CHAPTER 12: RANDOM EFFECTS

Transcription

1 CHAPTER 12: RANDOM EFFECTS AIRCRAFT COMPONENT STRENGTH(RANDOM EFFECTS) Composite materials used in the manufacture of aircraft components must be tested to determine tensile strength. A manufacturer tests five random specimens from five randomly selected batches. Is there evidence of that tensile strength varies by batch? What is the batch-to-batch variation? What is the variation with the batch? : = = = = or : = 0? batch fixed 5 1, 2, 3, 4, 5 Analysis of Variance for strength, using Adjusted SS for Tests DF Seq SS Adj SS Adj MS F P batch Error Total This first analysis considers batches as fixed. Are there differences among these batches. S = R-Sq = 72.51% R-Sq(adj) = 67.01% batch random 5 1, 2, 3, 4, 5 Analysis of Variance for strength, using Adjusted SS for Tests DF Seq SS Adj SS Adj MS F P batch Error Total S = R-Sq = 72.51% R-Sq(adj) = 67.01% Expected Mean Square for Each Term 1 batch (2) (1) 2 Error (2) 1 batch (2) batch Error This second analysis considers the batches to be a random sample from some larger population of batches. Is there significant variation among batches? With this model, we can decompose the variance in the data: that which is due to the random sample of batches, and that which is due to variation with the batch. We can even get estimates of these two variances. STAT 313/513 - Schaffner Ch12-1.Docx Page 1 of 4

2 VARIATION IN LABORATORIES (NESTED DESIGN) 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 equal is true. The experiment is thus really a study of the laboratories. Lab random 6 I, II, III, IV, V, VI Technician(Lab) random 12 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 Analysis of Variance for Fat_Content, using Adjusted SS for Tests DF Seq SS Adj SS Adj MS F P Lab Technician(Lab) Error Total We re considering both the Lab and Technicians in the labs as random. Why? S = R-Sq = 67.49% R-Sq(adj) = 57.56% Expected Mean Square for Each Term 1 Lab (3) (2) (1) 2 Technician(Lab) (3) (2) 3 Error (3) 1 Lab (2) 2 Technician(Lab) (3) Lab Technician(Lab) Error Note the EMSs and the denominators used for the tests. What are they telling us about the tests? What sources of variability are compared to what? What s the largest source of variation in the measurements? The second largest? The smallest? Fat_Content Individual Plot of Fat_Content What if we consider Lab as a fixed effect? What happens to these values? What are we testing? Technician Lab I II III IV V VI STAT 313/513 - Schaffner Ch12-1.Docx Page 2 of 4

3 MILK CONTAMINATION(TWO CROSSED RANDOM FACTORS) Milk is tested after Pasteurization to assure that Pasteurization was effective. This experiment was conducted to determine the variability in test results between laboratories, and to determine if the interlaboratory differences depend on the concentration of bacteria. Five contract laboratories were selected at random from those available in a large metropolitan area. Four levels of contamination are chosen at random by choosing milk from a collection of samples at various stages of spoilage. Laboratory random 5 1, 2, 3, 4, 5 Contamination random 4 1, 2, 3, 4 Analysis of Variance for ln(cfu), using Adjusted SS for Tests DF Seq SS Adj SS Adj MS F P Laboratory Contamination Laboratory*Contamination Error Total S = R-Sq = 99.45% R-Sq(adj) = 98.93% Expected Mean Square for Each Term 1 Laboratory (4) (3) (1) 2 Contamination (4) (3) (2) 3 Laboratory*Contamination (4) (3) 4 Error (4) Note that the interaction term is significant. Does it make sense to investigate the significance of the main effects? What do the different tests tell us about the labs and the effect of contamination on the variability? 1 Laboratory (3) 2 Contamination (3) 3 Laboratory*Contamination (4) Laboratory Contamination Laboratory*Contamination Error STAT 313/513 - Schaffner Ch12-1.Docx Page 3 of 4

4 SUNSCREENS (MIXED MODEL) A study was designed to evaluate the effectiveness of two different sunscreens. A random sample of 40 subjects each had a 1inch square marked off on their 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 square before application of the sunscreen and sun exposure was made. A second reading after two hours of exposure was made. There is concern that the measurement of the color is extremely variable and wanted to test for as well as account for this in the analysis sunscreen fixed 2 1, 2 tech random 10 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Analysis of Variance for y, using Adjusted SS for Tests DF Seq SS Adj SS Adj MS F P sunscreen tech sunscreen*tech Error Total Based on the p-values, what conclusions can we make about the sunscreens? The technicians? What does the interaction tell us? S = R-Sq = 99.50% R-Sq(adj) = 99.03% Expected Mean Square for Each Term 1 sunscreen (4) (3) + Q[1] 2 tech (4) (3) (2) 3 sunscreen*tech (4) (3) 4 Error (4) 1 sunscreen (3) 2 tech (3) 3 sunscreen*tech (4) tech sunscreen*tech Error STAT 313/513 - Schaffner Ch12-1.Docx Page 4 of 4

5 CHAPTER 12: SPLIT-PLOT One particular partially nested design that is popular is the split-plot design. It is commonly used in two-factor experiments for which one of the factors is difficult to vary. For example, it might be difficult to apply different fertilizers at the level of small plots (subplots) but it might be easy to apply fertilizers to large areas (whole plots). Our example will focus on an experiment that looks how 3 different varieties and two different fertilizers affect soybean yields. For this experiment we have 6 plots of land. We can only apply fertilizers to the whole plots, but can divide each plot in thirds (or even smaller if we want) to apply the different varieties. Fertilizer applied to whole plots Plot 1 Plot 2 Plot 3 Plot 4 Plot 5 Plot 6 F1 F2 F1 F2 F2 F1 Variety applied to subplots V1 V2 V3 V1 V1 V3 V2 V3 V2 V3 V2 V2 V3 V1 V1 V2 V3 V1 When comparing fertilizers we have to compare the variability due to fertilizer relative to the plot to plot variation (whole plot variation). When comparing the varieties and fertilizer variety interaction we compare that variation to the subplot variation. STAT 313/513 - Schaffner Ch12-2.Docx Page 1 of 2

6 Row Variety Fertlizer yield Plot Stat > ANOVA > General linear model yield Response Fertilizer Plot(Fertilizer) Variety Variety*Fertilizer Model Plot Random Factors Results Display expected mean squares Analysis of Variance for yield, using Adjusted SS for Tests DF Seq SS Adj SS Adj MS F P Between Wholeplots Fertilizer Plot(Fertilizer) Whole plot Error Between Subplots Variety Fertilizer*Variety Error Sub plot Error Total S = R-Sq = 99.57% R-Sq(adj) = 99.09% Expected Mean Square for Each Term 1 Fertilizer (5) (2) + Q[1, 4] 2 Plot(Fertilizer) (5) (2) 3 Variety (5) + Q[3, 4] 4 Fertilizer*Variety (5) + Q[4] 5 Error (5) 1 Fertilizer (2) 2 Plot(Fertilizer) (5) 3 Variety (5) 4 Fertilizer*Variety (5) Plot(Fertilizer) Error Model: Yield = Fertilizer + Variety + Fertilizer + Variety +Whole-Plot + Subplot There s a trick to get minitab to fit this model. We want minitab to carry out the tests using the right denominator, so we can do this through a careful use of nesting. Basically it forces minitab to split the error into two parts, one due to the whole plots and one due to subplots STAT 313/513 - Schaffner Ch12-2.Docx Page 2 of 2