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MANOVA

MANOVA. Multivariate Analysis of Variance. One way Analysis of Variance (ANOVA). Comparing k Populations. The F test – for comparing k means. Situation We have k normal populations Let m i and s denote the mean and standard deviation of population i . i = 1, 2, 3, … k .

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MANOVA

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  1. MANOVA Multivariate Analysis of Variance

  2. One way Analysis of Variance (ANOVA) Comparing k Populations

  3. The F test – for comparing k means Situation • We have k normal populations • Let miand s denote the mean and standard deviation of population i. • i = 1, 2, 3, … k. • Note: we assume that the standard deviation for each population is the same. s1 = s2 = … = sk = s

  4. We want to test against

  5. The F statistic where xij = the jth observation in the i th sample.

  6. The ANOVA table The ANOVA table is a tool for displaying the computations for the F test. It is very important when the Between Sample variability is due to two or more factors

  7. Computing Formulae: Compute 1) 2) 3) 4) 5)

  8. The data • Assume we have collected data from each of k populations • Let xi1, xi2 , xi3 , … denote the ni observations from population i. • i = 1, 2, 3, … k.

  9. Then 1) 2) 3)

  10. Anova Table

  11. Example In the following example we are comparing weight gains resulting from the following six diets • Diet 1 - High Protein , Beef • Diet 2 - High Protein , Cereal • Diet 3 - High Protein , Pork • Diet 4 - Low protein , Beef • Diet 5 - Low protein , Cereal • Diet 6 - Low protein , Pork

  12. Thus Thus since F > 2.386 we reject H0

  13. Anova Table * - Significant at 0.05 (not 0.01) ** - Significant at 0.01

  14. Equivalence of the F-test and the t-test when k = 2 the t-test

  15. the F-test

  16. Hence

  17. Factorial Experiments Analysis of Variance

  18. Dependent variable Y • k Categorical independent variables A, B, C, … (the Factors) • Let • a = the number of categories of A • b = the number of categories of B • c = the number of categories of C • etc.

  19. The Completely Randomized Design • We form the set of all treatment combinations – the set of all combinations of the k factors • Total number of treatment combinations • t = abc…. • In the completely randomized design n experimental units (test animals , test plots, etc. are randomly assigned to each treatment combination. • Total number of experimental units N = nt=nabc..

  20. The treatment combinations can thought to be arranged in a k-dimensional rectangular block B 1 2 b 1 2 A a

  21. C B A

  22. The Completely Randomized Design is called balanced • If the number of observations per treatment combination is unequal the design is called unbalanced. (resulting mathematically more complex analysis and computations) • If for some of the treatment combinations there are no observations the design is called incomplete. (In this case it may happen that some of the parameters - main effects and interactions - cannot be estimated.)

  23. Example In this example we are examining the effect of The level of protein A (High or Low) and the source of protein B (Beef, Cereal, or Pork) on weight gains (grams) in rats. We have n = 10 test animals randomly assigned to k = 6 diets

  24. The k = 6 diets are the 6 = 3×2 Level-Source combinations • High - Beef • High - Cereal • High - Pork • Low - Beef • Low - Cereal • Low - Pork

  25. Table Gains in weight (grams) for rats under six diets differing in level of protein (High or Low) and s ource of protein (Beef, Cereal, or Pork) Level of Protein High Protein Low protein Source of Protein Beef Cereal Pork Beef Cereal Pork Diet 1 2 3 4 5 6 73 98 94 90 107 49 102 74 79 76 95 82 118 56 96 90 97 73 104 111 98 64 80 86 81 95 102 86 98 81 107 88 102 51 74 97 100 82 108 72 74 106 87 77 91 90 67 70 117 86 120 95 89 61 111 92 105 78 58 82 Mean 100.0 85.9 99.5 79.2 83.9 78.7 Std. Dev. 15.14 15.02 10.92 13.89 15.71 16.55

  26. Treatment combinations Source of Protein Beef Cereal Pork Level of Protein High Diet 1 Diet 2 Diet 3 Low Diet 4 Diet 5 Diet 6

  27. Summary Table of Means Source of Protein Level of Protein Beef Cereal Pork Overall High 100.00 85.90 99.50 95.13 Low 79.20 83.90 78.70 80.60 Overall 89.60 84.90 89.10 87.87

  28. Profiles of the response relative to a factor A graphical representation of the effect of a factor on a reponse variable (dependent variable)

  29. Profile Y for A Y This could be for an individual case or averaged over a group of cases This could be for specific level of another factor or averaged levels of another factor a … 1 2 3 Levels of A

  30. Profiles of Weight Gain for Source and Level of Protein

  31. Profiles of Weight Gain for Source and Level of Protein

  32. Example – Four factor experiment Four factors are studied for their effect on Y (luster of paint film). The four factors are: 1) Film Thickness - (1 or 2 mils) 2) Drying conditions (Regular or Special) 3) Length of wash (10,30,40 or 60 Minutes), and 4) Temperature of wash (92 ˚C or 100 ˚C) Two observations of film luster (Y) are taken for each treatment combination

  33. The data is tabulated below: Regular Dry Special Dry Minutes 92 C 100 C 92C 100 C 1-mil Thickness 20 3.4 3.4 19.6 14.5 2.1 3.8 17.2 13.4 30 4.1 4.1 17.5 17.0 4.0 4.6 13.5 14.3 40 4.9 4.2 17.6 15.2 5.1 3.3 16.0 17.8 60 5.0 4.9 20.9 17.1 8.3 4.3 17.5 13.9 2-mil Thickness 20 5.5 3.7 26.6 29.5 4.5 4.5 25.6 22.5 30 5.7 6.1 31.6 30.2 5.9 5.9 29.2 29.8 40 5.5 5.6 30.5 30.2 5.5 5.8 32.6 27.4 60 7.2 6.0 31.4 29.6 8.0 9.9 33.5 29.5

  34. Definition: A factor is said to not affect the response if the profile of the factor is horizontal for all combinations of levels of the other factors: No change in the response when you change the levels of the factor (true for all combinations of levels of the other factors) Otherwise the factor is said to affect the response:

  35. Profile Y for A – A affects the response Y Levels of B a … 1 2 3 Levels of A

  36. Profile Y for A – no affect on the response Y Levels of B a … 1 2 3 Levels of A

  37. Definition: • Two (or more) factors are said to interact if changes in the response when you change the level of one factor depend on the level(s) of the other factor(s). • Profiles of the factor for different levels of the other factor(s) are not parallel • Otherwise the factors are said to be additive . • Profiles of the factor for different levels of the other factor(s) are parallel.

  38. Interacting factors A and B Y Levels of B a … 1 2 3 Levels of A

  39. Additive factors A and B Y Levels of B a … 1 2 3 Levels of A

  40. If two (or more) factors interact each factor effects the response. • If two (or more) factors are additive it still remains to be determined if the factors affect the response • In factorial experiments we are interested in determining • which factors effect the response and • which groups of factors interact.

  41. The testing in factorial experiments • Test first the higher order interactions. • If an interaction is present there is no need to test lower order interactions or main effects involving those factors. All factors in the interaction affect the response and they interact • The testing continues with for lower order interactions and main effects for factors which have not yet been determined to affect the response.

  42. Models for factorial Experiments

  43. The Single Factor Experiment Situation • We have t = a treatment combinations • Let miand s denote the mean and standard deviation of observations from treatment i. • i = 1, 2, 3, … a. • Note: we assume that the standard deviation for each population is the same. s1 = s2 = … = sa = s

  44. The data • Assume we have collected data for each of the a treatments • Let yi1, yi2 , yi3 , … , yindenote the nobservations for treatment i. • i = 1, 2, 3, … a.

  45. The model Note: where has N(0,s2) distribution (overall mean effect) (Effect of Factor A) Note: by their definition.

  46. yij (i = 1, … , a; j = 1, …, n) are independent Normal with mean mi and variance s2. Model 1: Model 2: where eij (i = 1, … , a; j = 1, …, n) are independent Normal with mean 0 and variance s2. Model 3: where eij (i = 1, … , a; j = 1, …, n) are independent Normal with mean 0 and variance s2 and

  47. The Two Factor Experiment Situation • We have t = ab treatment combinations • Let mijand s denote the mean and standard deviation of observations from the treatment combination when A = i and B = j. • i = 1, 2, 3, … a, j = 1, 2, 3, … b.

  48. The data • Assume we have collected data (n observations) for each of the t = ab treatment combinations. • Let yij1, yij2 , yij3 , … , yijndenote the nobservations for treatment combination - A = i, B = j. • i = 1, 2, 3, … a, j = 1, 2, 3, … b.

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