Design of Experiments and Data Analysis

1. Design of Experiments and Data Analysis

2. Let’s Work an Example • Data obtained from MS Thesis • Studied the “bioavailability” of metals in sediment cores • We’ll analyze chromium data

3. Pt. Mugu Marsh

5. Analytical Techniques • Sediment samples were taken with cores • Sliced into 1 cm slices • Sediment in each slice was extracted using a strong acid • Extracts were analyzed using an Inductively Coupled Plasma Mass Spectrometer (ICP-MS) • Calibrations were also conducted • Surfaces areas (SA) and organic carbon (OC) contents of sediment in each slice were also measured

6. Core processing

7. Objectives • To determine if there is a correlation between sediment surface area and organic carbon content • To determine if there is a relationship between concentration of a specific metal and sediment SA and/or OC • To determine if there is a relationship between or among metal concentrations

8. Example of Results

9. Example of Results

10. Data File • Create a folder entitled “REU” in the C:\My Documents folder • Create a folder entitled “2006” in this REU folder • Create a folder entitled “Data Analysis Workshop” in this 2006 folder • Download Excel File REU_dataanalysis_data.xls from instructional1.calstatela.edu/ckhachi into the Data Analysis Workshop folder • Open the file

11. Data File Structure • There should be 2 worksheets in the workbook: • Data: raw SA, OC, and metals concentration data • Calibration Curves: ICP-MS calibration data (relating raw metals concentrations to known calibration concentrations) • Data for the cores are separated by yellow bands

12. Data File Structure • Data Columns include: • ID: Random sample ID • Ave Depth: Ave depth of each slice • Solid Mass: Mass of sediments in each slice • Raw ICP-MS data for each of five metals • Calibration Columns include: • Conc: Concentration of standards in parts per billion (ppb) • ICP-MS responses for the 5 metals

13. Let’s Start with Calibration Curves • Most instruments over reasonable ranges have linear responses (i.e., calibration curves are straight lines) • We need to “model” the data – regression analysis to determine the best-fit line that relates ICP-MS response to concentrations • We will then use these calibration equations to calculate concentrations for our samples • Note: because we know that calibrations are usually linear, we will choose a linear regression model…if you don’t know the relationship b/w 2 variables, it sometimes helps to start with plots

14. Calibration Curve for Cr • Linear response • We know slope and intercept • R2 value provided • Best-fit line drawn (looks good to me) • Not enough statistical information provided to be able to conduct proper error analysis

15. Regression Analysis for Cr Rename Worksheet “Cr Analysis”

16. Assumptions • On average, errors are not consistently positive nor negative. • Linear Model: yi = mx + b + ei, where ei is the error associated with each observation • Line goes through the middle of data • Variance of error terms the same across all observations • Data are independent of each other • Error terms are normally distributed (not that important)

17. Residual Plot Look at data and linear fit carefully; points lie above the line for smaller values of concentration. If you delete the last point, you get a very different result

18. Regression Statistics • Multiple R (or just r) is the correlation: • +1  perfectly positively correlated (as x goes up, so does y) • 0  not correlated • -1  perfectly negatively correlated (as x goes up, y goes down)

19. Regression Statistics • R Square (R2): coefficient of determination • Between 0 and 1 • 0  no linear relationship • 1  perfect linear relationship (+ or -) • Square of the r value • Theoretically, as the number of data points  ∞, R2  1 (denominator is fixed) • Adjusted R Square: fixes this problem…is probably a better measure of how strong the linear relationship is (R2 more common) • Use 2 or 3 significant figures to report these #s

20. Regression Statistics • Standard Error: a measure of the amount of error in the prediction of y for an individual x. • Observations: # of data points

21. ANOVA • ANalysis Of VAriance (sometimes called an F test) • df: degrees of freedom • SS: sum of squares R2 = (1-SSresidual)/SStotal • MS: Mean squares = SS/df • F = MSregression/MSresidual larger  reject null hypothesis (no correlation) • Not very useful for single treatment

22. Correlation results • Linear Calibration: y = mx + b • Slope (m) = 259.0709 • Intercept (b) = 1787.2679 • Standard Error: used for hypothesis testing and confidence band formation

23. Correlation results • Confidence intervals • Intercept • Lower: 1787.2679 – 70.2724 (2.571) = 1606.597 • 2.571  standard two-tale t-test table with df = 5 and probability = 0.05 • Slope • Lower: 259.079 – 3.6280(2.571) = 249.74 • Upper: 259.079 + 3.6280(2.571) = 268.40 • t stat: = Coefficient/Standard Error

24. Correlation results • P-value: probability of wrongly rejecting the null hypothesis (Ho), in this case  no correlation, if it is in fact true • p > 0.10  null hypothesis maybe OK • 0.10 < p < 0.05  slight evidence against null hypothesis • p < 0.05  moderate evidence against null hypothesis • p < 0.01  strong evidence against null hypothesis

25. Consult statistical tables again: For df = 5 and t stat = 25.4, p < 0.000005 For df = 5 and t stat = 71.4, p < 0.0000001 Very, very strong evidence that Ho is false  the calibration curves are linear! Linear Model: Correlation results

26. Using Calibration Equations • Now we have an equation that relates the response of our equipment to concentrations • Let’s use this equation to determine concentrations in our samples

27. Raw Data Excel Sheet

28. Measurement Errors • Add 2 columns to the right of the Cr data • Assume instrument has a 3% error (in reality, you need to run sample 3 times to get the proper error)

29. Propagation of Errors • Let us assume that X is dependent upon the experimental variables p, q, and r, which fluctuate in a random and independent way. • Addition or Subtraction: X = p + q - r: • Where “s” is the standard deviation or error for each of the variables

30. Propagation of Errors (cont’d) • Multiplication or Division: X = p * (q/r) • Other equations exist for logs, etc. • Round +/- to the # of decimal places of the component number with the fewest number of decimal places • Round x/÷ to the number of significant digits of the component number with the fewest significant digits.

31. Let’s use the Calibration Eqn • Response  detector output • Concentration  what we are looking for in the column labeled “Cr Conc (ppb)”

32. Let’s use the Calibration Eqn • Let’s look at the first line: • Rearrange to solve for Conc: • Let’s look at the numerator

33. Let’s use the Calibration Eqn • Num = 8156.35-1787.27 = 6369.08 • Error in Num: • Recall for +/-: • Error in Conc = • So now:

34. Let’s use the Calibration Eqn • Conc = • Recall, for x/÷: or • So, ErrConc = • Final result  Conc = 24.58 ± 1.04

35. Final Results • Use error bars in the plots

36. Plotting Error Bars • Error bars can be: • 1-3 standard deviation(s) • Standard error • etc… • Just be clear in your figure caption what your error bar represents

37. Next Presentation • A little about design of experiments • A little more about errors, hypothesis testing, etc…