Overview

In this homework assignment, you will explore, analyze and model a data set containing approximately 2200 records. Each record represents a professional baseball team from the years 1871 to 2006 inclusive. Each record has the performance of the team for the given year, with all of the statistics adjusted to match the performance of a 162 game season.

Your objective is to build a multiple linear regression model on the training data to predict the number of wins for the team. You can only use the variables given to you (or variables that you derive from the variables provided). Below is a short description of the variables of interest in the data set:

Data Exploration

The dataset consists of 17 elements, with 2276 total cases. Out of those 17, 15 are explanatory variables, which can be broken down into four groups:

vars n mean sd median trimmed mad min max range skew kurtosis se na_count
TARGET_WINS 2 191 80.92670 12.115013 82 81.11765 13.3434 43 116 73 -0.1698314 -0.2952783 0.8766116 0
TEAM_BATTING_H 3 191 1478.62827 76.147869 1477 1477.42484 74.1300 1308 1667 359 0.1302702 -0.3710350 5.5098664 0
TEAM_BATTING_2B 4 191 297.19895 26.329335 296 296.62745 25.2042 201 373 172 0.0915189 0.4778716 1.9051238 0
TEAM_BATTING_3B 5 191 30.74346 9.043878 29 30.13072 8.8956 12 61 49 0.7007420 0.7446217 0.6543921 0
TEAM_BATTING_HR 6 191 178.05236 32.413243 175 176.81046 35.5824 116 260 144 0.2980673 -0.7172373 2.3453399 0
TEAM_BATTING_BB 7 191 543.31937 74.842133 535 541.31373 74.1300 365 775 410 0.3115199 -0.1474175 5.4153867 0
TEAM_BATTING_SO 8 191 1051.02618 104.156382 1050 1046.95425 97.8516 805 1399 594 0.3985050 0.3955105 7.5364913 102
TEAM_BASERUN_SB 9 191 90.90576 29.916401 87 89.06536 29.6520 31 177 146 0.5553966 -0.1414909 2.1646748 131
TEAM_BASERUN_CS 10 191 39.94241 11.898334 38 39.49020 11.8608 12 74 62 0.3468509 0.0006392 0.8609332 772
TEAM_BATTING_HBP 11 191 59.35602 12.967123 58 58.86275 11.8608 29 95 66 0.3185754 -0.1119828 0.9382681 2085
TEAM_PITCHING_H 12 191 1479.70157 75.788625 1480 1478.50327 72.6474 1312 1667 355 0.1279056 -0.3894781 5.4838725 0
TEAM_PITCHING_HR 13 191 178.17801 32.391678 175 176.93464 35.5824 116 260 144 0.2989191 -0.7190905 2.3437795 0
TEAM_PITCHING_BB 14 191 543.71728 74.916681 537 541.74510 72.6474 367 775 408 0.3144366 -0.1338563 5.4207808 0
TEAM_PITCHING_SO 15 191 1051.81675 104.347208 1052 1047.80392 97.8516 805 1399 594 0.3945586 0.3903991 7.5502990 102
TEAM_FIELDING_E 16 191 107.05236 16.632162 106 106.58170 17.7912 65 145 80 0.1780432 -0.3567367 1.2034610 0
TEAM_FIELDING_DP 17 191 152.33508 17.611682 152 152.04575 19.2738 113 204 91 0.2164822 -0.2115741 1.2743366 286

Looking at the data above, there are multiple variables with missing (NA) values, with TEAM-BATTING_HBP being the highest.

The boxplots below shows the spread of data within the dataset, and show various outliers. The variable, TEAM_PITCHING_H seems to have the highest spread with the most outliers.

## Warning: Removed 3478 rows containing non-finite values (stat_boxplot).

The graph below zooms into the other variables, so it becomes easier to see spread and outliers from the other variables.

In the Histograms below, the data shows multiple graphs with right skews while only a few have left-skew.

The above boxplots show all of the variables listed in the dataset. This visualization may assist in showing how the data is spread.

The correlation plot below shows how variables in the dataset are related to each other. Looking at the plot, we can see that certain variables are more related than others.

For this project, it makes sense to break down the correlation by wins - since that’s what we’re trying to predict.

x
TARGET_WINS 1.0000000
TEAM_BATTING_H 0.4699467
TEAM_BATTING_2B 0.3129840
TEAM_BATTING_3B -0.1243459
TEAM_BATTING_HR 0.4224168
TEAM_BATTING_BB 0.4686879
TEAM_BATTING_SO -0.2288927
TEAM_BASERUN_SB 0.0148364
TEAM_BASERUN_CS -0.1787560
TEAM_BATTING_HBP 0.0735042
TEAM_PITCHING_H 0.4712343
TEAM_PITCHING_HR 0.4224668
TEAM_PITCHING_BB 0.4683988
TEAM_PITCHING_SO -0.2293648
TEAM_FIELDING_E -0.3866880
TEAM_FIELDING_DP -0.1958660

Below is a visual representation of the correlation plot.

According to the coorelation graph, batting_h, batting_2b, batting_hr, batting_bb, pitching_h, pitching_hr, and pitching_bb are the most positively correlated.

Data Preparation

Removal of Data

The variable TEAM_BATTING_HBP is also missing over 90% of its values. That variable will be removed completely.

TEAM_PITCHING_HR and TEAM_BATTING_HR are also very closely correlated with each other. The TEAM_PITCHING_HR variable will be dropped from the dataset

Using the vifstep function variables determined to have collinearity issues will be discarded.

Imputation of Missing (NA) values

We can handle missing values in a number of: deleting the observations, deleting the variables, imputation with the mean/median/mode, or imputation with a prediction.

Imputation is the easy way, however it reduces the variance in the dataset and shrinks standard errors - which can invalidate hypothesis tests.

For this case, I will imputate data via prediction using the MICE (Multivariate Imputation) library using a random forest prediction method.

Variables that exceed the established threshold will be discarded to avoid collinearity issues.

vif(imputed)
##           Variables      VIF
## 1       TARGET_WINS 1.506621
## 2    TEAM_BATTING_H 3.993975
## 3   TEAM_BATTING_2B 2.458565
## 4   TEAM_BATTING_3B 3.010804
## 5   TEAM_BATTING_HR 4.885545
## 6   TEAM_BATTING_BB 5.534802
## 7   TEAM_BATTING_SO 5.195163
## 8   TEAM_BASERUN_SB 2.485101
## 9   TEAM_BASERUN_CS 2.302907
## 10  TEAM_PITCHING_H 3.713942
## 11 TEAM_PITCHING_BB 4.779953
## 12 TEAM_PITCHING_SO 2.985942
## 13  TEAM_FIELDING_E 4.743840
## 14 TEAM_FIELDING_DP 1.897401
v1 <- vifstep(imputed, th=10)

Output - The below table shows the results of above data manipulation.

The NA data has been iputed using the MICE prediction, using the Random Forest Method. Variables with collinearity as established by the vir/virstep package have been dropped.

vars n mean sd median trimmed mad min max range skew kurtosis se
TARGET_WINS 1 2276 80.79086 15.75215 82.0 81.31229 14.8260 0 146 146 -0.3987232 1.0274757 0.3301823
TEAM_BATTING_H 2 2276 1469.26977 144.59120 1454.0 1459.04116 114.1602 891 2554 1663 1.5713335 7.2785261 3.0307891
TEAM_BATTING_2B 3 2276 241.24692 46.80141 238.0 240.39627 47.4432 69 458 389 0.2151018 0.0061609 0.9810087
TEAM_BATTING_3B 4 2276 55.25000 27.93856 47.0 52.17563 23.7216 0 223 223 1.1094652 1.5032418 0.5856226
TEAM_BATTING_HR 5 2276 99.61204 60.54687 102.0 97.38529 78.5778 0 264 264 0.1860421 -0.9631189 1.2691285
TEAM_BATTING_BB 6 2276 501.55888 122.67086 512.0 512.18331 94.8864 0 878 878 -1.0257599 2.1828544 2.5713150
TEAM_BATTING_SO 7 2276 730.80448 245.12190 736.0 735.98134 278.7288 0 1399 1399 -0.2476563 -0.2978879 5.1380222
TEAM_BASERUN_SB 8 2276 130.74780 94.58052 104.0 115.27003 65.2344 0 697 697 1.8130000 4.1992661 1.9825107
TEAM_BASERUN_CS 9 2276 70.01098 43.20956 56.0 63.11032 26.6868 0 201 201 1.3918315 1.2849118 0.9057194
TEAM_PITCHING_H 10 2276 1779.21046 1406.84293 1518.0 1555.89517 174.9468 1137 30132 28995 10.3295111 141.8396985 29.4889618
TEAM_PITCHING_BB 11 2276 553.00791 166.35736 536.5 542.62459 98.5929 0 3645 3645 6.7438995 96.9676398 3.4870317
TEAM_PITCHING_SO 12 2276 811.93937 541.76116 802.5 791.09385 254.2659 0 19278 19278 22.5669238 697.4935725 11.3559046
TEAM_FIELDING_E 13 2276 246.48067 227.77097 159.0 193.43798 62.2692 65 1898 1833 2.9904656 10.9702717 4.7743279
TEAM_FIELDING_DP 14 2276 141.76889 28.59166 145.0 142.94182 26.6868 52 228 176 -0.3482886 -0.1600380 0.5993125

Build Models

From the training data provided, I will build 3 different linear regression models, to determine which will provide the best prediction for the number of wins for a baseball team. The tree approachs are: all variables, only significant variables, and backwards elimination of each variable.

Model 1: All Variables

After the data has been manipulated, all of the variables will be tested to determine the base model they provided. This will allow us to see which variables are significant in our dataset, and allow us to make other models based on that.

## 
## Call:
## lm(formula = TARGET_WINS ~ ., data = imputed)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.847  -8.427   0.127   8.336  52.379 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      30.1561308  5.2986485   5.691 1.42e-08 ***
## TEAM_BATTING_H    0.0443321  0.0036112  12.276  < 2e-16 ***
## TEAM_BATTING_2B  -0.0159758  0.0090339  -1.768  0.07712 .  
## TEAM_BATTING_3B   0.0436209  0.0167331   2.607  0.00920 ** 
## TEAM_BATTING_HR   0.0806287  0.0097035   8.309  < 2e-16 ***
## TEAM_BATTING_BB   0.0086114  0.0051717   1.665  0.09603 .  
## TEAM_BATTING_SO  -0.0136045  0.0024927  -5.458 5.35e-08 ***
## TEAM_BASERUN_SB   0.0388527  0.0044226   8.785  < 2e-16 ***
## TEAM_BASERUN_CS   0.0104644  0.0094740   1.105  0.26948    
## TEAM_PITCHING_H  -0.0001992  0.0003696  -0.539  0.58995    
## TEAM_PITCHING_BB -0.0010457  0.0035461  -0.295  0.76810    
## TEAM_PITCHING_SO  0.0023949  0.0008592   2.788  0.00536 ** 
## TEAM_FIELDING_E  -0.0308259  0.0024975 -12.343  < 2e-16 ***
## TEAM_FIELDING_DP -0.1036332  0.0128157  -8.086 9.91e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.87 on 2262 degrees of freedom
## Multiple R-squared:  0.3363, Adjusted R-squared:  0.3324 
## F-statistic: 88.15 on 13 and 2262 DF,  p-value: < 2.2e-16

Model 2: Highly Significant Variables Only

This model focusses only on the variables that are statistically significant - in order to see if only those variables allow for a better model. Variables will be chosen based on their significance level from the R output.

## 
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_3B + 
##     TEAM_BATTING_HR + TEAM_BATTING_SO + TEAM_BASERUN_SB + TEAM_PITCHING_SO + 
##     TEAM_FIELDING_E + TEAM_FIELDING_DP, data = imputed)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.269  -8.555   0.160   8.445  51.595 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      38.4071569  4.5938808   8.361  < 2e-16 ***
## TEAM_BATTING_H    0.0388120  0.0026595  14.593  < 2e-16 ***
## TEAM_BATTING_3B   0.0561732  0.0159408   3.524 0.000434 ***
## TEAM_BATTING_HR   0.0841740  0.0091128   9.237  < 2e-16 ***
## TEAM_BATTING_SO  -0.0144786  0.0023090  -6.270 4.30e-10 ***
## TEAM_BASERUN_SB   0.0439523  0.0037841  11.615  < 2e-16 ***
## TEAM_PITCHING_SO  0.0017983  0.0005797   3.102 0.001944 ** 
## TEAM_FIELDING_E  -0.0338963  0.0017925 -18.910  < 2e-16 ***
## TEAM_FIELDING_DP -0.1015798  0.0126642  -8.021 1.66e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.89 on 2267 degrees of freedom
## Multiple R-squared:  0.3332, Adjusted R-squared:  0.3309 
## F-statistic: 141.6 on 8 and 2267 DF,  p-value: < 2.2e-16

Model 3: Backwards Elimination and Significance

Variables are removed one by one to determine best fit model. After each variable is removed, the model re-run - until the most optimal output (r2, f-stat) are produced. Only the final output will be shown. .

## 
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BASERUN_SB + 
##     TEAM_FIELDING_E + TEAM_BATTING_HR, data = imputed)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -50.275  -9.079   0.046   8.547  51.820 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      5.687053   2.897573   1.963   0.0498 *  
## TEAM_BATTING_H   0.049527   0.002025  24.454  < 2e-16 ***
## TEAM_BASERUN_SB  0.049275   0.003441  14.320  < 2e-16 ***
## TEAM_FIELDING_E -0.026383   0.001637 -16.114  < 2e-16 ***
## TEAM_BATTING_HR  0.024049   0.005974   4.025 5.88e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 13.2 on 2271 degrees of freedom
## Multiple R-squared:  0.2985, Adjusted R-squared:  0.2973 
## F-statistic: 241.6 on 4 and 2271 DF,  p-value: < 2.2e-16

Select Models

From the three models, I decided to use model 3 for the predictions. While the first model had the highest R-squared, it had multiple variables that weren’t statistically significant, and some that had multicollinearity issues. The F-statistic in model 3 is also much higher than the other two.

A comparsion of the multiple linear regression models, based on: mean square error, R2, F-stat, and root MSE.

Model 1 Model 2 Model 3
Mean Squared Error: 164.620928943855 Mean Squared Error: 165.374013301175 Mean Squared Error: 173.988793693161
Root MSE: 12.8304687733479 Root MSE: 12.8597827859251 Root MSE: 13.1904811774689
Adjusted R-squared: 0.332448323150231 Adjusted R-squared: 0.330873562228461 Adjusted R-squared: 0.297256905782019
F-statistic: 88.1519892300165 F-statistic: 141.619416521762 F-statistic: 241.578479610191

Predictions

The evaulation data also needs some prep work. The evaluation data has had columns removed, and NA values imputed using the MICE - Random Forest method to predict what the NA values could be.

vars n mean sd median trimmed mad min max range skew kurtosis se
TEAM_BATTING_H 1 259 1469.38996 150.65523 1455 1463.68421 114.1602 819 2170 1351 0.5876139 3.6642947 9.361261
TEAM_BATTING_2B 2 259 241.32046 49.51612 239 242.32536 48.9258 44 376 332 -0.3273282 0.6693023 3.076782
TEAM_BATTING_3B 3 259 55.91120 27.14410 52 52.94737 26.6868 14 155 141 0.9790284 0.6987468 1.686652
TEAM_BATTING_HR 4 259 95.63320 56.33221 101 93.67943 66.7170 0 242 242 0.1712363 -0.9031262 3.500313
TEAM_BATTING_BB 5 259 498.95753 120.59215 509 505.98086 94.8864 15 792 777 -0.9209916 2.5265655 7.493232
TEAM_BATTING_SO 6 259 702.77220 240.15626 679 707.21531 259.4550 0 1268 1268 -0.1931048 -0.2332480 14.922584
TEAM_BASERUN_SB 7 259 130.22394 102.81396 95 113.03349 63.7518 0 580 580 1.8750357 4.1939456 6.388548
TEAM_BASERUN_CS 8 259 67.58687 37.40263 56 63.24402 28.1694 0 154 154 1.0180475 0.0418284 2.324086
TEAM_PITCHING_H 9 259 1813.46332 1662.91308 1515 1554.25359 173.4642 1155 22768 21613 9.2764797 102.0702914 103.328391
TEAM_PITCHING_BB 10 259 552.41699 172.95006 526 536.46411 97.8516 136 2008 1872 4.1113772 29.2127324 10.746594
TEAM_PITCHING_SO 11 259 798.13900 613.12180 745 766.58852 243.1464 0 9963 9963 12.8664835 189.8193449 38.097535
TEAM_FIELDING_E 12 259 249.74903 230.90260 163 197.36364 59.3040 73 1568 1495 3.0887263 10.8748551 14.347589
TEAM_FIELDING_DP 13 259 142.99614 27.37232 146 144.28230 25.2042 69 204 135 -0.4095434 -0.0454649 1.700834

The following are the predicted values for the test set of the data after imputing and cleaning the data, using the predict function and Model 3:

fit lwr upr
66.92268 40.999937 92.84541
67.37496 41.453558 93.29635
75.80523 49.898432 101.71203
89.75407 63.843475 115.66466
70.73069 44.809909 96.65147
69.64304 43.723411 95.56266
82.45832 56.520502 108.39613
75.53787 49.627803 101.44794
70.02338 44.099282 95.94749
73.97893 48.064187 99.89367
75.44152 49.526742 101.35631
82.13120 56.221969 108.04043
78.40153 52.484816 104.31825
80.26161 54.348725 106.17449
78.83370 52.928187 104.73921
79.34500 53.438119 105.25188
73.12605 47.215150 99.03694
82.05532 56.147393 107.96325
68.15105 42.228035 94.07407
91.72943 65.809262 117.64959
81.56106 55.642919 107.47920
85.70992 59.796247 111.62359
77.42335 51.516385 103.33032
73.43869 47.527756 99.34962
86.04064 60.134580 111.94671
89.96626 64.055599 115.87692
52.24112 26.185356 78.29688
76.30836 50.392998 102.22371
81.95801 56.034067 107.88195
77.98166 52.057315 103.90601
86.61157 60.702822 112.52031
84.04300 58.137041 109.94895
81.99959 56.092601 107.90658
82.69783 56.787017 108.60865
79.40377 53.496562 105.31098
80.73113 54.809317 106.65294
75.33815 49.431023 101.24528
87.88836 61.962980 113.81374
85.83796 59.928067 111.74785
87.28371 61.366039 113.20138
81.90652 55.999426 107.81362
87.12619 61.216677 113.03570
28.82696 2.567551 55.08637
100.87855 74.856949 126.90015
91.01457 65.063607 116.96554
90.58614 64.654928 116.51736
96.59429 70.652786 122.53580
73.53958 47.629081 99.45008
69.34761 43.425914 95.26931
76.59067 50.678585 102.50276
79.64748 53.736810 105.55815
87.57713 61.658375 113.49588
77.24140 51.334680 103.14812
73.02020 47.109076 98.93133
78.13481 52.232300 104.03732
79.30296 53.398444 105.20748
89.08135 63.152303 115.01039
74.03205 48.104835 99.95927
62.12591 36.177881 88.07393
77.58548 51.669466 103.50149
86.64667 60.725589 112.56775
76.05878 50.139631 101.97794
86.09616 60.189223 112.00310
86.06150 60.123467 111.99954
86.58163 60.659529 112.50373
100.89578 74.911726 126.87984
74.82751 48.920741 100.73428
83.08428 57.166430 109.00213
78.96080 53.033107 104.88849
85.31949 59.392277 111.24671
84.42904 58.494438 110.36364
76.95307 51.030840 102.87529
79.34882 53.441222 105.25642
83.69316 57.760297 109.62603
85.25025 59.340105 111.16040
86.42285 60.511490 112.33422
81.59128 55.684943 107.49762
82.72210 56.812553 108.63164
71.77843 45.856585 97.70028
78.40588 52.486185 104.32557
86.97827 61.063210 112.89334
89.00276 63.083922 114.92160
96.49687 70.558786 122.43495
81.34056 55.426550 107.25457
80.40719 54.499233 106.31515
82.15212 56.247502 108.05674
79.50539 53.596174 105.41460
82.54359 56.639333 108.44784
84.71414 58.810000 110.61828
90.36922 64.447176 116.29126
79.28654 53.367275 105.20581
86.04007 59.973958 112.10617
72.83668 46.925518 98.74784
82.64978 56.731051 108.56851
85.07111 59.143332 110.99888
80.08010 54.160695 105.99950
83.09778 57.178195 109.01736
96.05374 70.103277 122.00421
86.30262 60.374608 112.23063
88.87934 62.962240 114.79645
82.80816 56.892044 108.72428
72.67228 46.760962 98.58360
83.42855 57.517126 109.33998
78.20159 52.290206 104.11298
81.29989 55.377291 107.22250
92.34728 66.308368 118.38618
52.75695 26.781018 78.73288
83.70611 57.798418 109.61381
84.11533 58.201917 110.02874
60.71515 34.778944 86.65136
82.94621 57.042509 108.84992
87.48495 61.574205 113.39569
94.72611 68.809833 120.64239
91.45619 65.546208 117.36618
83.96714 58.064374 109.86991
83.19535 57.291689 109.09901
91.27144 65.360585 117.18229
82.60330 56.697531 108.50907
79.59740 53.692592 105.50221
77.53206 51.580585 103.48353
89.82395 63.898489 115.74940
67.06571 41.133676 92.99775
66.61299 40.686817 92.53917
59.92760 33.980299 85.87491
68.82271 42.897602 94.74782
87.18695 61.263663 113.11025
88.20367 62.267537 114.13981
74.77840 48.866903 100.68989
87.52618 61.615421 113.43694
93.00549 67.077565 118.93342
86.07974 60.162269 111.99720
80.46726 54.556454 106.37807
79.72906 53.821922 105.63619
85.28776 59.379189 111.19633
85.48319 59.563565 111.40281
73.22820 47.290067 99.16633
76.56396 50.657554 102.47037
79.02938 53.122371 104.93640
91.07672 65.151901 117.00153
82.66243 56.758791 108.56606
67.17663 41.250733 93.10253
69.66784 43.741766 95.59392
91.55086 65.624254 117.47746
76.31538 50.403335 102.22743
72.62282 46.703202 98.54244
71.99508 46.081797 97.90837
78.42951 52.522489 104.33653
81.58909 55.683422 107.49476
84.56701 58.659916 110.47411
81.22539 55.321016 107.12977
82.89474 56.977662 108.81182
84.35194 58.447392 110.25648
43.39417 17.208306 69.58004
73.50945 47.600100 99.41880
77.07875 51.172800 102.98470
76.42295 50.514476 102.33142
87.70229 61.779252 113.62533
56.96270 30.994598 82.93079
87.25032 61.331061 113.16957
72.28717 46.369388 98.20494
96.15019 70.218786 122.08159
99.57057 73.637171 125.50397
86.47203 60.563271 112.38078
97.71519 71.775163 123.65522
89.47279 63.552616 115.39296
84.21932 58.307868 110.13076
82.07300 56.166246 107.97975
81.68207 55.776407 107.58773
77.23639 51.325169 103.14762
82.90403 56.996919 108.81113
88.52659 62.601248 114.45194
86.19220 60.274053 112.11034
78.00279 52.091387 103.91418
90.45022 64.522541 116.37789
81.35685 55.451032 107.26266
73.48817 47.571778 99.40456
74.78003 48.872087 100.68797
75.01798 49.110867 100.92510
73.68057 47.769370 99.59178
79.89711 53.991234 105.80298
87.22675 61.271639 113.18187
84.54756 58.623773 110.47135
86.02771 60.120423 111.93500
82.00679 56.086670 107.92690
88.59053 62.542506 114.63856
100.80476 74.763279 126.84624
87.91094 61.984467 113.83741
63.40299 37.414286 89.39170
82.58361 56.527441 108.63979
114.18950 88.170689 140.20831
70.26129 44.347007 96.17557
79.73971 53.824864 105.65455
78.66191 52.757313 104.56651
80.96313 55.048436 106.87782
84.00568 58.084151 109.92722
69.98871 44.073800 95.90361
76.93274 51.028766 102.83672
75.98898 50.080979 101.89699
75.47013 49.563187 101.37708
82.07759 56.172316 107.98287
76.06528 50.156908 101.97365
79.99396 54.088941 105.89897
73.43833 47.528426 99.34823
87.57768 61.671863 113.48350
81.67512 55.771919 107.57832
78.68978 52.782933 104.59662
81.58848 55.684096 107.49286
79.08231 53.178272 104.98635
81.55775 55.641249 107.47426
71.86048 45.939767 97.78120
101.20446 75.251763 127.15716
90.39172 64.465149 116.31829
78.88545 52.978672 104.79224
67.77701 41.859664 93.69436
70.28390 44.369299 96.19850
84.73629 58.825283 110.64731
83.83116 57.926026 109.73629
94.63836 68.712399 120.56432
79.32932 53.425460 105.23318
77.50949 51.603740 103.41525
81.53938 55.635690 107.44306
81.49154 55.586608 107.39648
85.07000 59.161473 110.97854
80.65681 54.751815 106.56180
85.89885 59.769177 112.02853
74.77816 48.870707 100.68561
81.54431 55.641175 107.44745
82.17940 56.270810 108.08798
80.94609 55.042657 106.84953
81.32811 55.375891 107.28034
74.60742 48.684723 100.53011
92.11898 66.201029 118.03693
77.85023 51.943679 103.75677
85.85653 59.937677 111.77538
77.58977 51.685062 103.49448
73.72219 47.813103 99.63127
84.37142 58.469330 110.27350
77.52350 51.615871 103.43113
85.19092 59.276364 111.10548
72.88167 46.968670 98.79468
87.05422 61.142983 112.96546
85.52097 59.606156 111.43578
84.43803 58.518060 110.35799
86.98351 61.077051 112.88998
65.94902 40.017490 91.88055
89.85266 63.941649 115.76366
81.32921 55.426483 107.23194
85.08192 59.169123 110.99471
73.64528 47.733967 99.55659
89.25002 63.327865 115.17217
83.02056 57.107236 108.93387
56.66816 30.675035 82.66129
91.09554 65.166780 117.02430
35.98253 9.841609 62.12345
69.76125 43.845680 95.67683
74.80073 48.893928 100.70752
82.92322 57.012969 108.83347
85.49389 59.578752 111.40903
80.53982 54.629548 106.45010 
##       fit              lwr              upr        
##  Min.   : 28.83   Min.   : 2.568   Min.   : 55.09  
##  1st Qu.: 76.19   1st Qu.:50.275   1st Qu.:102.10  
##  Median : 81.59   Median :55.683   Median :107.49  
##  Mean   : 80.59   Mean   :54.664   Mean   :106.51  
##  3rd Qu.: 86.04   3rd Qu.:60.122   3rd Qu.:112.00  
##  Max.   :114.19   Max.   :88.171   Max.   :140.21
##        1 
## 81.08199
##        fit      lwr      upr
## 1 81.08199 55.17984 106.9841