Recipe 5: A Completely Randomized Design

NASA Weather: Analysis of Storm Data

Max Winkelman

Rensselaer Polytechnic Institute

October 23, 2014

Version 1

1. Setting

NASA Weather

The data in this package are atmospheric and geographic measurements from a 24 by 24 grid that encloses Central America. There are four sets of data. The data set to be analyzed in this recipe will be ‘storms.’

#load in the data
library("nasaweather", lib.loc="C:/Program Files/R/R-3.1.1/library")
#references 'fueleconomy' in R library
storm<-storms
#assigns the variables in the data set, storms, to the variable named 'storm'
storms = storm[-1666,]
storms = storms[-1604,]
#Removes specific two rows from the data set that contain the hour "1" and "15" and only appear once 
storms$name = as.factor(storms$name)
storms$year = as.factor(storms$year)
storms$month = as.factor(storms$month)
storms$day = as.factor(storms$day)
storms$hour = as.factor(storms$hour)
storms$type = as.factor(storms$type)
#assigns all of the factors in the data set as factors 

Factors and Levels

Factors: Year, Day, and Hour

Factor Levels: Years, 1995-2000, Days 1-31, and Hours 0, 6, 12, and 18

Blocking Factors: Type and Month

Blocking Factor Levels: Extratropical, Hurricane, Tropic Storm, and Tropical Depression, Months 6-12, and

head(storms)

##      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       4

#displays the first 6 sets of variables 
tail(storms)

##        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     144

#displays the last 6 sets of variables 
summary(storms)

##       name        year     month         day       hour          lat      
##  Alberto:  87   1995:724   6 : 83   24     : 117   0 :686   Min.   : 8.3  
##  Marilyn:  77   1996:536   7 :251   27     : 112   6 :671   1st Qu.:17.3  
##  Mitch  :  76   1997:186   8 :720   28     : 110   12:691   Median :25.0  
##  Edouard:  73   1998:481   9 :946   22     : 108   18:697   Mean   :26.7  
##  Fran   :  71   1999:411   10:568   25     : 107            3rd Qu.:33.9  
##  Felix  :  69   2000:407   11:170   19     : 106            Max.   :70.7  
##  (Other):2292              12:  7   (Other):2085                          
##       long           pressure         wind                        type    
##  Min.   :-107.3   Min.   : 905   Min.   : 15.0   Extratropical      :412  
##  1st Qu.: -77.6   1st Qu.: 980   1st Qu.: 35.0   Hurricane          :896  
##  Median : -60.9   Median : 995   Median : 50.0   Tropical Depression:511  
##  Mean   : -60.9   Mean   : 990   Mean   : 54.7   Tropical Storm     :926  
##  3rd Qu.: -45.8   3rd Qu.:1004   3rd Qu.: 70.0                            
##  Max.   :   1.0   Max.   :1019   Max.   :155.0                            
##                                                                           
##     seasday   
##  Min.   :  3  
##  1st Qu.: 84  
##  Median :103  
##  Mean   :103  
##  3rd Qu.:125  
##  Max.   :185  
## 

#displays a summary of the variables

Continuous Variables:

In the data set, “lat,” “long,” “pressure,” and “wind” can be considered continuous variables. Although several other variables may appear to be continuous, they can be grouped together and are technically categorical.

Response Variables:

The response variable for this recipe with by “pressure,” measured in millibars.

The Data: How is it organized and what does it look like?

The data sets of NASA weather were observed from January of 1995 to December of 2000 and were obtained from the NASA Langley Research Center Atmospheric Sciences Data Center. The variables are: “Name,” displays the name of the storm, “Year,” “Month,” and “Day” all describe the data on which the data was recorded, “Latitude” and “Longitude,” describe the location of the storm, “Pressure” shows the pressure in millibars, “Wind” shows the wind speed in mph, and “SeasDay” shows the day of the hurricane season that a storm occurred.

Randomization:

The data from “storms” is organized based on the variables within each column. However, it can be assumed that the original data was gathered with proper randomization methods utilized by NASA. In a real completely randomized design, data would be gathered and selected for analysis at random.

2. Experimental Design

How will the experiment be organized and conducted to test the hypothesis?

The results of weather testing conducted by NASA were not acquired with any one specific experiment or hypothesis in mind. This data is made publicly available for anyone to use to test their own hypothesis. For this recipe, a three-factor, multilevel experiment that blocks on two additional factors will be conducted. Multiple anovas will be performed to determine if the variation of pressure (millibar) can be attributed to storm year, day, and hour. Storm month and type will be the blocking factors in this experiment, giving a total of six anova tests performed. The hypothesis in each anova will be that the pressure means of all factor levels will be equal. If the null hypothesis is rejected, the alternative hypothesis, which states that the pressure means of all factor levels are not equal, will be accepted. A Tukey’s Honestly Significant Difference test will be conducted to determine specifically which pressure means are significantly different.

What is the rationale for this design?

The experiment that is performed in this recipe is a multi-factor, multi-level design. An anova is appropriate for determining the variation between the level means.

Randomize: What is the Randomization Scheme?

Since this data was gathered with no specific intention, the randomization scheme, if any, is unknown.

Replicate: Are there replicates and/or repeated measures?

Yes, measurements of pressure, wind, latitude, and longitude were recorded for the same storms at several different time points. No replicates were performed in this data gathering.

Block: Did you use blocking in the design?

The blocking factors in this recipe will be storm month and storm type.

#Boxplots
#Factors:
boxplot(pressure~year,data=storms, xlab="Year", ylab="Pressure (millibars)")

boxplot(pressure~day,data=storms, xlab="Day", ylab="Pressure (millibars)")

boxplot(pressure~hour,data=storms, xlab="Hour", ylab="Pressure (millibars)")

#displays the bloxplot of air pressure for storms based on the year, day, and hour that they occurred

#Blocking Factors
boxplot(pressure~month,data=storms, xlab="Month", ylab="Pressure (millibars)")

boxplot(pressure~type,data=storms, xlab="Storm Type", ylab="Pressure (millibars)")

#displays the bloxplot of air pressure for storms based on the month and the storm type

The boxplots above show the distribution of pressure means for the factors year, day, hour, month, and type. Due to the variation of pressure values between storms of different months and types, month and type have been selected as the blocking factors for this experiment.

An analysis of variance (ANOVA) will be used to determine the statistical significance between the pressure means. The null hypothesis for all three ANOVA tests is that the mean pressure of all samples are equal to each other. The anova tests will be performed as follows: Storm Year (Blocking for Storm Month), Storm Year (Blocking for Storm Type), Storm Day (Blocking for Storm Month), Storm Day (Blocking for Storm Type), Storm Hour (Blocking for Storm Month), and Storm Hour (Blocking for Storm Type). If the null hypothesis is rejected, the alternative hypothesis is accepted. Afterwards, a Tukey’s Honestly Significant Difference test will be performed to determine which means are significantly different.

# ANOVA
#Storm Year, blocking for Month
model_year_month = aov(pressure~year+month,data=storms) 
anova(model_year_month)

## Analysis of Variance Table
## 
## Response: pressure
##             Df Sum Sq Mean Sq F value Pr(>F)    
## year         5  22583    4517    14.2  1e-13 ***
## month        6  65274   10879    34.1 <2e-16 ***
## Residuals 2733 871216     319                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#Storm Year, blocking for Type
model_year_type = aov(pressure~year+type,data=storms) 
anova(model_year_type)

## Analysis of Variance Table
## 
## Response: pressure
##             Df Sum Sq Mean Sq F value Pr(>F)    
## year         5  22583    4517    30.2 <2e-16 ***
## type         3 527976  175992  1178.7 <2e-16 ***
## Residuals 2736 408515     149                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#Storm Day, blocking for Month
model_day_month = aov(pressure~day+month,data=storms) 
anova(model_day_month)

## Analysis of Variance Table
## 
## Response: pressure
##             Df Sum Sq Mean Sq F value  Pr(>F)    
## day         30  21120     704    2.18 0.00021 ***
## month        6  65283   10881   33.76 < 2e-16 ***
## Residuals 2708 872670     322                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#Storm Day, blocking for Type
model_day_type = aov(pressure~day+type,data=storms) 
anova(model_day_type)

## Analysis of Variance Table
## 
## Response: pressure
##             Df Sum Sq Mean Sq F value  Pr(>F)    
## day         30  21120     704    4.72 6.6e-16 ***
## type         3 533430  177810 1191.63 < 2e-16 ***
## Residuals 2711 404524     149                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#Storm Hour, blocking for Month
model_hour_month = aov(pressure~hour+month,data=storms) 
anova(model_hour_month)

## Analysis of Variance Table
## 
## Response: pressure
##             Df Sum Sq Mean Sq F value Pr(>F)    
## hour         3    146      49    0.15   0.93    
## month        6  62665   10444   31.87 <2e-16 ***
## Residuals 2735 896262     328                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#Storm Hour, blocking for Type
model_hour_type = aov(pressure~hour+type,data=storms) 
anova(model_hour_type)

## Analysis of Variance Table
## 
## Response: pressure
##             Df Sum Sq Mean Sq F value Pr(>F)    
## hour         3    146      49    0.32   0.81    
## type         3 537335  179112 1163.23 <2e-16 ***
## Residuals 2738 421592     154                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The anova for storm year (blocking for month), produced a p value of 1e-13, indicating that the probability that the variation of pressure can be attributed to the storm year, when blocking for month, is very high. The anova for storm year (blocking for type), produced a p value of less than 2e-16, indicating that the probability that the variation of pressure, when blocking for type, can be attributed to the storm year is very high. The anova for storm day (blocking for month), produced a p value of 0.00021, indicating that the probability that the variation of pressure, when blocking for month, can be attributed to the storm year is very high. The anova for storm day (blocking for type), produced a p value of 6.6e-16, indicating that the probability that the variation of pressure, when blocking for type, can be attributed to the storm day is very high. The anova for storm hour (blocking for month), produced a p value of 0.93, indicating that the probability that the variation of pressure, when blocking for month, can be attributed to the storm hour is very low. The anova for storm hour (blocking for type), produced a p value of 0.81, indicating that the probability that the variation of pressure, when blocking for type, can be attributed to the storm hour is very low.

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.

#Tukey's HSD
#Storm Year, blocking for Month
TukeyHSD(model_year_month, ordered = FALSE, conf.level = 0.95)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = pressure ~ year + month, data = storms)
## 
## $year
##                diff      lwr     upr  p adj
## 1996-1995   0.07629  -2.8250  2.9775 1.0000
## 1997-1995   7.65250   3.4670 11.8380 0.0000
## 1998-1995   0.33239  -2.6627  3.3274 0.9996
## 1999-1995  -3.63315  -6.7777 -0.4886 0.0128
## 2000-1995   4.29062   1.1362  7.4450 0.0015
## 1997-1996   7.57621   3.2433 11.9091 0.0000
## 1998-1996   0.25610  -2.9417  3.4539 0.9999
## 1999-1996  -3.70944  -7.0477 -0.3712 0.0193
## 2000-1996   4.21433   0.8668  7.5619 0.0045
## 1998-1997  -7.32011 -11.7164 -2.9238 0.0000
## 1999-1997 -11.28565 -15.7851 -6.7862 0.0000
## 2000-1997  -3.36188  -7.8682  1.1445 0.2733
## 1999-1998  -3.96554  -7.3857 -0.5454 0.0123
## 2000-1998   3.95823   0.5291  7.3874 0.0129
## 2000-1999   7.92377   4.3633 11.4843 0.0000
## 
## $month
##           diff      lwr     upr  p adj
## 7-6    -5.1317 -11.8019   1.539 0.2590
## 8-6   -12.7165 -18.8231  -6.610 0.0000
## 9-6   -18.3982 -24.4289 -12.368 0.0000
## 10-6  -12.2183 -18.4087  -6.028 0.0000
## 11-6   -8.6095 -15.6635  -1.555 0.0060
## 12-6  -17.8545 -38.5881   2.879 0.1453
## 8-7    -7.5848 -11.4463  -3.723 0.0000
## 9-7   -13.2665 -17.0068  -9.526 0.0000
## 10-7   -7.0866 -11.0794  -3.094 0.0000
## 11-7   -3.4778  -8.7104   1.755 0.4400
## 12-7  -12.7228 -32.9096   7.464 0.5076
## 9-8    -5.6817  -8.2871  -3.076 0.0000
## 10-8    0.4982  -2.4582   3.455 0.9989
## 11-8    4.1071  -0.3850   8.599 0.0992
## 12-8   -5.1380 -25.1456  14.870 0.9887
## 10-9    6.1799   3.3836   8.976 0.0000
## 11-9    9.7888   5.4004  14.177 0.0000
## 12-9    0.5437 -19.4408  20.528 1.0000
## 11-10   3.6089  -0.9966   8.214 0.2384
## 12-10  -5.6362 -25.6695  14.397 0.9819
## 12-11  -9.2451 -29.5619  11.072 0.8317

#Storm Year, blocking for Type
TukeyHSD(model_year_type, ordered = FALSE, conf.level = 0.95)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = pressure ~ year + type, data = storms)
## 
## $year
##                diff     lwr     upr  p adj
## 1996-1995   0.07629  -1.909  2.0619 1.0000
## 1997-1995   7.65250   4.788 10.5170 0.0000
## 1998-1995   0.33239  -1.717  2.3822 0.9974
## 1999-1995  -3.63315  -5.785 -1.4811 0.0000
## 2000-1995   4.29062   2.132  6.4495 0.0000
## 1997-1996   7.57621   4.611 10.5416 0.0000
## 1998-1996   0.25610  -1.932  2.4447 0.9995
## 1999-1996  -3.70944  -5.994 -1.4248 0.0001
## 2000-1996   4.21433   1.923  6.5054 0.0000
## 1998-1997  -7.32011 -10.329 -4.3113 0.0000
## 1999-1997 -11.28565 -14.365 -8.2063 0.0000
## 2000-1997  -3.36188  -6.446 -0.2778 0.0233
## 1999-1998  -3.96554  -6.306 -1.6249 0.0000
## 2000-1998   3.95823   1.611  6.3051 0.0000
## 2000-1999   7.92377   5.487 10.3605 0.0000
## 
## $type
##                                       diff    lwr     upr p adj
## Hurricane-Extratropical            -22.626 -24.50 -20.756     0
## Tropical Depression-Extratropical   12.745  10.67  14.825     0
## Tropical Storm-Extratropical         4.250   2.39   6.110     0
## Tropical Depression-Hurricane       35.371  33.63  37.113     0
## Tropical Storm-Hurricane            26.876  25.40  28.348     0
## Tropical Storm-Tropical Depression  -8.495 -10.23  -6.764     0

#Storm Day, blocking for Month
TukeyHSD(model_day_month, ordered = FALSE, conf.level = 0.95)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = pressure ~ day + month, data = storms)
## 
## $day
##           diff      lwr     upr  p adj
## 2-1   -2.02448 -12.0289  7.9800 1.0000
## 3-1   -2.05177 -12.5309  8.4274 1.0000
## 4-1   -1.68728 -12.0841  8.7095 1.0000
## 5-1    1.50253  -8.4097 11.4148 1.0000
## 6-1    0.72504  -9.2480 10.6981 1.0000
## 7-1   -1.07251 -11.5100  9.3650 1.0000
## 8-1   -1.29899 -11.6561  9.0581 1.0000
## 9-1   -4.64141 -15.3928  6.1099 0.9993
## 10-1  -6.65338 -17.9025  4.5957 0.9311
## 11-1  -7.44285 -18.6920  3.8063 0.7990
## 12-1  -7.44801 -18.1041  3.2081 0.6974
## 13-1  -6.77994 -17.3454  3.7855 0.8451
## 14-1  -3.65540 -13.8985  6.5877 1.0000
## 15-1  -2.15657 -12.0688  7.7557 1.0000
## 16-1  -1.94768 -11.8304  7.9351 1.0000
## 17-1  -1.73807 -11.6804  8.2043 1.0000
## 18-1  -2.70519 -12.6783  7.2679 1.0000
## 19-1   0.08529  -9.3709  9.5415 1.0000
## 20-1   0.41548  -9.0407  9.8717 1.0000
## 21-1   3.50230  -6.0202 13.0248 1.0000
## 22-1   1.41582  -7.9980 10.8297 1.0000
## 23-1   1.86831  -7.5879 11.3245 1.0000
## 24-1   2.29759  -6.9415 11.5366 1.0000
## 25-1  -3.35070 -12.7855  6.0841 1.0000
## 26-1  -4.79867 -14.3211  4.7238 0.9913
## 27-1  -3.20851 -12.5416  6.1246 1.0000
## 28-1  -5.12929 -14.5021  4.2435 0.9727
## 29-1  -3.24185 -12.7198  6.2361 1.0000
## 30-1  -0.64409 -10.1894  8.9012 1.0000
## 31-1  -1.92050 -12.8780  9.0370 1.0000
## 3-2   -0.02729 -10.8637 10.8091 1.0000
## 4-2    0.33720 -10.4196 11.0940 1.0000
## 5-2    3.52701  -6.7622 13.8162 1.0000
## 6-2    2.74952  -7.5983 13.0974 1.0000
## 7-2    0.95197  -9.8442 11.7481 1.0000
## 8-2    0.72549  -9.9929 11.4439 1.0000
## 9-2   -2.61693 -13.7168  8.4829 1.0000
## 10-2  -4.62890 -16.2115  6.9538 0.9998
## 11-2  -5.41837 -17.0010  6.1643 0.9973
## 12-2  -5.42353 -16.4311  5.5841 0.9938
## 13-2  -4.75546 -15.6753  6.1644 0.9992
## 14-2  -1.63092 -12.2393  8.9774 1.0000
## 15-2  -0.13209 -10.4213 10.1571 1.0000
## 16-2   0.07680 -10.1840 10.3376 1.0000
## 17-2   0.28641 -10.0318 10.6047 1.0000
## 18-2  -0.68071 -11.0286  9.6671 1.0000
## 19-2   2.10977  -7.7409 11.9604 1.0000
## 20-2   2.43996  -7.4107 12.2906 1.0000
## 21-2   5.52678  -4.3875 15.4411 0.9655
## 22-2   3.44031  -6.3697 13.2503 1.0000
## 23-2   3.89279  -5.9579 13.7434 0.9999
## 24-2   4.32207  -5.3203 13.9644 0.9987
## 25-2  -1.32622 -11.1564  8.5039 1.0000
## 26-2  -2.77419 -12.6885  7.1401 1.0000
## 27-2  -1.18403 -10.9166  8.5485 1.0000
## 28-2  -3.10481 -12.8754  6.6658 1.0000
## 29-2  -1.21737 -11.0889  8.6541 1.0000
## 30-2   1.38039  -8.5558 11.3166 1.0000
## 31-2   0.10398 -11.1956 11.4036 1.0000
## 4-3    0.36449 -10.8352 11.5642 1.0000
## 5-3    3.55429  -7.1971 14.3056 1.0000
## 6-3    2.77681  -8.0307 13.5843 1.0000
## 7-3    0.97926 -10.2582 12.2167 1.0000
## 8-3    0.75278 -10.4100 11.9156 1.0000
## 9-3   -2.58965 -14.1192  8.9399 1.0000
## 10-3  -4.60161 -16.5967  7.3934 0.9999
## 11-3  -5.39108 -17.3861  6.6040 0.9986
## 12-3  -5.39624 -16.8370  6.0445 0.9969
## 13-3  -4.72817 -16.0845  6.6282 0.9997
## 14-3  -1.60363 -12.6608  9.4535 1.0000
## 15-3  -0.10480 -10.8561 10.6466 1.0000
## 16-3   0.10409 -10.6200 10.8282 1.0000
## 17-3   0.31370 -10.4654 11.0928 1.0000
## 18-3  -0.65342 -11.4609 10.1540 1.0000
## 19-3   2.13705  -8.1954 12.4695 1.0000
## 20-3   2.46724  -7.8652 12.7996 1.0000
## 21-3   5.55407  -4.8390 15.9472 0.9800
## 22-3   3.46759  -6.8260 13.7612 1.0000
## 23-3   3.92007  -6.4123 14.2525 0.9999
## 24-3   4.34936  -5.7847 14.4834 0.9994
## 25-3  -1.29894 -11.6118  9.0139 1.0000
## 26-3  -2.74690 -13.1400  7.6462 1.0000
## 27-3  -1.15675 -11.3766  9.0631 1.0000
## 28-3  -3.07753 -13.3337  7.1786 1.0000
## 29-3  -1.19008 -11.5424  9.1622 1.0000
## 30-3   1.40768  -9.0064 11.8217 1.0000
## 31-3   0.13127 -11.5907 11.8533 1.0000
## 5-4    3.18980  -7.4813 13.8609 1.0000
## 6-4    2.41232  -8.3154 13.1400 1.0000
## 7-4    0.61477 -10.5459 11.7755 1.0000
## 8-4    0.38829 -10.6973 11.4738 1.0000
## 9-4   -2.95414 -14.4089  8.5006 1.0000
## 10-4  -4.96610 -16.8893  6.9571 0.9996
## 11-4  -5.75557 -17.6788  6.1676 0.9955
## 12-4  -5.76073 -17.1261  5.6047 0.9905
## 13-4  -5.09266 -16.3731  6.1878 0.9985
## 14-4  -1.96812 -12.9473  9.0110 1.0000
## 15-4  -0.46929 -11.1404 10.2019 1.0000
## 16-4  -0.26040 -10.9041 10.3833 1.0000
## 17-4  -0.05079 -10.7499 10.6483 1.0000
## 18-4  -1.01791 -11.7456  9.7098 1.0000
## 19-4   1.77257  -8.4764 12.0215 1.0000
## 20-4   2.10275  -8.1462 12.3517 1.0000
## 21-4   5.18958  -5.1205 15.4997 0.9914
## 22-4   3.10310  -7.1067 13.3129 1.0000
## 23-4   3.55558  -6.6933 13.8045 1.0000
## 24-4   3.98487  -6.0640 14.0338 0.9999
## 25-4  -1.66343 -11.8926  8.5658 1.0000
## 26-4  -3.11139 -13.4215  7.1987 1.0000
## 27-4  -1.52124 -11.6567  8.6142 1.0000
## 28-4  -3.44201 -13.6140  6.7300 1.0000
## 29-4  -1.55457 -11.8235  8.7144 1.0000
## 30-4   1.04319  -9.2880 11.3744 1.0000
## 31-4  -0.23322 -11.8817 11.4152 1.0000
## 6-5   -0.77748 -11.0362  9.4813 1.0000
## 7-5   -2.57503 -13.2858  8.1357 1.0000
## 8-5   -2.80152 -13.4339  7.8309 1.0000
## 9-5   -6.14394 -17.1608  4.8729 0.9653
## 10-5  -8.15590 -19.6590  3.3472 0.6674
## 11-5  -8.94537 -20.4485  2.5578 0.4541
## 12-5  -8.95053 -19.8744  1.9734 0.3342
## 13-5  -8.28247 -19.1180  2.5530 0.4949
## 14-5  -5.15793 -15.6794  5.3635 0.9943
## 15-5  -3.65909 -13.8587  6.5405 1.0000
## 16-5  -3.45020 -13.6211  6.7207 1.0000
## 17-5  -3.24060 -13.4695  6.9883 1.0000
## 18-5  -4.20772 -14.4665  6.0510 0.9997
## 19-5  -1.41724 -11.1743  8.3398 1.0000
## 20-5  -1.08705 -10.8441  8.6700 1.0000
## 21-5   1.99978  -7.8215 11.8210 1.0000
## 22-5  -0.08670  -9.8027  9.6293 1.0000
## 23-5   0.36578  -9.3912 10.1228 1.0000
## 24-5   0.79507  -8.7516 10.3418 1.0000
## 25-5  -4.85323 -14.5895  4.8831 0.9926
## 26-5  -6.30119 -16.1225  3.5201 0.8453
## 27-5  -4.71104 -14.3488  4.9267 0.9945
## 28-5  -6.63182 -16.3080  3.0444 0.7363
## 29-5  -4.74437 -14.5224  5.0337 0.9951
## 30-5  -2.14661 -11.9900  7.6968 1.0000
## 31-5  -3.42302 -14.6411  7.7951 1.0000
## 7-6   -1.79755 -12.5646  8.9695 1.0000
## 8-6   -2.02403 -12.7132  8.6651 1.0000
## 9-6   -5.36646 -16.4381  5.7052 0.9952
## 10-6  -7.37842 -18.9340  4.1772 0.8518
## 11-6  -8.16789 -19.7235  3.3877 0.6740
## 12-6  -8.17305 -19.1522  2.8061 0.5575
## 13-6  -7.50498 -18.3962  3.3862 0.7259
## 14-6  -4.38044 -14.9592  6.1984 0.9997
## 15-6  -2.88161 -13.1404  7.3772 1.0000
## 16-6  -2.67272 -12.9030  7.5575 1.0000
## 17-6  -2.46311 -12.7510  7.8247 1.0000
## 18-6  -3.43023 -13.7478  6.8873 1.0000
## 19-6  -0.63975 -10.4586  9.1791 1.0000
## 20-6  -0.30957 -10.1284  9.5093 1.0000
## 21-6   2.77726  -7.1054 12.6599 1.0000
## 22-6   0.69078  -9.0872 10.4688 1.0000
## 23-6   1.14326  -8.6756 10.9621 1.0000
## 24-6   1.57255  -8.0373 11.1824 1.0000
## 25-6  -4.07574 -13.8740  5.7225 0.9997
## 26-6  -5.52371 -15.4064  4.3589 0.9644
## 27-6  -3.93355 -13.6339  5.7667 0.9998
## 28-6  -5.85433 -15.5929  3.8842 0.9179
## 29-6  -3.96689 -13.8066  5.8728 0.9998
## 30-6  -1.36913 -11.2738  8.5355 1.0000
## 31-6  -2.64554 -13.9174  8.6263 1.0000
## 8-7   -0.22648 -11.3502 10.8972 1.0000
## 9-7   -3.56891 -15.0606  7.9228 1.0000
## 10-7  -5.58087 -17.5396  6.3778 0.9974
## 11-7  -6.37034 -18.3290  5.5883 0.9809
## 12-7  -6.37550 -17.7781  5.0271 0.9642
## 13-7  -5.70744 -17.0254  5.6105 0.9912
## 14-7  -2.58289 -13.6006  8.4348 1.0000
## 15-7  -1.08406 -11.7948  9.6267 1.0000
## 16-7  -0.87517 -11.5586  9.8083 1.0000
## 17-7  -0.66556 -11.4042 10.0731 1.0000
## 18-7  -1.63269 -12.3998  9.1344 1.0000
## 19-7   1.15779  -9.1324 11.4480 1.0000
## 20-7   1.48798  -8.8022 11.7782 1.0000
## 21-7   4.57481  -5.7763 14.9259 0.9990
## 22-7   2.48833  -7.7629 12.7396 1.0000
## 23-7   2.94081  -7.3494 13.2310 1.0000
## 24-7   3.37010  -6.7209 13.4611 1.0000
## 25-7  -2.27820 -12.5487  7.9924 1.0000
## 26-7  -3.72616 -14.0773  6.6250 1.0000
## 27-7  -2.13601 -12.3132  8.0411 1.0000
## 28-7  -4.05679 -14.2704  6.1568 0.9999
## 29-7  -2.16934 -12.4795  8.1408 1.0000
## 30-7   0.42842  -9.9437 10.8006 1.0000
## 31-7  -0.84799 -12.5328 10.8368 1.0000
## 9-8   -3.34242 -14.7611  8.0763 1.0000
## 10-8  -5.35439 -17.2430  6.5342 0.9986
## 11-8  -6.14386 -18.0324  5.7447 0.9874
## 12-8  -6.14902 -17.4781  5.1800 0.9755
## 13-8  -5.48095 -16.7248  5.7629 0.9947
## 14-8  -2.35641 -13.2980  8.5851 1.0000
## 15-8  -0.85758 -11.4900  9.7749 1.0000
## 16-8  -0.64869 -11.2536  9.9562 1.0000
## 17-8  -0.43908 -11.0996 10.2214 1.0000
## 18-8  -1.40620 -12.0954  9.2830 1.0000
## 19-8   1.38428  -8.8243 11.5929 1.0000
## 20-8   1.71447  -8.4941 11.9231 1.0000
## 21-8   4.80129  -5.4687 15.0713 0.9973
## 22-8   2.71481  -7.4545 12.8842 1.0000
## 23-8   3.16730  -7.0413 13.3759 1.0000
## 24-8   3.59658  -6.4112 13.6044 1.0000
## 25-8  -2.05171 -12.2405  8.1371 1.0000
## 26-8  -3.49968 -13.7697  6.7703 1.0000
## 27-8  -1.90952 -12.0042  8.1851 1.0000
## 28-8  -3.83030 -13.9617  6.3011 0.9999
## 29-8  -1.94286 -12.1716  8.2859 1.0000
## 30-8   0.65490  -9.6363 10.9461 1.0000
## 31-8  -0.62151 -12.2345 10.9915 1.0000
## 10-9  -2.01196 -14.2455 10.2216 1.0000
## 11-9  -2.80144 -15.0350  9.4322 1.0000
## 12-9  -2.80660 -14.4972  8.8840 1.0000
## 13-9  -2.13853 -13.7466  9.4695 1.0000
## 14-9   0.98601 -10.3295 12.3015 1.0000
## 15-9   2.48485  -8.5320 13.5017 1.0000
## 16-9   2.69374  -8.2966 13.6840 1.0000
## 17-9   2.90334  -8.1406 13.9473 1.0000
## 18-9   1.93622  -9.1354 13.0078 1.0000
## 19-9   4.72670  -5.8817 15.3351 0.9988
## 20-9   5.05689  -5.5515 15.6653 0.9963
## 21-9   8.14372  -2.5238 18.8112 0.4980
## 22-9   6.05724  -4.5134 16.6279 0.9519
## 23-9   6.50972  -4.0987 17.1181 0.8985
## 24-9   6.93901  -3.4763 17.3543 0.7875
## 25-9   1.29071  -9.2987 11.8801 1.0000
## 26-9  -0.15725 -10.8248 10.5103 1.0000
## 27-9   1.43290  -9.0659 11.9317 1.0000
## 28-9  -0.48788 -11.0220 10.0463 1.0000
## 29-9   1.39957  -9.2282 12.0273 1.0000
## 30-9   3.99733  -6.6906 14.6852 1.0000
## 31-9   2.72092  -9.2450 14.6869 1.0000
## 11-10 -0.78947 -13.4627 11.8838 1.0000
## 12-10 -0.79463 -12.9446 11.3553 1.0000
## 13-10 -0.12657 -12.1971 11.9440 1.0000
## 14-10  2.99798  -8.7915 14.7874 1.0000
## 15-10  4.49681  -7.0063 15.9999 0.9999
## 16-10  4.70570  -6.7720 16.1834 0.9997
## 17-10  4.91531  -6.6138 16.4444 0.9995
## 18-10  3.94818  -7.6074 15.5038 1.0000
## 19-10  6.73866  -4.3739 17.8512 0.9101
## 20-10  7.06885  -4.0437 18.1814 0.8568
## 21-10 10.15568  -1.0133 21.3247 0.1447
## 22-10  8.06920  -3.0073 19.1457 0.6075
## 23-10  8.52168  -2.5909 19.6342 0.4873
## 24-10  8.95097  -1.9774 19.8793 0.3350
## 25-10  3.30267  -7.7917 14.3971 1.0000
## 26-10  1.85471  -9.3143 13.0237 1.0000
## 27-10  3.44486  -7.5631 14.4528 1.0000
## 28-10  1.52408  -9.5176 12.5658 1.0000
## 29-10  3.41153  -7.7195 14.5426 1.0000
## 30-10  6.00929  -5.1792 17.1978 0.9786
## 31-10  4.73288  -7.6822 17.1480 0.9999
## 12-11 -0.00516 -12.1551 12.1448 1.0000
## 13-11  0.66291 -11.4076 12.7334 1.0000
## 14-11  3.78745  -8.0020 15.5769 1.0000
## 15-11  5.28628  -6.2168 16.7894 0.9980
## 16-11  5.49517  -5.9825 16.9729 0.9960
## 17-11  5.70478  -5.8243 17.2339 0.9934
## 18-11  4.73766  -6.8179 16.2932 0.9997
## 19-11  7.52814  -3.5844 18.6407 0.7581
## 20-11  7.85833  -3.2542 18.9709 0.6730
## 21-11 10.94515  -0.2239 22.1142 0.0643
## 22-11  8.85867  -2.2178 19.9352 0.3885
## 23-11  9.31116  -1.8014 20.4237 0.2866
## 24-11  9.74044  -1.1879 20.6688 0.1742
## 25-11  4.09215  -7.0022 15.1865 1.0000
## 26-11  2.64418  -8.5248 13.8132 1.0000
## 27-11  4.23434  -6.7736 15.2423 0.9999
## 28-11  2.31356  -8.7281 13.3552 1.0000
## 29-11  4.20100  -6.9300 15.3321 0.9999
## 30-11  6.79876  -4.3897 17.9873 0.9082
## 31-11  5.52235  -6.8928 17.9375 0.9988
## 13-12  0.66807 -10.8518 12.1879 1.0000
## 14-12  3.79261  -7.4324 15.0176 1.0000
## 15-12  5.29144  -5.6324 16.2153 0.9952
## 16-12  5.50033  -5.3968 16.3974 0.9911
## 17-12  5.70994  -5.2413 16.6612 0.9858
## 18-12  4.74282  -6.2363 15.7219 0.9993
## 19-12  7.53330  -2.9785 18.0451 0.6442
## 20-12  7.86349  -2.6483 18.3753 0.5461
## 21-12 10.95031   0.3788 21.5218 0.0311
## 22-12  8.86383  -1.6099 19.3375 0.2665
## 23-12  9.31632  -1.1955 19.8281 0.1832
## 24-12  9.74560  -0.5713 20.0625 0.0980
## 25-12  4.09731  -6.3953 14.5899 0.9999
## 26-12  2.64934  -7.9221 13.2208 1.0000
## 27-12  4.23950  -6.1617 14.6407 0.9998
## 28-12  2.31872  -8.1181 12.7556 1.0000
## 29-12  4.20616  -6.3252 14.7375 0.9998
## 30-12  6.80392  -3.7881 17.3960 0.8437
## 31-12  5.52751  -6.3529 17.4079 0.9975
## 14-13  3.12454  -8.0144 14.2635 1.0000
## 15-13  4.62338  -6.2121 15.4589 0.9995
## 16-13  4.83226  -5.9762 15.6408 0.9987
## 17-13  5.04187  -5.8212 15.9049 0.9976
## 18-13  4.07475  -6.8164 14.9659 1.0000
## 19-13  6.86523  -3.5547 17.2852 0.8058
## 20-13  7.19542  -3.2245 17.6154 0.7218
## 21-13 10.28225  -0.1979 20.7624 0.0634
## 22-13  8.19577  -2.1857 18.5773 0.4186
## 23-13  8.64825  -1.7717 19.0682 0.3064
## 24-13  9.07753  -1.1457 19.3008 0.1802
## 25-13  3.42924  -6.9713 13.8298 1.0000
## 26-13  1.98128  -8.4988 12.4614 1.0000
## 27-13  3.57143  -6.7369 13.8798 1.0000
## 28-13  1.65065  -8.6937 11.9950 1.0000
## 29-13  3.53810  -6.9016 13.9777 1.0000
## 30-13  6.13585  -4.3650 16.6367 0.9398
## 31-13  4.85945  -6.9398 16.6587 0.9997
## 15-14  1.49883  -9.0226 12.0203 1.0000
## 16-14  1.70772  -8.7859 12.2014 1.0000
## 17-14  1.91733  -8.6325 12.4672 1.0000
## 18-14  0.95021  -9.6286 11.5290 1.0000
## 19-14  3.74069  -6.3523 13.8337 1.0000
## 20-14  4.07088  -6.0221 14.1639 0.9998
## 21-14  7.15770  -2.9974 17.3128 0.6800
## 22-14  5.07123  -4.9821 15.1245 0.9912
## 23-14  5.52371  -4.5693 15.6167 0.9727
## 24-14  5.95299  -3.9368 15.8428 0.9168
## 25-14  0.30470  -9.7683 10.3777 1.0000
## 26-14 -1.14327 -11.2984  9.0118 1.0000
## 27-14  0.44689  -9.5308 10.4246 1.0000
## 28-14 -1.47389 -11.4888  8.5410 1.0000
## 29-14  0.41355  -9.6998 10.5269 1.0000
## 30-14  3.01131  -7.1652 13.1878 1.0000
## 31-14  1.73490  -9.7766 13.2464 1.0000
## 16-15  0.20889  -9.9621 10.3798 1.0000
## 17-15  0.41850  -9.8104 10.6474 1.0000
## 18-15 -0.54863 -10.8074  9.7101 1.0000
## 19-15  2.24185  -7.5152 11.9989 1.0000
## 20-15  2.57204  -7.1850 12.3291 1.0000
## 21-15  5.65887  -4.1624 15.4801 0.9487
## 22-15  3.57239  -6.1436 13.2883 1.0000
## 23-15  4.02487  -5.7322 13.7819 0.9997
## 24-15  4.45416  -5.0925 14.0009 0.9974
## 25-15 -1.19414 -10.9305  8.5422 1.0000
## 26-15 -2.64210 -12.4634  7.1792 1.0000
## 27-15 -1.05195 -10.6897  8.5858 1.0000
## 28-15 -2.97273 -12.6489  6.7035 1.0000
## 29-15 -1.08528 -10.8634  8.6928 1.0000
## 30-15  1.51248  -8.3309 11.3559 1.0000
## 31-15  0.23607 -10.9820 11.4542 1.0000
## 17-16  0.20961  -9.9907 10.4099 1.0000
## 18-16 -0.75751 -10.9877  9.4727 1.0000
## 19-16  2.03297  -7.6941 11.7600 1.0000
## 20-16  2.36315  -7.3639 12.0902 1.0000
## 21-16  5.44998  -4.3415 15.2414 0.9661
## 22-16  3.36350  -6.3223 13.0493 1.0000
## 23-16  3.81598  -5.9110 13.5430 0.9999
## 24-16  4.24527  -5.2708 13.7613 0.9988
## 25-16 -1.40302 -11.1093  8.3032 1.0000
## 26-16 -2.85099 -12.6425  6.9405 1.0000
## 27-16 -1.26083 -10.8682  8.3465 1.0000
## 28-16 -3.18161 -12.8276  6.4644 1.0000
## 29-16 -1.29417 -11.0423  8.4540 1.0000
## 30-16  1.30359  -8.5101 11.1173 1.0000
## 31-16  0.02718 -11.1648 11.2192 1.0000
## 18-17 -0.96712 -11.2550  9.3207 1.0000
## 19-17  1.82336  -7.9643 11.6110 1.0000
## 20-17  2.15355  -7.6341 11.9412 1.0000
## 21-17  5.24037  -4.6113 15.0920 0.9812
## 22-17  3.15390  -6.5928 12.9006 1.0000
## 23-17  3.60638  -6.1812 13.3940 1.0000
## 24-17  4.03566  -5.5423 13.6136 0.9996
## 25-17 -1.61263 -11.3796  8.1543 1.0000
## 26-17 -3.06060 -12.9123  6.7911 1.0000
## 27-17 -1.47044 -11.1392  8.1983 1.0000
## 28-17 -3.39122 -13.0983  6.3158 1.0000
## 29-17 -1.50378 -11.3124  8.3048 1.0000
## 30-17  1.09398  -8.7798 10.9677 1.0000
## 31-17 -0.18242 -11.4271 11.0623 1.0000
## 19-18  2.79048  -7.0283 12.6093 1.0000
## 20-18  3.12067  -6.6981 12.9395 1.0000
## 21-18  6.20750  -3.6752 16.0902 0.8727
## 22-18  4.12102  -5.6570 13.8990 0.9996
## 23-18  4.57350  -5.2453 14.3923 0.9975
## 24-18  5.00278  -4.6071 14.6126 0.9861
## 25-18 -0.64551 -10.4438  9.1527 1.0000
## 26-18 -2.09347 -11.9761  7.7892 1.0000
## 27-18 -0.50332 -10.2036  9.1970 1.0000
## 28-18 -2.42410 -12.1626  7.3144 1.0000
## 29-18 -0.53666 -10.3764  9.3031 1.0000
## 30-18  2.06110  -7.8436 11.9658 1.0000
## 31-18  0.78470 -10.4872 12.0566 1.0000
## 20-19  0.33019  -8.9632  9.6235 1.0000
## 21-19  3.41702  -5.9438 12.7778 1.0000
## 22-19  1.33054  -7.9197 10.5808 1.0000
## 23-19  1.78302  -7.5103 11.0764 1.0000
## 24-19  2.21230  -6.8600 11.2846 1.0000
## 25-19 -3.43599 -12.7076  5.8356 1.0000
## 26-19 -4.88395 -14.2447  4.4768 0.9856
## 27-19 -3.29380 -12.4619  5.8742 1.0000
## 28-19 -5.21458 -14.4231  3.9939 0.9582
## 29-19 -3.32713 -12.6426  5.9883 1.0000
## 30-19 -0.72937 -10.1134  8.6547 1.0000
## 31-19 -2.00578 -12.8230  8.8115 1.0000
## 21-20  3.08683  -6.2740 12.4476 1.0000
## 22-20  1.00035  -8.2499 10.2506 1.0000
## 23-20  1.45283  -7.8405 10.7462 1.0000
## 24-20  1.88212  -7.1902 10.9544 1.0000
## 25-20 -3.76618 -13.0378  5.5054 0.9998
## 26-20 -5.21414 -14.5749  4.1466 0.9658
## 27-20 -3.62399 -12.7920  5.5441 0.9999
## 28-20 -5.54477 -14.7533  3.6637 0.9165
## 29-20 -3.65732 -12.9728  5.6581 0.9999
## 30-20 -1.05956 -10.4436  8.3245 1.0000
## 31-20 -2.33597 -13.1532  8.4813 1.0000
## 22-21 -2.08648 -11.4045  7.2315 1.0000
## 23-21 -1.63400 -10.9948  7.7268 1.0000
## 24-21 -1.20471 -10.3461  7.9366 1.0000
## 25-21 -6.85301 -16.1922  2.4862 0.5909
## 26-21 -8.30097 -17.7287  1.1268 0.1939
## 27-21 -6.71082 -15.9472  2.5256 0.6135
## 28-21 -8.63160 -17.9081  0.6449 0.1150
## 29-21 -6.74415 -16.1269  2.6386 0.6376
## 30-21 -4.14639 -13.5972  5.3044 0.9991
## 31-21 -5.42280 -16.2980  5.4524 0.9926
## 23-22  0.45248  -8.7978  9.7027 1.0000
## 24-22  0.88177  -8.1463  9.9099 1.0000
## 25-22 -4.76653 -13.9949  4.4619 0.9875
## 26-22 -6.21449 -15.5325  3.1035 0.7857
## 27-22 -4.62434 -13.7487  4.5000 0.9905
## 28-22 -6.54512 -15.7101  2.6198 0.6519
## 29-22 -4.65767 -13.9301  4.6148 0.9917
## 30-22 -2.05991 -11.4012  7.2814 1.0000
## 31-22 -3.33632 -14.1165  7.4439 1.0000
## 24-23  0.42929  -8.6430  9.5016 1.0000
## 25-23 -5.21901 -14.4906  4.0526 0.9611
## 26-23 -6.66697 -16.0278  2.6938 0.6577
## 27-23 -5.07682 -14.2449  4.0912 0.9683
## 28-23 -6.99760 -16.2061  2.2109 0.5089
## 29-23 -5.11015 -14.4256  4.2053 0.9718
## 30-23 -2.51239 -11.8964  6.8716 1.0000
## 31-23 -3.78880 -14.6060  7.0284 1.0000
## 25-24 -5.64829 -14.6983  3.4017 0.8802
## 26-24 -7.09626 -16.2376  2.0451 0.4583
## 27-24 -5.50611 -14.4500  3.4378 0.8952
## 28-24 -7.42688 -16.4122  1.5585 0.3152
## 29-24 -5.53944 -14.6344  3.5555 0.9059
## 30-24 -2.94168 -12.1068  6.2235 1.0000
## 31-24 -4.21809 -14.8460  6.4098 0.9999
## 26-25 -1.44796 -10.7872  7.8912 1.0000
## 27-25  0.14219  -9.0038  9.2882 1.0000
## 28-25 -1.77859 -10.9651  7.4080 1.0000
## 29-25  0.10886  -9.1849  9.4026 1.0000
## 30-25  2.70662  -6.6559 12.0691 1.0000
## 31-25  1.43021  -9.3684 12.2288 1.0000
## 27-26  1.59015  -7.6462 10.8265 1.0000
## 28-26 -0.33063  -9.6072  8.9459 1.0000
## 29-26  1.55682  -7.8259 10.9395 1.0000
## 30-26  4.15458  -5.2962 13.6054 0.9991
## 31-26  2.87817  -7.9971 13.7534 1.0000
## 28-27 -1.92078 -11.0028  7.1612 1.0000
## 29-27 -0.03333  -9.2238  9.1571 1.0000
## 30-27  2.56443  -6.6955 11.8244 1.0000
## 31-27  1.28802  -9.4218 11.9978 1.0000
## 29-28  1.88745  -7.3433 11.1182 1.0000
## 30-28  4.48520  -4.8148 13.7852 0.9956
## 31-28  3.20880  -7.5356 13.9532 1.0000
## 30-29  2.59776  -6.8082 12.0037 1.0000
## 31-29  1.32135  -9.5149 12.1576 1.0000
## 31-30 -1.27641 -12.1716  9.6188 1.0000
## 
## $month
##          diff     lwr     upr  p adj
## 7-6    -1.041  -7.748   5.665 0.9993
## 8-6   -12.070 -18.209  -5.930 0.0000
## 9-6   -16.565 -22.629 -10.502 0.0000
## 10-6  -10.963 -17.187  -4.739 0.0000
## 11-6   -9.148 -16.241  -2.056 0.0028
## 12-6  -17.691 -38.538   3.156 0.1581
## 8-7   -11.028 -14.911  -7.146 0.0000
## 9-7   -15.524 -19.285 -11.763 0.0000
## 10-7   -9.922 -13.937  -5.907 0.0000
## 11-7   -8.107 -13.368  -2.846 0.0001
## 12-7  -16.650 -36.946   3.647 0.1903
## 9-8    -4.496  -7.115  -1.876 0.0000
## 10-8    1.106  -1.866   4.079 0.9288
## 11-8    2.921  -1.595   7.438 0.4748
## 12-8   -5.621 -25.738  14.495 0.9825
## 10-9    5.602   2.791   8.414 0.0000
## 11-9    7.417   3.005  11.829 0.0000
## 12-9   -1.126 -21.219  18.968 1.0000
## 11-10   1.815  -2.816   6.445 0.9102
## 12-10  -6.728 -26.870  13.415 0.9572
## 12-11  -8.543 -28.970  11.885 0.8810

#Storm Day, blocking for Type
TukeyHSD(model_day_type, ordered = FALSE, conf.level = 0.95)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = pressure ~ day + type, data = storms)
## 
## $day
##           diff       lwr      upr  p adj
## 2-1   -2.02448  -8.83215  4.78319 1.0000
## 3-1   -2.05177  -9.18245  5.07891 1.0000
## 4-1   -1.68728  -8.76195  5.38739 1.0000
## 5-1    1.50253  -5.24242  8.24747 1.0000
## 6-1    0.72504  -6.06130  7.51138 1.0000
## 7-1   -1.07251  -8.17485  6.02984 1.0000
## 8-1   -1.29899  -8.34662  5.74864 1.0000
## 9-1   -4.64141 -11.95733  2.67451 0.8603
## 10-1  -6.65338 -14.30800  1.00125 0.2162
## 11-1  -7.44285 -15.09747  0.21177 0.0706
## 12-1  -7.44801 -14.69909 -0.19693 0.0350
## 13-1  -6.77994 -13.96935  0.40947 0.0998
## 14-1  -3.65540 -10.62549  3.31469 0.9846
## 15-1  -2.15657  -8.90151  4.58838 1.0000
## 16-1  -1.94768  -8.67253  4.77718 1.0000
## 17-1  -1.73807  -8.50351  5.02737 1.0000
## 18-1  -2.70519  -9.49153  4.08115 0.9998
## 19-1   0.08529  -6.34933  6.51991 1.0000
## 20-1   0.41548  -6.01915  6.85010 1.0000
## 21-1   3.50230  -2.97741  9.98202 0.9768
## 22-1   1.41582  -4.98996  7.82161 1.0000
## 23-1   1.86831  -4.56632  8.30293 1.0000
## 24-1   2.29759  -3.98926  8.58444 1.0000
## 25-1  -3.35070  -9.77079  3.06938 0.9856
## 26-1  -4.79867 -11.27838  1.68105 0.5697
## 27-1  -3.20851  -9.55935  3.14233 0.9910
## 28-1  -5.12929 -11.50716  1.24858 0.3758
## 29-1  -3.24185  -9.69125  3.20755 0.9916
## 30-1  -0.64409  -7.13935  5.85118 1.0000
## 31-1  -1.92050  -9.37667  5.53568 1.0000
## 3-2   -0.02729  -7.40108  7.34650 1.0000
## 4-2    0.33720  -6.98244  7.65684 1.0000
## 5-2    3.52701  -3.47446 10.52847 0.9913
## 6-2    2.74952  -4.29183  9.79087 0.9999
## 7-2    0.95197  -6.39442  8.29837 1.0000
## 8-2    0.72549  -6.56802  8.01900 1.0000
## 9-2   -2.61693 -10.17001  4.93614 1.0000
## 10-2  -4.62890 -12.51049  3.25269 0.9362
## 11-2  -5.41837 -13.29996  2.46322 0.7305
## 12-2  -5.42353 -12.91381  2.06676 0.6213
## 13-2  -4.75546 -12.18606  2.67514 0.8488
## 14-2  -1.63092  -8.84953  5.58769 1.0000
## 15-2  -0.13209  -7.13355  6.86938 1.0000
## 16-2   0.07680  -6.90531  7.05891 1.0000
## 17-2   0.28641  -6.73480  7.30762 1.0000
## 18-2  -0.68071  -7.72206  6.36064 1.0000
## 19-2   2.10977  -4.59326  8.81279 1.0000
## 20-2   2.43996  -4.26307  9.14298 1.0000
## 21-2   5.52678  -1.21954 12.27311 0.3345
## 22-2   3.44031  -3.23504 10.11565 0.9879
## 23-2   3.89279  -2.81024 10.59581 0.9438
## 24-2   4.32207  -2.23923 10.88337 0.8061
## 25-2  -1.32622  -8.01529  5.36285 1.0000
## 26-2  -2.77419  -9.52051  3.97214 0.9997
## 27-2  -1.18403  -7.80667  5.43861 1.0000
## 28-2  -3.10481  -9.75338  3.54375 0.9974
## 29-2  -1.21737  -7.93458  5.49985 1.0000
## 30-2   1.38039  -5.38087  8.14166 1.0000
## 31-2   0.10398  -7.58502  7.79299 1.0000
## 4-3    0.36449  -7.25649  7.98547 1.0000
## 5-3    3.55429  -3.76163 10.87021 0.9950
## 6-3    2.77681  -4.57729 10.13091 0.9999
## 7-3    0.97926  -6.66742  8.62594 1.0000
## 8-3    0.75278  -6.84311  8.34866 1.0000
## 9-3   -2.58965 -10.43510  5.25580 1.0000
## 10-3  -4.60161 -12.76382  3.56060 0.9604
## 11-3  -5.39108 -13.55329  2.77113 0.8018
## 12-3  -5.39624 -13.18126  2.38878 0.7143
## 13-3  -4.72817 -12.45579  2.99944 0.9015
## 14-3  -1.60363  -9.12763  5.92037 1.0000
## 15-3  -0.10480  -7.42072  7.21112 1.0000
## 16-3   0.10409  -7.19331  7.40149 1.0000
## 17-3   0.31370  -7.02112  7.64851 1.0000
## 18-3  -0.65342  -8.00753  6.70068 1.0000
## 19-3   2.13705  -4.89379  9.16790 1.0000
## 20-3   2.46724  -4.56360  9.49809 1.0000
## 21-3   5.55407  -1.51807 12.62621 0.4309
## 22-3   3.46759  -3.53687 10.47205 0.9933
## 23-3   3.92007  -3.11077 10.95092 0.9654
## 24-3   4.34936  -2.54650 11.24522 0.8677
## 25-3  -1.29894  -8.31648  5.71861 1.0000
## 26-3  -2.74690  -9.81904  4.32524 0.9999
## 27-3  -1.15675  -8.11099  5.79750 1.0000
## 28-3  -3.07753 -10.05647  3.90142 0.9990
## 29-3  -1.19008  -8.23445  5.85429 1.0000
## 30-3   1.40768  -5.67871  8.49407 1.0000
## 31-3   0.13127  -7.84513  8.10767 1.0000
## 5-4    3.18980  -4.07154 10.45115 0.9991
## 6-4    2.41232  -4.88749  9.71213 1.0000
## 7-4    0.61477  -6.97971  8.20925 1.0000
## 8-4    0.38829  -7.15504  7.93162 1.0000
## 9-4   -2.95414 -10.74872  4.84044 0.9999
## 10-4  -4.96610 -13.07942  3.14723 0.9011
## 11-4  -5.75557 -13.86890  2.35775 0.6663
## 12-4  -5.76073 -13.49449  1.97302 0.5561
## 13-4  -5.09266 -12.76863  2.58330 0.7945
## 14-4  -1.96812  -9.43906  5.50282 1.0000
## 15-4  -0.46929  -7.73063  6.79205 1.0000
## 16-4  -0.26040  -7.50308  6.98228 1.0000
## 17-4  -0.05079  -7.33117  7.22959 1.0000
## 18-4  -1.01791  -8.31772  6.28190 1.0000
## 19-4   1.77257  -5.20147  8.74660 1.0000
## 20-4   2.10275  -4.87128  9.07679 1.0000
## 21-4   5.18958  -1.82608 12.20525 0.5723
## 22-4   3.10310  -3.84433 10.05054 0.9988
## 23-4   3.55558  -3.41845 10.52962 0.9897
## 24-4   3.98487  -2.85306 10.82280 0.9415
## 25-4  -1.66343  -8.62405  5.29720 1.0000
## 26-4  -3.11139 -10.12705  3.90428 0.9989
## 27-4  -1.52124  -8.41804  5.37557 1.0000
## 28-4  -3.44201 -10.36372  3.47969 0.9929
## 29-4  -1.55457  -8.54224  5.43310 1.0000
## 30-4   1.04319  -5.98684  8.07322 1.0000
## 31-4  -0.23322  -8.15959  7.69315 1.0000
## 6-5   -0.77748  -7.75821  6.20324 1.0000
## 7-5   -2.57503  -9.86334  4.71328 1.0000
## 8-5   -2.80152 -10.03651  4.43348 0.9999
## 9-5   -6.14394 -13.64053  1.35265 0.3336
## 10-5  -8.15590 -15.98338 -0.32842 0.0286
## 11-5  -8.94537 -16.77285 -1.11790 0.0065
## 12-5  -8.95053 -16.38386 -1.51721 0.0024
## 13-5  -8.28247 -15.65565 -0.90929 0.0087
## 14-5  -5.15793 -12.31741  2.00156 0.6326
## 15-5  -3.65909 -10.59958  3.28140 0.9834
## 16-5  -3.45020 -10.37117  3.47076 0.9926
## 17-5  -3.24060 -10.20100  3.71981 0.9975
## 18-5  -4.20772 -11.18844  2.77301 0.9156
## 19-5  -1.41724  -8.05655  5.22208 1.0000
## 20-5  -1.08705  -7.72636  5.55226 1.0000
## 21-5   1.99978  -4.68325  8.68281 1.0000
## 22-5  -0.08670  -6.69807  6.52467 1.0000
## 23-5   0.36578  -6.27353  7.00509 1.0000
## 24-5   0.79507  -5.70113  7.29126 1.0000
## 25-5  -4.85323 -11.47845  1.77200 0.5949
## 26-5  -6.30119 -12.98422  0.38184 0.1000
## 27-5  -4.71104 -11.26919  1.84711 0.6390
## 28-5  -6.63182 -13.21615 -0.04749 0.0456
## 29-5  -4.74437 -11.39801  1.90927 0.6552
## 30-5  -2.14661  -8.84472  4.55149 1.0000
## 31-5  -3.42302 -11.05654  4.21050 0.9987
## 7-6   -1.79755  -9.12418  5.52909 1.0000
## 8-6   -2.02403  -9.29764  5.24957 1.0000
## 9-6   -5.36646 -12.90031  2.16740 0.6575
## 10-6  -7.37842 -15.24159  0.48476 0.1053
## 11-6  -8.16789 -16.03107 -0.30472 0.0299
## 12-6  -8.17305 -15.64396 -0.70214 0.0136
## 13-6  -7.50498 -14.91605 -0.09392 0.0424
## 14-6  -4.38044 -11.57894  2.81806 0.9068
## 15-6  -2.88161  -9.86233  4.09912 0.9997
## 16-6  -2.67272  -9.63404  4.28860 0.9999
## 17-6  -2.46311  -9.46364  4.53742 1.0000
## 18-6  -3.43023 -10.45096  3.59050 0.9945
## 19-6  -0.63975  -7.32112  6.04161 1.0000
## 20-6  -0.30957  -6.99093  6.37180 1.0000
## 21-6   2.77726  -3.94754  9.50207 0.9997
## 22-6   0.69078  -5.96281  7.34438 1.0000
## 23-6   1.14326  -5.53810  7.82463 1.0000
## 24-6   1.57255  -4.96662  8.11172 1.0000
## 25-6  -4.07574 -10.74311  2.59162 0.9024
## 26-6  -5.52371 -12.24851  1.20110 0.3287
## 27-6  -3.93355 -10.53427  2.66716 0.9252
## 28-6  -5.85433 -12.48106  0.77239 0.1884
## 29-6  -3.96689 -10.66249  2.72871 0.9298
## 30-6  -1.36913  -8.10892  5.37066 1.0000
## 31-6  -2.64554 -10.31566  5.02459 1.0000
## 8-7   -0.22648  -7.79578  7.34281 1.0000
## 9-7   -3.56891 -11.38862  4.25080 0.9982
## 10-7  -5.58087 -13.71834  2.55660 0.7351
## 11-7  -6.37034 -14.50781  1.76713 0.4384
## 12-7  -6.37550 -14.13458  1.38358 0.3280
## 13-7  -5.70744 -13.40892  1.99404 0.5680
## 14-7  -2.58289 -10.08005  4.91426 1.0000
## 15-7  -1.08406  -8.37237  6.20425 1.0000
## 16-7  -0.87517  -8.14489  6.39455 1.0000
## 17-7  -0.66556  -7.97284  6.64171 1.0000
## 18-7  -1.63269  -8.95932  5.69395 1.0000
## 19-7   1.15779  -5.84431  8.15990 1.0000
## 20-7   1.48798  -5.51413  8.49009 1.0000
## 21-7   4.57481  -2.46876 11.61838 0.8278
## 22-7   2.48833  -4.48729  9.46395 1.0000
## 23-7   2.94081  -4.06130  9.94292 0.9996
## 24-7   3.37010  -3.49646 10.23665 0.9942
## 25-7  -2.27820  -9.26695  4.71055 1.0000
## 26-7  -3.72616 -10.76973  3.31741 0.9826
## 27-7  -2.13601  -9.06120  4.78919 1.0000
## 28-7  -4.05679 -11.00678  2.89321 0.9405
## 29-7  -2.16934  -9.18503  4.84635 1.0000
## 30-7   0.42842  -6.62946  7.48629 1.0000
## 31-7  -0.84799  -8.79907  7.10309 1.0000
## 9-8   -3.34242 -11.11247  4.42762 0.9994
## 10-8  -5.35439 -13.44414  2.73537 0.7984
## 11-8  -6.14386 -14.23362  1.94590 0.5103
## 12-8  -6.14902 -13.85805  1.56001 0.3947
## 13-8  -5.48095 -13.13200  2.17010 0.6451
## 14-8  -2.35641  -9.80175  5.08893 1.0000
## 15-8  -0.85758  -8.09257  6.37742 1.0000
## 16-8  -0.64869  -7.86496  6.56758 1.0000
## 17-8  -0.43908  -7.69319  6.81502 1.0000
## 18-8  -1.40620  -8.67981  5.86740 1.0000
## 19-8   1.38428  -5.56232  8.33088 1.0000
## 20-8   1.71447  -5.23213  8.66107 1.0000
## 21-8   4.80129  -2.18710 11.78969 0.7317
## 22-8   2.71481  -4.20508  9.63471 0.9999
## 23-8   3.16730  -3.77930 10.11390 0.9982
## 24-8   3.59658  -3.21336 10.40653 0.9830
## 25-8  -2.05171  -8.98485  4.88142 1.0000
## 26-8  -3.49968 -10.48807  3.48872 0.9921
## 27-8  -1.90952  -8.77859  4.95954 1.0000
## 28-8  -3.83030 -10.72437  3.06376 0.9669
## 29-8  -1.94286  -8.90315  5.01744 1.0000
## 30-8   0.65490  -6.34791  7.65771 1.0000
## 31-8  -0.62151  -8.52375  7.28074 1.0000
## 10-9  -2.01196 -10.33649  6.31257 1.0000
## 11-9  -2.80144 -11.12597  5.52309 1.0000
## 12-9  -2.80660 -10.76164  5.14845 1.0000
## 13-9  -2.13853 -10.03740  5.76034 1.0000
## 14-9   0.98601  -6.71377  8.68580 1.0000
## 15-9   2.48485  -5.01174  9.98144 1.0000
## 16-9   2.69374  -4.78478 10.17225 1.0000
## 17-9   2.90334  -4.61169 10.41837 0.9999
## 18-9   1.93622  -5.59763  9.47008 1.0000
## 19-9   4.72670  -2.49195 11.94535 0.8155
## 20-9   5.05689  -2.16176 12.27554 0.6928
## 21-9   8.14372   0.88484 15.40259 0.0089
## 22-9   6.05724  -1.13572 13.25019 0.2764
## 23-9   6.50972  -0.70893 13.72837 0.1565
## 24-9   6.93901  -0.14824 14.02625 0.0650
## 25-9   1.29071  -5.91498  8.49640 1.0000
## 26-9  -0.15725  -7.41613  7.10162 1.0000
## 27-9   1.43290  -5.71117  8.57697 1.0000
## 28-9  -0.48788  -7.65599  6.68023 1.0000
## 29-9   1.39957  -5.83226  8.63139 1.0000
## 30-9   3.99733  -3.27543 11.27008 0.9711
## 31-9   2.72092  -5.42151 10.86334 1.0000
## 11-10 -0.78947  -9.41318  7.83424 1.0000
## 12-10 -0.79463  -9.06224  7.47297 1.0000
## 13-10 -0.12657  -8.34014  8.08700 1.0000
## 14-10  2.99798  -5.02432 11.02027 1.0000
## 15-10  4.49681  -3.33067 12.32429 0.9504
## 16-10  4.70570  -3.10447 12.51587 0.9159
## 17-10  4.91531  -2.92984 12.76045 0.8757
## 18-10  3.94818  -3.91499 11.81136 0.9917
## 19-10  6.73866  -0.82305 14.30037 0.1744
## 20-10  7.06885  -0.49286 14.63056 0.1095
## 21-10 10.15568   2.55556 17.75580 0.0002
## 22-10  8.06920   0.53201 15.60639 0.0192
## 23-10  8.52168   0.95997 16.08339 0.0083
## 24-10  8.95097   1.51460 16.38733 0.0024
## 25-10  3.30267  -4.24667 10.85202 0.9991
## 26-10  1.85471  -5.74541  9.45483 1.0000
## 27-10  3.44486  -4.04568 10.93541 0.9980
## 28-10  1.52408  -5.98940  9.03756 1.0000
## 29-10  3.41153  -4.16276 10.98582 0.9986
## 30-10  6.00929  -1.60409 13.62267 0.4191
## 31-10  4.73288  -3.71518 13.18094 0.9633
## 12-11 -0.00516  -8.27276  8.26244 1.0000
## 13-11  0.66291  -7.55066  8.87648 1.0000
## 14-11  3.78745  -4.23485 11.80975 0.9968
## 15-11  5.28628  -2.54119 13.11376 0.7637
## 16-11  5.49517  -2.31500 13.30534 0.6837
## 17-11  5.70478  -2.14036 13.54992 0.6116
## 18-11  4.73766  -3.12552 12.60083 0.9159
## 19-11  7.52814  -0.03357 15.08985 0.0529
## 20-11  7.85833   0.29662 15.42004 0.0297
## 21-11 10.94515   3.34503 18.54528 0.0000
## 22-11  8.85867   1.32149 16.39586 0.0038
## 23-11  9.31116   1.74945 16.87287 0.0015
## 24-11  9.74044   2.30407 17.17681 0.0004
## 25-11  4.09215  -3.45720 11.64149 0.9759
## 26-11  2.64418  -4.95594 10.24430 1.0000
## 27-11  4.23434  -3.25621 11.72488 0.9591
## 28-11  2.31356  -5.19992  9.82704 1.0000
## 29-11  4.20100  -3.37329 11.77529 0.9676
## 30-11  6.79876  -0.81462 14.41214 0.1712
## 31-11  5.52235  -2.92570 13.97041 0.8181
## 13-12  0.66807  -7.17079  8.50692 1.0000
## 14-12  3.79261  -3.84559 11.43081 0.9930
## 15-12  5.29144  -2.14188 12.72477 0.6588
## 16-12  5.50033  -1.91477 12.91543 0.5659
## 17-12  5.70994  -1.74198 13.16186 0.4892
## 18-12  4.74282  -2.72809 12.21372 0.8594
## 19-12  7.53330   0.38037 14.68622 0.0245
## 20-12  7.86349   0.71056 15.01641 0.0126
## 21-12 10.95031   3.75679 18.14384 0.0000
## 22-12  8.86383   1.73684 15.99083 0.0012
## 23-12  9.31632   2.16339 16.46924 0.0004
## 24-12  9.74560   2.72531 16.76589 0.0001
## 25-12  4.09731  -3.04255 11.23716 0.9510
## 26-12  2.64934  -4.54418  9.84287 1.0000
## 27-12  4.23950  -2.83816 11.31715 0.9210
## 28-12  2.31872  -4.78320  9.42064 1.0000
## 29-12  4.20616  -2.96006 11.37239 0.9367
## 30-12  6.80392  -0.40361 14.01145 0.0987
## 31-12  5.52751  -2.55670 13.61173 0.7409
## 14-13  3.12454  -4.45514 10.70423 0.9997
## 15-13  4.62338  -2.74980 11.99656 0.8747
## 16-13  4.83226  -2.52254 12.18707 0.8102
## 17-13  5.04187  -2.35006 12.43380 0.7455
## 18-13  4.07475  -3.33631 11.48582 0.9710
## 19-13  6.86523  -0.22517 13.95563 0.0742
## 20-13  7.19542   0.10501 14.28582 0.0413
## 21-13 10.28225   3.15089 17.41360 0.0000
## 22-13  8.19577   1.13152 15.26001 0.0049
## 23-13  8.64825   1.55784 15.73865 0.0018
## 24-13  9.07753   2.12096 16.03411 0.0004
## 25-13  3.42924  -3.64797 10.50645 0.9952
## 26-13  1.98128  -5.15008  9.11263 1.0000
## 27-13  3.57143  -3.44303 10.58589 0.9899
## 28-13  1.65065  -5.38829  8.68959 1.0000
## 29-13  3.53810  -3.56572 10.64191 0.9927
## 30-13  6.13585  -1.00963 13.28134 0.2384
## 31-13  4.85945  -3.16950 12.88840 0.9119
## 15-14  1.49883  -5.66065  8.65832 1.0000
## 16-14  1.70772  -5.43284  8.84829 1.0000
## 17-14  1.91733  -5.26147  9.09613 1.0000
## 18-14  0.95021  -6.24829  8.14871 1.0000
## 19-14  3.74069  -3.12723 10.60861 0.9743
## 20-14  4.07088  -2.79704 10.93880 0.9294
## 21-14  7.15770   0.24752 14.06789 0.0312
## 22-14  5.07123  -1.76968 11.91213 0.5673
## 23-14  5.52371  -1.34421 12.39163 0.3757
## 24-14  5.95299  -0.77668 12.68266 0.1863
## 25-14  0.30470  -6.54961  7.15900 1.0000
## 26-14 -1.14327  -8.05346  5.76692 1.0000
## 27-14  0.44689  -6.34260  7.23638 1.0000
## 28-14 -1.47389  -8.28867  5.34089 1.0000
## 29-14  0.41355  -6.46822  7.29532 1.0000
## 30-14  3.01131  -3.91346  9.93608 0.9992
## 31-14  1.73490  -6.09826  9.56807 1.0000
## 16-15  0.20889  -6.71208  7.12986 1.0000
## 17-15  0.41850  -6.54191  7.37890 1.0000
## 18-15 -0.54863  -7.52935  6.43210 1.0000
## 19-15  2.24185  -4.39746  8.88117 1.0000
## 20-15  2.57204  -4.06727  9.21135 0.9999
## 21-15  5.65887  -1.02416 12.34190 0.2654
## 22-15  3.57239  -3.03898 10.18376 0.9769
## 23-15  4.02487  -2.61444 10.66418 0.9104
## 24-15  4.45416  -2.04204 10.95035 0.7356
## 25-15 -1.19414  -7.81936  5.43109 1.0000
## 26-15 -2.64210  -9.32513  4.04093 0.9999
## 27-15 -1.05195  -7.61010  5.50620 1.0000
## 28-15 -2.97273  -9.55706  3.61160 0.9985
## 29-15 -1.08528  -7.73892  5.56836 1.0000
## 30-15  1.51248  -5.18563  8.21058 1.0000
## 31-15  0.23607  -7.39745  7.86959 1.0000
## 17-16  0.20961  -6.73133  7.15055 1.0000
## 18-16 -0.75751  -7.71883  6.20380 1.0000
## 19-16  2.03297  -4.58594  8.65187 1.0000
## 20-16  2.36315  -4.25575  8.98206 1.0000
## 21-16  5.44998  -1.21277 12.11273 0.3379
## 22-16  3.36350  -3.22737  9.95437 0.9895
## 23-16  3.81598  -2.80292 10.43489 0.9483
## 24-16  4.24527  -2.23006 10.72061 0.8136
## 25-16 -1.40302  -8.00779  5.20175 1.0000
## 26-16 -2.85099  -9.51374  3.81176 0.9994
## 27-16 -1.26083  -7.79832  5.27665 1.0000
## 28-16 -3.18161  -9.74536  3.38213 0.9952
## 29-16 -1.29417  -7.92744  5.33910 1.0000
## 30-16  1.30359  -5.37428  7.98146 1.0000
## 31-16  0.02718  -7.58859  7.64296 1.0000
## 18-17 -0.96712  -7.96765  6.03341 1.0000
## 19-17  1.82336  -4.83677  8.48349 1.0000
## 20-17  2.15355  -4.50658  8.81368 1.0000
## 21-17  5.24037  -1.46333 11.94408 0.4418
## 22-17  3.15390  -3.47838  9.78617 0.9964
## 23-17  3.60638  -3.05375 10.26651 0.9762
## 24-17  4.03566  -2.48181 10.55313 0.8890
## 25-17 -1.61263  -8.25872  5.03345 1.0000
## 26-17 -3.06060  -9.76430  3.64311 0.9982
## 27-17 -1.47044  -8.04966  5.10878 1.0000
## 28-17 -3.39122  -9.99654  3.21410 0.9886
## 29-17 -1.50378  -8.17819  5.17063 1.0000
## 30-17  1.09398  -5.62476  7.81272 1.0000
## 31-17 -0.18242  -7.83406  7.46921 1.0000
## 19-18  2.79048  -3.89088  9.47184 0.9996
## 20-18  3.12067  -3.56070  9.80203 0.9973
## 21-18  6.20750  -0.51731 12.93230 0.1248
## 22-18  4.12102  -2.53258 10.77461 0.8888
## 23-18  4.57350  -2.10787 11.25486 0.7387
## 24-18  5.00278  -1.53638 11.54195 0.4929
## 25-18 -0.64551  -7.31288  6.02185 1.0000
## 26-18 -2.09347  -8.81828  4.63133 1.0000
## 27-18 -0.50332  -7.10404  6.09739 1.0000
## 28-18 -2.42410  -9.05083  4.20263 1.0000
## 29-18 -0.53666  -7.23225  6.15894 1.0000
## 30-18  2.06110  -4.67868  8.80089 1.0000
## 31-18  0.78470  -6.88543  8.45482 1.0000
## 20-19  0.33019  -5.99362  6.65400 1.0000
## 21-19  3.41702  -2.95267  9.78671 0.9790
## 22-19  1.33054  -4.96392  7.62500 1.0000
## 23-19  1.78302  -4.54079  8.10683 1.0000
## 24-19  2.21230  -3.96108  8.38569 1.0000
## 25-19 -3.43599  -9.74501  2.87302 0.9743
## 26-19 -4.88395 -11.25364  1.48574 0.4876
## 27-19 -3.29380  -9.53234  2.94474 0.9831
## 28-19 -5.21458 -11.48063  1.05147 0.3008
## 29-19 -3.32713  -9.66598  3.01171 0.9844
## 30-19 -0.72937  -7.11488  5.65613 1.0000
## 31-19 -2.00578  -9.36654  5.35497 1.0000
## 21-20  3.08683  -3.28286  9.45652 0.9952
## 22-20  1.00035  -5.29411  7.29481 1.0000
## 23-20  1.45283  -4.87098  7.77664 1.0000
## 24-20  1.88212  -4.29127  8.05550 1.0000
## 25-20 -3.76618 -10.07519  2.54284 0.9238
## 26-20 -5.21414 -11.58383  1.15555 0.3362
## 27-20 -3.62399  -9.86253  2.61455 0.9436
## 28-20 -5.54477 -11.81082  0.72129 0.1857
## 29-20 -3.65732  -9.99617  2.68152 0.9479
## 30-20 -1.05956  -7.44507  5.32594 1.0000
## 31-20 -2.33597  -9.69673  5.02479 1.0000
## 22-21 -2.08648  -8.42704  4.25408 1.0000
## 23-21 -1.63400  -8.00369  4.73569 1.0000
## 24-21 -1.20471  -7.42509  5.01566 1.0000
## 25-21 -6.85301 -13.20801 -0.49801 0.0171
## 26-21 -8.30097 -14.71621 -1.88573 0.0005
## 27-21 -6.71082 -12.99586 -0.42578 0.0200
## 28-21 -8.63160 -14.94395 -2.31924 0.0001
## 29-21 -6.74415 -13.12877 -0.35953 0.0235
## 30-21 -4.14639 -10.57734  2.28455 0.8384
## 31-21 -5.42280 -12.82301  1.97741 0.5941
## 23-22  0.45248  -5.84198  6.74694 1.0000
## 24-22  0.88177  -5.26155  7.02508 1.0000
## 25-22 -4.76653 -11.04613  1.51307 0.5116
## 26-22 -6.21449 -12.55505  0.12606 0.0642
## 27-22 -4.62434 -10.83313  1.58445 0.5563
## 28-22 -6.54512 -12.78156 -0.30868 0.0258
## 29-22 -4.65767 -10.96724  1.65190 0.5771
## 30-22 -2.05991  -8.41636  4.29653 1.0000
## 31-22 -3.33632 -10.67188  3.99924 0.9983
## 24-23  0.42929  -5.74410  6.60267 1.0000
## 25-23 -5.21901 -11.52802  1.09001 0.3135
## 26-23 -6.66697 -13.03666 -0.29728 0.0268
## 27-23 -5.07682 -11.31536  1.16172 0.3492
## 28-23 -6.99760 -13.26365 -0.73154 0.0097
## 29-23 -5.11015 -11.44900  1.22869 0.3704
## 30-23 -2.51239  -8.89790  3.87311 0.9999
## 31-23 -3.78880 -11.14956  3.57196 0.9881
## 25-24 -5.64829 -11.80652  0.50993 0.1330
## 26-24 -7.09626 -13.31663 -0.87589 0.0067
## 27-24 -5.50611 -11.59211  0.57990 0.1518
## 28-24 -7.42688 -13.54109 -1.31268 0.0020
## 29-24 -5.53944 -11.72822  0.64935 0.1677
## 30-24 -2.94168  -9.17825  3.29489 0.9969
## 31-24 -4.21809 -11.45002  3.01385 0.9410
## 26-25 -1.44796  -7.80297  4.90704 1.0000
## 27-25  0.14219  -6.08135  6.36573 1.0000
## 28-25 -1.77859  -8.02972  4.47254 1.0000
## 29-25  0.10886  -6.21523  6.43295 1.0000
## 30-25  2.70662  -3.66424  9.07747 0.9995
## 31-25  1.43021  -5.91784  8.77826 1.0000
## 27-26  1.59015  -4.69489  7.87519 1.0000
## 28-26 -0.33063  -6.64298  5.98173 1.0000
## 29-26  1.55682  -4.82780  7.94144 1.0000
## 30-26  4.15458  -2.27637 10.58552 0.8356
## 31-26  2.87817  -4.52204 10.27838 0.9999
## 28-27 -1.92078  -8.10077  4.25921 1.0000
## 29-27 -0.03333  -6.28712  6.22045 1.0000
## 30-27  2.56443  -3.73665  8.86550 0.9998
## 31-27  1.28802  -5.99961  8.57565 1.0000
## 29-28  1.88745  -4.39379  8.16868 1.0000
## 30-28  4.48520  -1.84311 10.81352 0.6682
## 31-28  3.20880  -4.10240 10.52000 0.9991
## 30-29  2.59776  -3.80264  8.99816 0.9998
## 31-29  1.32135  -6.05233  8.69503 1.0000
## 31-30 -1.27641  -8.69024  6.13742 1.0000
## 
## $type
##                                       diff     lwr     upr p adj
## Hurricane-Extratropical            -23.283 -25.152 -21.414     0
## Tropical Depression-Extratropical   11.706   9.627  13.785     0
## Tropical Storm-Extratropical         3.563   1.703   5.422     0
## Tropical Depression-Hurricane       34.989  33.248  36.730     0
## Tropical Storm-Hurricane            26.846  25.374  28.317     0
## Tropical Storm-Tropical Depression  -8.143  -9.874  -6.413     0

#Storm Hour, blocking for Month
TukeyHSD(model_hour_month, ordered = FALSE, conf.level = 0.95)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = pressure ~ hour + month, data = storms)
## 
## $hour
##          diff    lwr   upr  p adj
## 6-0   -0.3401 -2.867 2.186 0.9858
## 12-0   0.1608 -2.347 2.669 0.9984
## 18-0   0.2731 -2.230 2.776 0.9923
## 12-6   0.5009 -2.021 3.023 0.9566
## 18-6   0.6133 -1.903 3.130 0.9236
## 18-12  0.1124 -2.386 2.611 0.9994
## 
## $month
##            diff     lwr     upr  p adj
## 7-6    -1.49118  -8.254   5.272 0.9950
## 8-6   -11.61859 -17.810  -5.427 0.0000
## 9-6   -16.55968 -22.674 -10.445 0.0000
## 10-6  -10.50805 -16.785  -4.232 0.0000
## 11-6   -8.41215 -15.564  -1.260 0.0095
## 12-6  -16.54314 -37.565   4.479 0.2337
## 8-7   -10.12741 -14.043  -6.212 0.0000
## 9-7   -15.06850 -18.861 -11.276 0.0000
## 10-7   -9.01687 -13.065  -4.969 0.0000
## 11-7   -6.92097 -12.226  -1.616 0.0023
## 12-7  -15.05196 -35.519   5.415 0.3124
## 9-8    -4.94109  -7.583  -2.300 0.0000
## 10-8    1.11054  -1.887   4.108 0.9303
## 11-8    3.20645  -1.348   7.761 0.3666
## 12-8   -4.92455 -25.210  15.361 0.9917
## 10-9    6.05163   3.216   8.887 0.0000
## 11-9    8.14753   3.698  12.597 0.0000
## 12-9    0.01654 -20.246  20.279 1.0000
## 11-10   2.09591  -2.574   6.765 0.8407
## 12-10  -6.03508 -26.347  14.277 0.9760
## 12-11  -8.13099 -28.730  12.468 0.9073

#Storm Hour, blocking for Type
TukeyHSD(model_hour_type, ordered = FALSE, conf.level = 0.95)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = pressure ~ hour + type, data = storms)
## 
## $hour
##          diff    lwr   upr  p adj
## 6-0   -0.3401 -2.072 1.392 0.9579
## 12-0   0.1608 -1.558 1.880 0.9951
## 18-0   0.2731 -1.442 1.989 0.9769
## 12-6   0.5009 -1.228 2.230 0.8789
## 18-6   0.6133 -1.112 2.338 0.7975
## 18-12  0.1124 -1.600 1.825 0.9983
## 
## $type
##                                       diff     lwr     upr p adj
## Hurricane-Extratropical            -23.511 -25.410 -21.612     0
## Tropical Depression-Extratropical   12.194  10.082  14.306     0
## Tropical Storm-Extratropical         3.753   1.864   5.642     0
## Tropical Depression-Hurricane       35.705  33.937  37.473     0
## Tropical Storm-Hurricane            27.264  25.769  28.759     0
## Tropical Storm-Tropical Depression  -8.441 -10.199  -6.683     0

Tukey’s HSD returns a matrix that contains statistical parameters for the interaction of an individual sample mean with every other sample mean being statistically analyzed. An adj p value of less than 0.05 indicates that the variation of pressure can be explained by something other than randomization, with a confidence level of 95%.

Diagnostics/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
#Storm Year, blocking for Month
qqnorm(residuals(model_year_month), main="Normal Q-Q Plot for Year (Month Blocked)", ylab="Pressure (millibars) Residuals")
qqline(residuals(model_year_month))

#Storm Year, blocking for Type
qqnorm(residuals(model_year_type), main="Normal Q-Q Plot for Year (Type Blocked)", ylab="Pressure (millibars) Residuals")
qqline(residuals(model_year_type))

#Storm Day, blocking for Month
qqnorm(residuals(model_day_month), main="Normal Q-Q Plot for Day (Month Blocked)", ylab="Pressure (millibars) Residuals")
qqline(residuals(model_day_month))

#Storm Day, blocking for Type
qqnorm(residuals(model_day_type), main="Normal Q-Q Plot for Day (Type Blocked)", ylab="Pressure (millibars) Residuals")
qqline(residuals(model_day_type))

#Storm Hour, blocking for Month
qqnorm(residuals(model_hour_month), main="Normal Q-Q Plot for Hour (Month Blocked)", ylab="Pressure (millibars) Residuals")
qqline(residuals(model_hour_month))

#Storm Hour, blocking for Type
qqnorm(residuals(model_hour_type), main="Normal Q-Q Plot for Hour (Type Blocked)", ylab="Pressure (millibars) Residuals")
qqline(residuals(model_hour_type))

All of the Q-Q graphs above produce plots with linear regions and non-linear tails. Despite the non-linear regions of the plots, the graphs above indicate that the population of pressure means that the individual samples were gathered from is normally distributed. The use of anova is justified in this experiment.

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 Plots
#Day-Year
interaction.plot(storms$day,storms$year,storms$pressure, xlab="Storm Day", ylab="Means of Pressure (millibars)", trace.label="Storm Year", main="Storm Day-Year Interaction Plot")

#Day-Hour
interaction.plot(storms$day,storms$hour,storms$pressure, xlab="Storm Day", ylab="Means of Pressure (millibars)", trace.label="Storm Hour", main="Storm Day-Hour Interaction Plot")

#Hour-Year
interaction.plot(storms$hour,storms$year,storms$pressure, xlab="Storm hour", ylab="Means of Pressure (millibars)", trace.label="Storm Year", main="Storm Hour-Year Interaction Plot")

The day-year interaction plot produced lines that are not parallel and contain many intersections, indicating that there is interaction between the two factors. The day-hour interaction plot produced lines that appear to have slopes that are similar but also contain a lot of intersections, indicating that there is interaction between the two factors. The hour-year interaction plot produced lines of similar slope that contain only a few intersections, indicating very little interaction between the factors.

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.

#Residual vs Fit Plot
#Storm Year, blocking for Month
plot(fitted(model_year_month),residuals(model_year_month), main="Residual vs Fitted Plot for Storm Year: Month Blocked")

#Storm Year, blocking for Type
plot(fitted(model_year_type),residuals(model_year_type), main="Residual vs Fitted Plot for Storm Year: Type Blocked")

#Storm Day, blocking for Month
plot(fitted(model_day_month),residuals(model_day_month), main="Residual vs Fitted Plot for Storm Day: Month Blocked") 

#Storm Day, blocking for Type
plot(fitted(model_day_type),residuals(model_day_type), main="Residual vs Fitted Plot for Storm Day: Type Blocked")

#Storm Hour, blocking for Month
plot(fitted(model_hour_month),residuals(model_hour_month), main="Residual vs Fitted Plot for Storm Hour: Month Blocked") 

#Storm Hour, blocking for Type
plot(fitted(model_hour_type),residuals(model_hour_type), main="Residual vs Fitted Plot for Storm Hour: Type Blocked")

All plots above contain distributions that are slightly skewed negatively. However, the distributions do not contain any extreme outliers and therefore indicate that the use of anova in this experiment was appropriate.

4. References to the Literature

There are no literature references in this recipe

5. Appendices

The raw data used in this statistical analysis are results of vehicle testing conducted by NASA. It can be readily accessed using R or RStudio. It is available as a downloadable package and can be found online at https://github.com/hadley/nasaweather