/BE Many calculators, spreadsheets & other math tools are programmed to perform lnear least-squares fttng, as well as fts to more comple equatons followng a smlar premse. Many nonlnear equatons can be lnearzed by takng the log of both sdes e.g., m y b becomes ln y mln + ln b or y' m' + b' References ) D.C. Bard, Epermentaton: An Introducton to Measurement Theory and Eperment Desgn, nd Ed., Prentce Hall, Englewood Clffs, J (988). ) D.C. Montgomery, Desgn and Analyss of Eperments, 3 rd Ed., John Wley and ons, ew York, Y (99). 3) A. Goldsten, Bostatstcs: An Introductory Tet, MacMllan Co., ew York, Y (964). 4) C.I. Blss, tatstcs n Bology, Volume, McGraw-Hll, Inc. ew York, Y (970). 5) R.J. Larson and M.L. Mar, An Intro. to Mathematcal tatstcs and t Applcatons, nd ed., Prentce-Hall, Englewood, J (986).