Experiments, Two‐factor ANOVA, and Regression 1. Read the following excerpt from the news and then identify the characteristics of the experiment that follow. Power of Placebos (Ivanhoe Newswire) ‐‐ What is it about the very act of taking a pill ‐‐ even if it contains no medicine ‐‐ that makes some people feel better? It's commonly called the placebo effect and researchers studying depression have gathered some new insight into this response to treatment. [Janis] Schonfeld was clinically depressed so she enrolled in a study at UCLA where she received pills for her depression. She says, "With each passing week, I just felt that I was really getting better." But her pills actually contained no medicine, just sugar. "We knew that some patients had a transient relief of symptoms. What we didn't know was that we could actually alter the way the brain worked," says neuropsychiatrist Andrew Leuchter, M.D., of the UCLA Neuropsychiatric Institute. In the eight‐week study, some patients received anti‐depressants, while others got a placebo. Researchers periodically monitored patients' brainwaves. Fifty‐two percent of those taking the medication showed improvement, but 38 percent of those taking the placebo said they felt better, too. What stunned researchers is that the brains of people in the placebo group actually showed the change. Dr. Leuchter says, "We can't say whether people feel better because their brain function changed, or whether their brain function changed because they felt better." After the study, doctors told the placebo patients they hadn't been on medication. Schonfeld says, "I said to him, 'I really think that you should check your records. I really think that I was on medication.' And he laughed." Today, her depression is gone, and researchers have more evidence to back up the power of suggestion. Experimental Units/Subjects: Response: Factor: Treatment: From this article, what is the placebo effect? Page 1 of 7 Experiments, Two‐factor ANOVA, and Regression 2. Output from a linear regression of the relationship between the value (in dollars) of a home and the size (in square feet) is provided below. Simple linear regression results: Dependent Variable: value Independent Variable: size value = ‐75819.84 + 165.8999 size Sample size: 99 R (correlation coefficient) = 0.8236 R‐sq = 0.67824453 Estimate of error standard deviation: 82959.67 Parameter estimates: Parameter Estimate Intercept Std. Err.
‐75819.84 Slope Alternative DF
21764.1
T‐Stat
≠ 0 97 ‐3.4837115
165.8999 11.601926
P‐Value 0.0007 ≠ 0 97 14.299342 <0.0001 Analysis of variance table for regression model: Source DF Model SS MS
F‐stat
P‐value 1 1.40723356E12 1.40723356E12 204.47119 <0.0001 Total 98 2.07481746E12 a. Use the regression equation to predict the value of a home that is 2,000 square feet. b. A home owner would like to know what the expected increase in value would be for building an addition of 100 square feet on her home. Provide your best estimate of this number. Error 97 6.6758384E11 6.8823076E9
c. Is the relationship between home value and size positive or negative? Does it appear to be a strong relationship? Explain. Page 2 of 7 Experiments, Two‐factor ANOVA, and Regression d. Is there a significant linear relationship between value and square footage of a home? Complete the hypothesis test, reporting the appropriate values from the output above where necessary. Ho: ____________________________ vs. Ha: _____________________________ Test statistic = ___________________________ p‐value = ___________________________ Conclusion: There (circle one) is is not a significant linear relationship. 3. Raspberries have a high content of many beneficial compounds like vitamins C and E, folic and ellagic acid, calcium, selenium, etc. As a result, researchers have investigated their anti‐cancer properties. From a group of 20 mice, each with a tumor growing just under the skin on their backs, researchers assigned 10 mice to have raspberries added in their diet. The remaining 10 ate a normal diet without the raspberries. The size of the tumors was measured both at the beginning and end of the study, so that the change in their size could be examined. Identify the following characteristics of this experiment. Experimental units/Subjects: Response: Factor: Treatments: Number of replicates per group: Did this experiment use matching and/or blocking? Page 3 of 7 Experiments, Two‐factor ANOVA, and Regression 4. A consumer advocacy group recorded several variables on 140 models of cars. The resulting information was used to produce the following regression output that relates the city gas mileage (in mpg) and the engine displacement (in cubic inches). The regression equation is
mpg:city = 33.4 - 0.0624 displacement
Predictor
Constant
displacement
S = 3.13923
Coef
33.4
-0.0624
SE Coef
0.7762
0.003810
R-Sq = 66.0%
T
43.00
______
P
0.000
0.000
R-Sq(adj) = 65.8%
a. We have a car that has an engine with 150 cubic inches. Based on this output, what city gas mileage would you predict for this car? b. Based on this output what is the correlation between city gas mileage and displacement? The group also recorded the power of the engine (in horsepower) for each car. The following regression output was produced. The regression equation is
mpg:city = 32.2 - 0.0572 horsepower
Predictor
Constant
horsepower
Coef
32.2
-0.0572
S = 3.86688
R-Sq = 48.4%
SE Coef
1.000
0.005022
T
32.19
-11.39
P
0.000
0.000
R-Sq(adj) = 48.1%
c. Which of the variables (horsepower or displacement) would be the better predictor of city gas mileage? Explain. Page 4 of 7 Experiments, Two‐factor ANOVA, and Regression 5. An article in Industrial Quality Control (1956, pp. 5‐8) describes an experiment to investigate the effect of the type of glass and the type of phosphor on the brightness of a television tube. The response variable is the current necessary (in microamps) to obtain a specified brightness level. The data and table of means are as follows: Data in Stacked Form Glass Type 1 2 Table of Treatment Means Phosphor Type 1 280 290 285 230 235 240 2 300 310 295 260 240 235 Phosphor Type
3
290
285
290
220
225
230
Glass Type
1
2
1
285.0
235.0
2 301.7 245.0 3
288.3
225.0
a. Compute the estimated main effects. b. Fill in the 6 missing values in the ANOVA table. Show your work at the right. Source Glass SS DF MS
14450.00 (2) 14450.00
Phosophor 933.33
(3) 466.67
Interaction 133.33
Error 2 12 (1) Total 16150.00 (4) F‐statistic
p‐value <0.0001 (6)
8.84
0.0044 1.26
0.3178 52.78
‐‐‐
‐‐‐
‐‐‐
‐‐‐
‐‐‐
(5)
(1) (2) (3) (4) (5) (6) c. Which of the p‐values in the above ANOVA table should you look at first? List its value here: ______________. What hypotheses are being tested by this p‐value? Are the findings significant at the .05 level? (circle one):
Yes No Page 5 of 7 Experiments, Two‐factor ANOVA, and Regression 6. Johnson and Leone (Statistics and Experimental Design in Engineering and the Physical Sciences, Wiley 1977) describe an experiment to investigate the warping of copper plates. The two factors studied were the temperature and the copper content of the plates. The response variable was a measure of the amount of warping. The technician keeping the data knew to use ANOVA to analyze the data, but was unsure how to interpret the results. Consequently, he fit many different ANOVA tables. Use these to answer the questions on the next page. ANOVA table 1 df SS
MS
F‐Stat
P‐value Temp 3 156.09375
52.03125
7.672811
0.0021 Copper 3 698.34375
232.78125
34.32719
<0.0001 Interaction 9 113.78125
12.642361
1.8643113
0.1327 Error 16 108.5
6.78125
Total 31 1076.7188
Source ANOVA table 2 df SS MS
F‐Stat
P‐value temp 3 156.09375
52.03125
5.851961
0.0036 copper 3 698.34375
232.78125
26.180937
<0.0001 Error 25 222.28125
8.89125
Total 31 1076.7188
df SS MS
F‐Stat
P‐value temp 3 156.09375
52.03125
1.5824847
0.2157 Error 28 920.625
32.879463
Total 31 1076.7188
Source df SS MS
F‐Stat
copper 3 698.34375
232.78125
17.225965
Error 28 378.375
13.513392
Total 31 1076.7188
Source ANOVA table 3 Source ANOVA table 4 P‐value <0.0001 Page 6 of 7 Experiments, Two‐factor ANOVA, and Regression a. How many levels of each factor were fit? # of levels for Temperature: ___________ # of levels for Copper: ___________ b. How many replicates were there per treatment? c. What would be the final model for this data? Explain why you chose this model. 7. The factors that influence the breaking strength of a synthetic fiber are being studied. Four production machines and three operators are chosen and a factorial experiment is run using fiber from the same production batch. The data and table of means are as follows: Data in Stacked Form Machine Operator 1 Table of Means 2 Machine Operator
1 2 . 1 109 110 110 115 1 109.5 112.5 111.0 2 110 112 110 111 2 111.0 110.5 110.8 110.3 111.5 .. =110.9
.
Calculate the total sums of squares. Page 7 of 7