Chapter 9 Factorial ANOVA with Two Between-Group Factors 10/22/2001 1 Factorial ANOVA with Two Between-Group Factors Recall that in one-way ANOVA we study the relation between one criterion variable and one independent variable. In factorial design, we study the relation between one criterion and two or more independent variables. In practice at most three or four independent variables are analyzed In this chapter we learn about Two-WayANOVA in which the effect of two independent variable on a criterion variable is investigated. 10/22/2001 2 Chapter 9 1
The Two Way ANOVA Model: Yij, k = µ + αi + βj + eijk i = 1,, r j = 1,, c k = 1,, nij Factor B Factor A Level B1 Level B2 Level Bc Level A1 n11 n12 n1c Level A2 n21 n22 n2c : : : : Level Ar nr1 nr2 nrc Each cell contains nij observations 10/22/2001 3 Example: If there are 9 observations in cell (1,2), then n12 = 9 and you have observed values y12,1, y12,2,, y12,9 µ : represents the overall mean αi : effect of factor A (row effect) βj : effect of factor B (column effect) 10/22/2001 4 Chapter 9 2
Two Assumptions: The mathematical form µ + αi + βj implies that row and column effects are additive. For example, the difference in effect (apart from the errors eijk) between row2 and row1 in column j is: (µ + α2 + βj) (µ + α1 + βj) = α2 - α1 That is, the row differences are the same in all columns. (This assumption may or may not hold.) 10/22/2001 5 The errors eijk are assumed to come from a normal distribution with mean 0 and variance σ² eijk : may represent natural variation among the units to which treatments are applied, or they may represent errors of measurements of the yijk. If each cell has more than 30 observations the assumption of normality can be relaxed. If max(nij) 1.5 min(nij) equality of variance (σ²) can be relaxed. 10/22/2001 6 Chapter 9 3
Let s Consider a Specific Example: You are interested in conducting a study that investigates aggression in eight-year-old children. You want to test the following two hypotheses: Boys will display a higher level of aggression than girls. The amount of sugar consumed will have a positive effect on levels ofaggression. 10/22/2001 7 Experiment: Take a sample of size 60 (30 boys and 30 girls) from eight-year-olds. Assign 20 children to 0 grams of sugar treatment condition (10 boys, 10 girls). Assign 20 children to 20 grams of sugar treatment condition. Assign 20 children to 40 grams of sugar treatment condition. For each child measure the level of aggression. 10/22/2001 8 Chapter 9 4
The Factorial Design Matrix: Factor A: Sex Factor B: Amount of Sugar B1: 0g B2: 20g A1: male n11 = 10 n12 = 10 A2: female n21 = 10 n22 = 10 B3: 40g n13 = 10 n23 = 10 Columns represent subjects in different levels of amount of sugar consumed. Rows represents subjects in levels of subject sex ; 30 boys and 30 girls. 10/22/2001 9 Factorial designs allow you to test for several different types of effects in a single investigation. Results from factorial designs are generally more difficult to interpret, as compared to one way ANOVA. Plots will help greatly in interpreting results from a factorial design. 10/22/2001 10 Chapter 9 5
Significant Main Effects In the mathematical model Yij, k = µ + αi + βj + eijk We say that there is a significant main effect if at least one of the αi s is different from the other αi s (i.e. at least one level of factor A is different from other levels of factor A). or at least one of the βi s is different from the other βi s (i.e. at least one level of factor B is different from other levels of factor B). 10/22/2001 11 This concept, of course, can be generalized to models with more than two predictor variables. For example, in the study on aggression it is possible to get any of the following outcomes related to main effects: Sugar consumption affects aggression (A significant main effect for factor A). Level of aggression is significantly different in boys as compared to girls (A significant main effect for factor B). 10/22/2001 12 Chapter 9 6
Both gender and sugar consumption effect the level of aggression (A significant main effect for both factor A and B). There are no significant effects from either gender or sugar consumption on aggression (No significant main effect). 10/22/2001 13 Using Graphs: Aggression males females mean level of aggression for females who consumed 20g of sugar 0g 20g 40g Amount of sugar consumed (Factor A) Horizontal Axis : levels of factor A Vertical Axis : Mean level of aggression Body : Two levels of factor B (Subject Sex) 10/22/2001 14 Chapter 9 7
Interpreting the Graph: There is a significant main effect for factor A if the plot has both of the following conditions: The lines for various groups are parallel At least one line segment displays a relatively steep angle. Parallel lines ensures that the two predictor variables are not involved in an interaction. 10/22/2001 15 It does not make sense to interpret main effects in presence of interactions (We will discuss this later). A large slope (steep angle) in at least one line segment indicated that there is a difference in at least two levels of predictor. How steep in steep? Statistical tests will tell you. 10/22/2001 16 Chapter 9 8
A significant main effect for factor B Aggression male female 0g 20g 40g Amount of sugar consumed (FACTOR A) There is significant main effect for factor B if: The lines for various groups are parallel At least two of the lines are significantly separated from each other 10/22/2001 17 Aggression male female 0g 20g 40g Amount of sugar consumed (FACTOR A) There is significant main effect for both predictor variables if: The lines for various groups are parallel At least one line segment displays a steep angle At least two of the lines are relatively separated from 10/22/2001 18 each other Chapter 9 9
No main effects Aggression male female 0g 20g 40g Amount of sugar consumed (FACTOR A) 10/22/2001 19 Significant Interaction An interaction is a condition in which the effect of one independent variable is different at different levels of the second independent variable male Aggression female 0g 20g 40g Amount of sugar consumed (FACTOR A) 10/22/2001 20 Chapter 9 10
Non-parallel lines indicate interaction. The effect of sugar consumptionis different levels of sugar (male or female) Consuming larger amounts of sugar results in dramatic increase in aggression in boys. Consuming larger amount of sugar results in moderate increase in aggression in girls. 10/22/2001 21 Why does it not make sense to interpret main effects in presence of interactions? Consider following figure: male Aggression 0g 20g 40g female Does it make sense to say that there is a main effect in sugar consumption? No, because this effect is different for boys and girls (No effect for girls. Significant effect for boys) 10/22/2001 22 Chapter 9 11
Example: The Egg story The performance of six laboratories in measuring fat contents of eggs was to be studied. A single can of dried eggs was stirred well. Samples were drawn and a pair of samples (claimed to be of two "types"), was sent to each of six commercial laboratories to be analyzed for fat content. Each laboratory assigned two technicians, who each analyzed both "types". Since the data were all drawn from a single well-mixed can, the null hypothesis for ANOVA that the mean fat content of each sample is equalis true. LINK: Example1. SAS 10/22/2001 23 DATA EGGS; infile 'c:\classes\sta5206\notes\chapter9\sas_ files\eggs.dat'; input Fat_Content Technician Sample$; proc print; run; Lab$ 10/22/2001 24 Chapter 9 12
Fat_ Obs Content Lab Technician Sample 1 0.62 I 1 G 2 0.55 I 1 G 3 0.34 I 1 H 4 0.24 I 1 H 5 0.80 I 2 G 6 0.68 I 2 G 7 0.76 I 2 H 8 0.65 I 2 H 9 0.30 II 1 G 10 0.40 II 1 G 11 0.33 II 1 H 12 0.43 II 1 H 13 0.39 II 2 G 14 0.40 II 2 G 10/22/2001 15 0.29 II 2 H 25 15 0.29 II 2 H 16 0.18 II 2 H 17 0.46 III 1 G 18 0.38 III 1 G 19 0.27 III 1 H 20 0.37 III 1 H 21 0.37 III 2 G 22 0.42 III 2 G 23 0.45 III 2 H 24 0.54 III 2 H 25 0.18 IV 1 G 26 0.47 IV 1 G 27 0.53 IV 1 H 28 0.32 IV 1 H 29 0.40 IV 2 G 30 0.37 IV 2 G 31 0.31 IV 2 H 10/22/2001 26 Chapter 9 13
32 0.43 IV 2 H 33 0.35 V 1 G 34 0.39 V 1 G 35 0.37 V 1 H 36 0.33 V 1 H 37 0.42 V 2 G 38 0.36 V 2 G 39 0.20 V 2 H 40 0.41 V 2 H 41 0.37 VI 1 G 42 0.43 VI 1 G 43 0.28 VI 1 H 44 0.36 VI 1 H 45 0.18 VI 2 G 46 0.20 VI 2 G 47 0.26 VI 2 H 10/22/2001 48 0.06 VI 2 H 27 This experiment we first examine the effect of Laboratory and Sample type on the response Fat content. LINK: Example1A.SAS 10/22/2001 28 Chapter 9 14
DATA EGGS; infile 'c:\classes\sta5206\notes\chapter9\sas_files\eggs.dat'; input Fat_Content Lab$ Technician proc GLM; CLASS Lab Sample; Sample$; Model Fat_Content = Lab Sample Lab*Sample; Means Lab Sample Lab*Sample; Fits the model Output means for lab, run; Means Lab Sample / Tukey; sample, and each cell(lab*sample) Performs multiple comparison test 10/22/2001 29 Check class levels and number of observations: The GLM Procedure Class Level Information Class Levels Values Lab 6 I II III IV V VI Sample 2 G H Number of observations 48 10/22/2001 30 Chapter 9 15
The first step is to check whether there are any significant interactions. You check Type III SS and the p-value corresponding to Lab* Sample Source DF Type III SS Mean Square F Value Pr > F Lab 5 0.44302500 0.08860500 6.35 0.0003 Sample 1 0.02900833 0.02900833 2.08 0.1579 Lab*Sample 5 0.04906667 0.00981333 0.70 0.6243 10/22/2001 31 If there are no interactions, then you can check and interpret the main effects. For the Egg data, there is no significant interactionbetween Lab and Sample (p-value =.62) 10/22/2001 32 Chapter 9 16
Graph : To produce a graph, you refer to mean values. Level of ---------Fat_Content--------- Lab N Mean Std Dev I 8 0.58000000 0.19690462 II 8 0.34000000 0.08246211 III 8 0.40750000 0.07995534 IV 8 0.37625000 0.10822430 V 8 0.35375000 0.06906260 VI 8 0.26750000 0.12008925 Level of ---------Fat_Content--------- Sample N Mean Std Dev G 24 0.41208333 0.14298006 H 24 0.36291667 0.15092409 10/22/2001 33 Level of Level of ---------Fat_Content--------- Lab Sample N Mean Std Dev I G 4 0.66250000 0.10594810 I H 4 0.49750000 0.24716728 II G 4 0.37250000 0.04856267 II H 4 0.30750000 0.10340052 III G 4 0.40750000 0.04112988 III H 4 0.40750000 0.11500000 IV G 4 0.35500000 0.12396236 IV H 4 0.39750000 0.10372239 V G 4 0.38000000 0.03162278 V H 4 0.32750000 0.09105859 VI G 4 0.29500000 0.12396236 VI H 4 0.24000000 0.12754084 10/22/2001 34 Chapter 9 17
Fat Content sample G sample H I ll lll iv v vi Laboratory We expect No significant sample effect A significant lab effect (The first line segment has a sharp angle) 10/22/2001 35 The GLM Procedure Dependent Variable: Fat_Content sum of Source DF Squares Mean Square F Value Pr > F Model 11 0.52110000 0.04737273 3.40 0.0027 Error 36 0.50200000 0.01394444 Corrected Total 47 1.02310000 R-Square Coeff Var Root MSE Fat_Content Mean 0.509334 30.47396 0.118087 0.387500 Source DF Type I SS Mean Square F Value Pr > F Lab 5 0.44302500 0.08860500 6.35 0.0003 Sample 1 0.02900833 0.02900833 2.08 0.1579 Lab*Sample 5 0.04906667 0.00981333 0.70 0.6243 Source DF Type III SS Mean Square F Value Pr > F Lab 5 0.44302500 0.08860500 6.35 0.0003 Sample 1 0.02900833 0.02900833 2.08 0.1579 Lab*Sample 10/22/2001 5 0.04906667 0.00981333 0.70 0.6243 36 Chapter 9 18
Preparing Your ANOVA Table: Source SS DF MS F LAB.443 5.0886 6.35* Sample.029 1.029 2.08** Lab*Sample.049 5.0098 0.70*** Within.502 36.014 Total 1.023 47 * p-value =.0003 ** p-value =.1579 *** p-value =.6243 10/22/2001 37 Only significant main effect is Laboratory. Which laboratories are similar, and which are different. Perform Tukey HSD multiple comparison test. 10/22/2001 38 Chapter 9 19
The GLM Procedure Tukey's Studentized Range (HSD) Test for Fat_Content NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ. Alpha 0.05 Error Degrees of Freedom 36 Error Mean Square 0.013944 Critical Value of Studentized Range 4.25477 Minimum Significant Difference 0.1776 Means with the same letter are not significantly different. Tukey Grouping Mean N Lab A 0.58000 8 I A B A 0.40750 8 II B B 0.37625 8 IV B B 0.35375 8 V B B 0.34000 8 II B 10/22/2001 B 0.26750 8 VI 39 The GLM Procedure Tukey's Studentized Range (HSD) Test for Fat_Content NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ. Alpha 0.05 Error Degrees of Freedom 36 Error Mean Square 0.013944 Critical Value of Studentized Range 2.86818 Minimum Significant Difference 0.0691 Means with the same letter are not significantly different. Tukey Grouping Mean N Sample A 0.41208 24 G A A 0.36292 24 H 10/22/2001 40 Chapter 9 20
From the multiple comparison test, it is clear that Lab I is significantly different from other labs. You can obtain R² for each given effect. For example, R² for the Lab effect is R² = type III SS for Lab = 0.443 = 0.443 Total 1.023 44% of the variation in fat content is explained by Lab. Note: The output reports an overall R² value of 0.509 10/22/2001 41 Next, we might be interested to examine the effects of Lab and Technician on fat contents. LINK: Example1B.SAS 10/22/2001 42 Chapter 9 21
DATA EGGS; infile 'c:\classes\sta5206\notes\chapter9\sas_files\eggs.dat'; input Fat_Content proc GLM; CLASS Lab Technician; Lab$ Technician Sample$; Model Fat_Content = Lab Technician Lab*Technician; run; Means Lab Technician Lab*Technician; 10/22/2001 43 The GLM Procedure Class Level Information Class Levels Values Lab 6 I II III IV V VI Technician 2 1 2 Number of observations 48 Source DF Type III SS Mean Square F Value Pr > F Lab 5 0.44302500 0.08860500 9.59 <.0001 Technician 1 0.00440833 0.00440833 0.48 0.4941 Lab*Technician 5 0.24306667 0.04861333 5.26 0.0010 10/22/2001 44 Chapter 9 22
There is a significant interaction in this example. To understand it we plot the means. Fat Content technician l technician ll l ll lll iv v vi laboratory 10/22/2001 45 The effect of Technician on fat content is different at different levels of Laboratory. In particular, the effect of technician in Lab I on fat content is quite different from that effect in Labs II VI Conclusion : May be Lab Technician II in Laboratory I Screwed up! 10/22/2001 46 Chapter 9 23
Simple Effect We say that there is a Simple effect for independent variable A at a given level of independent variable B, when there is a significant relationship between independent variable A at that level of independent variable B. Fat Content technician l technician ll l ll lll iv v vi Laboratory 10/22/2001 47 Example: Consider the pervious example Dependent : Fat Content Factor A : Lab Factor B : Technician The plot indicates that there may be a significant relationship between Laboratory and Fat Content for technicians in group II. (a similar statement do not seem to be true about technician I) 10/22/2001 48 Chapter 9 24
When testing for simple effect, we perform a one way ANOVA. In our example, MODEL Fat Content = Lab ; We do this, however, only for part of the data. In our example for Technician I & II BY Technician ; LINK : Example1C. SAS 10/22/2001 49 DATA EGGS; infile 'c:\classes\sta5206\notes\chapter9\sas_files\eggs.dat'; input Fat_Content Lab$ Technician Sample$; proc GLM; CLASS Lab Technician; Model Fat_Content = Lab Technician Lab*Technician; Means Lab Technician Lab*Technician; run; proc SORT Data=eggs; by Technician; run; proc GLM Data=eggs; Class Lab; model FAt_Content = Lab; Means Lab; by Technician; 10/22/2001run; 50 Chapter 9 25
The SAS System 23:49 Monday, October 30, 2000 1 The GLM Procedure Class Level Information Class Levels Values Lab 6 I II III IV V VI Technician 2 1 2 Number of observations 48 10/22/2001 51 The GLM Procedure Dependent Variable: Fat_Content Sum of Source DF Squares Mean Square F Value Pr > F Model 11 0.69050000 0.06277273 6.79 <.0001 Error 36 0.33260000 0.00923889 Corrected Total 47 1.02310000 R-Square Coeff Var Root MSE Fat_Content Mean 0.674910 24.80494 0.096119 0.387500 Source DF Type I SS Mean Square F Value Pr > F Lab 5 0.44302500 0.08860500 9.59 <.0001 Technician 1 0.00440833 0.00440833 0.48 0.4941 Lab*Technician 5 0.24306667 0.04861333 5.26 0.0010 Source DF Type III SS Mean Square F Value Pr > F Lab 5 0.44302500 0.08860500 9.59 <.0001 Technician 1 0.00440833 0.00440833 0.48 0.4941 10/22/2001 Lab*Technician 5 0.24306667 0.04861333 5.26 0.001 52 Chapter 9 26
The GLM Procedure Level of ---------Fat_Content--------- Lab N Mean Std Dev I 8 0.58000000 0.19690462 II 8 0.34000000 0.08246211 III 8 0.40750000 0.07995534 IV 8 0.37625000 0. 10822430 V 8 0.35375000 0. 06906260 VI 8 0.26750000 0. 12008925 Level of ---------Fat_Content--------- Technician N Mean Std Dev 1 24 0.37791667 0.09973788 2 24 0.39708333 0.18531945 10/22/2001 53 Level of Level of ---------Fat_Content--------- Lab Technician N Mean Std Dev I 1 4 0.43750000 0.17745892 I 2 4 0.72250000 0.06946222 II 1 4 0.36500000 0.06027714 II 2 4 0.31500000 0.10279429 III 1 4 0.37000000 0.07788881 III 2 4 0.44500000 0.07141428 IV 1 4 0.37500000 0.15716234 IV 2 4 0.37750000 0.05123475 V 1 4 0.36000000 0.02581989 V 2 4 0.34750000 0.10177590 VI 1 4 0.36000000 0.06164414 VI 2 4 0.17500000 0.08386497 10/22/2001 54 Chapter 9 27
The SAS System 23:49 Monday, October 30, 2000 4 --------------------------- Technician=1 --------------------------- The GLM Procedure Class Level Information Class Levels Values Lab 6 I II III IV V VI Number of observations 24 10/22/2001 55 - Technician=1 ------------------------------------------- The GLM Procedure Dependent Variable: Fat_Content Sum of Source DF Squares Mean Square F Value Pr > F Model 5 0.01772083 0.00354417 0.30 0.9051 Error 18 0.21107500 0.01172639 Corrected Total 23 0.22879583 R-Square Coeff Var Root MSE Fat_Content Mean 0.077453 28.65406 0.108288 0.377917 Source DF Type I SS Mean Square F Value Pr > F Lab 5 0.01772083 0.00354417 0.30 0.9051 Source DF Type III SS Mean Square F Value Pr > F Lab 5 0.01772083 0.00354417 0.30 0.9051 10/22/2001 56 Chapter 9 28
The SAS System 23:49 Monday, October 30, 2000 6 ------------------------------- Technician=1 ----------------------------- The GLM Procedure Level of ---------Fat_Content--------- Lab N Mean Std Dev I 4 0.43750000 0.17745892 II 4 0.36500000 0.06027714 III 4 0.37000000 0.07788881 IV 4 0.37500000 0.15716234 V 4 0.36000000 0.02581989 VI 4 0.36000000 0.06164414 10/22/2001 57 Technician=2 ------------------------------------------- The GLM Procedure Class Level Information Class Levels Values Lab 6 I II III IV V VI Number of observations 24 10/22/2001 58 Chapter 9 29
Technician=2 ------------------------------------------- The GLM Procedure Dependent Variable: Fat_Content Sum of Source DF Squares Mean Square F Value Pr > F Model 5 0.66837083 0.13367417 19.80 <.0001 Error 18 0.12152500 0.00675139 Corrected Total 23 0.78989583 R-Square Coeff Var Root MSE Fat_Content Mean 0.846151 20.69259 0.082167 0.397083 Source DF Type I SS Mean Square F Value Pr > F Lab 5 0.66837083 0.13367417 19.80 <.0001 Source DF Type III SS Mean Square F Value Pr > F Lab 5 0.66837083 0.13367417 19.80 <.0001 10/22/2001 59 - Technician=2 ------------------------------------------- The GLM Procedure Level of ---------Fat_Content--------- Lab N Mean Std Dev I 4 0.72250000 0.06946222 II 4 0.31500000 0.10279429 III 4 0.44500000 0.07141428 IV 4 0.37750000 0.05123475 V 4 0.34750000 0.10177590 VI 4 0.17500000 0.08386497 10/22/2001 60 Chapter 9 30
To test whether there is a simple lab effect at levels of Technician II we compute F = Type III MS (from Technician II) =.1337 @ 14.5 Error MS (from two way ANOVA).00923 Now Compare this to 2.45 < F.05 (5, 36) < 2.53 There is a significant simple main effect of Lab at Technician II level 10/22/2001 61 Doing the same test at Technician I level: F =.0177 @ 1.92.00923 As expected, this is NOT significant. 10/22/2001 62 Chapter 9 31