Recipe 5: Blocked Designs with Mutiple Explanatory and Nuisance Factors
Caroline Hsia
October 23, 2014
Recipe 5: Blocked Designs with Mutiple Explanatory and Nuisance Factors
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Recipes for the Design of Experiments: Recipe Outline
as of August 28, 2014, superceding the version of August 24. Always use the most recent version.
Storm Analysis
Caroline Hsia
Rensselaer Polytechnic Institute
October 23, 2014 V1
1. Setting
System under test
This study takes a look at storm data from National Hurricane Center. It tracks different tropical cyclones through the Atlantic Ocean, Carribean Sea, and Gulf of Mexico from 1995 to 2005. It includes various metadata about each storm including name, year, month, date, hour, latitude, longitude, type, air pressure, maximum wind speeds, and day of the hurricane season. This specific recipe will be looking into the Year and type of storms.
remove(list=ls())
install.packages("nasaweather", repos='http://cran.us.r-project.org')## package 'nasaweather' successfully unpacked and MD5 sums checked
##
## The downloaded binary packages are in
## C:\Users\Caroline\AppData\Local\Temp\Rtmp4CPkYb\downloaded_packagesrequire(nasaweather)## Loading required package: nasaweatherlibrary("nasaweather", lib.loc="~/R/win-library/3.0/")
x<-storms
storm = x[-1666,]
storm = storm[-1604,]
head(x)## name year month day hour lat long pressure wind type
## 1 Allison 1995 6 3 0 17.4 -84.3 1005 30 Tropical Depression
## 2 Allison 1995 6 3 6 18.3 -84.9 1004 30 Tropical Depression
## 3 Allison 1995 6 3 12 19.3 -85.7 1003 35 Tropical Storm
## 4 Allison 1995 6 3 18 20.6 -85.8 1001 40 Tropical Storm
## 5 Allison 1995 6 4 0 22.0 -86.0 997 50 Tropical Storm
## 6 Allison 1995 6 4 6 23.3 -86.3 995 60 Tropical Storm
## seasday
## 1 3
## 2 3
## 3 3
## 4 3
## 5 4
## 6 4str(x)## Classes 'tbl_df', 'tbl' and 'data.frame': 2747 obs. of 11 variables:
## $ name : chr "Allison" "Allison" "Allison" "Allison" ...
## $ year : int 1995 1995 1995 1995 1995 1995 1995 1995 1995 1995 ...
## $ month : int 6 6 6 6 6 6 6 6 6 6 ...
## $ day : int 3 3 3 3 4 4 4 4 5 5 ...
## $ hour : int 0 6 12 18 0 6 12 18 0 6 ...
## $ lat : num 17.4 18.3 19.3 20.6 22 23.3 24.7 26.2 27.6 28.5 ...
## $ long : num -84.3 -84.9 -85.7 -85.8 -86 -86.3 -86.2 -86.2 -86.1 -85.6 ...
## $ pressure: int 1005 1004 1003 1001 997 995 987 988 988 990 ...
## $ wind : int 30 30 35 40 50 60 65 65 65 60 ...
## $ type : chr "Tropical Depression" "Tropical Depression" "Tropical Storm" "Tropical Storm" ...
## $ seasday : int 3 3 3 3 4 4 4 4 5 5 ...Factors and Levels
There are five factors in this dataset: year, month, day, hour, and type. The factors that were used in this analysis was Storm Type and year. The levels for year were 1995 to 2000. The levels for Day were 1 - 31. The levels for hour were 0, 6, 12, or 18. The levels for month were 6 - 12. The levels for type were: extratropical, hurricane, tropic storm, and tropical depression.
head(x)## name year month day hour lat long pressure wind type
## 1 Allison 1995 6 3 0 17.4 -84.3 1005 30 Tropical Depression
## 2 Allison 1995 6 3 6 18.3 -84.9 1004 30 Tropical Depression
## 3 Allison 1995 6 3 12 19.3 -85.7 1003 35 Tropical Storm
## 4 Allison 1995 6 3 18 20.6 -85.8 1001 40 Tropical Storm
## 5 Allison 1995 6 4 0 22.0 -86.0 997 50 Tropical Storm
## 6 Allison 1995 6 4 6 23.3 -86.3 995 60 Tropical Storm
## seasday
## 1 3
## 2 3
## 3 3
## 4 3
## 5 4
## 6 4tail(x)## name year month day hour lat long pressure wind type
## 2742 Nadine 2000 10 21 6 33.3 -53.5 1000 50 Tropical Storm
## 2743 Nadine 2000 10 21 12 34.1 -52.3 1000 50 Tropical Storm
## 2744 Nadine 2000 10 21 18 34.8 -51.3 1000 45 Tropical Storm
## 2745 Nadine 2000 10 22 0 35.7 -50.5 1004 40 Extratropical
## 2746 Nadine 2000 10 22 6 37.0 -49.0 1005 40 Extratropical
## 2747 Nadine 2000 10 22 12 39.0 -47.0 1005 35 Extratropical
## seasday
## 2742 143
## 2743 143
## 2744 143
## 2745 144
## 2746 144
## 2747 144summary(x)## name year month day
## Length:2747 Min. :1995 Min. : 6.0 Min. : 1
## Class :character 1st Qu.:1995 1st Qu.: 8.0 1st Qu.: 9
## Mode :character Median :1997 Median : 9.0 Median :18
## Mean :1997 Mean : 8.8 Mean :17
## 3rd Qu.:1999 3rd Qu.:10.0 3rd Qu.:25
## Max. :2000 Max. :12.0 Max. :31
## hour lat long pressure
## Min. : 0.00 Min. : 8.3 Min. :-107.3 Min. : 905
## 1st Qu.: 3.50 1st Qu.:17.2 1st Qu.: -77.6 1st Qu.: 980
## Median :12.00 Median :25.0 Median : -60.9 Median : 995
## Mean : 9.06 Mean :26.7 Mean : -60.9 Mean : 990
## 3rd Qu.:18.00 3rd Qu.:33.9 3rd Qu.: -45.8 3rd Qu.:1004
## Max. :18.00 Max. :70.7 Max. : 1.0 Max. :1019
## wind type seasday
## Min. : 15.0 Length:2747 Min. : 3
## 1st Qu.: 35.0 Class :character 1st Qu.: 84
## Median : 50.0 Mode :character Median :103
## Mean : 54.7 Mean :103
## 3rd Qu.: 70.0 3rd Qu.:125
## Max. :155.0 Max. :185#Set all of the factors in the dataset to as.factor
storm$name = as.factor(storm$name)
storm$year = as.factor(storm$year)
storm$month = as.factor(storm$month)
storm$day = as.factor(storm$day)
storm$hour = as.factor(storm$hour)
storm$type = as.factor(storm$type)Continuous variables (if any)
The continuous variables in this dataset are longitude, latitude, air pressure, and wind speed.
Response variables
The response variables in this dataset are air pressure and wind speed.
The Data: How is it organized and what does it look like?
The data from ‘storms’ describes data about the tropical cyclones that are tracked through the Atlantic Ocean, Carribean Sea, and Gulf of Mexico from 1995 to 2005. The information about storms include various metadata about each storm including name, year, month, date, hour, latitude, longitude, type, air pressure, maximum wind speeds, and day of the hurricane season. There are four levels to type factor which includes Extratropical, Tropical Depression, Hurricane, and Tropical Storm.
Randomization
This data originated from the National Hurricane Center’s archive of Tropical Cyclone reports, handscraped from track tables of individual tropical cyclone reports. We can assume that this data was collected using proper randomization techniques.
2. (Experimental) Design
How will the experiment be organized and conducted to test the hypothesis?
This data was publically available for anyone to use and perform analysis on. In this analysis, a multifactor, multilevel experiment blocking on two factors will be performed. Multiple ANOVA tests will be conducted in order to determine the relationship between wind pressure and storm month, day, and hour. The storm year and type will be the blocking factor. ANOVA tests will be performed between each of the factors and response to compare the means of the factor levels.
The null hypothesis that will be tested is:
The variance in storm month, day, and hour of a storm has no significant impact on the variance of its storm wind pressure.
This makes the alternate hypothesis:
The variance in storm month, day, and hour of a storm has a significant effect on the variance storm wind pressure.
What is the rationale for this design?
The rationale for the collection of data was just for the National Hurricane Center to gather information on the tropical cyclones that travel through the Atlantic Ocean, Carribean Sea, and Gulf of Mexico from 1995 to 2005. The analysis performed for this dataset involved a multifactor, multilevel design. An ANOVA test was conducted in order to determine the variation between the level means.
Randomize: What is the Randomization Scheme?
This data was collected with no intention, just for data collection.
Replicate: Are there replicates and/or repeated measures?
No, there were no replicates. Repeated measures included taking measurements for the same storm at different points of time.
Block: Did you use blocking in the design?
The original dataset was organized without experimental groups, with measurements recorded based on certain variables. The blocking factors used in this recipe were storm year and storm type.
3. (Statistical) Analysis
(Exploratory Data Analysis) Graphics and descriptive summary
#Here we will just look at the boxplots of the blocking factors.
boxplot(wind~year, data = storm, xlab = "Year", ylab = "Wind Pressure")
boxplot(wind~type, data = storm, xlab = "Type", ylab = "Wind Pressure")
From these boxplots, we can see the variability between the data in these categories. It is clear that the variability between the year is less than the type, but they both make good candidates for blocking in our analysis.
#Here we will examine the histogram of the response variable.
par(mfrow=c(1,1))
hist(storm$wind, xlim = c(0,180))
#Here we will just look at the boxplots of the 3 factors.
boxplot(wind~month, data = storm, xlab = "Month", ylab = "Wind Pressure")
boxplot(wind~day, data = storm, xlab = "Day", ylab = "Wind Pressure")
boxplot(wind~hour, data = storm, xlab = "Hour", ylab = "Wind Pressure")
It can be seen from the boxplot above, that there are two rows of data that contain the hour 1 and 15, so we will take those out for the purpose of our analysis.
Testing
ANOVA tests will be performed.
#ANOVA
#The following 3 tests will be blocking on year.
modely_month <- aov(wind~year+as.factor(month),data = storm)
anova(modely_month)## Analysis of Variance Table
##
## Response: wind
## Df Sum Sq Mean Sq F value Pr(>F)
## year 5 31268 6254 10.0 1.7e-09 ***
## as.factor(month) 6 93355 15559 24.9 < 2e-16 ***
## Residuals 2733 1708610 625
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This ANOVA test produced a p-value of 2e-16, blocked for month, which showed us that the probability that the variation of wind can be attributed to month, when blocked for year. This p-value indicates that we reject the null hypothesis.
modely_day <- aov(wind~year+as.factor(day),data = storm)
anova(modely_day)## Analysis of Variance Table
##
## Response: wind
## Df Sum Sq Mean Sq F value Pr(>F)
## year 5 31268 6254 9.65 3.8e-09 ***
## as.factor(day) 30 46325 1544 2.38 3.7e-05 ***
## Residuals 2709 1755640 648
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This ANOVA test produced a p-value of 3.7e-05, blocked for month, which showed us that the probability that the variation of wind can be attributed to day, when blocked for year. This p-value indicates that we reject the null hypothesis.
modely_hour <- aov(wind~year+as.factor(hour),data = storm)
anova(modely_hour)## Analysis of Variance Table
##
## Response: wind
## Df Sum Sq Mean Sq F value Pr(>F)
## year 5 31268 6254 9.5 5.4e-09 ***
## as.factor(hour) 3 207 69 0.1 0.96
## Residuals 2736 1801758 659
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This ANOVA test produced a p-value of 0.96, blocked for month, which showed us that the probability that the variation of wind cannot be attributed to month, when blocked for year. This p-value indicates that we fail to reject the null hypothesis.
#ANOVA
#The following 3 tests will be blocking on type.
modelt_month = aov(wind~type+as.factor(month),data = storm)
anova(modelt_month)## Analysis of Variance Table
##
## Response: wind
## Df Sum Sq Mean Sq F value Pr(>F)
## type 3 1324947 441649 2414.91 < 2e-16 ***
## as.factor(month) 6 8098 1350 7.38 7.5e-08 ***
## Residuals 2735 500188 183
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This ANOVA test produced a p-value of 7.5e-08, blocked for month, which showed us that the probability that the variation of wind can be attributed to month, when blocked for year. This p-value indicates that we reject the null hypothesis.
modelt_day = aov(wind~type+as.factor(day),data = storm)
anova(modelt_day)## Analysis of Variance Table
##
## Response: wind
## Df Sum Sq Mean Sq F value Pr(>F)
## type 3 1324947 441649 2445.36 < 2e-16 ***
## as.factor(day) 30 18661 622 3.44 9.2e-10 ***
## Residuals 2711 489625 181
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This ANOVA test produced a p-value of 9.2e-10, blocked for day, which showed us that the probability that the variation of wind can be attributed to day, when blocked for type. This p-value indicates that we reject the null hypothesis.
modelt_hour = aov(wind~type+as.factor(hour),data = storm)
anova(modelt_hour)## Analysis of Variance Table
##
## Response: wind
## Df Sum Sq Mean Sq F value Pr(>F)
## type 3 1324947 441649 2379.34 <2e-16 ***
## as.factor(hour) 3 64 21 0.11 0.95
## Residuals 2738 508223 186
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This ANOVA test produced a p-value of 0.9, blocked for type, which showed us that the probability that the variation of wind cannot be attributed to hour, when blocked for type. This p-value indicates that we fail to reject the null hypothesis.
#ANOVA
#These following ANOVA test for all factors within each block as well as for both block factors.
modely_all = aov(wind~year+(as.factor(hour)*as.factor(month)*as.factor(day)),data = storm)
anova(modely_all)## Analysis of Variance Table
##
## Response: wind
## Df Sum Sq Mean Sq
## year 5 31268 6254
## as.factor(hour) 3 207 69
## as.factor(month) 6 93365 15561
## as.factor(day) 30 41360 1379
## as.factor(hour):as.factor(month) 18 987 55
## as.factor(hour):as.factor(day) 90 5377 60
## as.factor(month):as.factor(day) 137 195519 1427
## as.factor(hour):as.factor(month):as.factor(day) 401 24764 62
## Residuals 2054 1440387 701
## F value Pr(>F)
## year 8.92 2.2e-08 ***
## as.factor(hour) 0.10 0.9610
## as.factor(month) 22.19 < 2e-16 ***
## as.factor(day) 1.97 0.0014 **
## as.factor(hour):as.factor(month) 0.08 1.0000
## as.factor(hour):as.factor(day) 0.09 1.0000
## as.factor(month):as.factor(day) 2.04 9.9e-11 ***
## as.factor(hour):as.factor(month):as.factor(day) 0.09 1.0000
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1modelt_all = aov(wind~type+(as.factor(hour)*as.factor(month)*as.factor(day)),data = storm)
anova(modely_all)## Analysis of Variance Table
##
## Response: wind
## Df Sum Sq Mean Sq
## year 5 31268 6254
## as.factor(hour) 3 207 69
## as.factor(month) 6 93365 15561
## as.factor(day) 30 41360 1379
## as.factor(hour):as.factor(month) 18 987 55
## as.factor(hour):as.factor(day) 90 5377 60
## as.factor(month):as.factor(day) 137 195519 1427
## as.factor(hour):as.factor(month):as.factor(day) 401 24764 62
## Residuals 2054 1440387 701
## F value Pr(>F)
## year 8.92 2.2e-08 ***
## as.factor(hour) 0.10 0.9610
## as.factor(month) 22.19 < 2e-16 ***
## as.factor(day) 1.97 0.0014 **
## as.factor(hour):as.factor(month) 0.08 1.0000
## as.factor(hour):as.factor(day) 0.09 1.0000
## as.factor(month):as.factor(day) 2.04 9.9e-11 ***
## as.factor(hour):as.factor(month):as.factor(day) 0.09 1.0000
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1model_all = aov(wind~year+type+(as.factor(hour)*as.factor(month)*as.factor(day)),data = storm)
anova(modely_all)## Analysis of Variance Table
##
## Response: wind
## Df Sum Sq Mean Sq
## year 5 31268 6254
## as.factor(hour) 3 207 69
## as.factor(month) 6 93365 15561
## as.factor(day) 30 41360 1379
## as.factor(hour):as.factor(month) 18 987 55
## as.factor(hour):as.factor(day) 90 5377 60
## as.factor(month):as.factor(day) 137 195519 1427
## as.factor(hour):as.factor(month):as.factor(day) 401 24764 62
## Residuals 2054 1440387 701
## F value Pr(>F)
## year 8.92 2.2e-08 ***
## as.factor(hour) 0.10 0.9610
## as.factor(month) 22.19 < 2e-16 ***
## as.factor(day) 1.97 0.0014 **
## as.factor(hour):as.factor(month) 0.08 1.0000
## as.factor(hour):as.factor(day) 0.09 1.0000
## as.factor(month):as.factor(day) 2.04 9.9e-11 ***
## as.factor(hour):as.factor(month):as.factor(day) 0.09 1.0000
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This ANOVA test produced a p-value less than 0.05 for all except when the factor hour was involved. This showed us that the probability that the variation of wind can be attributed to month and day, but not hour when blocked for year and type.
Tukey’s Test
A Tukey’s HSD test is performed in order to avoid the chance of discovering false positives in our analysis. These Tukey’s tests show us that an adjusted p-value of less than 0.05 at a 95% confidence interval indicates that the variation in wind pressure can be explained by something other than randomization.
#The following Tukey's HSD test were performed on the block Type.
#Block Type, Factor month
TukeyHSD(modelt_month, ordered=FALSE, conf.level =0.95)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = wind ~ type + as.factor(month), data = storm)
##
## $type
## diff lwr upr p adj
## Hurricane-Extratropical 44.599 42.530 46.67 0
## Tropical Depression-Extratropical -12.703 -15.004 -10.40 0
## Tropical Storm-Extratropical 7.261 5.202 9.32 0
## Tropical Depression-Hurricane -57.301 -59.229 -55.37 0
## Tropical Storm-Hurricane -37.338 -38.967 -35.71 0
## Tropical Storm-Tropical Depression 19.964 18.048 21.88 0
##
## $`as.factor(month)`
## diff lwr upr p adj
## 7-6 -1.1988 -6.251 3.853 0.9926
## 8-6 -0.3546 -4.980 4.271 1.0000
## 9-6 3.2603 -1.308 7.828 0.3493
## 10-6 0.3113 -4.378 5.000 1.0000
## 11-6 1.2267 -4.116 6.570 0.9938
## 12-6 6.5232 -9.181 22.228 0.8844
## 8-7 0.8441 -2.081 3.769 0.9793
## 9-7 4.4591 1.626 7.292 0.0001
## 10-7 1.5101 -1.514 4.534 0.7608
## 11-7 2.4255 -1.538 6.389 0.5443
## 12-7 7.7219 -7.568 23.012 0.7509
## 9-8 3.6149 1.642 5.588 0.0000
## 10-8 0.6660 -1.573 2.905 0.9759
## 11-8 1.5814 -1.821 4.984 0.8172
## 12-8 6.8778 -8.277 22.032 0.8335
## 10-9 -2.9490 -5.067 -0.831 0.0008
## 11-9 -2.0336 -5.357 1.290 0.5446
## 12-9 3.2629 -11.874 18.400 0.9956
## 11-10 0.9154 -2.573 4.404 0.9874
## 12-10 6.2118 -8.962 21.386 0.8913
## 12-11 5.2964 -10.092 20.685 0.9506#Block Type, Factor month
TukeyHSD(modelt_day, ordered=FALSE, conf.level =0.95)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = wind ~ type + as.factor(day), data = storm)
##
## $type
## diff lwr upr p adj
## Hurricane-Extratropical 44.599 42.543 46.655 0
## Tropical Depression-Extratropical -12.703 -14.990 -10.415 0
## Tropical Storm-Extratropical 7.261 5.215 9.307 0
## Tropical Depression-Hurricane -57.301 -59.217 -55.386 0
## Tropical Storm-Hurricane -37.338 -38.957 -35.719 0
## Tropical Storm-Tropical Depression 19.964 18.060 21.867 0
##
## $`as.factor(day)`
## diff lwr upr p adj
## 2-1 -0.658069 -8.14766 6.83152 1.0000
## 3-1 0.578026 -7.26693 8.42298 1.0000
## 4-1 4.913101 -2.87024 12.69644 0.8666
## 5-1 1.070733 -6.34985 8.49132 1.0000
## 6-1 0.960995 -6.50513 8.42712 1.0000
## 7-1 2.961853 -4.85193 10.77564 0.9999
## 8-1 3.761207 -3.99238 11.51479 0.9951
## 9-1 5.420605 -2.62815 13.46936 0.7687
## 10-1 3.108272 -5.31311 11.52966 1.0000
## 11-1 2.454570 -5.96682 10.87596 1.0000
## 12-1 8.602940 0.62552 16.58036 0.0171
## 13-1 9.192708 1.28313 17.10228 0.0047
## 14-1 2.584963 -5.08332 10.25325 1.0000
## 15-1 0.523579 -6.89701 7.94416 1.0000
## 16-1 -1.078927 -8.47741 6.31956 1.0000
## 17-1 3.892184 -3.55095 11.33531 0.9852
## 18-1 1.573632 -5.89250 9.03976 1.0000
## 19-1 1.374107 -5.70507 8.45328 1.0000
## 20-1 3.634746 -3.44443 10.71392 0.9886
## 21-1 -0.289601 -7.41839 6.83919 1.0000
## 22-1 0.001549 -7.04590 7.04900 1.0000
## 23-1 -1.005253 -8.08443 6.07392 1.0000
## 24-1 -1.124189 -8.04079 5.79241 1.0000
## 25-1 4.301661 -2.76152 11.36484 0.9060
## 26-1 1.430225 -5.69857 8.55901 1.0000
## 27-1 3.395226 -3.59178 10.38223 0.9950
## 28-1 4.221654 -2.79509 11.23840 0.9172
## 29-1 -0.715009 -7.81045 6.38043 1.0000
## 30-1 -1.922675 -9.06857 5.22322 1.0000
## 31-1 -0.988809 -9.19187 7.21425 1.0000
## 3-2 1.236095 -6.87633 9.34852 1.0000
## 4-2 5.571170 -2.48168 13.62402 0.7181
## 5-2 1.728802 -5.97400 9.43160 1.0000
## 6-2 1.619064 -6.12762 9.36575 1.0000
## 7-2 3.619922 -4.46236 11.70220 0.9987
## 8-2 4.419276 -3.60482 12.44337 0.9704
## 9-2 6.078675 -2.23099 14.38834 0.5981
## 10-2 3.766341 -4.90475 12.43743 0.9992
## 11-2 3.112639 -5.55845 11.78373 1.0000
## 12-2 9.261009 1.02042 17.50160 0.0087
## 13-2 9.850778 1.67585 18.02570 0.0023
## 14-2 3.243032 -4.69867 11.18473 0.9998
## 15-2 1.181648 -6.52115 8.88445 1.0000
## 16-2 -0.420858 -8.10237 7.26065 1.0000
## 17-2 4.550253 -3.17427 12.27477 0.9341
## 18-2 2.231701 -5.51498 9.97838 1.0000
## 19-2 2.032176 -5.34229 9.40664 1.0000
## 20-2 4.292815 -3.08165 11.66728 0.9423
## 21-2 0.368468 -7.05364 7.79057 1.0000
## 22-2 0.659618 -6.68440 8.00363 1.0000
## 23-2 -0.347184 -7.72165 7.02728 1.0000
## 24-2 -0.466120 -7.68466 6.75242 1.0000
## 25-2 4.959730 -2.39938 12.31885 0.7674
## 26-2 2.088294 -5.33381 9.51040 1.0000
## 27-2 4.053295 -3.23274 11.33933 0.9664
## 28-2 4.879723 -2.43483 12.19428 0.7852
## 29-2 -0.056940 -7.44702 7.33314 1.0000
## 30-2 -1.264606 -8.70314 6.17393 1.0000
## 31-2 -0.330740 -8.78995 8.12847 1.0000
## 4-3 4.335075 -4.04930 12.71945 0.9873
## 5-3 0.492707 -7.55605 8.54146 1.0000
## 6-3 0.382968 -7.70779 8.47373 1.0000
## 7-3 2.383827 -6.02882 10.79647 1.0000
## 8-3 3.183180 -5.17358 11.53994 0.9999
## 9-3 4.842579 -3.78875 13.47391 0.9627
## 10-3 2.530246 -6.44957 11.51006 1.0000
## 11-3 1.876544 -7.10327 10.85636 1.0000
## 12-3 8.024914 -0.53993 16.58976 0.1069
## 13-3 8.614682 0.11299 17.11637 0.0421
## 14-3 2.006937 -6.27074 10.28461 1.0000
## 15-3 -0.054447 -8.10320 7.99431 1.0000
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## 22-21 0.291150 -6.68454 7.26684 1.0000
## 23-21 -0.715652 -7.72339 6.29209 1.0000
## 24-21 -0.834588 -7.67805 6.00888 1.0000
## 25-21 4.591262 -2.40032 11.58284 0.8110
## 26-21 1.719826 -5.33803 8.77768 1.0000
## 27-21 3.684827 -3.22979 10.59944 0.9808
## 28-21 4.511255 -2.43341 11.45592 0.8276
## 29-21 -0.425408 -7.44957 6.59876 1.0000
## 30-21 -1.633074 -8.70821 5.44206 1.0000
## 31-21 -0.699208 -8.84070 7.44228 1.0000
## 23-22 -1.006802 -7.93178 5.91818 1.0000
## 24-22 -1.125738 -7.88443 5.63296 1.0000
## 25-22 4.300112 -2.60852 11.20874 0.8833
## 26-22 1.428676 -5.54701 8.40436 1.0000
## 27-22 3.393677 -3.43705 10.22440 0.9930
## 28-22 4.220105 -2.64104 11.08125 0.8962
## 29-22 -0.716559 -7.65816 6.22504 1.0000
## 30-22 -1.924224 -8.91739 5.06894 1.0000
## 31-22 -0.990358 -9.06072 7.08000 1.0000
## 24-23 -0.118936 -6.91071 6.67283 1.0000
## 25-23 5.306914 -1.63407 12.24790 0.4944
## 26-23 2.435478 -4.57226 9.44322 1.0000
## 27-23 4.400478 -2.46297 11.26393 0.8463
## 28-23 5.226907 -1.66682 12.12063 0.5142
## 29-23 0.290243 -6.68356 7.26405 1.0000
## 30-23 -0.917422 -7.94256 6.10772 1.0000
## 31-23 0.016444 -8.08164 8.11453 1.0000
## 25-24 5.425850 -1.34925 12.20095 0.3854
## 26-24 2.554414 -4.28905 9.39788 1.0000
## 27-24 4.519415 -2.17623 11.21506 0.7647
## 28-24 5.345843 -1.38082 12.07251 0.4031
## 29-24 0.409179 -6.39954 7.21790 1.0000
## 30-24 -0.798486 -7.65977 6.06280 1.0000
## 31-24 0.135380 -7.82097 8.09173 1.0000
## 26-25 -2.871437 -9.86302 4.12015 0.9997
## 27-25 -0.906436 -7.75339 5.94052 1.0000
## 28-25 -0.080007 -6.95731 6.79729 1.0000
## 29-25 -5.016671 -11.97424 1.94090 0.6307
## 30-25 -6.224337 -13.23336 0.78469 0.1800
## 31-25 -5.290470 -13.37458 2.79363 0.8164
## 27-26 1.965001 -4.94961 8.87962 1.0000
## 28-26 2.791430 -4.15323 9.73609 0.9998
## 29-26 -2.145234 -9.16940 4.87893 1.0000
## 30-26 -3.352900 -10.42803 3.72223 0.9966
## 31-26 -2.419034 -10.56052 5.72246 1.0000
## 28-27 0.826429 -5.97261 7.62547 1.0000
## 29-27 -4.110235 -10.99046 2.76999 0.9232
## 30-27 -5.317901 -12.25015 1.61435 0.4864
## 31-27 -4.384035 -12.40167 3.63360 0.9730
## 29-28 -4.936664 -11.84709 1.97376 0.6512
## 30-28 -6.144330 -13.10655 0.81789 0.1901
## 31-28 -5.210463 -13.25402 2.83310 0.8317
## 30-29 -1.207666 -8.24919 5.83386 1.0000
## 31-29 -0.273800 -8.38610 7.83850 1.0000
## 31-30 0.933866 -7.22261 9.09034 1.0000#Block Type, Factor month
TukeyHSD(modelt_hour, ordered=FALSE, conf.level =0.95)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = wind ~ type + as.factor(hour), data = storm)
##
## $type
## diff lwr upr p adj
## Hurricane-Extratropical 44.599 42.514 46.684 0
## Tropical Depression-Extratropical -12.703 -15.021 -10.384 0
## Tropical Storm-Extratropical 7.261 5.187 9.335 0
## Tropical Depression-Hurricane -57.301 -59.243 -55.360 0
## Tropical Storm-Hurricane -37.338 -38.979 -35.697 0
## Tropical Storm-Tropical Depression 19.964 18.034 21.894 0
##
## $`as.factor(hour)`
## diff lwr upr p adj
## 6-0 0.09522 -1.806 1.997 0.9992
## 12-0 -0.31553 -2.203 1.572 0.9734
## 18-0 -0.05000 -1.934 1.834 0.9999
## 12-6 -0.41076 -2.309 1.487 0.9449
## 18-6 -0.14523 -2.039 1.749 0.9973
## 18-12 0.26553 -1.615 2.146 0.9836Diagnostics/Model Adequacy Checking
The adequacy of the ANOVA model can be checked through a Shapiro-Wilk normality test.
shapiro.test(x$wind)##
## Shapiro-Wilk normality test
##
## data: x$wind
## W = 0.9273, p-value < 2.2e-16The results from the Shapiro-Wilk NOrmality test show us that the data is normally distributed because of the p-value being less than 0.1.
A QQ plot is used in order to test the normality of the data. The following QQ plots were plotted in order for us to confirm our assumption of normality.
qqnorm(residuals(modely_month), main = "Normal QQ Plot for Month (Blocked by Year)", ylab = "Wind Pressure")
qqline(residuals(modely_month))
qqnorm(residuals(modely_day), main = "Normal QQ Plot for Day (Blocked by Year)", ylab = "Wind Pressure")
qqline(residuals(modely_day))
qqnorm(residuals(modely_hour), main = "Normal QQ Plot for Hour (Blocked by Year)", ylab = "Wind Pressure")
qqline(residuals(modely_hour))
qqnorm(residuals(modelt_month), main = "Normal QQ Plot for Month (Blocked by Type)", ylab = "Wind Pressure")
qqline(residuals(modelt_month))
qqnorm(residuals(modelt_day), main = "Normal QQ Plot for Day (Blocked by Type)", ylab = "Wind Pressure")
qqline(residuals(modelt_day))
qqnorm(residuals(modelt_hour), main = "Normal QQ Plot for Hour (Blocked by Type)", ylab = "Wind Pressure")
qqline(residuals(modelt_hour))
Our QQ plots are fairly linear with a slight non-linear tail. These graphs indicate that the population of the samples were normally distributed. The use of ANOVA testing was correct for our model.
plot(fitted(modely_month), residuals(modely_month),main="Residual vs Fitted Plot for Month: Year Blocked")
plot(fitted(modely_day), residuals(modely_day), main="Residual vs Fitted Plot for Day: Year Blocked")
plot(fitted(modely_hour), residuals(modely_hour), main="Residual vs Fitted Plot for Hour: Year Blocked")
plot(fitted(modelt_month), residuals(modelt_month), main="Residual vs Fitted Plot for Month: Type Blocked")
plot(fitted(modelt_day), residuals(modelt_day), main="Residual vs Fitted Plot for Day: Type Blocked")
plot(fitted(modelt_hour), residuals(modelt_hour), main="Residual vs Fitted Plot for Hour: Type Blocked")
Residuals vs. Fitted Plots are a graph that is used for residual analysis. They help identify linearity, outliers, and error variances.
From our data plots, it can be seen that there is a slight skew positively. However, there were no extreme outliers or error variances, which indicate that the ANOVA testing was appropriate.
#The following are interaction plots of the three factors
interaction.plot(storm$day, storm$month, storm$wind, xlab="Month", ylab="Wind Pressure", trace.label="Day", main="Interaction Plot: Month vs. Day")
interaction.plot(storm$month, storm$hour, storm$wind, xlab="Month", ylab="Wind Pressure", trace.label="Day", main="Interaction Plot: Month vs. Hour")
interaction.plot(storm$day, storm$hour, storm$wind, xlab="Month", ylab="Wind Pressure", trace.label="Day", main="Interaction Plot: Day vs. Hour")
Interaction plots are used in order to display the mean of the response for two-way combinations of factors. They indicate the interaction between these factors through visual plots. Interactions are displayed through changes in slope and intersections of the factors.
It can be seen in our three interaction plots that there are many interactions between the factors. The interactions plots of Month vs Day and Hour vs Day show there are more interactions than the Month vs Hour plot.
4. References to the literature
No literature was used.
5. Appendices
A summary of, or pointer to, the raw data
complete and documented R code
The data originated from the National Hurricane Center’s archive of Tropical Cyclone Reports (http://www.nhc.noaa.gov/). This dataset was hand-scraped from best track tables in the individual tropical cyclone reports (PDF, HTML and Microsoft Word) by Jon Hobbs and is publically available at: https://github.com/hadley/nasaweather.
The Tropical Cyclone Reports had a variety of storm type designations and there appeared to be no consistent naming convention for cyclones that were not hurricanes, tropical depressions, or tropical storm. Many of these designations have been combined into the “Extratropical” category in this dataset.