Recipe 3: Two Factor, Multi-Level Analysis
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Recipes for the Design of Experiments: Recipe Outline
as of August 28, 2014, superseding the version of August 24. Always use the most recent version.
Cars: Analysis of Horsepower
Max Winkelman
Rensselaer Polytechnic Institute
October 9 2014
Version 1
1. Setting
Cars
The data analyzed in this recipe is a csv file that contains various measured parameters from vehicles.
Install the ‘Cars.csv’ file
cars <- read.csv("~/RPI/Classes/Design of Experiments/R/Cars.csv", header=FALSE)
#reads in the data from the csv file 'Cars.csv' and assigns it to the dataframe 'cars'Factors and Levels
Factor: Country of Origin and Model Year
Levels: Countries 1, 2, and 3 and Model Years 1970-1982
#Summary of Data (Columns are assigned variable names V1 to V6)
head(cars)## V1 V2 V3 V4 V5 V6
## 1 70 1 130 18 307 8
## 2 70 1 165 15 350 8
## 3 70 1 150 18 318 8
## 4 70 1 150 16 304 8
## 5 70 1 140 17 302 8
## 6 70 1 198 15 429 8#displays the first 6 sets of variables
tail(cars)## V1 V2 V3 V4 V5 V6
## 392 82 1 90 36 135 4
## 393 82 1 86 27 151 4
## 394 82 2 52 27 140 4
## 395 82 1 84 44 97 4
## 396 82 1 79 32 135 4
## 397 82 1 82 28 120 4#displays the last 6 sets of variables
summary(cars)## V1 V2 V3 V4 V5
## Min. :70 Min. :1.00 Min. : 46 Min. : 9.0 Min. : 4
## 1st Qu.:73 1st Qu.:1.00 1st Qu.: 75 1st Qu.:17.5 1st Qu.:101
## Median :76 Median :1.00 Median : 92 Median :23.0 Median :146
## Mean :76 Mean :1.57 Mean :104 Mean :23.5 Mean :193
## 3rd Qu.:79 3rd Qu.:2.00 3rd Qu.:125 3rd Qu.:29.0 3rd Qu.:262
## Max. :82 Max. :3.00 Max. :230 Max. :46.6 Max. :455
## NA's :1
## V6
## Min. :3.00
## 1st Qu.:4.00
## Median :4.00
## Mean :5.45
## 3rd Qu.:8.00
## Max. :8.00
## #displays a summary of the variablesContinuous variables:
The csv file, ‘Cars,’ was created for no specific experiment. Certain variables such as “Miles per Gallon,” “Engine Size (cube in),” “Model Year,” and “Horse Power” can be considered continuous variables. The “Country of Origin” and “Number of Cylinders” can be considered categorical variables.
Response variables:
Depending on the analysis preformed on the csv file, ‘Cars,’ several variables can be considered response variables. For this sample recipe, “Horse Power” will be considered as a response variable.
The Data: How is it organized and what does it look like?
The vehicle data of “Cars”" were observed from 1970 to 1982 for Countries “1,” “2,” and “3.” The csv file is for educational purposes and the methods of how that data were gathered are unknown. The data in “Cars” are organized into columns: “Miles per Gallon” (V4) designates the city gas mileage, “Engine Size (cube in)” (V5) represents the volume of the engine in cubic inches, “Model Year” (V1) shows the year that the vehicle was produced, “Horse Power” (V3) displays the horse power, “Country of Origin” (V2) reveals the country in which the vehicle was made, and “Number of Cylinders” (V6) shows the number of cylinders in each vehicle’s engine.
Randomization
The data from “Cars” is organized based on the variable name in each column; however, it can be assumed that the original data was gathered with proper randomization methods.
2. Experimental Design
How will the experiment be organized and conducted to test the hypothesis?
The vehicle data in “Cars” were not acquired with any specific experiment or hypothesis. This data is made publicly available for anyone to test their own hypothesis. For this case, the data from “Cars” will be analyzed to determine if the variation in a vehicle’s “Horse Power”" can be attributed to the variation in the “Year Model” (1970 to 1982) or the “Country of Origin” (Countries 1, 2, and 3). An analysis of variance test will be performed to determine the relationship between the sample horse power means. The null hypothesis of this experiment is that the horse power means across all vehicles from different model years and countries of origin are the same.
What is the rationale for this design?
The rationale behind the vehicle observations was to gather various parameters from different storms and make them readily available. With no specific aim in mind, the data can be analyzed be anyone wishing to test a hypothesis relating to the data.
Randomize: What is the Randomization Scheme?
Since this data was gathered with no specific intention, the randomization scheme, if any, is unknown. The data used in this example will be selected based solely on the year model and the country of origin of the vehicle. Horse power will be selected as the response variable.
Replicate: Are there replicates and/or repeated measures?
There are no replicates or repeated measures in this data set.
Block: Did you use blocking in the design?
The original data set, while organized into variables, was never arranged into experimental groups because no experiment was conducted. For this specific example, the vehicle data will be broken up into groups based on the year model and the country of origin of the vehicle
3. Statistical Analysis
Exploratory Data Analysis: Graphics and Descriptive Summary
#Assign the data types
cars$V1=as.factor(cars$V1)
#makes the variable "V1" (Model Year) a factor
cars$V2=as.factor(cars$V2)
#makes the variable "V2" (Country of Origin) a factor
#Boxplot
boxplot(V3~V1,data=cars, xlab="Model Year", ylab="Horse Power")
title("Model Year Plots of Horse Power")
#boxplot of the horse power data from each model year
boxplot(V3~V2,data=cars, xlab="Country of Origin", ylab="Horse Power")
title("Country of Origin Plots of Horse Power")
#boxplot of the horse power data from each country of originThe boxplots above display the distribution of the variation of horse power that can be attributed to the model year and the vehicle country of origin. No statistical inference can be made from this boxplot without performing a statistical test. By visual inspection, the organization of the first boxplot suggests that the variation of horse power can be attributed to the variation in model year. The second boxplot reveals that vehicles from countries 2 and 3 may have similar distributions of horse power.
ANOVA Testing
An analysis of variance (ANOVA) will be used to determine the statistical significance between the horse power means. The null hypothesis for all three ANOVA tests is that the mean horse power vectors of all samples are equal to each other. The first anova test will analyze the horse power variance as a result of the vehicle model year. The second anova test will analyze the horse power variance as a result of the country of origin. The third anova test will analyze the horse power variance as a result of the interaction between the model year and the country of origin. If the null hypothesis is rejected, the alternative hypothesis, which states that the mean vectors are not equal to each other, is accepted.
# ANOVA
#Model Year
model_year = aov(V3~V1,data=cars)
anova(model_year)## Analysis of Variance Table
##
## Response: V3
## Df Sum Sq Mean Sq F value Pr(>F)
## V1 12 141853 11821 10.4 <2e-16 ***
## Residuals 383 433289 1131
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#performs an anova test
#Country of Origin
model_country = aov(V3~V2,data=cars)
anova(model_country)## Analysis of Variance Table
##
## Response: V3
## Df Sum Sq Mean Sq F value Pr(>F)
## V2 2 136334 68167 61 <2e-16 ***
## Residuals 393 438808 1117
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#performs an anova test
#Model Year and Country of Origin
model_year_country = aov(V3~V1*V2,data=cars)
anova(model_year_country)## Analysis of Variance Table
##
## Response: V3
## Df Sum Sq Mean Sq F value Pr(>F)
## V1 12 141853 11821 14.03 <2e-16 ***
## V2 2 98373 49186 58.39 <2e-16 ***
## V1:V2 24 34202 1425 1.69 0.023 *
## Residuals 357 300714 842
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#performs an anova testANOVA Results: The anova test that analyzed the variation in horse power as a result of the variation in the model year produced a p-value of 2e-16. This indicates that there is a very small probability that the variation of horse power data can be attributed to solely randomization. It is highly likely that the model year during which a vehicle was made has an effect on the horse power mean. The anova test that analyzed the variation in horse power as a result of the variation in the country of origin also produced a p-value of 2e-16. This indicates that there is a very small probability that the variation of horse power data can be attributed to solely randomization. It is highly likely that the country of origin in which a vehicle was made has an effect on the horse power mean. The anova test that analyzed the variation in horse power as a result of the interaction of both the model year and the country of origin produced a p-value of 0.023. This indicates that there is a small probability that the variation of horse power data can be attributed to solely randomization. It is likely that the interaction of the model year and country origin have an effect on the horse power mean. However, without a post-hoc analysis, it is impossible to determine precisely which horse power means are significantly distinct from the others.
Post-Hoc Analysis
Tukey’s Honestly Significantly Difference is a multiple comparison procedure that is used after an ANOVA to determine which specific sample means are significantly different from the others. In this recipe, Tukey’s HSD is used to determine which model year and country of origin produced significantly different horse power means.
#Tukey's HSD
#Model Year
TukeyHSD(model_year, ordered = FALSE, conf.level = 0.95)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = V3 ~ V1, data = cars)
##
## $V1
## diff lwr upr p adj
## 71-70 -39.17725 -69.423 -8.9315 0.0014
## 72-70 -26.03571 -56.005 3.9338 0.1650
## 73-70 -15.73929 -43.370 11.8912 0.7952
## 74-70 -51.91799 -82.164 -21.6723 0.0000
## 75-70 -45.14762 -74.613 -15.6818 0.0000
## 76-70 -45.09664 -73.713 -16.4798 0.0000
## 77-70 -41.14286 -71.112 -11.1733 0.0005
## 78-70 -46.51984 -74.775 -18.2643 0.0000
## 79-70 -45.00739 -74.717 -15.2974 0.0001
## 80-70 -69.35222 -99.062 -39.6422 0.0000
## 81-70 -65.42118 -95.131 -35.7112 0.0000
## 82-70 -64.47235 -93.708 -35.2369 0.0000
## 72-71 13.14153 -17.104 43.3873 0.9659
## 73-71 23.43796 -4.492 51.3679 0.2076
## 74-71 -12.74074 -43.260 17.7787 0.9751
## 75-71 -5.97037 -35.717 23.7763 1.0000
## 76-71 -5.91939 -34.825 22.9866 1.0000
## 77-71 -1.96561 -32.211 28.2801 1.0000
## 78-71 -7.34259 -35.891 21.2057 0.9997
## 79-71 -5.83014 -35.819 24.1585 1.0000
## 80-71 -30.17497 -60.164 -0.1863 0.0469
## 81-71 -26.24393 -56.233 3.7447 0.1567
## 82-71 -25.29510 -54.814 4.2234 0.1812
## 73-72 10.29643 -17.334 37.9270 0.9904
## 74-72 -25.88228 -56.128 4.3635 0.1829
## 75-72 -19.11190 -48.578 10.3539 0.6192
## 76-72 -19.06092 -47.678 9.5559 0.5767
## 77-72 -15.10714 -45.077 14.8624 0.9014
## 78-72 -20.48413 -48.740 7.7714 0.4343
## 79-72 -18.97167 -48.682 10.7384 0.6434
## 80-72 -43.31650 -73.027 -13.6065 0.0001
## 81-72 -39.38547 -69.096 -9.6754 0.0009
## 82-72 -38.43664 -67.672 -9.2012 0.0011
## 74-73 -36.17870 -64.109 -8.2488 0.0014
## 75-73 -29.40833 -56.492 -2.3250 0.0199
## 76-73 -29.35735 -55.514 -3.2002 0.0130
## 77-73 -25.40357 -53.034 2.2270 0.1075
## 78-73 -30.78056 -56.542 -5.0192 0.0054
## 79-73 -29.26810 -56.617 -1.9192 0.0237
## 80-73 -53.61293 -80.962 -26.2641 0.0000
## 81-73 -49.68190 -77.031 -22.3330 0.0000
## 82-73 -48.73306 -75.566 -21.9005 0.0000
## 75-74 6.77037 -22.976 36.5171 0.9999
## 76-74 6.82135 -22.085 35.7273 0.9999
## 77-74 10.77513 -19.471 41.0209 0.9935
## 78-74 5.39815 -23.150 33.9465 1.0000
## 79-74 6.91060 -23.078 36.8992 0.9999
## 80-74 -17.43423 -47.423 12.5544 0.7717
## 81-74 -13.50319 -43.492 16.4855 0.9553
## 82-74 -12.55436 -42.073 16.9642 0.9711
## 76-75 0.05098 -28.038 28.1398 1.0000
## 77-75 4.00476 -25.461 33.4706 1.0000
## 78-75 -1.37222 -29.093 26.3485 1.0000
## 79-75 0.14023 -29.062 29.3421 1.0000
## 80-75 -24.20460 -53.406 4.9972 0.2241
## 81-75 -20.27356 -49.475 8.9283 0.5077
## 82-75 -19.32473 -48.044 9.3941 0.5600
## 77-76 3.95378 -24.663 32.5706 1.0000
## 78-76 -1.42320 -28.240 25.3933 1.0000
## 79-76 0.08925 -28.256 28.4342 1.0000
## 80-76 -24.25558 -52.601 4.0894 0.1829
## 81-76 -20.32454 -48.669 8.0204 0.4529
## 82-76 -19.37571 -47.223 8.4714 0.5040
## 78-77 -5.37698 -33.633 22.8785 1.0000
## 79-77 -3.86453 -33.575 25.8455 1.0000
## 80-77 -28.20936 -57.919 1.5007 0.0818
## 81-77 -24.27833 -53.988 5.4317 0.2438
## 82-77 -23.32949 -52.565 5.9060 0.2785
## 79-78 1.51245 -26.468 29.4926 1.0000
## 80-78 -22.83238 -50.813 5.1478 0.2458
## 81-78 -18.90134 -46.881 9.0788 0.5535
## 82-78 -17.95251 -45.428 9.5232 0.6076
## 80-79 -24.34483 -53.793 5.1034 0.2276
## 81-79 -20.41379 -49.862 9.0345 0.5103
## 82-79 -19.46496 -48.434 9.5044 0.5624
## 81-80 3.93103 -25.517 33.3793 1.0000
## 82-80 4.87987 -24.090 33.8493 1.0000
## 82-81 0.94883 -28.021 29.9182 1.0000#performs a THSD test for the model year
#Country of Origin
TukeyHSD(model_country, ordered = FALSE, conf.level = 0.95)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = V3 ~ V2, data = cars)
##
## $V2
## diff lwr upr p adj
## 2-1 -38.069 -48.71 -27.42 0.0000
## 3-1 -38.505 -48.67 -28.34 0.0000
## 3-2 -0.436 -13.34 12.47 0.9965#performs a THSD test for the country of origin
#Model Year and Country of Origin
TukeyHSD(model_year_country, ordered = FALSE, conf.level = 0.95)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = V3 ~ V1 * V2, data = cars)
##
## $V1
## diff lwr upr p adj
## 71-70 -39.17725 -65.2880 -13.0665 0.0001
## 72-70 -26.03571 -51.9080 -0.1635 0.0469
## 73-70 -15.73929 -39.5923 8.1137 0.5912
## 74-70 -51.91799 -78.0287 -25.8073 0.0000
## 75-70 -45.14762 -70.5850 -19.7102 0.0000
## 76-70 -45.09664 -69.8011 -20.3922 0.0000
## 77-70 -41.14286 -67.0151 -15.2706 0.0000
## 78-70 -46.51984 -70.9124 -22.1273 0.0000
## 79-70 -45.00739 -70.6556 -19.3591 0.0000
## 80-70 -69.35222 -95.0005 -43.7040 0.0000
## 81-70 -65.42118 -91.0694 -39.7729 0.0000
## 82-70 -64.47235 -89.7109 -39.2338 0.0000
## 72-71 13.14153 -12.9692 39.2522 0.9020
## 73-71 23.43796 -0.6735 47.5494 0.0659
## 74-71 -12.74074 -39.0877 13.6063 0.9251
## 75-71 -5.97037 -31.6503 19.7095 0.9999
## 76-71 -5.91939 -30.8735 19.0347 0.9999
## 77-71 -1.96561 -28.0763 24.1451 1.0000
## 78-71 -7.34259 -31.9880 17.3028 0.9988
## 79-71 -5.83014 -31.7189 20.0586 0.9999
## 80-71 -30.17497 -56.0637 -4.2862 0.0078
## 81-71 -26.24393 -52.1327 -0.3552 0.0435
## 82-71 -25.29510 -50.7780 0.1878 0.0538
## 73-72 10.29643 -13.5566 34.1495 0.9674
## 74-72 -25.88228 -51.9930 0.2284 0.0546
## 75-72 -19.11190 -44.5493 6.3255 0.3737
## 76-72 -19.06092 -43.7654 5.6435 0.3304
## 77-72 -15.10714 -40.9794 10.7651 0.7658
## 78-72 -20.48413 -44.8767 3.9085 0.2064
## 79-72 -18.97167 -44.6199 6.6766 0.3997
## 80-72 -43.31650 -68.9647 -17.6683 0.0000
## 81-72 -39.38547 -65.0337 -13.7372 0.0000
## 82-72 -38.43664 -63.6752 -13.1981 0.0000
## 74-73 -36.17870 -60.2902 -12.0672 0.0001
## 75-73 -29.40833 -52.7890 -6.0277 0.0024
## 76-73 -29.35735 -51.9384 -6.7763 0.0013
## 77-73 -25.40357 -49.2566 -1.5505 0.0251
## 78-73 -30.78056 -53.0200 -8.5411 0.0004
## 79-73 -29.26810 -52.8780 -5.6582 0.0030
## 80-73 -53.61293 -77.2228 -30.0031 0.0000
## 81-73 -49.68190 -73.2918 -26.0720 0.0000
## 82-73 -48.73306 -71.8972 -25.5689 0.0000
## 75-74 6.77037 -18.9095 32.4503 0.9996
## 76-74 6.82135 -18.1327 31.7754 0.9995
## 77-74 10.77513 -15.3356 36.8858 0.9772
## 78-74 5.39815 -19.2472 30.0435 0.9999
## 79-74 6.91060 -18.9782 32.7994 0.9996
## 80-74 -17.43423 -43.3230 8.4545 0.5580
## 81-74 -13.50319 -39.3920 12.3856 0.8770
## 82-74 -12.55436 -38.0373 12.9286 0.9150
## 76-75 0.05098 -24.1977 24.2997 1.0000
## 77-75 4.00476 -21.4326 29.4422 1.0000
## 78-75 -1.37222 -25.3031 22.5586 1.0000
## 79-75 0.14023 -25.0693 25.3497 1.0000
## 80-75 -24.20460 -49.4141 1.0049 0.0739
## 81-75 -20.27356 -45.4831 4.9360 0.2666
## 82-75 -19.32473 -44.1173 5.4678 0.3143
## 77-76 3.95378 -20.7507 28.6583 1.0000
## 78-76 -1.42320 -24.5735 21.7271 1.0000
## 79-76 0.08925 -24.3805 24.5590 1.0000
## 80-76 -24.25558 -48.7254 0.2142 0.0546
## 81-76 -20.32454 -44.7943 4.1452 0.2209
## 82-76 -19.37571 -43.4157 4.6643 0.2634
## 78-77 -5.37698 -29.7696 19.0156 0.9999
## 79-77 -3.86453 -29.5128 21.7837 1.0000
## 80-77 -28.20936 -53.8576 -2.5611 0.0170
## 81-77 -24.27833 -49.9266 1.3699 0.0840
## 82-77 -23.32949 -48.5680 1.9091 0.1028
## 79-78 1.51245 -22.6424 25.6673 1.0000
## 80-78 -22.83238 -46.9872 1.3225 0.0850
## 81-78 -18.90134 -43.0562 5.2535 0.3082
## 82-78 -17.95251 -41.6719 5.7669 0.3616
## 80-79 -24.34483 -49.7671 1.0774 0.0757
## 81-79 -20.41379 -45.8361 5.0085 0.2688
## 82-79 -19.46496 -44.4738 5.5439 0.3166
## 81-80 3.93103 -21.4912 29.3533 1.0000
## 82-80 4.87987 -20.1290 29.8887 1.0000
## 82-81 0.94883 -24.0600 25.9577 1.0000
##
## $V2
## diff lwr upr p adj
## 2-1 -35.061 -44.310 -25.81 0.0000
## 3-1 -27.440 -36.269 -18.61 0.0000
## 3-2 7.621 -3.591 18.83 0.2471
##
## $`V1:V2`
## diff lwr upr p adj
## 71:1-70:1 -4.587e+01 -81.802 -9.94241 0.0006
## 72:1-70:1 -2.694e+01 -63.386 9.51347 0.6021
## 73:1-70:1 -1.909e+01 -51.609 13.42166 0.9536
## 74:1-70:1 -5.465e+01 -93.010 -16.28513 0.0000
## 75:1-70:1 -5.701e+01 -92.469 -21.55923 0.0000
## 76:1-70:1 -5.521e+01 -89.834 -20.59452 0.0000
## 77:1-70:1 -4.733e+01 -83.775 -10.87542 0.0004
## 78:1-70:1 -5.844e+01 -93.061 -23.82180 0.0000
## 79:1-70:1 -5.628e+01 -90.530 -22.02926 0.0000
## 80:1-70:1 -7.957e+01 -129.097 -30.04566 0.0000
## 81:1-70:1 -8.118e+01 -121.223 -41.12892 0.0000
## 82:1-70:1 -7.861e+01 -114.069 -43.15923 0.0000
## 70:2-70:1 -7.951e+01 -135.982 -23.04622 0.0000
## 71:2-70:1 -9.171e+01 -153.621 -29.80708 0.0000
## 72:2-70:1 -8.611e+01 -142.582 -29.64622 0.0000
## 73:2-70:1 -8.386e+01 -133.383 -34.33138 0.0000
## 74:2-70:1 -9.155e+01 -144.078 -39.01761 0.0000
## 75:2-70:1 -7.621e+01 -128.744 -23.68428 0.0000
## 76:2-70:1 -7.809e+01 -125.236 -30.94216 0.0000
## 77:2-70:1 -8.471e+01 -146.621 -22.80708 0.0001
## 78:2-70:1 -6.655e+01 -119.078 -14.01761 0.0007
## 79:2-70:1 -9.371e+01 -155.621 -31.80708 0.0000
## 80:2-70:1 -9.894e+01 -144.147 -53.72588 0.0000
## 81:2-70:1 -8.971e+01 -151.621 -27.80708 0.0000
## 82:2-70:1 -1.027e+02 -186.689 -18.73926 0.0016
## 70:3-70:1 -7.421e+01 -158.189 9.76074 0.1941
## 71:3-70:1 -8.646e+01 -148.371 -24.55708 0.0001
## 72:3-70:1 -7.191e+01 -128.382 -15.44622 0.0007
## 73:3-70:1 -6.721e+01 -129.121 -5.30708 0.0150
## 74:3-70:1 -9.321e+01 -145.744 -40.68428 0.0000
## 75:3-70:1 -8.546e+01 -147.371 -23.55708 0.0001
## 76:3-70:1 -8.921e+01 -151.121 -27.30708 0.0000
## 77:3-70:1 -8.455e+01 -137.078 -32.01761 0.0000
## 78:3-70:1 -8.646e+01 -133.611 -39.31716 0.0000
## 79:3-70:1 -1.007e+02 -184.689 -16.73926 0.0024
## 80:3-70:1 -8.687e+01 -126.915 -46.82122 0.0000
## 81:3-70:1 -8.738e+01 -128.446 -46.31636 0.0000
## 82:3-70:1 -9.171e+01 -136.925 -46.50365 0.0000
## 72:1-71:1 1.894e+01 -18.389 56.26057 0.9948
## 73:1-71:1 2.678e+01 -6.715 60.27171 0.4076
## 74:1-71:1 -8.775e+00 -47.970 30.41931 1.0000
## 75:1-71:1 -1.114e+01 -47.496 25.21182 1.0000
## 76:1-71:1 -9.342e+00 -44.882 26.19767 1.0000
## 77:1-71:1 -1.453e+00 -38.778 35.87168 1.0000
## 78:1-71:1 -1.257e+01 -48.109 22.97039 1.0000
## 79:1-71:1 -1.041e+01 -45.587 24.77259 1.0000
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## 76:3-79:2 4.500e+00 -75.741 84.74091 1.0000
## 77:3-79:2 9.167e+00 -64.083 82.41626 1.0000
## 78:3-79:2 7.250e+00 -62.241 76.74067 1.0000
## 79:3-79:2 -7.000e+00 -105.275 91.27465 1.0000
## 80:3-79:2 6.846e+00 -58.037 71.72956 1.0000
## 81:3-79:2 6.333e+00 -59.183 71.84976 1.0000
## 82:3-79:2 2.000e+00 -66.192 70.19166 1.0000
## 81:2-80:2 9.222e+00 -58.969 77.41388 1.0000
## 82:2-80:2 -3.778e+00 -92.487 84.93189 1.0000
## 70:3-80:2 2.472e+01 -63.987 113.43189 1.0000
## 71:3-80:2 1.247e+01 -55.719 80.66388 1.0000
## 72:3-80:2 2.702e+01 -36.273 90.31711 0.9998
## 73:3-80:2 3.172e+01 -36.469 99.91388 0.9989
## 74:3-80:2 5.722e+00 -54.086 65.53027 1.0000
## 75:3-80:2 1.347e+01 -54.719 81.66388 1.0000
## 76:3-80:2 9.722e+00 -58.469 77.91388 1.0000
## 77:3-80:2 1.439e+01 -45.419 74.19693 1.0000
## 78:3-80:2 1.247e+01 -42.668 67.61251 1.0000
## 79:3-80:2 -1.778e+00 -90.487 86.93189 1.0000
## 80:3-80:2 1.207e+01 -37.139 61.27565 1.0000
## 81:3-80:2 1.156e+01 -38.483 61.59456 1.0000
## 82:3-80:2 7.222e+00 -46.272 60.71616 1.0000
## 82:2-81:2 -1.300e+01 -111.275 85.27465 1.0000
## 70:3-81:2 1.550e+01 -82.775 113.77465 1.0000
## 71:3-81:2 3.250e+00 -76.991 83.49091 1.0000
## 72:3-81:2 1.780e+01 -58.323 93.92321 1.0000
## 73:3-81:2 2.250e+01 -57.741 102.74091 1.0000
## 74:3-81:2 -3.500e+00 -76.750 69.74960 1.0000
## 75:3-81:2 4.250e+00 -75.991 84.49091 1.0000
## 76:3-81:2 5.000e-01 -79.741 80.74091 1.0000
## 77:3-81:2 5.167e+00 -68.083 78.41626 1.0000
## 78:3-81:2 3.250e+00 -66.241 72.74067 1.0000
## 79:3-81:2 -1.100e+01 -109.275 87.27465 1.0000
## 80:3-81:2 2.846e+00 -62.037 67.72956 1.0000
## 81:3-81:2 2.333e+00 -63.183 67.84976 1.0000
## 82:3-81:2 -2.000e+00 -70.192 66.19166 1.0000
## 70:3-82:2 2.850e+01 -84.978 141.97779 1.0000
## 71:3-82:2 1.625e+01 -82.025 114.52465 1.0000
## 72:3-82:2 3.080e+01 -64.142 125.74233 1.0000
## 73:3-82:2 3.550e+01 -62.775 133.77465 1.0000
## 74:3-82:2 9.500e+00 -83.154 102.15422 1.0000
## 75:3-82:2 1.725e+01 -81.025 115.52465 1.0000
## 76:3-82:2 1.350e+01 -84.775 111.77465 1.0000
## 77:3-82:2 1.817e+01 -74.488 110.82089 1.0000
## 78:3-82:2 1.625e+01 -73.462 105.96207 1.0000
## 79:3-82:2 2.000e+00 -111.478 115.47779 1.0000
## 80:3-82:2 1.585e+01 -70.346 102.03872 1.0000
## 81:3-82:2 1.533e+01 -71.337 102.00342 1.0000
## 82:3-82:2 1.100e+01 -77.710 99.70967 1.0000
## 71:3-70:3 -1.225e+01 -110.525 86.02465 1.0000
## 72:3-70:3 2.300e+00 -92.642 97.24233 1.0000
## 73:3-70:3 7.000e+00 -91.275 105.27465 1.0000
## 74:3-70:3 -1.900e+01 -111.654 73.65422 1.0000
## 75:3-70:3 -1.125e+01 -109.525 87.02465 1.0000
## 76:3-70:3 -1.500e+01 -113.275 83.27465 1.0000
## 77:3-70:3 -1.033e+01 -102.988 82.32089 1.0000
## 78:3-70:3 -1.225e+01 -101.962 77.46207 1.0000
## 79:3-70:3 -2.650e+01 -139.978 86.97779 1.0000
## 80:3-70:3 -1.265e+01 -98.846 73.53872 1.0000
## 81:3-70:3 -1.317e+01 -99.837 73.50342 1.0000
## 82:3-70:3 -1.750e+01 -106.210 71.20967 1.0000
## 72:3-71:3 1.455e+01 -61.573 90.67321 1.0000
## 73:3-71:3 1.925e+01 -60.991 99.49091 1.0000
## 74:3-71:3 -6.750e+00 -80.000 66.49960 1.0000
## 75:3-71:3 1.000e+00 -79.241 81.24091 1.0000
## 76:3-71:3 -2.750e+00 -82.991 77.49091 1.0000
## 77:3-71:3 1.917e+00 -71.333 75.16626 1.0000
## 78:3-71:3 2.700e-13 -69.491 69.49067 1.0000
## 79:3-71:3 -1.425e+01 -112.525 84.02465 1.0000
## 80:3-71:3 -4.038e-01 -65.287 64.47956 1.0000
## 81:3-71:3 -9.167e-01 -66.433 64.59976 1.0000
## 82:3-71:3 -5.250e+00 -73.442 62.94166 1.0000
## 73:3-72:3 4.700e+00 -71.423 80.82321 1.0000
## 74:3-72:3 -2.130e+01 -90.014 47.41421 1.0000
## 75:3-72:3 -1.355e+01 -89.673 62.57321 1.0000
## 76:3-72:3 -1.730e+01 -93.423 58.82321 1.0000
## 77:3-72:3 -1.263e+01 -81.348 56.08088 1.0000
## 78:3-72:3 -1.455e+01 -79.242 50.14229 1.0000
## 79:3-72:3 -2.880e+01 -123.742 66.14233 1.0000
## 80:3-72:3 -1.495e+01 -74.670 44.76212 1.0000
## 81:3-72:3 -1.547e+01 -75.870 44.93650 1.0000
## 82:3-72:3 -1.980e+01 -83.095 43.49488 1.0000
## 74:3-73:3 -2.600e+01 -99.250 47.24960 1.0000
## 75:3-73:3 -1.825e+01 -98.491 61.99091 1.0000
## 76:3-73:3 -2.200e+01 -102.241 58.24091 1.0000
## 77:3-73:3 -1.733e+01 -90.583 55.91626 1.0000
## 78:3-73:3 -1.925e+01 -88.741 50.24067 1.0000
## 79:3-73:3 -3.350e+01 -131.775 64.77465 1.0000
## 80:3-73:3 -1.965e+01 -84.537 45.22956 1.0000
## 81:3-73:3 -2.017e+01 -85.683 45.34976 1.0000
## 82:3-73:3 -2.450e+01 -92.692 43.69166 1.0000
## 75:3-74:3 7.750e+00 -65.500 80.99960 1.0000
## 76:3-74:3 4.000e+00 -69.250 77.24960 1.0000
## 77:3-74:3 8.667e+00 -56.850 74.18310 1.0000
## 78:3-74:3 6.750e+00 -54.535 68.03501 1.0000
## 79:3-74:3 -7.500e+00 -100.154 85.15422 1.0000
## 80:3-74:3 6.346e+00 -49.661 62.35290 1.0000
## 81:3-74:3 5.833e+00 -50.906 62.57223 1.0000
## 82:3-74:3 1.500e+00 -58.308 61.30804 1.0000
## 76:3-75:3 -3.750e+00 -83.991 76.49091 1.0000
## 77:3-75:3 9.167e-01 -72.333 74.16626 1.0000
## 78:3-75:3 -1.000e+00 -70.491 68.49067 1.0000
## 79:3-75:3 -1.525e+01 -113.525 83.02465 1.0000
## 80:3-75:3 -1.404e+00 -66.287 63.47956 1.0000
## 81:3-75:3 -1.917e+00 -67.433 63.59976 1.0000
## 82:3-75:3 -6.250e+00 -74.442 61.94166 1.0000
## 77:3-76:3 4.667e+00 -68.583 77.91626 1.0000
## 78:3-76:3 2.750e+00 -66.741 72.24067 1.0000
## 79:3-76:3 -1.150e+01 -109.775 86.77465 1.0000
## 80:3-76:3 2.346e+00 -62.537 67.22956 1.0000
## 81:3-76:3 1.833e+00 -63.683 67.34976 1.0000
## 82:3-76:3 -2.500e+00 -70.692 65.69166 1.0000
## 78:3-77:3 -1.917e+00 -63.202 59.36834 1.0000
## 79:3-77:3 -1.617e+01 -108.821 76.48756 1.0000
## 80:3-77:3 -2.321e+00 -58.327 53.68623 1.0000
## 81:3-77:3 -2.833e+00 -59.572 53.90556 1.0000
## 82:3-77:3 -7.167e+00 -66.975 52.64138 1.0000
## 79:3-78:3 -1.425e+01 -103.962 75.46207 1.0000
## 80:3-78:3 -4.038e-01 -51.396 50.58836 1.0000
## 81:3-78:3 -9.167e-01 -52.712 50.87862 1.0000
## 82:3-78:3 -5.250e+00 -60.390 49.89029 1.0000
## 80:3-79:3 1.385e+01 -72.346 100.03872 1.0000
## 81:3-79:3 1.333e+01 -73.337 100.00342 1.0000
## 82:3-79:3 9.000e+00 -79.710 97.70967 1.0000
## 81:3-80:3 -5.128e-01 -45.940 44.91465 1.0000
## 82:3-80:3 -4.846e+00 -54.053 44.36112 1.0000
## 82:3-81:3 -4.333e+00 -54.372 45.70567 1.0000#performs a THSD test for the interaction of the model year and the country of originTukey’s HSD returns a large matrix that contains statistical parameters for the interaction of an individual sample mean with every other sample mean being statistically analyzed. For the post-hoc analysis of the model year and the interaction of the model year and the country of origin, large matrices are returned due to the large number of comparisons that are made. Since the confidence level selected for these post-hoc test was 0.95, any p adj value less than 0.05 indicates that there is a high probability that the variation between the two horse power means being compared is not due solely to randomization. This is easier to visualize in the post-hoc analysis of the horse power means of the country of origin. The comparison made between country 2 and country 1 and country 3 and country 1 both returned p adj values of 0.00 and produced confidence intervals that do not contain the number 0. This indicates that there is a high probability that the variation of horse power means observed between country 2 and 1 and country 3 and 1 cannot be attributed solely to randomization. It is very likely that the difference in horse power variation between those vehicles is a result of the country of origin. The post-hoc test between country 2 and country 3 returned a p adj value of 0.25 and produced a confidence interval that contains the number 0. This indicates that the variation of horse power means observed between country 2 and country 3 can be attributed to randomization.
Estimation of Parameters
Summary of all factors and levels
#Summaries
seventy<-cars$V1=="70"
#assigns all of the '70' data to the vector, seventy
summary(cars[seventy,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 46 95 150 146 198 225#displays a summary of the horse power data for the year of 1970
seventy_one<-cars$V1=="71"
#assigns all of the '71' data to the vector, seventy_one
summary(cars[seventy_one,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 60 81 95 107 130 180 1#displays a summary of the horse power data for the year of 1971
seventy_two<-cars$V1=="72"
#assigns all of the '72' data to the vector, seventy_two
summary(cars[seventy_two,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 54.0 86.8 104.0 120.0 151.0 208.0#displays a summary of the horse power data for the year of 1972
seventy_three<-cars$V1=="73"
#assigns all of the '73' data to the vector, seventy_three
summary(cars[seventy_three,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 46.0 93.2 130.0 130.0 160.0 230.0#displays a summary of the horse power data for the year of 1973
seventy_four<-cars$V1=="74"
#assigns all of the '74' data to the vector, seventy_four
summary(cars[seventy_four,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 52.0 75.0 93.0 94.3 102.0 150.0#displays a summary of the horse power data for the year of 1974
seventy_five<-cars$V1=="75"
#assigns all of the '75' data to the vector, seventy_five
summary(cars[seventy_five,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 53.0 84.2 97.0 101.0 110.0 170.0#displays a summary of the horse power data for the year of 1975
seventy_six<-cars$V1=="76"
#assigns all of the '76' data to the vector, seventy_six
summary(cars[seventy_six,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 52.0 78.2 93.5 101.0 120.0 180.0#displays a summary of the horse power data for the year of 1976
seventy_seven<-cars$V1=="77"
#assigns all of the '77' data to the vector, seventy_seven
summary(cars[seventy_seven,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 58.0 78.0 97.5 105.0 115.0 190.0#displays a summary of the horse power data for the year of 1977
seventy_eight<-cars$V1=="78"
#assigns all of the '78' data to the vector, seventy_eight
summary(cars[seventy_eight,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 48.0 82.5 97.0 99.7 116.0 165.0#displays a summary of the horse power data for the year of 1978
seventy_nine<-cars$V1=="79"
#assigns all of the '79' data to the vector, seventy_nine
summary(cars[seventy_nine,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 65 77 90 101 125 155#displays a summary of the horse power data for the year of 1979
eighty<-cars$V1=="80"
#assigns all of the '80' data to the vector, eighty
summary(cars[eighty,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 48.0 67.0 72.0 76.9 90.0 132.0#displays a summary of the horse power data for the year of 1980
eighty_one<-cars$V1=="81"
#assigns all of the '81' data to the vector, eighty_one
summary(cars[eighty_one,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 58.0 65.0 75.0 80.8 88.0 120.0#displays a summary of the horse power data for the year of 1981
eighty_two<-cars$V1=="82"
#assigns all of the '82' data to the vector, eighty_two
summary(cars[eighty_two,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 52.0 70.0 84.0 81.7 89.0 112.0#displays a summary of the horse power data for the year of 1982
#Country of Origin
country_one<-cars$V2=="1"
#assigns all of the '1' to the vector, country_one
summary(cars[country_one,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 52 88 105 118 150 230 1#displays a summary of the horse power data for country 1
country_two<-cars$V2=="2"
#assigns all of the '2' to the vector, country_two
summary(cars[country_two,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 46.0 69.2 76.0 80.3 90.0 133.0#displays a summary of the horse power data for country 2
country_three<-cars$V2=="3"
#assigns all of the '3' to the vector, country_three
summary(cars[country_three,"V3"])## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 52.0 67.0 75.0 79.8 95.0 132.0#displays a summary of the horse power data for country 3Diagnostics/Model Adequacy Checking
Quantile-Quantile (Q-Q) plots are graphs used to verify the distributional assumption for a set of data. Based on the theoretical distribution, the expected value for each datum is determined. If the data values in a set follow the theoretical distribution, then they will appear as a straight line on a Q-Q plot. When an anova is performed, it is done so with the assumption that the test statistic follows a normal distribution. Visualization of a Q-Q plot will further confirm if that assumption is correct for the anova tests that were performed.
#Q-Q Plots
#Model Year
qqnorm(residuals(model_year), main="Normal Q-Q Plot for Model Year", ylab="Horse Power Residuals")
qqline(residuals(model_year))
#produces a Q-Q normal plot for the model Year residuals with a normal fit line
#Country of Origin
qqnorm(residuals(model_country),main="Normal Q-Q Plot for Country of Origin", ylab="Horse Power Residuals")
qqline(residuals(model_country))
#produces a Q-Q normal plot for the model Year residuals with a normal fit line
#CModel Year and Country of Origin
qqnorm(residuals(model_year_country), main="Normal Q-Q Plot for the Interaction of Model Year and Country of Origin", ylab="Horse Power Residuals")
qqline(residuals(model_year_country))
#produces a Q-Q normal plot for the model Year residuals with a normal fit lineThe Normal Q-Q plots for the model year and the interaction of model year and country of origin returned relatively linear relationships between the horse power values and the theoretical quantities, indicating that they follow the theoretical distribution. The Normal Q-Q plot for the country of origin also produced a relatively linear line, but contained tails that deviated from the linear line on the Q-Q plot. This could indicate that the data is not normally distributed; however, no certain conclusion can be made by visual inspection. It is likely that the small quantity of countries made any deviation from the linear plot look more pronounced. It can be assumed that the tail deviations do not indicate that the data set is not normally distributed. For an ANOVA, the data sets must follow a normal distribution. The relatively linear relationship for all three data sets justifies the use of ANOVA to test for the significant difference.
Two Way Interaction Plots display the mean of the response for two-way combinations of factors, and can indicate if there is any interactions between them through visual inspection. Data sets that do not have any interaction will appear as perfectly parallel lines. Changes in slope and intersections are good indications of interactions.
interaction.plot(cars$V1,cars$V2,cars$V3, xlab="Model Year", ylab="Means of Horse Power", main="Interaction Plot", trace.label="Country")
#creates a plot that shows the interaction of the model and country on a vehicles's horsepowerIn the interaction plot above, there are clear observations of changes in slope and plot intersections, indicating some degree of interaction between the model year and the country of origin of the vehicle, in regards to the response variable, horse power. This supports the results observed from the anova and the post-hoc analysis.
A Residuals vs. Fits Plot is a common graph used in residual analysis. It is a scatter plot of residuals as a function of fitted values, or the estimated responses. These plots are used to identify linearity, outliers, and error variances.
#Model Year
plot(fitted(model_year),residuals(model_year), main="Residual vs Fitted Plot for Model Year") 
#Country of Origin
plot(fitted(model_country),residuals(model_country), main="Residual vs Fitted Plot for Country of Origin")
#Model Year and Country
plot(fitted(model_year_country),residuals(model_year_country), main="Residual vs Fitted Plot for Interaction") 