1 / 58

Experimental Statistics - week 6

Experimental Statistics - week 6. Chapter 15: Randomized Complete Block Design (15.3) Factorial Models (15.5). Caution: Chapter 15 introduces some new notation - i.e. changes notation already defined. Recall: Sum-of-Squares Identity 1-Factor ANOVA.

zuzela
Télécharger la présentation

Experimental Statistics - week 6

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Experimental Statistics - week 6 Chapter 15: Randomized Complete Block Design (15.3) Factorial Models (15.5)

  2. Caution: Chapter 15 introduces some new notation - i.e. changes notation already defined

  3. Recall: Sum-of-Squares Identity 1-Factor ANOVA In words: Total SS = SS between samples + within sample SS

  4. Recall: Sum-of-Squares Identity 1-Factor ANOVA - new notation for Chapter 15

  5. Recall: Sum-of-Squares Identity 1-Factor ANOVA - new notation for Chapter 15

  6. Recall: Sum-of-Squares Identity 1-Factor ANOVA - new notation for Chapter 15 In words: Total SS = SS for “treatments” + SS for “error”

  7. Revised ANOVA Table for 1-Factor ANOVA(Ch. 15 terminology - p.857) Source SS df MS F Treatments SST t -1 Error SSE N -t Total TSS N -1

  8. Recall CRD Model for Gasoline Data yij = mi+ eij or yij = m + ai+ eij observed octane mean for ith gasoline unexplained part -- car-to-car differences -- temperature -- etc.

  9. Gasoline Data Question: What if car differences are obscuring gasoline differences? Similar to diet t-test example: Recall: person-to-person differences obscured effect of diet

  10. Possible Alternative Design: Test all 5 gasolines on the same car - in essence we test the gasoline effect directly and remove effect of car-to-car variation Question: How would you randomize an experiment with 4 cars?

  11. Blocking an Experiment - dividing the observations into groups (called blocks) where the observations in each block are collected under relatively similar conditions - comparisons can many times be made more precisely this way

  12. Terminology is based on Agricultural Experiments Consider the problem of testingfertilizers on a crop - t fertilizers - n observations on each

  13. Completely Randomized Design B A C A B B A C A C C B C A t = 3 fertilizers n = 5 replications B - randomly select 15 plots - randomly assign fertilizers to the 15 plots

  14. Randomized Complete Block Strategy A | C | B B | A | C C | B | A A | B | C t = 3 fertilizers C | A | B - select 5 “blocks” - randomly assign the 3 treatments to each block Note:The 3 “plots” within each block are similar - similar soil type, sun, water, etc

  15. Randomized Complete Block Design Randomly assign each treatment once to every block Car Example Car 1: randomly assign each gas to this car Car 2: .... etc. Agricultural Example Randomly assign each fertilizer to one of the 3 plots within each block

  16. Model For Randomized Complete Block (RCB) Design yij = m + ai+ bj+ eij effect of ith treatment effect of jth block unexplained error (gasoline) (car) -- temperature -- etc.

  17. Back to CAR data: Suppose that instead of 20 cars, there were only 4 cars, and we tested each gasoline on each car. “Restructured” CAR Data Car Old Data Format 1 2 3 4 A 91.7 91.2 90.9 90.6 B 91.7 91.9 90.9 90.9 C 92.4 91.2 91.6 91.0 D 91.8 92.2 92.0 91.4 E 93.1 92.9 92.4 92.4 A 91.7 91.2 90.9 90.6 B 91.7 91.9 90.9 90.9 C 92.4 91.2 91.6 91.0 D 91.8 92.2 92.0 91.4 E 93.1 92.9 92.4 92.4 Gas Gas

  18. Back to CAR data: Suppose that instead of 20 cars, there were only 4 cars, and we tested each gasoline on each car. “Restructured” CAR Data Car Old Data Format 1 2 3 4 A 91.7 91.2 90.9 90.6 B 91.7 91.9 90.9 90.9 C 92.4 91.2 91.6 91.0 D 91.8 92.2 92.0 91.4 E 93.1 92.9 92.4 92.4 A 91.7 91.2 90.9 90.6 B 91.7 91.9 90.9 90.9 C 92.4 91.2 91.6 91.0 D 91.8 92.2 92.0 91.4 E 93.1 92.9 92.4 92.4 Gas Gas

  19. Recall: Sum-of-Squares Identity 1-Factor ANOVA - new notation for Chapter 15 In words: Total SS = SS for “treatments” + SS for “error”

  20. A New Sum-of-Squares Identity In words: Total SS = SS for treatments + SS for blocks + SS for error

  21. Hypotheses: To test for treatmenteffects - i.e. gas differences we test To test for block effects - i.e. car differences (not usually the research hypothesis) we test

  22. Randomized Complete Block Design ANOVA Table Source SS df MS F Treatments SST t -1 Blocks SSB Error SSE Total TSS bt -1 See page 866

  23. Test for Treatment Effects Note:

  24. Test for Block Effects

  25. “Restructured” CAR Data - SAS Format A B1 91.7 A B2 91.2 A B3 90.9 A B4 90.6 B B1 91.7 B B2 91.9 B B3 90.9 B B4 90.9 C B1 92.4 C B2 91.2 C B3 91.6 C B4 91.0 D B1 91.8 D B2 92.2 D B3 92.0 D B4 91.4 E B1 93.1 E B2 92.9 E B3 92.4 E B4 92.4 The first variable (A - E) indicates gas as it did with the Completely Randomized Design. The second variable (B1 - B4) indicates car.

  26. SAS file - Randomized Complete Block Design for CAR Data INPUT gas$ block$ octane; PROC GLM; CLASS gas block; MODEL octane=gas block; TITLE 'Gasoline Example -Randomized Complete Block Design'; MEANS gas/LSD; RUN;

  27. CRD ANOVA Table Output - car data Source SS df MS F p-value Gas 6.108 4 1.527 6.80 0.0025 (treatments) Error 3.370 15 0.225  Totals 9.478 19

  28. RCB ANOVA Table Output - car data Source SS df MS F p-value Gas 6.108 4 1.527 15.58 0.0001 (treatments) Cars 2.194 3 0.731 7.46 0.0044 (blocks) Error 1.176 12 0.098 Totals 9.478 19

  29. SAS Output -- RCB CAR Data Dependent Variable: OCTANE Sum of Mean Source DF Squares Square F Value Pr > F Model 7 8.30200000 1.18600000 12.10 0.0001 Error 12 1.17600000 0.09800000 Corrected Total 19 9.47800000 R-Square C.V. Root MSE OCTANE Mean 0.875923 0.341347 0.3130495 91.710000 Source DF Anova SS Mean Square F Value Pr > F GAS 4 6.10800000 1.52700000 15.58 0.0001 BLOCK 3 2.19400000 0.73133333 7.46 0.0044

  30. Multiple Comparisons in RCB Analysis

  31. CAR Data -- LSD Results CRD Analysis t Grouping Mean N gas A 92.7000 4 E B 91.8500 4 D B C B 91.5500 4 C C B C B 91.3500 4 B C C 91.1000 4 A RCB Analysis t Grouping Mean N gas A 92.7000 4 E B 91.8500 4 D B C B 91.5500 4 C C C 91.3500 4 B C C 91.1000 4 A 

  32. CAR Data -- Bonferroni Results CRD Analysis   Bon Grouping Mean N gas A 92.7000 4 E A B A 91.8500 4 D B B 91.5500 4 C B B 91.3500 4 B B B 91.1000 4 A RCB Analysis Bon Grouping Mean N gas A 92.7000 4 E B 91.8500 4 D B B 91.5500 4 C B B 91.3500 4 B B B 91.1000 4 A

  33. STIMULUS EXAMPLE: Personal computer presents stimulus, and person responds. Study of how RESPONSE TIME is effected by a WARNING given prior to the stimulus: 2-factors of interest: Warning Type --- auditory or visual Time between warning and stimulus -- 5 sec, 10 sec, or 15 sec. 

  34. Auditory Visual .204 .257 .170 .279 .181 .269 .167 .283 .182 .235 .187 .260 .202 .256 .198 .281 .236 .258 5 sec WarningTime 10 sec 15 sec Note:“Sort of like RCB” -- what is the difference? Question: How would you randomize? - 18 subjects - 1 subject

  35. Stimulus Data Observed data (response time) Replication Level of Factor A Level of Factor B (warning type) (time)

  36. 2-Factor ANOVA Data Factor B FactorA

  37. A Possible Model for STIMULUS Data i = type, j = time Note: so according to this model Note: The model assumes that the difference between types is the same for all times

  38. Hypothetical Cell Means Auditory Visual 5 10 15

  39. Similarly i.e. the model says We may not want to make these assumptions!!

  40. Hypothetical Cell Means Auditory Visual 5 10 15 Auditory Visual 5 10 15

  41. Model for 2-factor Design where

  42. Sum-of-Squares Breakdown (2-factor ANOVA) SSA SSB SSAB SSE

  43. 2-Factor ANOVA Table(2-Factor Completely Randomized Design) Source SS df MS F Main Effects A SSA a -1 B SSB b- 1 Interaction AB SSAB (a -1)(b- 1) Error SSE ab(n -1) Total TSS abn -1 See page 900

  44. ***************************************************** * Two-Way ANOVA using PROC GLM * * showing Interaction Plots * ***************************************************** ; data stimulus; input type$ time response; datalines; A 5 .204 A 5 .170 A 5 .181 A 10 .167 A 10 .182 A 10 .187 A 15 .202 A 15 .198 A 15 .236 V 5 .257 V 5 .279 V 5 .269 V 10 .283 V 10 .235 V 10 .260 V 15 .256 V 15 .281 V 15 .258 ; PROCGLM; CLASSES type time; MODEL response=type time type*time; TITLE‘Stimulus Data'; run; PROCSORT;BY type time; PROCMEANS; BY type time; OUTPUTOUT=cells MEAN=response; RUN; * OUTPUT MEAN INTERACTION PLOTS; PROCGPLOT; PLOT response*type=time; SYMBOL1V=CIRCLE I=JOIN C=BLACK; SYMBOL2V=DOT I=JOIN C=BLACK; symbol3V=BOX I=JOIN C=BLACK; RUN; PROCGPLOT; PLOT response*time=type; SYMBOL1V=CIRCLE I=JOIN C=BLACK; SYMBOL2V=DOT I=JOIN C=BLACK; RUN; PROCPRINT; RUN;

  45. ***************************************************** * Two-Way ANOVA using PROC GLM * showing Interaction Plots ***************************************************** ; data stimulus; input type$ time response; datalines; A 5 .204 A 5 .170 A 5 .181 A 10 .167 A 10 .182 A 10 .187 A 15 .202 A 15 .198 A 15 .236 V 5 .257 V 5 .279 V 5 .269 V 10 .283 V 10 .235 V 10 .260 V 15 .256 V 15 .281 V 15 .258 ; PROCGLM; CLASSES type time; MODEL response=type time type*time; TITLE‘Stimulus Data'; run;

  46. GLM Output Stimulus Data The GLM Procedure Dependent Variable: response Sum of Source DF Squares Mean Square F Value Pr > F Model 5 0.02554894 0.00510979 17.66 <.0001 Error 12 0.00347200 0.00028933 Corrected Total 17 0.02902094 R-Square Coeff Var Root MSE response Mean 0.880362 7.458622 0.017010 0.228056 Source DF Type I SS Mean Square F Value Pr > F type 1 0.02354450 0.02354450 81.38 <.0001 time 2 0.00115811 0.00057906 2.00 0.1778 type*time 2 0.00084633 0.00042317 1.46 0.2701

  47. PROCSORT;BY type time; PROCMEANS; BY type time; OUTPUTOUT=cells MEAN=response; RUN;

More Related