1. ChE 551 Lecture 04 Statistical Tests Of Rate Equations 1

2. Last Time Considered Paramecium Example 2 Errorr2r2 Lineweaver Burke 94540.910 Eadie Hofstee56470.344 Nonlinear Least Squares 49190.905 r 2 does not indicate goodness of fit

3. Today: Statistical Analysis Of Rate Data Can we do a calculation to tell if one model fits the data better than another model? Is the result statistically significant? 3

4. Method: Calculate A Variance Usually model with the lowest variance works best! 4 (3.B.1) substituting in equations (3.A.7) yields (3.B.2)

5. Limitations Of Using Variance To Assess Which Model Fits Best Assumes error in data Follows a “  2 distribution” (i.e. error is random) Usually good assumption in direct rate data It is not good to assume 1/rate follows  2 distribution, so one needs to be careful about linearizing data. 5

6. For Our Example 6 (3.B.3)

7. For Our Example Continued 7 (3.B.4) (3.B.5) Eadie-Hofstee: while for the Lineweaver-Burk Plot: The non-linear least squares fit the data best.

8. Next: Using An F-Test To Tell If the Difference Is Statistically Significant 8 Method: Compute F inverse, given by (3.B.6) If F inverse is large enough, the model is statistically better.

9. Statistics: Gives A Value of F inverse That Is “Large Enough” 9 Table 3.B.2 Values of F inverse as a function of nf when both models have the same value of nf nf =Significance Level 90%95%99%99.5% 139.86161.5405216212 29.01999199 35.399.2829.4647 44.116.3915.9823 nf=number of data points - parameters in the model (3.B.8) To read the table, if nf=4, you need F inverse to be at least 15.98 to be 99% sure that the better model really is better. There will still be 1% chance that the differences caused by random errors in the data

10. Assumptions In Using the Values Of F In Table 3.B.2 Models are independent (non-nested)  2 distribution of errors Not mathematically rigorous in our example since models not independent! (Gives small error in F inverse ) 10

11. F dist Gives The Probability That A Given Model Is Better % confidence=1-FDIST (F inverse, nf for better model, nf for worse model) (3.B.9) Not mathematically rigorous, but close. 11

12. Example: Is The Non-Linear Least Squares Better Than LineWeaver-Burke Variance Lineweaver-Burke = 321 Variance non-linear = 185 nf=30 12 I used Excel to calculate 1-FDIST (1.92, 30, 30)=0.96 96% sure non-linear least squares fits better 4% chance difference due to noise in data.

13. Another Example: Comparing Two Models Previously fit data to 13 (3.A.1) Does the following work better? Is the difference statistically significant?

14. The Spreadsheet Is The Same As In Problem 3.A: 14

15. F Test To Determine Which Model Is Better V 3.A.1 the variance of equation 3.A.1 is 15 V 3.C.1 the variance of equation 3.C.1 is The ratio of variance is

16. Calculate Probability Second Model Is Better From FDIST probability=1-FDIST (1.07,30,30)=0.58. 58% chance second model is better 42% probability first model is better Note: Not rigorous number 16

17. Next: Multiple Variable Analysis Rates of reaction usually strongly effected by many variables Temperature: concentration, solvents, inpurities, catalysts, …… So far only consider one variable: Concentration 17

18. Example: Develop A Rate Equation For The Growth Of Grass Variables Sunlight Rain Amount of grass seed Number of birds and insects Fertilizer Soil type Soil bacteria How do we proceed to measure a rate? 18

19. Usual Technique: Initial Rate Method 19 Start with multiple parallel reactors Fill each with a different concentration Let reaction go & measure conversion vs time Get rate from slope extrapolated to zero

20. If We Have Several Variables, What Do We Measure? General approach Take some preliminary data to determine what variables are important Usually requires multiple iterations Take more detailed measurements on the variables that are most important 20

21. Design Of Experiments To Determine Which Variables Are Important 2 n designs Pick two values of each of the variables Look at two possibilities for each variable Do experiments for all combinations Do analysis to decide which variables are important 21

22. Example: How Does Temperature And Concentration Affect Selectivity Of A Reaction Pick two values of each variable Temperature + = higher temperature Temperature - = lower temperature Concentration + = higher concentration Concentration - = lower concentration Look at all possibilities 22

23. Table of All Possibilities Run #TCResult 1++30% 2+-40% 3-+60% 4--50% 23

24. How Do We Analyze The Data? Look at the deviation from the mean Calculate row averages 24

25. For Our Example, Mean=45% Run #TCDeviation 1++-15% 2+--5% 3-++15% 4--+5% 25

26. Calculate Row Averages Run #TCDeviation 1++-15% 2+--5% 3-++15% 4--+5% =+(-15%) +(-5%) -(+15%) -(+5%) =-40% =+(-15%) -(-5%) +(+15%) -(+5%) =0% +5% 26

27. First Conclusion Want temperature to be low Cannot tell about concentration 27

28. Calculate Row Averages Run #TCDeviation 1++-15% 2+--5% 3-++15% 4--+5% =+(-15%) +(-5%) -(+15%) -(+5%) =-40% =+(-15%) -(-5%) +(+15%) -(+5%) =0% +5% 28

29. Is It True That We Do Not Care About Concentration? Run #TCDeviation 1++-15% 2+--5% 3-++15% 4--+5% 29 Answer no: If the temperature is low, can improve conversion by keeping the concentration high – it is just that the opposite effect occurs when the temperature is high

30. Lets Examine The Effect Of TC (Simultaneous Variation of T+C) Run #TCTCDeviation 1+++-15% 2+---5% 3-+-+15% 4--++5% -40%0-20+5% 30 Want T – and TC -

31. Can Extend Process To Several Variables RunABCD 1++++ 2+++- 3++-+ 4++-- 5+-++ 6+-+- 7+--+ 8+--- 9-+++ 10-++- 11-+-+ 12-+-- 13--++ 14--+- 15---+ 16---- 31 Gives too many runs

32. Software To Help Concept: we usually want to fit the data to a simple function: Response=C 1 +C 2 A+C 3 B+… Only need enough runs to fit constants accurately 32

33. Echip Software Example 33

34. Software Setup 34

35. Number of Runs Substantially Reduced 4 variables, 4 values with 3 replicates gives (4) 4 + 3*4 = 268 runs Echip achieves almost the same accuracy with 23 runs! 35

36. Summary Single variables use ANOVA to check models Multivariable problems Use design of experiments to see which variables are important (2 n ) designs Software can simplify runs Use variances to fit models (automatic in software) 36

37. Class Question What did you learn new today? 37