2. statistics lod lecture

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Handling Data and Figures of Merit Data comes in different formats time Histograms Lists But…. Can contain the same information about quality What is meant by quality? (figures of merit) Precision, separation (selectivity), limits of detection, Linear range

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My weight Plot as a function of time data was acquired:

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Do not use curved lines to connect data points – that assumes you know more about the relationship of the data than you really do Comments: background is white (less ink); Font size is larger than Excel default (use 14 or 16)

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Bin refers to what groups of weight to cluster. Like A grade curve which lists number of students who got between 95 and 100 pts 95-100 would be a bin

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Assume my weight is a single, random, set of similar data Make a frequency chart (histogram) of the data Create a “model” of my weight and determine average Weight and how consistent my weight is

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= measure of the consistency, or similarity, of weights average 143.11 s = 1.4 lbs Inflection pt s = standard deviation

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Characteristics of the Model Population (Random, Normal) Peak height, A Peak location (mean or average), m Peak width, W, at baseline Peak width at half height, W 1/2 Standard deviation, s, estimates the variation in an infinite population, s Related concepts

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Width is measured At inflection point = s W 1/2 Triangulated peak: Base width is 2s < W < 4s

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+/- 1s Area +/- 2s = 95.4% Area +/- 3s = 99.74 % Pp = peak to peak – or – largest separation of measurements Peak to peak is sometimes Easier to “see” on the data vs time plot Area = 68.3%

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Peak to peak 139.5 144.9 s~ pp/6 = (144.9-139.5)/6~0.9 (Calculated s= 1.4)

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Scale up the first derivative and second derivative to see better There are some other important characteristics of a normal (random) population 1 st derivative 2 nd derivative

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Population, 0 th derivative 1 st derivative, Peak is at the inflection Determines the std. dev. 2 nd derivative Peak is at the inflection Of first derivative – should Be symmetrical for normal Population; goes to zero at Std. dev.

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Asymmetry can be determined from principle component analysis A. F. ( ≠Alanah Fitch) = asymmetric factor

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Is there a difference between my “baseline” weight and school weight? Can you “detect” a difference? Can you “quantitate” a difference? Comparing TWO populations of measurements

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Exact same information displayed differently, but now we divide The data into different measurement populations baseline school Model of the data as two normal populations

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Average Baseline weight Average school weight Standard deviation Of baseline weight Standard deviation Of the school weight

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We have two models to describe the population of measurements Of my weight. In one we assume that all measurements fall into a single population. In the second we assume that the measurements Have sampled two different populations. Which is the better model? How to we quantify “better”?

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Compare how close The measured data Fits the model Did I gain weight? The red bars represent the difference Between the two population model and The data The purple lines represent The difference between The single population Model and the data Which model Has less summed differences?

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This process (summing of the squares of the differences) Is essentially what occurs in an ANOVA An alysis o f va riance Normally sum the square of the difference in order to account for Both positive and negative differences. In the bad old days you had to work out all the sums of squares. In the good new days you can ask Excel program to do it for you.

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Test: is F<F critical ? If true = hypothesis true, single population if false = hypothesis false, can not be explained by a single population at the 5% certainty level

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In an Analysis of Variance you test the hypothesis that the sample is Best described as a single population. Create the expected frequency (Gaussian from normal error curve) Measure the deviation between the histogram point and the expected frequency Square to remove signs SS = sum squares Compare to expected SS which scales with population size If larger than expected then can not explain deviations assuming a single population

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The square differences For an assumption of A single population Is larger than for The assumption of Two individual populations

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There are other measurements which describe the two populations Resolution of two peaks Mean or average Baseline width

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x a x b In this example Peaks are baseline resolved when R > 1

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x a x b In this example Peaks are just baseline resolved when R = 1

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x a x b In this example Peaks are not baseline resolved when R < 1

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2008 Data What is the R for this data?

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Visually less resolved Visually better resolved Anonymous 2009 student analysis of Needleman data

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Visually less resolved Visually better resolved Anonymous 2009 student analysis of Needleman data

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Other measures of the quality of separation of the Peaks Limit of detection Limit of quantification Signal to noise (S/N)

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X blank X limit of detection 99.74% Of the observations Of the blank will lie below the mean of the First detectable signal (LOD)

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Two peaks are visible when all the data is summed together

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Estimate the LOD (signal) of this data

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Other measures of the quality of separation of the Peaks Limit of detection Limit of quantification Signal to noise (S/N)

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Your book suggests 10 Limit of quantification requires absolute Certainty that no blank is part of the measurement

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Estimate the LOQ (signal) of this data

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Other measures of the quality of separation of the Peaks Limit of detection Limit of quantification Signal to noise (S/N) S ignal = x sample - x blank N oise = N = standard deviation, s

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Estimate the S/N of this data (This assumes pp school ~ pp baseline)

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Can you “tell” where the switch between Red and white potatoes begins? What is the signal (length of white)? What is the background (length of red)? What is the S/N ?

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Effect of sample size on the measurement

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Error curve Peak height grows with # of measurements. + - 1 s always has same proportion of total number of measurements However, the actual value of s decreases as population grows

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2008 Data

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Calibration Curve A calibration curve is based on a selected measurement as linear In response to the concentration of the analyte. Or… a prediction of measurement due to some change Can we predict my weight change if I had spent a longer time on Vacation?

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5 days The calibration curve contains information about the sampling Of the population

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Can get this by using “trend line”

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This is just a trendline From “format” data Using the analysis Data pack Get an error Associated with The intercept

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In the best of all worlds you should have a series of blanks That determine you’re the “noise” associated with the background Sometimes you forget, so to fall back and punt, estimate The standard deviation of the “blank” from the linear regression But remember, in doing this you are acknowledging A failure to plan ahead in your analysis

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Extrapolation of the associated error Can be obtained from the Linear Regression data Sensitivity (slope) The concentration LOD depends on BOTH Stdev of blank and sensitivity Signal LOD !!Note!! Signal LOD ≠ Conc LOD We want Conc. LOD

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Difference in slope is one measure selectivity In a perfect method the sensing device would have zero Slope for the interfering species Selectivity Pb 2+ H +

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Limit of linearity 5% deviation

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Summary: Figures of Merit Thus far R = resolution S/N LOD = both signal and concentration LOQ LOL Sensitivity (calibration curve slope) Selectivity (essentially difference in slopes) Can be expressed in terms of signal, but better Expression is in terms of concentration Tests: Anova Why is the limit of detection important? Why has the limit of detection changed so much in the Last 20 years?

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The End

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Which of these two data sets would be likely To have better numerical value for the Ability to distinguish between two different Populations? Needleman’s data

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2008 Data Height for normalized Bell curve <1 Which population is more variable? How can you tell?

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Increasing the sample size decreases the std dev and increases separation Of the populations, notice that the means also change, will do so until We have a reasonable sample of the population

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