Unit 8: A Mixed Two-Factor Design

Transcription

1 Minitab Notes for STAT 3503 Dept. of Statistics CSU Hayward Unit 8: A Mixed Two-Factor Design 8.1. The Data We use data quoted in Brownlee: Statistical Theory and Methodology in Science and Engineering, 2nd ed., page 502, on the rupture strength of concrete beams. Six beams were cast from each of five batches of concrete. Of the six beams from each batch, two were treated with each of three different "capping" formulas (a process intended to add strength). The relevant measurement was the breaking strength of each beam in psi (lbs. per square inch). The results are shown below. Breaking Strength (psi) of Concrete Beams (Results for 3 capping formulas with each of 5 batches of concrete) Batch (B) Formula (A) Problems: Source: George Werner, "The effect of capping material on the compressive strength of concrete cylinders," Proceedings of the American Society for Testing and Materials, 58 (1958}, (Somewhat abridged here.) For your convenience the breaking strength measurements in the table above have been repeated below, reading across each row of the table. Put them into c1 of a Minitab worksheet; label it PSI. Then use the patterned data feature to make subscript columns c2 and c3 for A and B, respectively. What "pattern codes" can be used to generate these columns? Finally, look in the worksheet or print it out to verify that its rows contain the information shown at the top of the next page Use Minitab to make a table similar to the one shown above. Also make a table that shows the means for each cell, row, and column Make box plots or dot plots of these data broken out by Formula. Even though these plots do not account for the fact that the data come from five different batches of concrete, do they suggest that one capping formula may result in beams that are significantly stronger (or weaker) than the others? Is there any evidence of outliers or skewness among the observations for any of the Formulas? Based on graphical displays similar to those in the previous problem, does it seem that there is significant variability among the batches?

2 Minitab Notes for STAT 3503 Unit 8-2 c1 c2 c3 c1 c2 c3 c1 c2 c3 ROW PSI A B ROW PSI A B ROW PSI A B The Model In this experiment there are two factors that may influence the strength of a particular concrete beam: the capping Formula used to enhance its strength and the Batch of concrete from which it was made. We consider Formula to be a fixed effect. Only these three capping formulas are currently under study. We would like to know if they differ significantly as to the strength they impart to the beams to which they are applied. If, so we would like to establish the pattern of significant differences. In particular, we would like to know if one capping formula is better than the others. We consider Batch to be a random effect. The five batches are of interest only because variability among them may contribute to variability in strength of the beams. We would like to know if the variability among batches is noticeable over and above the inevitable variability in strength from beam to beam, even ones made in the same way from the same batch. In addition, there might be interaction between these two factors. (As a purely speculative scenario, some batches may be mixed in such a way that voids from air bubbles show at the surface of beams cast from them. Unfilled, these voids may weaken a beam. One capping formula may not be as good as the others overall, but it may be particularly effective in filling such voids.) We are in a position to test for interaction here because there is more than one observation per cell, so that interaction and error effects are not confounded. Formally, this is a two-way mixed-model analysis of variance (mixed because one effect is fixed and the other is random). There are a = 3 levels of the fixed factor, b = 5 levels of the random factor, and n = 2 replications within each cell. The model is: Y ijk = µ + α i + B j + (αb) ij + e ijk, for i = 1, 2,3, j = 1,..., 5, and k = 1, 2. The parameters α i correspond to the fixed effects of the 3 levels of Formula, the random variables B j correspond to the random effects of the 5 levels of Batch, and the random variables (αb) ij correspond to 15 random "non-additive" adjustments, one for each cell. (Non-additive means that for a given k, the value of Y ijk cannot be expressed as the sum of mean and effects µ + α i + B j.) These quantities are subject to the following restrictions: Σ i α i = 0; and Σ i (αb) ij = 0, for each value of j; where each sum is taken over the full range of the subscript i.

3 Minitab Notes for STAT 3503 Unit 8-3 Moreover, the distributions of the random variables in this model are as follows: B j are identically distributed according to N(0, σ 2 B ), (αb) ij are distributed identically according to N(0, σ 2 αb ), and e ijk are identically distributed according to N(0, σ 2 ). Moreover, all of these random variables are mutually independent, except that Cov[(αB) ij, (αb) i'j ] = σ 2 αb /(a 1), for i i'. For a given j and different i, the random variables (αb) ij cannot be independent because of the restriction that they sum to 0 over the index i. It may seem strange to put restrictions on the random variables (αb) ij that represent the interaction effect, and some authors do not do so. The advantage is that these restrictions make for a more straightforward interpretation of the main random effects B j. One speaks of the distinction between restricted ANOVA models and unrestricted ones. We always use restricted models in these units. (See the note in Problem ) In our ANOVA we shall perform tests of three null hypotheses: No Formula effect [H 0 : Σ i α 2 i = 0], no Batch effect [H 0 : σ 2 B = 0], and no Interaction [H 0 : σ 2 αb = 0]. For each test the alternative is that the relevant sum of squared parameters or variance component is positive. Interaction Plot (data means) for PSI A A B B Problems: Look at the interaction plot that has Batches on the horizontal axis, so that there are three broken-line traces representing the three capping formulas. If you had to guess just from looking at this plot, would you say that there is significant interaction? If interaction were significant, would it be disorderly with respect to the Formula effect?

4 Minitab Notes for STAT 3503 Unit (a) Because we are interested in the randomly chosen batches only as a possible source of variability, we will not be interested in determining specific patterns of differences among the batches, even if the Batch effect turns out to be significant. But use the profile plot with Formulas on the horizontal axis to speculate whether some Batches seem to produce stronger (weaker) beams than others. (b) If the Formula effect turns out to be significant, is there a formula that seems better (or worse) than the others? Suppose that you were given the 15 cell means rather than the 30 individual observations. Write the model you would use to analyze the results Analysis and Interpretation Here we use Minitab to make the ANOVA table and to interpret the results of this mixed two-factor model. The model for this ANOVA is more complex than for those that have gone before. It is very important to include every key feature of the formal model (Section 2) in specifying the model in Minitab. Notice the following requirements for using Minitab correctly: You must include the interaction term in specifying the model on the command line. There are two ways to do this: either include the term A * B in the model or put a vertical bar between the designations of the two main effects: A B. Minitab assumes that effects are fixed unless declared as random. You must declare Batch as a random effect, either using the subcommand random Batch or, if using menus, by putting Batch in the "text box" for random factors. As we shall see later, it is important to declare that you are using the restricted model. Do this either with the subcommand restrict or, if using menus, by marking the "check box" for the restricted model under Options. A proper interpretation of the resulting ANOVA table is based on an understanding of the expected mean squares (EMS table). Include the subcommand ems or by marking the check box for EMS under Results. As always, we will want to look at a normal probability plot of the residuals. This requires the subcommand residuals cx, where cx denotes an empty worksheet column. Alternatively, in the Balanced ANOVA menu, check the box for residuals under Storage, in which case Minitab selects an empty column automatically. (In menus, you can also get a normal probability plot of the residuals by marking the appropriate check box under Graphs, but this plot will not have confidence bands.) STAT ANOVA Balanced, Declare B random, Options to restrict, Results to show EMS, Storage of residuals [or Graph normal probability plot of residuals]. MTB > anova PSI = A B SUBC> random B; SUBC> restrict; SUBC> ems: SUBC> resid c4.

5 Minitab Notes for STAT 3503 Unit 8-5 ANOVA: PSI versus A, B Factor Type Levels Values A fixed 3 1, 2, 3 B random 5 1, 2, 3, 4, 5 Analysis of Variance for PSI Source DF SS MS F P A B A*B Error Total S = R-Sq = 90.01% R-Sq(adj) = 80.69% Expected Mean Square Variance Error for Each Term (using Source component term restricted model) 1 A 3 (4) + 2 (3) + 10 Q[1] 2 B (4) + 6 (2) 3 A*B (4) + 2 (3) 4 Error (4) In all of the models we have seen to date, MS(Error) has been used in the denominator of the F-statistic for each test of significance. The situation is now somewhat more complex. The following table interprets Minitab's EMS table in terms of our symbols above, where we define θ α = [Σ i α i 2 ]/(a 1) and where "Error Term" denotes which MS is used in the denominator of the F-statistic for the corresponding row. Two-Way Mixed Model (Restricted) Source Error Term Expected Mean Square Formula (A) Interaction σ 2 + 2σ 2 αb + 10θ α Batch (B) Error σ σ B Interaction Error σ σ αb Error σ 2 We begin our interpretation of the ANOVA table by noticing that the Interaction effect is not significant, P = Thus we can make straightforward interpretations of the main effects. Both of the main effects are very highly significant (with very small P-values). A final interpretation as to which capping formula is best or worst awaits an analysis of multiple comparisons (see Problem 8.3.1). As for the Batch, we now know that batch-to-batch variation is a significant source of variability in the strengths of the beams. Of course there is also variability within each cell of the data table because no two beams will ever be exactly alike and because of measurement errors. But the component of variance due to Batches significantly stands out above this within-cell variability. There is no point in determining the exact pattern of differences among the Batches because we will never see these particular batches again. It may, however, be worth pondering whether something can be done to decrease this component of variance in order to get a more uniform product.

6 Minitab Notes for STAT 3503 Unit 8-6 The important thing to notice about the tests of the main effects is that the F-ratio for testing the Formula effect is MS(A)/MS(A*B), whereas the F-ratio for testing the Batch effect is MS(B)/MS(Error). The rationale for using MS(A*B) in the denominator of F(A) is as follows. If there is no Formula effect, then all of the α i are 0 so that θ α = 0. In that case, we have EMS(A) = EMS(A*B) = σ 2 + 2σ 2 αb, so that MS(A)/MS(A*B) has an F-distribution with 4 and 8 degrees of freedom. Similar arguments based on the EMS table lead us to conclude that the Batch effect and Interaction should each be "tested against Error." That is, the F-ratios used to test them should have MS(Error) in the denominator. Problems: Look at the normal probability plot of the residuals. What is your interpretation? What is the P-value of the Anderson-Darling test for normality? Because Interaction is not significant one could remove the interaction term from the ANOVA model before testing the main effects. (a) What happens to the df for Interaction when you "pool" in this fashion? What is the correct "error term" for testing the main effects in this model? (b) Use a calculator to perform Tukey's HSD procedure to determine the pattern of differences among Formulas. Use the "pooled" MS(Error) and df(error) in your computations. Recall that each Formula mean is based on 10 observations. Verify your results using Tukey comparisons in Minitab's general linear models (GLM) procedure Perform the ANOVA including the interaction term, but not specifying the restricted model. What important difference in the formation of F-ratios is implied by Minitab's EMS table? Note: The difference in error terms (between the restricted and unrestricted model) that you see here has been the cause of a controversy among statisticians for several decades in the interpretation of mixed-model ANOVAs. The question has been: Which model is correct: restricted or unrestricted? A paper by Daniel T. Voss in The American Statistician, 53/4, (Nov. 1999) pages , claims to have resolved this controversy. Voss claims that the true interpretation of the terms in the EMSs of the unrestricted model is not obvious, that the F-ratios given by Minitab (and other software packages) for the unrestricted model are incorrect, and that the EMS tables resulting from the restricted model give the correct F-ratios. whether the restricted or unrestricted model is used. If Voss's view were to become generally accepted, some textbooks and much statistical software would need to be rewritten. For example, SAS uses unrestricted models as the default and does not even appear to have an automated way to produce "restricted" EMSs or F-ratios. Not surprisingly, a statistician from SAS wrote a letter to the editor of The American Statistician critical of Voss's interpretation. Some texts suggest that the restricted model is appropriate in some applications and the unrestricted model in others. The texts by Brownlee and Snedecor and Cochran discuss the restricted model exclusively. The text by Ott/Longnecker does not explicitly state the restrictions necessary for the restricted model, but all of the EMSs and F-ratios found there seem to be consistent with the restricted model. Thus, using the restricted model in Minitab for every ANOVA with any fixed effect will give the same results as in the books by Ott / Longnecker, Brownlee, and Snedecor / Cochran. Situations in which the distinction between restricted and unrestricted models will lead to different practical interpretations may be rare. For example, in the current two-way mixed

7 Minitab Notes for STAT 3503 Unit 8-7 model ANOVA, the importance of the distinction evaporates if interaction is nowhere near significant and the interaction term is removed from the model (as in Problem 8.3.2). Even when the interaction term remains, for the practical interpretation of main effects it may not make much difference whether MS(Error) or MS(Interaction) is put in the denominator of F Suppose that the 30 observations given at the beginning of this unit were collected collected exactly in order reading across the rows of the original data table, with Formula 1 tests being run on Monday, Formula 2 on Tuesday, and Formula 3 on Wednesday. Furthermore, suppose that on each of these days the Day shift ran Batch 1, the Swing shift ran Batches 2, 3, and 4, and the Night shift ran Batch 5. What effect would this knowledge have on the report you would write about your conclusions from the experiment? Suppose that the beams that produced the values 613 and 656 in the first two cells with Formula 1 had been damaged in handling so that these two values are missing. Further suppose that for each of the cells in Batch 3 a third beam was tested with results: 649, 612, 614 for Formulas 1, 2, 3, respectively. How would you analyze the resulting unbalanced design? Do the missing and extra observations change the overall interpretation of the results? 8.4. Comparison with Fixed and Random Models In this section we compare the mixed model just analyzed with a model in which both factors are fixed and one in which both factors are random. We do this by "changing the story" behind the data on strengths of concrete beams. These fantasy models are incorrect for the data given, but they are instructive from a computational point of view. If Both Factors Were Fixed. First, suppose that we have access to five major suppliers of concrete, and that Factor B represents "brands" of concrete rather than batches from the same supplier. Suppose that Factor A remains capping formulas as in the true story. In this case both factors would be fixed effects. Because fixed factors are the default for Minitab's general ANOVA procedure, it is not necessary to declare them as fixed. So that you can compare the EMS table with the results of the procedures in Section 17.5 of Ott/Longnecker (or other discussions of the Bennett-Franklin Algorithm), we use restricted models. We do not capture residuals because they are based on cell means, which do not depend on whether effects are fixed or random. INCORRECT FIXED-EFFECTS MODEL MTB > anova PSI = A B; SUBC> restrict; SUBC> ems. Factor Type Levels Values A fixed 3 1, 2, 3 B fixed 5 1, 2, 3, 4, 5 Analysis of Variance for PSI Source DF SS MS F P A B A*B Error Total

8 Minitab Notes for STAT 3503 Unit 8-8 Source Variance Error Expected Mean Square component term (using restricted model) 1 A 4 (4) + 10Q[1] 2 B 4 (4) + 6Q[2] 3 A*B 4 (4) + 2Q[3] 4 Error (4) Notice that MS(Error) is used in the denominator for F(A*B), F(A) and F(B) when both effects are fixed. There is not much point in giving interpretations of the F-ratios here because our phony model was not used to collect the data. A legitimate example of a fixed two-factor ANOVA with interaction was discussed in a prior unit. If Both Factors Were Random. Second, pretend that Factor B refers to randomly chosen batches as in the true story. However, imagine that Factor A is also a random factor perhaps corresponding to randomly chosen to batches of a single capping formula. The issue would be how much of the variability in measured strength of the beams is due to random concrete batch variability, how much to random variability in batches of the capping formula, how much to any interaction between these two random effects, and how much to unexplained random error. To use Minitab here, it is necessary to declare both factors as random. We declare the restricted model here just for consistency. Restriction is not an issue unless there is at least one fixed effect; the only result of the restrict subcommand is in the heading of the EMS table. (The residuals are the same as for the previous two models.) INCORRECT RANDOM-EFFECTS MODEL MTB > anova PSI = A B; SUBC> random A B; SUBC> restrict; SUBC> ems. Factor Type Levels Values A random 3 1, 2, 3 B random 5 1, 2, 3, 4, 5 Analysis of Variance for PSI Source DF SS MS F P A B A*B Error Total Source Variance Error Expected Mean Square component term (using restricted model) 1 A (4) + 2(3) + 10(1) 2 B (4) + 2(3) + 6(2) 3 A*B (4) + 2(3) 4 Error (4) In this case, the EMS information tells us to use MS(Error) as the denominator for testing interaction. If the variance component due to interaction, designated as (3) by Minitab, is negligible, then the ratio MS(A*B)/MS(Error) will have the distribution F(8, 15). However, the denominator of F for testing either main effect is MS(A*B). For example if the variance component due to Factor A, designated (1), is negligible, then the ratio MS(A)/MS(A*B) will be distributed

9 Minitab Notes for STAT 3503 Unit 8-9 as F(2, 8). The argument for Factor B is similar. Notice that Minitab has used this information to compute F-ratios that are different from those in the fixed model. Again here, we do not attempt to interpret the results from our make-believe model. Problems: Table in Ott/Longnecker and similar tables in other books give general formulas for EMSs in the two-factor fixed, mixed, and random models. The coefficients of the terms in each EMS depend on the number of levels of a and b of the two factors and the number n of replications in each cell. Compare these results with those given in the Minitab (restricted) EMS tables in this unit to convince yourself that they agree. By hand, make a table similar to the one in section 8.3 for the case a = 3, b = 4, and n = Compare two-way ANOVA tables for fixed, mixed, and random models. Which of the following columns are unchanged, regardless of which factors are declared as random: DF, SS, MS, F? In Problem we saw that, for the mixed model, Minitab's F-ratios are different depending on whether the restricted model is declared or whether the default unrestricted model is used. Is there a similar difference in F-ratios for the two-way model with both effects random? How about when both effects are fixed? By hand, make a table similar to the one in section 8.3 for each case (both factors fixed, both factors random). Note: If interaction is not significant and the reduced model (leaving out interaction) is used, then all two-way models (fixed/ mixed/ random, restricted/ unrestricted) have the same F- ratios: both main effects are tested against error. The EMS tables are somewhat different, but not in ways that change the F-ratios For the data in this unit, the most fundamental interpretation has been the same whether with the true story (mixed model), or with the two imaginary scenarios (both fixed, both random): Interaction is not significant, A-effect highly significant, and B-effect highly significant. Thus, with these data an untrained "statistician" who does not know the difference between fixed and random effects would get the same results for the three F-tests no matter which model he or she happened to choose. Under what circumstances would the results of these F-tests depend on getting the correct model? (Of course, the interpretation of profile plots, use of multiple comparison procedures for significant main effects, and explanation of the findings would depend on knowing the difference between fixed and random effects in any case.) Minitab Notes for Statistics 3503 by Bruce E. Trumbo, Department of Statistics, CSU Hayward, Hayward CA, 94542, btrumbo@csuhayward.edu. Comments and corrections welcome. Copyright 1991, 1995, 1997, 2000, 2001, 2002, 2004 by Bruce E. Trumbo. All rights reserved. These notes are intended primarily for use at California State University, Hayward. For other uses, please contact the author. Preparation of earlier versions of this document was partially supported by NSF grant USE Modified: 1/04