Recipe 3
Matthew Macchi
Rensselaer Polytechnic Institute
10/9/14 Version 1
1. Setting
System under test
This recipe will conduct an experiment on the ecdat dataset. The experiment will attempt to investigate the Air Quality for Californian Metropolitan Areas dataset and examine the analysis of variance between the amount of rain (in inches) and coastal location (yes or no) on air quality in hopes of supporting or refuting the claim that the amount of rain and location on air quality do not have much variance.
install.packages("Ecdat", repos='http://cran.us.r-project.org')##
## The downloaded binary packages are in
## /var/folders/55/ql66yz5j3jzgkn6dmnb9sk1c0000gn/T//RtmpbCNfUt/downloaded_packageslibrary("Ecdat", lib.loc="/Library/Frameworks/R.framework/Versions/3.1/Resources/library")## Loading required package: Ecfun
##
## Attaching package: 'Ecdat'
##
## The following object is masked from 'package:datasets':
##
## Orangey<-Airq
head(y)## airq vala rain coas dens medi
## 1 104 2734 12.63 yes 1815.9 4397
## 2 85 2479 47.14 yes 804.9 5667
## 3 127 4845 42.77 yes 1907.9 15817
## 4 145 19734 33.18 no 1876.1 32698
## 5 84 4094 34.55 yes 340.9 6250
## 6 135 1850 14.81 no 335.5 4705Factors and Levels
A factor of an experiment is a controlled independent variable; a variable whose levels are set by the experimenter. In this instance, I am conducting a two-factor analysis.
The term level is also used for categorical variables. In this case, this is a multi-level analysis.
The first factor that this experiment will examine is the amount of rain in a certain area.
The second factor that I will consider is whether or not the location is coastal.
head(y)## airq vala rain coas dens medi
## 1 104 2734 12.63 yes 1815.9 4397
## 2 85 2479 47.14 yes 804.9 5667
## 3 127 4845 42.77 yes 1907.9 15817
## 4 145 19734 33.18 no 1876.1 32698
## 5 84 4094 34.55 yes 340.9 6250
## 6 135 1850 14.81 no 335.5 4705tail(y)## airq vala rain coas dens medi
## 25 74 5609 42.36 yes 2649.1 8947
## 26 124 3700 29.51 no 9642.9 5952
## 27 69 1396 42.92 yes 1105.5 4146
## 28 118 3023 41.32 no 910.8 3207
## 29 129 1515 31.22 no 379.6 853
## 30 129 1879 30.95 no 455.9 853summary(y)## airq vala rain coas dens
## Min. : 59 Min. : 993 Min. :12.6 no : 9 Min. : 272
## 1st Qu.: 81 1st Qu.: 1536 1st Qu.:31.0 yes:21 1st Qu.: 365
## Median :114 Median : 2630 Median :36.7 Median : 796
## Mean :105 Mean : 4188 Mean :36.1 Mean : 1729
## 3rd Qu.:126 3rd Qu.: 4141 3rd Qu.:42.7 3rd Qu.: 1635
## Max. :165 Max. :19734 Max. :68.1 Max. :12958
## medi
## Min. : 853
## 1st Qu.: 3340
## Median : 4858
## Mean : 9477
## 3rd Qu.: 8715
## Max. :59460Continuous variables (if any)
If a variable can take on any value between its minimum value and its maximum value, it is called a continuous variable; otherwise, it is called a discrete variable.
In this instance, only one variable can be considered continuous. Since air quality is not a categorical variable, it is continuous.
Response variables
A response variable is defined as the outcome of a study. It is a variable you would be interested in predicting or forecasting. It is often called a dependent variable or predicted variable. In this instance, a response variable is city gas mileage, since it will attempt to describe the difference between levels of the two factors of interst.
The Data: How is it organized and what does it look like?
The data is organized initially into an 6 column table: The columns are titled as follows: airq, vala, rain, coas, dens, medi. All data is numeric minus coas which is textual.
Randomization
This data comes from a collection from Yves Croissant. Since this is the only information available in regards to background information about the data collection, it is entirely possible that this data might not be completely randomized or the experiment had a completely randomized design. It is also published on 9/4/2014, so this data is very recent.
2. (Experimental) Design
How will the experiment be organized and conducted to test the hypothesis?
In order to conduct this experiment, I will conduct two separate analysis of the factors at hand. First, I will analyze multiple levels of the rain (rain) of the data. I will then look at the Air Quality (airq) values to see if an obvious difference or pattern can be seen.
Second, I will analyze multiple levels of the location (coas) of the data, which is the second factor. Again, I will then look at the Airq values to see if an obvious difference or pattern can be seen.
What is the rationale for this design?
I have chosen to use this type of experimental design to demonstrate proper experimentation with a data set with at least two factors and at least two levels of each factor.
Randomize: What is the Randomization Scheme?
Like I previously stated, since there is no credible proof this data is randomized, the only randomization involved with this experiments lies in the fact that the factors and their corresponding levels were chosen completely randomly by myself, the experiment conductor.
Replicate: Are there replicates and/or repeated measures?
There are no replicates, but repeated measures do occur between the factors and levels.
Block: Did you use blocking in the design?
The only blocking that I performed in this experimental data analysis is seen in the blocking of vehicles into the different levels of their respective factors.
3. (Statistical) Analysis
(Exploratory Data Analysis) Graphics and descriptive summary
At this point, I must define the amount of rain (rain) and the location (coas) as the factors for analysis.
y$rain=as.factor(y$rain)
y$coas=as.factor(y$coas)Below are the boxplots of the Air Quality of all levels of the two factors of interest.
par(mfrow=c(1,1))
hist(y$airq)
par(mfrow=c(1,1))
boxplot(y$airq, main="Boxplot of Air Quality for Metropolitan California", xlab="Air Quality", ylab=" Air Quality Metric", names=c("Cty"))
boxplot(airq~rain, data=y)
boxplot(airq~coas, data=y)
Testing
At this point, I am introducitng the Analysis of Variance (ANOVA) test. The ANOVA test is used to analyze the differences in the mean Air Qualities of the data with varying number of inches of rain and locations. A third ANOVA test analyzes the interaction effect between the two factors.
model_rain=aov(airq~rain,data=y)
model_coas=aov(airq~coas,data=y)
model_rain_coas=aov(airq~rain*coas,data=y)anova(model_rain)## Analysis of Variance Table
##
## Response: airq
## Df Sum Sq Mean Sq F value Pr(>F)
## rain 24 22348 931 10.7 0.0075 **
## Residuals 5 434 87
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(model_coas)## Analysis of Variance Table
##
## Response: airq
## Df Sum Sq Mean Sq F value Pr(>F)
## coas 1 5474 5474 8.85 0.006 **
## Residuals 28 17309 618
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(model_rain_coas)## Analysis of Variance Table
##
## Response: airq
## Df Sum Sq Mean Sq F value Pr(>F)
## rain 24 22348 931 10.7 0.0075 **
## Residuals 5 434 87
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1ANOVA Results
The ANOVA test that analyzed the variation in air quality as a result in variation of the amount of rainreturned a p-value of 0.007506. This small p-value translates to the fact that there is a small probability that the variations in air quality with regards to amount of rain in the area is a result of randomization. Thus the conclusion may be drawn that the change in air quality is a result in the change of the inches of rain in a location.
The ANOVA test that analyzed the variation in air quality as a result in variation of the location returned a p-value of 0.005965. This small p-value translates to the fact that there is a small probability that the variations in air quality with regards to location changes, specifically distance from the coast, is a result of randomization. Thus the conclusion may be drawn that the change in air quality is a result in the change of the distance from the coast.
Because both ANOVAs alluded to the fact that both factors can effect the air quality of the Metropolitan area of California, I then performed an ANOVA to analyze the interaction effect of the two factors. The resulting p-value was once again 0.007506 which indicates that when the two factors work together there is a very small probability that the changes in the air quality is a result of randomization.
Diagnostics/Model Adequacy Checking
To check the adequacy of using the ANOVA as a means of analyzing this set of data I performed Quantile-Quantile (Q-Q) tests on the residual error to determine if the residuals followed a normal distribution. I also created an interaction plot to see if there was an interaction effect between the two factors.
The nearly linear fit of the residuals in the first QQ plot in reference to ‘rain’ is an indication that the model is adequate for this analysis.
The nearly linear fit of the residuals in the second QQ plot in refernece to ‘coas’ is an indication that the model is adequate for this analysis.
The interaction plot following the QQ plots shows that the two factors are interacting with eachother to create an effect in the response variable whenever there is an intersection of curves on the plot.
The third type of plot is a Residuals vs.Fits plot which is used to identify the linearity of the residual values and to detemrine if there are any outlying values. Because there are slightly more outliers in the ‘rain’ response variable than in the ‘coas’ response variables it can be reasoned that the model is slightly less adequate to model the ‘coas’ data.
qqnorm(residuals(model_rain))
qqline(residuals(model_rain))
qqnorm(residuals(model_coas))
qqline(residuals(model_coas))
interaction.plot(y$rain, y$coas, y$airq)
plot(fitted(model_rain),residuals(model_rain))
plot(fitted(model_coas),residuals(model_coas))
4. Post-Hoc Test
Tukey’s HSD test is a post-hoc test, meaning that it is performed after an analysis of variance (ANOVA) test. This means that to maintain integrity, a statistician should not perform Tukey’s HSD test unless she has first performed an ANOVA analysis. In statistics, post-hoc tests are used only for further data analysis; these types of tests are not pre-planned. In other words, you should have no plans to use Tukey’s HSD test before you collect and analyze the data once.
The purpose of Tukey’s HSD test is to determine which groups in the sample differ. While ANOVA can tell the researcher whether groups in the sample differ, it cannot tell the researcher which groups differ. That is, if the results of ANOVA are positive in the sense that they state there is a significant difference among the groups, the obvious question becomes: Which groups in this sample differ significantly? It is not likely that all groups differ when compared to each other, only that a handful have significant differences. Tukey’s HSD can clarify to the researcher which groups among the sample in specific have significant differences.
TukeyHSD(model_rain_coas, "rain")## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = airq ~ rain * coas, data = y)
##
## $rain
## diff lwr upr p adj
## 14.81-12.63 3.900e+01 -26.2579 104.25788 0.3031
## 18.69-12.63 -3.700e+01 -88.5909 14.59088 0.1728
## 29.51-12.63 2.800e+01 -37.2579 93.25788 0.6350
## 30.95-12.63 3.300e+01 -32.2579 98.25788 0.4621
## 31.22-12.63 3.300e+01 -32.2579 98.25788 0.4621
## 33.18-12.63 4.900e+01 -16.2579 114.25788 0.1478
## 34.55-12.63 -1.200e+01 -77.2579 53.25788 0.9986
## 35.08-12.63 2.400e+01 -41.2579 89.25788 0.7829
## 35.35-12.63 1.900e+01 -32.5909 70.59088 0.7817
## 36.14-12.63 6.900e+01 3.7421 134.25788 0.0398
## 37.18-12.63 -1.200e+01 -77.2579 53.25788 0.9986
## 39.25-12.63 2.200e+01 -43.2579 87.25788 0.8504
## 41.32-12.63 2.200e+01 -43.2579 87.25788 0.8504
## 42.36-12.63 -2.200e+01 -73.5909 29.59088 0.6414
## 42.37-12.63 3.500e+01 -30.2579 100.25788 0.4027
## 42.48-12.63 3.300e+01 -32.2579 98.25788 0.4621
## 42.77-12.63 3.100e+01 -34.2579 96.25788 0.5276
## 42.92-12.63 -2.700e+01 -92.2579 38.25788 0.6723
## 43.05-12.63 2.200e+01 -43.2579 87.25788 0.8504
## 45.94-12.63 -8.000e+00 -73.2579 57.25788 1.0000
## 47.14-12.63 -1.100e+01 -76.2579 54.25788 0.9995
## 53.9-12.63 -2.100e+01 -86.2579 44.25788 0.8806
## 59.76-12.63 -3.200e+01 -97.2579 33.25788 0.4941
## 68.13-12.63 2.400e+01 -41.2579 89.25788 0.7829
## 18.69-14.81 -7.600e+01 -145.2164 -6.78357 0.0340
## 29.51-14.81 -1.100e+01 -90.9243 68.92425 1.0000
## 30.95-14.81 -6.000e+00 -85.9243 73.92425 1.0000
## 31.22-14.81 -6.000e+00 -85.9243 73.92425 1.0000
## 33.18-14.81 1.000e+01 -69.9243 89.92425 1.0000
## 34.55-14.81 -5.100e+01 -130.9243 28.92425 0.2505
## 35.08-14.81 -1.500e+01 -94.9243 64.92425 0.9982
## 35.35-14.81 -2.000e+01 -89.2164 49.21643 0.9343
## 36.14-14.81 3.000e+01 -49.9243 109.92425 0.7651
## 37.18-14.81 -5.100e+01 -130.9243 28.92425 0.2505
## 39.25-14.81 -1.700e+01 -96.9243 62.92425 0.9937
## 41.32-14.81 -1.700e+01 -96.9243 62.92425 0.9937
## 42.36-14.81 -6.100e+01 -130.2164 8.21643 0.0823
## 42.37-14.81 -4.000e+00 -83.9243 75.92425 1.0000
## 42.48-14.81 -6.000e+00 -85.9243 73.92425 1.0000
## 42.77-14.81 -8.000e+00 -87.9243 71.92425 1.0000
## 42.92-14.81 -6.600e+01 -145.9243 13.92425 0.1052
## 43.05-14.81 -1.700e+01 -96.9243 62.92425 0.9937
## 45.94-14.81 -4.700e+01 -126.9243 32.92425 0.3170
## 47.14-14.81 -5.000e+01 -129.9243 29.92425 0.2657
## 53.9-14.81 -6.000e+01 -139.9243 19.92425 0.1479
## 59.76-14.81 -7.100e+01 -150.9243 8.92425 0.0799
## 68.13-14.81 -1.500e+01 -94.9243 64.92425 0.9982
## 29.51-18.69 6.500e+01 -4.2164 134.21643 0.0643
## 30.95-18.69 7.000e+01 0.7836 139.21643 0.0478
## 31.22-18.69 7.000e+01 0.7836 139.21643 0.0478
## 33.18-18.69 8.600e+01 16.7836 155.21643 0.0200
## 34.55-18.69 2.500e+01 -44.2164 94.21643 0.7981
## 35.08-18.69 6.100e+01 -8.2164 130.21643 0.0823
## 35.35-18.69 5.600e+01 -0.5150 112.51498 0.0519
## 36.14-18.69 1.060e+02 36.7836 175.21643 0.0078
## 37.18-18.69 2.500e+01 -44.2164 94.21643 0.7981
## 39.25-18.69 5.900e+01 -10.2164 128.21643 0.0934
## 41.32-18.69 5.900e+01 -10.2164 128.21643 0.0934
## 42.36-18.69 1.500e+01 -41.5150 71.51498 0.9621
## 42.37-18.69 7.200e+01 2.7836 141.21643 0.0425
## 42.48-18.69 7.000e+01 0.7836 139.21643 0.0478
## 42.77-18.69 6.800e+01 -1.2164 137.21643 0.0537
## 42.92-18.69 1.000e+01 -59.2164 79.21643 0.9999
## 43.05-18.69 5.900e+01 -10.2164 128.21643 0.0934
## 45.94-18.69 2.900e+01 -40.2164 98.21643 0.6595
## 47.14-18.69 2.600e+01 -43.2164 95.21643 0.7645
## 53.9-18.69 1.600e+01 -53.2164 85.21643 0.9868
## 59.76-18.69 5.000e+00 -64.2164 74.21643 1.0000
## 68.13-18.69 6.100e+01 -8.2164 130.21643 0.0823
## 30.95-29.51 5.000e+00 -74.9243 84.92425 1.0000
## 31.22-29.51 5.000e+00 -74.9243 84.92425 1.0000
## 33.18-29.51 2.100e+01 -58.9243 100.92425 0.9647
## 34.55-29.51 -4.000e+01 -119.9243 39.92425 0.4728
## 35.08-29.51 -4.000e+00 -83.9243 75.92425 1.0000
## 35.35-29.51 -9.000e+00 -78.2164 60.21643 1.0000
## 36.14-29.51 4.100e+01 -38.9243 120.92425 0.4474
## 37.18-29.51 -4.000e+01 -119.9243 39.92425 0.4728
## 39.25-29.51 -6.000e+00 -85.9243 73.92425 1.0000
## 41.32-29.51 -6.000e+00 -85.9243 73.92425 1.0000
## 42.36-29.51 -5.000e+01 -119.2164 19.21643 0.1686
## 42.37-29.51 7.000e+00 -72.9243 86.92425 1.0000
## 42.48-29.51 5.000e+00 -74.9243 84.92425 1.0000
## 42.77-29.51 3.000e+00 -76.9243 82.92425 1.0000
## 42.92-29.51 -5.500e+01 -134.9243 24.92425 0.1979
## 43.05-29.51 -6.000e+00 -85.9243 73.92425 1.0000
## 45.94-29.51 -3.600e+01 -115.9243 43.92425 0.5840
## 47.14-29.51 -3.900e+01 -118.9243 40.92425 0.4993
## 53.9-29.51 -4.900e+01 -128.9243 30.92425 0.2819
## 59.76-29.51 -6.000e+01 -139.9243 19.92425 0.1479
## 68.13-29.51 -4.000e+00 -83.9243 75.92425 1.0000
## 31.22-30.95 -2.842e-14 -79.9243 79.92425 1.0000
## 33.18-30.95 1.600e+01 -63.9243 95.92425 0.9965
## 34.55-30.95 -4.500e+01 -124.9243 34.92425 0.3562
## 35.08-30.95 -9.000e+00 -88.9243 70.92425 1.0000
## 35.35-30.95 -1.400e+01 -83.2164 55.21643 0.9961
## 36.14-30.95 3.600e+01 -43.9243 115.92425 0.5840
## 37.18-30.95 -4.500e+01 -124.9243 34.92425 0.3562
## 39.25-30.95 -1.100e+01 -90.9243 68.92425 1.0000
## 41.32-30.95 -1.100e+01 -90.9243 68.92425 1.0000
## 42.36-30.95 -5.500e+01 -124.2164 14.21643 0.1210
## 42.37-30.95 2.000e+00 -77.9243 81.92425 1.0000
## 42.48-30.95 -2.842e-14 -79.9243 79.92425 1.0000
## 42.77-30.95 -2.000e+00 -81.9243 77.92425 1.0000
## 42.92-30.95 -6.000e+01 -139.9243 19.92425 0.1479
## 43.05-30.95 -1.100e+01 -90.9243 68.92425 1.0000
## 45.94-30.95 -4.100e+01 -120.9243 38.92425 0.4474
## 47.14-30.95 -4.400e+01 -123.9243 35.92425 0.3774
## 53.9-30.95 -5.400e+01 -133.9243 25.92425 0.2099
## 59.76-30.95 -6.500e+01 -144.9243 14.92425 0.1112
## 68.13-30.95 -9.000e+00 -88.9243 70.92425 1.0000
## 33.18-31.22 1.600e+01 -63.9243 95.92425 0.9965
## 34.55-31.22 -4.500e+01 -124.9243 34.92425 0.3562
## 35.08-31.22 -9.000e+00 -88.9243 70.92425 1.0000
## 35.35-31.22 -1.400e+01 -83.2164 55.21643 0.9961
## 36.14-31.22 3.600e+01 -43.9243 115.92425 0.5840
## 37.18-31.22 -4.500e+01 -124.9243 34.92425 0.3562
## 39.25-31.22 -1.100e+01 -90.9243 68.92425 1.0000
## 41.32-31.22 -1.100e+01 -90.9243 68.92425 1.0000
## 42.36-31.22 -5.500e+01 -124.2164 14.21643 0.1210
## 42.37-31.22 2.000e+00 -77.9243 81.92425 1.0000
## 42.48-31.22 0.000e+00 -79.9243 79.92425 1.0000
## 42.77-31.22 -2.000e+00 -81.9243 77.92425 1.0000
## 42.92-31.22 -6.000e+01 -139.9243 19.92425 0.1479
## 43.05-31.22 -1.100e+01 -90.9243 68.92425 1.0000
## 45.94-31.22 -4.100e+01 -120.9243 38.92425 0.4474
## 47.14-31.22 -4.400e+01 -123.9243 35.92425 0.3774
## 53.9-31.22 -5.400e+01 -133.9243 25.92425 0.2099
## 59.76-31.22 -6.500e+01 -144.9243 14.92425 0.1112
## 68.13-31.22 -9.000e+00 -88.9243 70.92425 1.0000
## 34.55-33.18 -6.100e+01 -140.9243 18.92425 0.1396
## 35.08-33.18 -2.500e+01 -104.9243 54.92425 0.8969
## 35.35-33.18 -3.000e+01 -99.2164 39.21643 0.6245
## 36.14-33.18 2.000e+01 -59.9243 99.92425 0.9752
## 37.18-33.18 -6.100e+01 -140.9243 18.92425 0.1396
## 39.25-33.18 -2.700e+01 -106.9243 52.92425 0.8490
## 41.32-33.18 -2.700e+01 -106.9243 52.92425 0.8490
## 42.36-33.18 -7.100e+01 -140.2164 -1.78357 0.0451
## 42.37-33.18 -1.400e+01 -93.9243 65.92425 0.9992
## 42.48-33.18 -1.600e+01 -95.9243 63.92425 0.9965
## 42.77-33.18 -1.800e+01 -97.9243 61.92425 0.9895
## 42.92-33.18 -7.600e+01 -155.9243 3.92425 0.0612
## 43.05-33.18 -2.700e+01 -106.9243 52.92425 0.8490
## 45.94-33.18 -5.700e+01 -136.9243 22.92425 0.1760
## 47.14-33.18 -6.000e+01 -139.9243 19.92425 0.1479
## 53.9-33.18 -7.000e+01 -149.9243 9.92425 0.0843
## 59.76-33.18 -8.100e+01 -160.9243 -1.07575 0.0474
## 68.13-33.18 -2.500e+01 -104.9243 54.92425 0.8969
## 35.08-34.55 3.600e+01 -43.9243 115.92425 0.5840
## 35.35-34.55 3.100e+01 -38.2164 100.21643 0.5900
## 36.14-34.55 8.100e+01 1.0757 160.92425 0.0474
## 37.18-34.55 -1.421e-14 -79.9243 79.92425 1.0000
## 39.25-34.55 3.400e+01 -45.9243 113.92425 0.6439
## 41.32-34.55 3.400e+01 -45.9243 113.92425 0.6439
## 42.36-34.55 -1.000e+01 -79.2164 59.21643 0.9999
## 42.37-34.55 4.700e+01 -32.9243 126.92425 0.3170
## 42.48-34.55 4.500e+01 -34.9243 124.92425 0.3562
## 42.77-34.55 4.300e+01 -36.9243 122.92425 0.3997
## 42.92-34.55 -1.500e+01 -94.9243 64.92425 0.9982
## 43.05-34.55 3.400e+01 -45.9243 113.92425 0.6439
## 45.94-34.55 4.000e+00 -75.9243 83.92425 1.0000
## 47.14-34.55 1.000e+00 -78.9243 80.92425 1.0000
## 53.9-34.55 -9.000e+00 -88.9243 70.92425 1.0000
## 59.76-34.55 -2.000e+01 -99.9243 59.92425 0.9752
## 68.13-34.55 3.600e+01 -43.9243 115.92425 0.5840
## 35.35-35.08 -5.000e+00 -74.2164 64.21643 1.0000
## 36.14-35.08 4.500e+01 -34.9243 124.92425 0.3562
## 37.18-35.08 -3.600e+01 -115.9243 43.92425 0.5840
## 39.25-35.08 -2.000e+00 -81.9243 77.92425 1.0000
## 41.32-35.08 -2.000e+00 -81.9243 77.92425 1.0000
## 42.36-35.08 -4.600e+01 -115.2164 23.21643 0.2211
## 42.37-35.08 1.100e+01 -68.9243 90.92425 1.0000
## 42.48-35.08 9.000e+00 -70.9243 88.92425 1.0000
## 42.77-35.08 7.000e+00 -72.9243 86.92425 1.0000
## 42.92-35.08 -5.100e+01 -130.9243 28.92425 0.2505
## 43.05-35.08 -2.000e+00 -81.9243 77.92425 1.0000
## 45.94-35.08 -3.200e+01 -111.9243 47.92425 0.7049
## 47.14-35.08 -3.500e+01 -114.9243 44.92425 0.6137
## 53.9-35.08 -4.500e+01 -124.9243 34.92425 0.3562
## 59.76-35.08 -5.600e+01 -135.9243 23.92425 0.1866
## 68.13-35.08 1.421e-14 -79.9243 79.92425 1.0000
## 36.14-35.35 5.000e+01 -19.2164 119.21643 0.1686
## 37.18-35.35 -3.100e+01 -100.2164 38.21643 0.5900
## 39.25-35.35 3.000e+00 -66.2164 72.21643 1.0000
## 41.32-35.35 3.000e+00 -66.2164 72.21643 1.0000
## 42.36-35.35 -4.100e+01 -97.5150 15.51498 0.1662
## 42.37-35.35 1.600e+01 -53.2164 85.21643 0.9868
## 42.48-35.35 1.400e+01 -55.2164 83.21643 0.9961
## 42.77-35.35 1.200e+01 -57.2164 81.21643 0.9993
## 42.92-35.35 -4.600e+01 -115.2164 23.21643 0.2211
## 43.05-35.35 3.000e+00 -66.2164 72.21643 1.0000
## 45.94-35.35 -2.700e+01 -96.2164 42.21643 0.7299
## 47.14-35.35 -3.000e+01 -99.2164 39.21643 0.6245
## 53.9-35.35 -4.000e+01 -109.2164 29.21643 0.3324
## 59.76-35.35 -5.100e+01 -120.2164 18.21643 0.1577
## 68.13-35.35 5.000e+00 -64.2164 74.21643 1.0000
## 37.18-36.14 -8.100e+01 -160.9243 -1.07575 0.0474
## 39.25-36.14 -4.700e+01 -126.9243 32.92425 0.3170
## 41.32-36.14 -4.700e+01 -126.9243 32.92425 0.3170
## 42.36-36.14 -9.100e+01 -160.2164 -21.78357 0.0156
## 42.37-36.14 -3.400e+01 -113.9243 45.92425 0.6439
## 42.48-36.14 -3.600e+01 -115.9243 43.92425 0.5840
## 42.77-36.14 -3.800e+01 -117.9243 41.92425 0.5267
## 42.92-36.14 -9.600e+01 -175.9243 -16.07575 0.0232
## 43.05-36.14 -4.700e+01 -126.9243 32.92425 0.3170
## 45.94-36.14 -7.700e+01 -156.9243 2.92425 0.0581
## 47.14-36.14 -8.000e+01 -159.9243 -0.07575 0.0498
## 53.9-36.14 -9.000e+01 -169.9243 -10.07575 0.0305
## 59.76-36.14 -1.010e+02 -180.9243 -21.07575 0.0186
## 68.13-36.14 -4.500e+01 -124.9243 34.92425 0.3562
## 39.25-37.18 3.400e+01 -45.9243 113.92425 0.6439
## 41.32-37.18 3.400e+01 -45.9243 113.92425 0.6439
## 42.36-37.18 -1.000e+01 -79.2164 59.21643 0.9999
## 42.37-37.18 4.700e+01 -32.9243 126.92425 0.3170
## 42.48-37.18 4.500e+01 -34.9243 124.92425 0.3562
## 42.77-37.18 4.300e+01 -36.9243 122.92425 0.3997
## 42.92-37.18 -1.500e+01 -94.9243 64.92425 0.9982
## 43.05-37.18 3.400e+01 -45.9243 113.92425 0.6439
## 45.94-37.18 4.000e+00 -75.9243 83.92425 1.0000
## 47.14-37.18 1.000e+00 -78.9243 80.92425 1.0000
## 53.9-37.18 -9.000e+00 -88.9243 70.92425 1.0000
## 59.76-37.18 -2.000e+01 -99.9243 59.92425 0.9752
## 68.13-37.18 3.600e+01 -43.9243 115.92425 0.5840
## 41.32-39.25 0.000e+00 -79.9243 79.92425 1.0000
## 42.36-39.25 -4.400e+01 -113.2164 25.21643 0.2534
## 42.37-39.25 1.300e+01 -66.9243 92.92425 0.9997
## 42.48-39.25 1.100e+01 -68.9243 90.92425 1.0000
## 42.77-39.25 9.000e+00 -70.9243 88.92425 1.0000
## 42.92-39.25 -4.900e+01 -128.9243 30.92425 0.2819
## 43.05-39.25 -1.421e-14 -79.9243 79.92425 1.0000
## 45.94-39.25 -3.000e+01 -109.9243 49.92425 0.7651
## 47.14-39.25 -3.300e+01 -112.9243 46.92425 0.6743
## 53.9-39.25 -4.300e+01 -122.9243 36.92425 0.3997
## 59.76-39.25 -5.400e+01 -133.9243 25.92425 0.2099
## 68.13-39.25 2.000e+00 -77.9243 81.92425 1.0000
## 42.36-41.32 -4.400e+01 -113.2164 25.21643 0.2534
## 42.37-41.32 1.300e+01 -66.9243 92.92425 0.9997
## 42.48-41.32 1.100e+01 -68.9243 90.92425 1.0000
## 42.77-41.32 9.000e+00 -70.9243 88.92425 1.0000
## 42.92-41.32 -4.900e+01 -128.9243 30.92425 0.2819
## 43.05-41.32 -1.421e-14 -79.9243 79.92425 1.0000
## 45.94-41.32 -3.000e+01 -109.9243 49.92425 0.7651
## 47.14-41.32 -3.300e+01 -112.9243 46.92425 0.6743
## 53.9-41.32 -4.300e+01 -122.9243 36.92425 0.3997
## 59.76-41.32 -5.400e+01 -133.9243 25.92425 0.2099
## 68.13-41.32 2.000e+00 -77.9243 81.92425 1.0000
## 42.37-42.36 5.700e+01 -12.2164 126.21643 0.1062
## 42.48-42.36 5.500e+01 -14.2164 124.21643 0.1210
## 42.77-42.36 5.300e+01 -16.2164 122.21643 0.1380
## 42.92-42.36 -5.000e+00 -74.2164 64.21643 1.0000
## 43.05-42.36 4.400e+01 -25.2164 113.21643 0.2534
## 45.94-42.36 1.400e+01 -55.2164 83.21643 0.9961
## 47.14-42.36 1.100e+01 -58.2164 80.21643 0.9997
## 53.9-42.36 1.000e+00 -68.2164 70.21643 1.0000
## 59.76-42.36 -1.000e+01 -79.2164 59.21643 0.9999
## 68.13-42.36 4.600e+01 -23.2164 115.21643 0.2211
## 42.48-42.37 -2.000e+00 -81.9243 77.92425 1.0000
## 42.77-42.37 -4.000e+00 -83.9243 75.92425 1.0000
## 42.92-42.37 -6.200e+01 -141.9243 17.92425 0.1318
## 43.05-42.37 -1.300e+01 -92.9243 66.92425 0.9997
## 45.94-42.37 -4.300e+01 -122.9243 36.92425 0.3997
## 47.14-42.37 -4.600e+01 -125.9243 33.92425 0.3361
## 53.9-42.37 -5.600e+01 -135.9243 23.92425 0.1866
## 59.76-42.37 -6.700e+01 -146.9243 12.92425 0.0995
## 68.13-42.37 -1.100e+01 -90.9243 68.92425 1.0000
## 42.77-42.48 -2.000e+00 -81.9243 77.92425 1.0000
## 42.92-42.48 -6.000e+01 -139.9243 19.92425 0.1479
## 43.05-42.48 -1.100e+01 -90.9243 68.92425 1.0000
## 45.94-42.48 -4.100e+01 -120.9243 38.92425 0.4474
## 47.14-42.48 -4.400e+01 -123.9243 35.92425 0.3774
## 53.9-42.48 -5.400e+01 -133.9243 25.92425 0.2099
## 59.76-42.48 -6.500e+01 -144.9243 14.92425 0.1112
## 68.13-42.48 -9.000e+00 -88.9243 70.92425 1.0000
## 42.92-42.77 -5.800e+01 -137.9243 21.92425 0.1661
## 43.05-42.77 -9.000e+00 -88.9243 70.92425 1.0000
## 45.94-42.77 -3.900e+01 -118.9243 40.92425 0.4993
## 47.14-42.77 -4.200e+01 -121.9243 37.92425 0.4230
## 53.9-42.77 -5.200e+01 -131.9243 27.92425 0.2362
## 59.76-42.77 -6.300e+01 -142.9243 16.92425 0.1245
## 68.13-42.77 -7.000e+00 -86.9243 72.92425 1.0000
## 43.05-42.92 4.900e+01 -30.9243 128.92425 0.2819
## 45.94-42.92 1.900e+01 -60.9243 98.92425 0.9834
## 47.14-42.92 1.600e+01 -63.9243 95.92425 0.9965
## 53.9-42.92 6.000e+00 -73.9243 85.92425 1.0000
## 59.76-42.92 -5.000e+00 -84.9243 74.92425 1.0000
## 68.13-42.92 5.100e+01 -28.9243 130.92425 0.2505
## 45.94-43.05 -3.000e+01 -109.9243 49.92425 0.7651
## 47.14-43.05 -3.300e+01 -112.9243 46.92425 0.6743
## 53.9-43.05 -4.300e+01 -122.9243 36.92425 0.3997
## 59.76-43.05 -5.400e+01 -133.9243 25.92425 0.2099
## 68.13-43.05 2.000e+00 -77.9243 81.92425 1.0000
## 47.14-45.94 -3.000e+00 -82.9243 76.92425 1.0000
## 53.9-45.94 -1.300e+01 -92.9243 66.92425 0.9997
## 59.76-45.94 -2.400e+01 -103.9243 55.92425 0.9177
## 68.13-45.94 3.200e+01 -47.9243 111.92425 0.7049
## 53.9-47.14 -1.000e+01 -89.9243 69.92425 1.0000
## 59.76-47.14 -2.100e+01 -100.9243 58.92425 0.9647
## 68.13-47.14 3.500e+01 -44.9243 114.92425 0.6137
## 59.76-53.9 -1.100e+01 -90.9243 68.92425 1.0000
## 68.13-53.9 4.500e+01 -34.9243 124.92425 0.3562
## 68.13-59.76 5.600e+01 -23.9243 135.92425 0.1866plot(TukeyHSD(model_rain_coas, "rain"))
5. References to the literature
See course canvas site. Also http://cran.r-project.org/web/packages/Ecdat/index.html