Two-Factor Analysis of Variance Here we look at applying two-factor analysis of variance where we interested in studying the effects of two factors on a response variable. Example A horticulturist wants to study the effects of four different pesticides on the yield of fruit from three different varieties of a citrus tree. Two Factors: Pesticide and Variety The factor pesticide has four levels and factor variety has three levels Response variable: fruit yield Example A store manager wishes to study the effects of height of shelf display (bottom, middle, top) and width of the shelf display ( regular, wide) on sales of the bakery's bread. Two Factors: Height and Width The factor height has three levels and the factor width has two levels. Response variable: bread sales The Castle Bakery Company supplies wrapped Italian Bread to a large number of supermarkets in a metropolitan area. An experimental study was made of the effects of height of the shelf display (bottom, middle, top) and the width of the shelf display (regular, wide) on sales of this bakery’s bread (measured in cases) during the experimental period. Twelve supermarkets, similar in terms of sales volumes and clientale, were utilized in the study. Two stores were assigned at random to each of the factor level combinations and the sales of the bread were recorded. Display Height Bottom Middle Top Display Width Regular Wide 47, 43 46, 40 62, 68 67, 71 41, 39 42, 46 Questions of Interest 1. Are the mean sales different for the three heights? 2. Are the mean sales different for the two widths? 3. Does the effect of display width on sales depend on display height? In other words, do the two factors interact? What is meant by interaction? Two Factors A and B are said to interact if the difference in population mean responses for two levels of one factor is not constant across levels of the second factor. When the effect of one factor on a response variable y depends on the level of a second factor, we say that the two factors interact. Examine the following scenarios. In which of the scenarios does there appear to be interaction between Factors A and B? B1 n=10 =84 n=10 =92 x x A1 A2 A1 A2 A1 A2 n=10 n=10 n=10 n=10 x x B1 =84 =74 x x B1 =74 =76 B2 n=10 =74 n=10 =83 x x n=10 n=10 n=10 n=10 B2 =76 =90 Row Mean 80 82 B2 =75 =90 Row Mean 74.5 83 x x x x Two-Factor Anova Hypothesis Tests 1. H0: There is no interaction between the two factors Ha: There is interaction between the two factors 2. H0: The population average response is the same for each level of Factor A Ha: The population means for the different levels of factor A are not all equal. 3. H0: The population average response is the same for each level of Factor B Ha: The population means for the different levels of factor B are not all equal. Assumptions required for F-tests: The observations on any particular treatment are independently selected from a normal distribution with variance 2 (the same variance for each treatment), and samples from different treatments are independent of one another. Examples of Analysis performed in Two-Factor ANOVA A company was interested in comparing three different display panels for use by air traffic controllers. Each display panel was to be examined under five different simulated emergency conditions. Thirty highly trained air traffic controllers with similar work experience were enlisted for the study. A random assignment of controllers to display-panel-emergency conditions was made, with two controllers assigned to each factor-level combination. The time (in seconds) required to stabilize the emergency situation was recorded for each controller at a panel-emergency condition. These data appear below. Display Panel 1 2 3 1 18 13 24 16 15 28 Emergency Condition 2 3 31 35 22 27 33 30 24 21 42 46 40 37 4 39 35 52 5 36 38 57 15 10 28 12 16 24 Analysis of Variance for Time Source Cond Panel Cond*Panel Error Total DF 4 2 8 15 29 Seq SS 2850.13 1227.80 44.87 106.00 4228.80 Adj SS 2850.13 1227.80 44.87 106.00 Adj MS 712.53 613.90 5.61 7.07 F 100.83 86.87 0.79 P 0.000 0.000 0.617 Means with the same letter are not significantly different. Tukey Grouping Mean N panel A 37.800 10 3 B B B 25.100 10 1 23.500 10 2 Means with the same letter are not significantly different. Tukey Grouping Mean N conditin A 42.833 6 4 B 36.167 6 2 C 28.500 6 3 D D D 19.000 6 1 17.500 6 5 EXAMPLE An experiment was conducted to examine the effects of different levels of reinforcement and different levels of isolation on children’s ability to recall. A single analyst was to work with a random sample of 36 children. Two levels of reinforcement (none and verbal) and three levels of isolation (20, 40, and 60 minutes) were to be used. Students were randomly assigned to the six treatment groups, with a to total of 6 students being assigned to each group. Each student was to spend a 30-minute session with the analyst. During this time, the student was to memorize a specific passage, with reinforcement provided as dictated by the group to which the student was assigned. Following the 30-minute session, the student was isolated for the time specified for his or her group and then tested for recall of the memorized passage. These data appear in the accompanying table. t.00833, 30=2.536 Level of reinforcement None Verbal 20 20 18 15 22 19 28 16 25 Time of Isolation (Minutes) 40 23 25 24 21 30 28 24 27 ANOVA TABLE: Source C1 DF 2 SS 181.56 MS 90.778 F 5.80 P 0.007 36 27 26 23 25 24 29 21 6 14 31 34 60 10 17 38 35 11 19 29 30 1 2 30 35 225.00 1016.67 469.33 1892.56 225.000 508.333 15.644 14.38 32.49 0.001 0.000 Interaction Plot - Means for C3 C2 'None' 'Verbal' 'None' 'Verbal' 32 Mean 22 12 '20' '40' '60' C1 Normal Probability Plot for Assessing Normality Normal Probability Plot of the Residuals (response is C3) 99 95 90 80 Percent C2 Interaction Error Total 70 60 50 40 30 20 10 5 1 -10 -5 0 Residual 5 10 Plot of Residuals vs. Predicted Values for Assessing Equal Variance Assumption Residuals Versus the Fitted Values (response is C3) 8 6 Residual 4 2 0 -2 -4 -6 -8 10 15 20 25 30 35 Fitted Value Fixed Versus Random Factors A random factor is a factor whose levels may be regarded as a sample from some large population of levels. A fixed factor is a factor whose levels are the only ones of interest. When a fixed factor is used, the inferences from the data are for the levels of the factor actually used in the experiment. When a random factor is used, the inferences from the data in the experiment are for all levels of the factor in the population from which the levels were selected and not only the levels used in the experiment. In ANOVA situations involving two or more factors, the F tests required for making inferences differ depending on whether all factors are fixed, all factors are random, or some of both are present. In this class, we look at examples that involve only fixed factors. Example A study was designed to evaluate the effectiveness of two different sunscreens for protecting the skin of persons who want to avoid burning or additional tanning while exposed to the sun. A random sample of 40 subjects (ages 20-25) agreed to participate in the study. For each subject a 1-inch square was marked off on their back, under the shoulder but above the small of the back. Twenty subjects were randomly assigned to each of the two types of sunscreen. A reading based on the color of the skin in the designated square was made prior to the application of a fixed amount of the assigned sunscreen, and then again after application and exposure to the sun for a 2-hour period. The company was concerned that the measurement of color is extremely variable, and wanted to assess the variability in the readings due to the technician taking the readings. Thus, the company randomly selected ten technicians from their worldwide staff to participate in the study. Four subjects, two having sunscreen 1 and 2 having sunscreen 2 were randomly assigned to each technician for evaluation. The data recorded were differences (postexposure-preexposure) for the subjects in the study. Fixed factor – sunscreen type Random Factor - technician