MIT 3 051J - Statistical Analysis in Biomaterials Research

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1 3.051J/20.340J Lecture 15 Statistical Analysis in Biomaterials Research 1. Rationale: Why is statistical analysis needed in biomaterials research?  Many error sources in measurements on living systems! Examples of Measured Values and Error in Biomaterials Data Graphs removed for copyright reasons.2 3.051J/20.340J Case Example: cell sedimentation assay of % of cells adhering to a biomaterial vs. control (the simplest cell assay) Sources of data variation: ¾ Contamination of surface ⇒ cytotoxicity or modified surface chemistry ¾ Variations in the number of cells seeded on each surface ¾ Variations in biomaterial synthesis (reagent amounts, T, time, etc.) ¾ Variations in cells (different cell passages) ¾ Variations in media (e.g., different concentrations, proteins agglomerated) ¾ Variations in sterilization procedure ¾ Researcher error (e.g., sneezing on samples)  Behavior is best characterized by a population or distibution of values Case Example: Titration of HCl solution with equimolar NaOH solution. Graphs removed for copyright reasons.3 3.051J/20.340J Data converge to a population as N → ∞: Graph removed for copyright reasons.  Our Goal: to make enough measurements to accurately characterize the distribution of behaviors, via ¾ The mean of the data distribution = <x> from N measurements N ∑ xi x = i=1 N ¾ The width of the distribution, or standard deviation, S: N 2∑(xi −< x >) Note that the standard deviation, S, is i=1 divided by N-1 to avoid biasing, sinceS = we do not know the “true” mean, N −1 µ.. Τhe universal standard deviation, σ, is divided by N.+σ−σ4 3.051J/20.340J Equivalently the variance, S2, is defined by: N N ∑xi 2 ⎛ N ⎞2 −⎜∑xi ⎟ S 2 = i=1 ⎝ i=1 ⎠ ( NN −1) Ellipsometer Measurements 1670 1680 1690 1700 1710 1720 17300 0.5 1 1.5 2 2.5 3 3.5 mean -S +S Film Thickness (Angstroms) 2. Important Distribution Functions A. Gaussian (or normal) distribution ¾ describes processes dominated by diffusive forces Examples: cell migration ¾ processes subject to random error or fluctuation: (+ or – equally likely) Example: cell number seeded onto surface5 3.051J/20.340J For a Gaussian distribution, the probability P of measuring a value x is given by: ( x−µ)2 − e 2σ2 () = Px σ 2π where µ is the true (universal) mean of the distribution and σis the universal standard deviation. 68% of values fall within µ±σ P95% of values fall within µ±2σ Real Datasets vs. Distribution 2σ 1σ µ 1σ 2σ ¾ involve a finite set of points ¾ approach the theoretical distribution only as N→ ∞ ¾ must account for standard deviation of measured mean (<x>) (<x> → µ as N → ∞) ⇒ standard deviation of the mean, Sm (also known as the standard error): S = S 68% of measured <x> values m N fall within µ±Sm Use as “confidence95% of measured <x> values interval” for datasetsfall within µ±2Sm with large N x6 3.051J/20.340J B. Student’s t Distribution Named for W.S. Gossett, who published work on statistics under¾ Appropriate for small datasets (N < 30) pseudonym “Student” in early 1900s.(can be applied to any size) P0P = t2 N/2(1 + )N−1 where P0is a constant chosen so that the area under the probability curve ≡ 1 (obtained by integrating P) and t is the statistic: < x>−µt = N −1S Y NormalNormalNormalNormalNormalNormalNormalNormaln = 4 n = 1n = 4n = 1n = 4n = 1n = 4n = 1n = 4n = 1n = 4n = 1n = 4n = 1n = 4n = 10.40.40.0.30.0.20.0.1tt-4 -3 -2 -1 01 2 3 44¾ Uses of the t Distribution i) Calculate confidence interval for a mean derived from small datasets: ⎛ ⎞ ⎛ ⎞ ⎜ x t S− 1+ Pm⎟ <µ< ⎜ x t S+ 1+ Pm⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ Interval in x over which you are P% confident of finding the “universal” mean µ. Student's t distribution in various values of ν.Figure by MIT OCW.73.051J/20.340J t1+P is the critical t value for a given confidence level P (e.g., 90%, 95%, 99%) 2 t and a dataset with degrees of freedom ν = N-1 1+P is obtained from tp Distribution Percentile chart (see handout)2 Case Example: 11 measurements of migration speed of cells are made. What is the 95% confidence interval for the mean migration speed? measurement # migration speed (µm/min) 1 62 2 52 3 68 4 23 5 34 6 45 7 27 8 42 9 83 10 56 11 40 tν=10 .5(1+.95)= t.975= 2.23 (from table) 1. Calculate <x> N x = ∑xi i=1 = 48.4N8 3.051J/20.340J 2. Calculate Sm ⎞∑ −⎜⎛∑S = i=1 ⎝ i=1 ⎠⎟ 11(29, 000) − (532)2 NN−1) = 11(10) = 18.1( SS =m N = 18.1/(11)0.5 = 5.46 The true mean (µ) migration speed is 95% likely to be found in the interval: ⎛ ⎞ ⎛ ⎞ ⎜ 22 N N i ixN xx tS − 1+ Pm⎟ <µ< ⎜ x tS + 1+ Pm⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ 48.4-2.23(5.46) < µ < 48.4+2.23(5.46) 36.2 < µ < 60.6 ii) Evaluate whether sample populations are statistically different For a two-sample t-test: x − x ' − (µ −µ )x x' t = 1 1 test that µx ≠ µx’ to σ+ some confidence level p NN '9 3.051J/20.340J where σp = standard deviation of pooled population N N ' ∑(<>− ' >− 2xx )2 +∑(< x xj ') (NS N S i 2 σ2 = i=1 j=1 =−1) 2 + ( '−1)x x' p 1)(N −+(N '−1) N + N '− 2 Case Example: Does a surface modification change the % cells adhered? fraction of cells adhered fraction of cells adhered to control surface to modified surface 0.225 0.209 0.262 0.205 0.217 0.196 0.240 0.210 0.230 0.202 0.229 0.207 0.235 0.224 0.217 0.223 0.220 0.201 1. Calculate <x>, <x’>, Sx, Sx’ NN ∑xi ∑xi ' <x> = i=1 = 0.232 <x’> = i=1 = 0.210N N10 x 3.051J/20.340J N N 2 ∑⎞Nx i 2 −⎜⎛∑xi⎟ 8(0.4316) − (1.8552)2 S= i=1 ⎝i=1 ⎠ = NN−1) 8(7) = 0.0145( N N 2 ∑⎞Nxi '2 −⎜⎛∑xi ' ⎟ 10(0.4406) − (2.097)2 S = i=1 ⎝i=1 ⎠ x' '( = NN'−1) 10(9) = 0.00977 2. Calculate σp 2(NS N S 2 7(0.0145)2 + 9(0.00977)−1) 2 + ( '−1)x x'σ= = = 0.012p NN'2 810 − 2+ −+ 3. Calculate two-sample t value for µx = µx’ ' ' ( ) 1 1 ' x x p t N N µ µ σ − − − = + xx <>−< x ' >xt = = 0.232 − 0.210 = 3.86 σ 1 +11 +10.012pN N ' 810 4. Test µx ≠ µx’ at 99% confidence level (P<0.01):11 3.051J/20.340J tt−1+P << t1+P ⇒ µx = µx’for 2 2 tν = N + N’ – 2 = 16 .5(1+.99)= t.995= 2.92 (from table) -2.92 ≤t ≤ 2.92 t=3.86 does not fall within this interval. The means are statistically different with P<0.01, so a change in % cells adhered was observed. Reject µx = µx’ Area = 0.005 Reject µx = µx’t axis


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