(1) The explanatory or predictor variables may be qualitative. (We ll focus on examples where this is the case.)

Transcription

1 Introduction to Analysis of Variance Analysis of variance models are similar to regression models, in that we re interested in learning about the relationship between a dependent variable (a response) and one or more independent variables. They differ from regression models in two ways: (1) The explanatory or predictor variables may be qualitative. (We ll focus on examples where this is the case.) (2) Even if the predictor variables are quantitative, no assumption is made regarding the nature of the statistical relationship between these variables and the response variable. Regression: E{Y} increases (or decreases) linearly as X increases, etc. Y X 1

2 ANOVA: E{Y} is different at different values of X. Y X = Group Number The numbers in the horizontal axis have no meaning here: they could just as well be labels like Group 1, etc. In analysis of variance, the explanatory or predictor variables are called factors or treatments (we ll use factors ). The value of a factor is called a factor level, and could be either quantitative or qualitative. Read pgs. 666 through 671. Seriously. Know the terms & concepts. That s a hint. Some analysis of variance notation: r The number of levels of the factor under study Y ij The value of the jth observation at factor level i µ i Mean response at factor level i n i The number of observations (cases) at factor level i n T The total number of observations: n T = n1 + n nr The Model I or fixed effects ANOVA Model: Yij = µ + ε i ij 2

3 Analysis of Variance Example Fifteen students enrolled in an honors course in mathematics were divided at random into three groups of five. Each group was randomly assigned one of three instructional modes which augmented the traditional course materials: (1) programmed text; (2) video tapes; and (3) interactive computer programs. At the end of the course each student was given the same achievement test. The results are as follows: Factor Level (i) Observation (j) Programmed Text Video Tapes Software Sum Average Boxplots of Response by Factor_L (means are indicated by solid circles) 90 Response Factor_Level ProgTxt Software VideoTape 3

4 One-way Analysis of Variance Analysis of Variance for Response Source DF SS MS F P Factor_L Error Total Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ProgTxt ( * ) Software ( * ) VideoTap ( * ) Pooled StDev =

5 Analysis of Variance Example II A senior partner in a brokerage firm wishes to determine if there is really any difference between long-run performance of different categories of people hired as customers' representatives. The junior members of the firm are classified into four groups: professionals who have changed careers, recent business school graduates, former salespersons (PC!), and brokers hired from competing firms. A random sample of five individuals is selected from each of these categories and a "detailed performance score" is obtained. The scores are as follows: Professionals Business School Grads Salespersons Brokers To address the partner's question we test: H 0 : µ 1 = µ 2 = µ 3 = µ 4 least one vs. H 1 : At Minitab Data Layout & Results: Score Background 85 Professional 95 Professional 96 Professional 91 Professional 88 Professional 73 Grads 54 Grads 72 Grads 81 Grads 69 Grads 67 Salespersons 74 Salespersons 65 Salespersons 68 Salespersons 77 Salespersons 87 Brokers 90 Brokers 84 Brokers 92 Brokers 94 Brokers 5

6 Boxplots of Score by Backgrou (means are indicated by solid circles) Score Background Brokers Grads Professional Salespersons One-way Analysis of Variance Analysis of Variance Source DF SS MS F P Factor Error Total Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev Professi (-----*-----) Grads (-----*-----) Salesper (-----*-----) Brokers (-----*-----) Pooled StDev =

7 Tukey's pairwise comparisons Family error rate = Individual error rate = Critical value = 4.05 Intervals for (column level mean) - (row level mean) Brokers Grads Professi Grads Professi Salesper We re 95% confident that ALL of these intervals contain the true differences in population means! (This is a family confidence statement.) Homogeneity of Variance Test for Score 95% Confidence Intervals for Sigmas Factor Levels Brokers Bartlett's Test Grads Test Statistic: P-Value : Professional Lev ene's Test Test Statistic: P-Value : Salespersons

8 Analysis of Variance vs. Two-Sample t-test (r = 2) Response = score on a general knowledge test given to high school students Factor = type of high school attended: public or private Boxplots of B_Score by School (means are indicated by solid circles) B_Score School Priv_School Pub_School One-way Analysis of Variance Analysis of Variance for B_Score Source DF SS MS F P School Error Total Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev Priv_Sch ( * ) Pub_Scho ( * ) Pooled StDev = Two Sample T-Test and Confidence Interval (Equal variances assumed!) Two sample T for B_Score School N Mean StDev SE Mean Priv_Schoo Pub_School % CI for mu (Priv_Schoo) - mu (Pub_School): ( 1.11, 4.59) T-Test mu (Priv_Schoo) = mu (Pub_School) (vs not =): T = 3.28 P = DF = 57 Both use Pooled StDev =

9 Regression approach to analysis of variance (Section 16.11) (Not on test!) Consider the brokerage firm example and set up indicator (dummy) variables for each factor level: Score Background Prof_Ind Grad_Ind Sales_Ind Broker_Ind 85 Professional Professional Professional Professional Professional Grads Grads Grads Grads Grads Salespersons Salespersons Salespersons Salespersons Salespersons Brokers Brokers Brokers Brokers Brokers

10 One-way Analysis of Variance Analysis of Variance for Score Source DF SS MS F P Backgrou Error Total Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev Brokers (-----*-----) Grads (-----*-----) Professi (-----*-----) Salesper (-----*-----) Pooled StDev = Regression Analysis (X s are the four indicator variables) * Broker_Ind is highly correlated with other X variables * Broker_Ind has been removed from the equation The regression equation is Score = Prof_Ind Grad_Ind Sales_Ind Predictor Coef StDev T P Constant Prof_Ind Grad_Ind Sales_In S = R-Sq = 76.1% R-Sq(adj) = 71.6% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS Prof_Ind Grad_Ind Sales_In Unusual Observations Obs Prof_Ind Score Fit StDev Fit Residual St Resid R R denotes an observation with a large standardized residual 10

11 Selected ANOVA Topics, Concepts, & Comments (NOT ON TEST!) Alternative model formulation (Section 16.10): Yij = µ. + τ + ε What do the parameters represent? What are the appropriate null and alternative hypotheses? Estimation (17.3): Estimating a specific factor level mean (original formulation) µ i : i ij Y ± t(1 α / 2, n r) i. T MSE n i Estimating a difference D between two factor level means: D = µ µ i i' ( Y Y ) ± t(1 α / 2; n i. i'. T 1 r) MSE ni 1 + n i' 1/ 2 You can use Bonnferroni s logic if you need to perform simultaneous estimation. Fixed (Model I) vs. Random (Model II) Effects Models (16.6) Fixed effects: A company produces widgets on five different machines. The question of interest is whether the mean strength of the widgets is the same for all machines. To test, we obtain responses from each machine and perform ANOVA. Random effects: A company produces widgets on 100 different machines, and once again the question of interest is whether the mean strength of the widgets is the same for all machines. To test, we randomly select five machines, obtain responses from these machines, and perform ANOVA. In the first case we have data for all five possible factor levels, and our interest is restricted to these five. In the second case for have data for five factor levels that represent a sample from a larger population of factor levels. Our interest is in the larger population. 11

12 Power and Sample Size (Section 26.4) ni ( µ i µ 1. ) i= 1 Power is a function of the noncentrality parameter: φ =, a σ r measure of how much the µ s vary from their mean (when H 0 is true, this value is zero). Sample sizes can be chosen so that the test will have a desired power value. As an example, suppose that r = 4, σ = 10, and we d like a 90% chance of rejecting H 0 when H 1 is true and, specifically, µ 1 = µ 2 = µ 3 but µ 4 = µ Minitab tells us that we d need a sample size of n 1 = n 2 = n 3 = n 4 = 20 to achieve this power level. Here the output: Power and Sample Size One-way ANOVA Sigma = 10 Alpha = 0.05 Number of Levels = 4 Corrected Sum of Squares of Means = 75 Means = 10, 10, 10, 20 Sample Target Actual Size Power Power Note that the choice of a base mean of 10 is arbitrary and irrelevant. The key factor is the relationship among the means. Now suppose we want the power to be 0.90 when H 1 is true and, specifically, µ 1 = µ 2 = 10 and µ 3 = µ 4 = (again assume that r = 4, σ = 10). Minitab says: Power and Sample Size One-way ANOVA Sigma = 10 Alpha = 0.05 Number of Levels = 4 Corrected Sum of Squares of Means = Means = 10, 10, 18.66, Sample Target Actual Size Power Power This is the same result because the µ-values in H 1 were chosen so that the noncentrality parameter ϕ is the same in both cases. r 2 12

13 Two Factor (Two-Way) Analysis of Variance (Chapter 19) This is, conceptually, the same as going from simple linear regression to multiple regression with two independent variables. There are now two factors of interest in the study. Here s an example (text Problem 19.20). The response is the prediction error obtained when a programmer at a large computer software firm is asked to predict the number of programmer-days required to complete a large project. The two factors are: Sys_Exp: The type of experience that the programmer has. A 1 means that the programmer s experience is limited to working with small systems, a 2 means that the programmer has experience on small and large systems. Yrs_Exp: The number of years of experience for the programmer. A 1 means under 5 years., a 2 means >= 5 but < 10 years, and a 3 means 10 or more years. The experiment was performed with 24 programmers (4 per cell). Here s the Minitab printout: Two-way Analysis of Variance Analysis of Variance for Error Source DF SS MS F P Sys_Expe Yrs_Exp Error Total Individual 95% CI Sys_Expe Mean (-----*------) (-----*------) Individual 95% CI Yrs_Exp Mean (------*------) 2 76 (------*------) 3 50 (------*------) As is the case for multiple regression, we could also add an interaction term to the model! 13

14 A Famous Two-Factor Design: the randomized block design (Chapter 27) Research Question: Is there a difference in night vision effectiveness for four different headlamp designs (these are the treatments or factor levels )? To test this, we obtain 12 test subjects and desire a balanced design. In the completely randomized design approach, we would randomly assign three subjects to each treatment (to each headlamp design). Now consider the fact that our subjects vary in age. Let Y = young driver; M = middle age driver, O = old driver. Here s a possible result with the completely randomized approach: Treatment (Headlamp Design) Y M Y O Y O M Y O O Y M Response will be the distance (in feet) at which a sign can be read. There's a possible problem here. If, for example, test results indicate that Design 1 is the best design, is it because it really is best or because it was tested by younger drivers who had better night vision? The solution: a randomized block design which will equalize the effects of age, if any exist. Arrange subjects into similar groups called blocks (in this case "similar" means in the same age group), then randomly assign treatments within the blocks. Treatment (Headlamp Design) Y Y Y Y M M M M O O O O The goal is to choose blocks which have minimal variation within blocks (they re as homogenous as possible) and maximal variation between blocks. 14

15 The model looks like: Yij =.. µ + ρ + τ + ε i j ij The ρ s are block effects and the τ s are treatment effects. Here s a Minitab result: Two-way Analysis of Variance Analysis of Variance for Distance Source DF SS MS F P Age Grou Treatmen Error Total Individual 95% CI Age Grou Mean M (---*--) O (---*---) Y (---*--) Individual 95% CI Treatmen Mean ( * ) ( * ) ( * ) ( * )