Multiple Linear Regression and Factors
Adam J Sullivan
Assistant Professor of Biostatistics
Brown University
Multiple Variables
Multiple Variables
library(readr)
library(tidyverse)
bikes <- read_csv("../Notes/Data/bike_sharing.csv") %>%
mutate(season = as.factor(season)) %>%
mutate(weather=as.factor(weather))
Multiple Variables
mod.final <- lm(count~season+weather+humidity+windspeed, data=bikes)
tidy(mod.final)[-1,-c(3:4)]
glance(mod.final)
Multiple Variables
| term | estimate | p.value |
|---|---|---|
| season2 | 115.8007186 | 0.0000000 |
| season3 | 148.3532069 | 0.0000000 |
| season4 | 118.4943844 | 0.0000000 |
| weather2 | 19.9875113 | 0.0000001 |
| weather3 | 0.1237865 | 0.9844830 |
| weather4 | 162.2596870 | 0.3185115 |
| humidity | -3.4929513 | 0.0000000 |
| windspeed | 0.6328680 | 0.0020498 |
Multiple Variables
| r.squared | adj.r.squared | sigma | statistic | p.value | df | logLik | AIC | BIC | deviance | df.residual |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.1949699 | 0.1943778 | 162.5889 | 329.2869 | 0 | 9 | -70865.13 | 141750.3 | 141823.2 | 287534958 | 10877 |
Inference on Linear Regressions
Inference on Linear Regressions
- Overall F Test of Model
- Individual Coefficient Tests
- Testing Groups of Variables
Overall Model F test
- We can perform an overall F Test for a model.
- When we do this we test the following Hypothesis \[H_0:\beta_1=\beta_2=\cdots=\beta_p=0\] \[H_1=\text{ at least one }\beta_i\ne0\]
Overall Model F test: Bike Sharing
kable(glance(mod.final))
Overall Model F test: Bike Sharing
| r.squared | adj.r.squared | sigma | statistic | p.value | df | logLik | AIC | BIC | deviance | df.residual |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.1949699 | 0.1943778 | 162.5889 | 329.2869 | 0 | 9 | -70865.13 | 141750.3 | 141823.2 | 287534958 | 10877 |
Overall Model F test: Bike Sharing
- We have an F Statistic of 3329.3
- This yields a p-value of \(<0.0001\)
- We can reject the null in favor of the alternative hypothesis.
- This suggests that at least one \(\beta_I\) is not 0.
Individual Coefficients \(t\)-test
- We can test each individual coefficients.
- The hypothesis we test is that: \[H_0:\beta_i=0\] \[H_1=\beta_i\ne0\]
- We do this with a t-test.
Individual Coefficients \(t\)-test
- With the t-test we have that: \[ t_i=\dfrac{\beta_i}{se(\beta_i)}\]
- Then we can test this with the \(t\)-distribution.
Individual Coefficients \(t\)-test
- Consider out Bike model:
\[ \begin{align} E[count] &= \beta_0 + \beta_1 season(Summer) + \beta_2 season(Fall) + \\ & \beta_3 season(Winter) + \beta_4 weather(2) + \beta_5 weather(3) + \\ & \beta_6 weather(4) + \beta_7 humidity + \beta_8 windspeed\\ \end{align} \]
tidy(mod.final)
Individual Coefficients \(t\)-Test
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | 298.3348913 | 7.3616043 | 40.5257985 | 0.0000000 |
| season2 | 115.8007186 | 4.4387984 | 26.0883030 | 0.0000000 |
| season3 | 148.3532069 | 4.5043842 | 32.9352918 | 0.0000000 |
| season4 | 118.4943844 | 4.5112582 | 26.2663719 | 0.0000000 |
| weather2 | 19.9875113 | 3.7920390 | 5.2709140 | 0.0000001 |
| weather3 | 0.1237865 | 6.3645757 | 0.0194493 | 0.9844830 |
| weather4 | 162.2596870 | 162.6554195 | 0.9975670 | 0.3185115 |
| humidity | -3.4929513 | 0.0960939 | -36.3493686 | 0.0000000 |
| windspeed | 0.6328680 | 0.2052323 | 3.0836662 | 0.0020498 |
F-test for Groups of Coefficients
- Many times we want to be able to test the significance of groups of coefficients.
- We can do this with an F-test as well.
- For example we may want to test that: \[H_0:\beta_1=\beta_2=0\] \[H_1:\text{ at least 1 }\beta_i\ne0\]
Groups of Coefficients Example
- Consider
Seasonin our bike example. - Only the first coefficient is significant.
- We may want to know if we the whole variable is worth having in the model.
- We will use the
anova()function in R.
Groups of Coefficients Example
mod1 <- lm(count~season+weather+humidity+windspeed, data=bikes)
mod2 <- lm(count~weather+humidity+windspeed, data=bikes)
anova(mod1, mod2)
Groups of Coefficients Example
## Analysis of Variance Table
##
## Model 1: count ~ season + weather + humidity + windspeed
## Model 2: count ~ weather + humidity + windspeed
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 10877 287534958
## 2 10880 320760441 -3 -33225483 418.96 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Groups of Coefficients Example 2
- Consider
weatherin our bike example. - Only the first coefficient is significant.
- We may want to know if we the whole variable is worth having in the model.
- We will use the
anova()function in R.
Groups of Coefficients Example 2
mod1 <- lm(count~season+weather+humidity+windspeed, data=bikes)
mod2 <- lm(count~season+humidity+windspeed, data=bikes)
anova(mod1, mod2)
Groups of Coefficients Example 2
## Analysis of Variance Table
##
## Model 1: count ~ season + weather + humidity + windspeed
## Model 2: count ~ season + humidity + windspeed
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 10877 287534958
## 2 10880 288348337 -3 -813379 10.256 9.704e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Factors
What are Factors?
- Factors are categorical data.
- Factors contain
- Levels
- Can be numerical or character data
Why do we use them?
- Factors allow us to group things by category.
- Factors create dummy variables or indicator variables in our regressions.
What is an indicator variable?
- Consider the scenario where we have 3 treatments: A, B, & C
- We could have two indicator variables:
- I(Treat_A) is
- 1 if patient is on treatment A
- 0 if patient is not on treatment A
- I(Treat_B) is
- 1 if patient is on treatment B
- 0 if patient is not on treatment B
- Treatment C would be both:
- I(Treat_A) = 0
- I(Treat_B) = 0
- I(Treat_A) is
What does this mean in regressions?
- Indicator variables change the regression: \[Outcome = \beta_0 + \beta_1 Age + \beta_2 I(Treat_A) + \beta_3 I(Treat_B)\]
- For a person on Treatment A: \[Outcome = (\beta_0 + \beta_2 ) + \beta_1 Age\]
- For a person on Treatment B: \[Outcome = (\beta_0 + \beta_3 ) + \beta_1 Age\]
- For a person on Treatment C: \[Outcome = \beta_0 + \beta_1 Age\]
What does this mean in Regression?
- We can see that a factor leads to multiple different regression lines.
- Each line then has a different intercept than the others.
- In this regression age has the same effect, just the baseline is different.
Are there different types of factors?
- We can have different types of factors
- Nominal
- Ordinal
Nominal Factors
- Nominal factors are factors that represent named categories.
- These are categories that do not have an intrinsic ordering.
- Examples:
- Gender
- Sex
- Race/ethnicity
- We must treat these as indicator variables in models.
Ordinal Factors
- Ordinal factors are factors that represent some ordered categories.
- These factors have an intrinsic ordering.
- Examples:
- Likert Scales (Poor, Neutral, Good)
- BMI (Underweight, Normal, Overweight, Obese)
- Age Groups (under 18, 18-25, 25-35, 35+)
- In regression models can be indicator variables or a trend.
Indicator Variables vs Trends
- We saw with indicator variables that we have multiple variables to represent the factor.
- Each category leads to a different regression.
- Consider this: \[ Outcome = \beta_0 + \beta_1 age + \beta_2 I(BMI = underweight) + \beta_3 I(BMI=Overweight+)\]
- We then have 3 different regressions:
- 1 for normal BMI
- 1 for underweight BMI
- 1 for overweight+ BMI
Our 3 regressions
- Normal BMI \[Outcome =\beta_0 + \beta_1 age\]
- Underweight BMI \[Outcome =(\beta_0 + \beta_2) + \beta_1 age\]
- Overweight+ BMI \[Outcome =(\beta_0+ \beta_3) + \beta_1 age\]
Indicator Variables vs Trends
- With a trend we allow the factor to have one slope.
- Instead of 1 category leading to a new regression, each category leads to a further increase.
- Our model \[Outcome =\beta_0 + \beta_1 age + \beta_2 BMI\]
Our Regressions
- Normal BMI \[Outcome =\beta_0 + \beta_1 age\]
- Underweight BMI \[Outcome =(\beta_0 + \beta_2) + \beta_1 age\]
- Overweight+ BMI \[Outcome =(\beta_0+ 2\beta_2) + \beta_1 age\]
What is the difference?
- You can see that it appears that we still have 3 regressions.
- indicator variable regression, each group can have a unique change from the baseline.
- \(\beta_{group=2}\ne\beta_{group=3}\)
- trend regression, each group has the same difference between them
An example: PBC Data
- This data is from the Mayo Clinic trial in primary biliary cirrhosis (PBC) of the liver conducted between 1974 and 1984.
- A total of 424 PBC patients, referred to Mayo Clinic during that ten-year interval, met eligibility criteria for the randomized placebo controlled trial of the drug D-penicillamine.
- The first 312 cases in the data set participated in the randomized trial and contain largely complete data.
- The additional 112 cases did not participate in the clinical trial, but consented to have basic measurements recorded and to be followed for survival.
PBC Data
| Variable | Description |
|---|---|
| age | in years |
| albumin | serum albumin (g/dl) |
| alk.phos | alkaline phosphotase (U/liter) |
| ascites | presence of ascites |
| ast | aspartate aminotransferase, once called SGOT (U/ml) |
| bili | serum bilirunbin (mg/dl) |
PBC Data
| Variable | Description |
|---|---|
| chol | serum cholesterol (mg/dl) |
| copper | urine copper (ug/day) |
| edema | 0 no edema, 0.5 untreated or successfully treated |
| 1 edema despite diuretic therapy | |
| hepato | presence of hepatomegaly or enlarged liver |
| id | case number |
PBC Data
| Variable | Description |
|---|---|
| platelet | platelet count |
| protime | standardised blood clotting time |
| sex | m/f |
| spiders | blood vessel malformations in the skin |
| stage | histologic stage of disease (needs biopsy) |
| status | status at endpoint, 0/1/2 for censored, transplant, dead |
PBC Data
| Variable | Description |
|---|---|
| time | number of days between registration and the earlier of death, transplantion, or study analysis in July, 1986 |
| trt | 1/2/NA for D-penicillmain, placebo, not randomised |
| trig | triglycerides (mg/dl) |
Data
library(survival)
pbc
## id time status trt age sex ascites hepato spiders edema bili
## 1 1 400 2 1 58.76523 f 1 1 1 1.0 14.5
## 2 2 4500 0 1 56.44627 f 0 1 1 0.0 1.1
## 3 3 1012 2 1 70.07255 m 0 0 0 0.5 1.4
## 4 4 1925 2 1 54.74059 f 0 1 1 0.5 1.8
## 5 5 1504 1 2 38.10541 f 0 1 1 0.0 3.4
## 6 6 2503 2 2 66.25873 f 0 1 0 0.0 0.8
## 7 7 1832 0 2 55.53457 f 0 1 0 0.0 1.0
## 8 8 2466 2 2 53.05681 f 0 0 0 0.0 0.3
## 9 9 2400 2 1 42.50787 f 0 0 1 0.0 3.2
## 10 10 51 2 2 70.55989 f 1 0 1 1.0 12.6
## 11 11 3762 2 2 53.71389 f 0 1 1 0.0 1.4
## 12 12 304 2 2 59.13758 f 0 0 1 0.0 3.6
## 13 13 3577 0 2 45.68925 f 0 0 0 0.0 0.7
## 14 14 1217 2 2 56.22177 m 1 1 0 1.0 0.8
## 15 15 3584 2 1 64.64613 f 0 0 0 0.0 0.8
## 16 16 3672 0 2 40.44353 f 0 0 0 0.0 0.7
## 17 17 769 2 2 52.18344 f 0 1 0 0.0 2.7
## 18 18 131 2 1 53.93018 f 0 1 1 1.0 11.4
## 19 19 4232 0 1 49.56057 f 0 1 0 0.5 0.7
## 20 20 1356 2 2 59.95346 f 0 1 0 0.0 5.1
## 21 21 3445 0 2 64.18891 m 0 1 1 0.0 0.6
## 22 22 673 2 1 56.27652 f 0 0 1 0.0 3.4
## 23 23 264 2 2 55.96715 f 1 1 1 1.0 17.4
## 24 24 4079 2 1 44.52019 m 0 1 0 0.0 2.1
## 25 25 4127 0 2 45.07324 f 0 0 0 0.0 0.7
## 26 26 1444 2 2 52.02464 f 0 1 1 0.0 5.2
## 27 27 77 2 2 54.43943 f 1 1 1 0.5 21.6
## 28 28 549 2 2 44.94730 f 1 1 1 1.0 17.2
## 29 29 4509 0 2 63.87680 f 0 0 0 0.0 0.7
## 30 30 321 2 2 41.38535 f 0 1 1 0.0 3.6
## 31 31 3839 2 2 41.55236 f 0 1 0 0.0 4.7
## 32 32 4523 0 2 53.99589 f 0 1 0 0.0 1.8
## 33 33 3170 2 2 51.28268 f 0 0 0 0.0 0.8
## 34 34 3933 0 1 52.06023 f 0 0 0 0.0 0.8
## 35 35 2847 2 2 48.61875 f 0 0 0 0.0 1.2
## 36 36 3611 0 2 56.41068 f 0 0 0 0.0 0.3
## 37 37 223 2 1 61.72758 f 1 1 0 1.0 7.1
## 38 38 3244 2 2 36.62697 f 0 1 1 0.0 3.3
## 39 39 2297 2 1 55.39220 f 0 1 0 0.0 0.7
## 40 40 4467 0 1 46.66940 f 0 0 0 0.0 1.3
## 41 41 1350 2 1 33.63450 f 0 1 0 0.0 6.8
## 42 42 4453 0 2 33.69473 f 0 1 1 0.0 2.1
## 43 43 4556 0 1 48.87064 f 0 0 0 0.0 1.1
## 44 44 3428 2 2 37.58248 f 0 1 1 1.0 3.3
## 45 45 4025 0 2 41.79329 f 0 0 0 0.0 0.6
## 46 46 2256 2 1 45.79877 f 0 1 0 0.0 5.7
## 47 47 2576 0 2 47.42779 f 0 0 0 0.0 0.5
## 48 48 4427 0 2 49.13621 m 0 0 0 0.0 1.9
## 49 49 708 2 2 61.15264 f 0 1 0 0.0 0.8
## 50 50 2598 2 1 53.50856 f 0 1 0 0.0 1.1
## 51 51 3853 2 2 52.08761 f 0 0 0 0.0 0.8
## 52 52 2386 2 1 50.54073 m 0 0 0 0.0 6.0
## 53 53 1000 2 1 67.40862 f 0 1 0 0.0 2.6
## 54 54 1434 2 1 39.19781 f 1 1 1 1.0 1.3
## 55 55 1360 2 1 65.76318 m 0 0 0 0.0 1.8
## 56 56 1847 2 2 33.61807 f 0 1 1 0.0 1.1
## 57 57 3282 2 1 53.57153 f 0 1 0 0.5 2.3
## 58 58 4459 0 1 44.56947 m 0 0 0 0.0 0.7
## 59 59 2224 2 1 40.39425 f 0 1 1 0.0 0.8
## 60 60 4365 0 1 58.38193 f 0 0 0 0.0 0.9
## 61 61 4256 0 2 43.89870 m 0 0 0 0.0 0.6
## 62 62 3090 2 2 60.70637 f 1 0 0 0.0 1.3
## 63 63 859 2 2 46.62834 f 0 0 1 1.0 22.5
## 64 64 1487 2 2 62.90760 f 0 1 0 0.0 2.1
## 65 65 3992 0 1 40.20260 f 0 0 0 0.0 1.2
## 66 66 4191 2 1 46.45311 m 0 1 0 0.0 1.4
## 67 67 2769 2 2 51.28816 f 0 0 0 0.0 1.1
## 68 68 4039 0 1 32.61328 f 0 0 0 0.0 0.7
## 69 69 1170 2 1 49.33881 f 0 1 1 0.5 20.0
## 70 70 3458 0 1 56.39973 f 0 0 0 0.0 0.6
## 71 71 4196 0 2 48.84600 f 0 1 0 0.0 1.2
## 72 72 4184 0 2 32.49281 f 0 0 0 0.0 0.5
## 73 73 4190 0 2 38.49418 f 0 0 0 0.0 0.7
## 74 74 1827 2 1 51.92060 f 0 1 1 0.0 8.4
## 75 75 1191 2 1 43.51814 f 1 1 1 0.5 17.1
## 76 76 71 2 1 51.94251 f 0 1 1 0.5 12.2
## 77 77 326 2 2 49.82615 f 0 1 1 0.5 6.6
## 78 78 1690 2 1 47.94524 f 0 1 0 0.0 6.3
## 79 79 3707 0 1 46.51608 f 0 1 0 0.0 0.8
## 80 80 890 2 2 67.41136 m 0 1 0 0.0 7.2
## 81 81 2540 2 1 63.26352 f 0 1 1 0.0 14.4
## 82 82 3574 2 1 67.31006 f 0 0 0 0.0 4.5
## 83 83 4050 0 1 56.01369 f 0 1 0 0.5 1.3
## 84 84 4032 0 2 55.83025 f 0 0 0 0.0 0.4
## 85 85 3358 2 2 47.21697 f 0 1 0 0.0 2.1
## 86 86 1657 2 1 52.75838 f 0 1 1 0.0 5.0
## 87 87 198 2 1 37.27858 f 0 0 0 0.0 1.1
## 88 88 2452 0 2 41.39357 f 0 0 0 0.5 0.6
## 89 89 1741 2 1 52.44353 f 0 1 0 0.0 2.0
## 90 90 2689 2 1 33.47570 m 0 0 0 0.0 1.6
## 91 91 460 2 2 45.60712 f 0 1 1 0.5 5.0
## 92 92 388 2 1 76.70910 f 1 0 0 1.0 1.4
## 93 93 3913 0 1 36.53388 f 0 0 0 0.0 1.3
## 94 94 750 2 1 53.91650 f 0 1 1 0.0 3.2
## 95 95 130 2 2 46.39014 f 1 1 1 1.0 17.4
## 96 96 3850 0 1 48.84600 f 0 0 0 0.0 1.0
## 97 97 611 2 2 71.89322 m 0 1 0 0.5 2.0
## 98 98 3823 0 1 28.88433 f 0 0 0 0.0 1.0
## 99 99 3820 0 2 48.46817 m 0 0 0 0.0 1.8
## 100 100 552 2 2 51.46886 m 0 1 0 0.0 2.3
## 101 101 3581 0 2 44.95003 f 0 0 0 0.0 0.9
## 102 102 3099 0 1 56.56947 f 0 0 0 0.0 0.9
## 103 103 110 2 2 48.96372 f 1 1 1 1.0 2.5
## 104 104 3086 2 1 43.01711 f 0 0 0 0.0 1.1
## 105 105 3092 1 2 34.03970 f 0 1 0 0.0 1.1
## 106 106 3222 2 1 68.50924 f 1 1 0 0.0 2.1
## 107 107 3388 0 2 62.52156 f 0 0 0 0.0 0.6
## 108 108 2583 2 1 50.35729 f 0 0 0 0.0 0.4
## 109 109 2504 0 2 44.06297 f 0 0 0 0.0 0.5
## 110 110 2105 2 1 38.91034 f 0 1 1 0.0 1.9
## 111 111 2350 1 1 41.15264 f 0 0 0 0.0 5.5
## 112 112 3445 2 2 55.45791 f 0 1 1 0.0 2.0
## 113 113 980 2 1 51.23340 f 0 1 1 0.0 6.7
## 114 114 3395 2 2 52.82683 m 0 0 0 0.0 3.2
## 115 115 3422 0 2 42.63929 f 0 0 1 0.0 0.7
## 116 116 3336 0 1 61.07050 f 0 0 1 0.5 3.0
## 117 117 1083 2 1 49.65640 f 0 1 1 0.0 6.5
## 118 118 2288 2 1 48.85421 f 0 1 0 0.0 3.5
## 119 119 515 2 1 54.25599 f 0 0 1 0.0 0.6
## 120 120 2033 1 1 35.15127 m 0 0 0 0.0 3.5
## 121 121 191 2 2 67.90691 m 1 1 0 1.0 1.3
## 122 122 3297 0 1 55.43600 f 0 0 0 0.0 0.6
## 123 123 971 2 1 45.82067 f 0 1 1 1.0 5.1
## 124 124 3069 0 1 52.88980 m 0 1 0 0.0 0.6
## 125 125 2468 1 2 47.18138 f 0 1 0 0.0 1.3
## 126 126 824 2 1 53.59890 f 1 1 1 0.0 1.2
## 127 127 3255 0 2 44.10404 f 0 0 0 0.0 0.5
## 128 128 1037 2 1 41.94935 f 0 1 1 0.0 16.2
## 129 129 3239 0 1 63.61396 f 0 1 0 0.0 0.9
## 130 130 1413 2 2 44.22724 f 0 1 1 0.0 17.4
## 131 131 850 2 2 62.00137 f 0 1 1 0.0 2.8
## 132 132 2944 0 1 40.55305 f 0 0 0 0.0 1.9
## 133 133 2796 2 2 62.64476 m 0 0 0 0.0 1.5
## 134 134 3149 0 2 42.33539 f 0 0 0 0.0 0.7
## 135 135 3150 0 1 42.96783 f 0 0 0 0.0 0.4
## 136 136 3098 0 1 55.96167 f 0 0 0 0.0 0.8
## 137 137 2990 0 1 62.86105 f 0 0 0 0.0 1.1
## 138 138 1297 2 1 51.24983 m 0 1 0 0.0 7.3
## 139 139 2106 0 2 46.76249 f 0 1 0 0.0 1.1
## 140 140 3059 0 1 54.07529 f 0 1 0 0.0 1.1
## 141 141 3050 0 1 47.03628 f 0 0 0 0.0 0.9
## 142 142 2419 2 2 55.72621 f 0 1 0 0.0 1.0
## 143 143 786 2 2 46.10267 f 0 1 0 0.0 2.9
## 144 144 943 2 2 52.28747 f 0 1 0 0.5 28.0
## 145 145 2976 0 2 51.20055 f 0 0 1 0.0 0.7
## 146 146 2615 0 2 33.86448 f 0 0 0 0.5 1.2
## 147 147 2995 0 1 75.01164 f 0 0 0 0.5 1.2
## 148 148 1427 2 2 30.86379 f 0 1 0 0.0 7.2
## 149 149 762 2 1 61.80424 m 0 1 1 0.5 3.0
## 150 150 2891 0 2 34.98700 f 0 0 1 0.0 1.0
## 151 151 2870 0 1 55.04175 f 0 0 0 0.0 0.9
## 152 152 1152 2 1 69.94114 m 0 1 0 0.0 2.3
## 153 153 2863 0 1 49.60438 f 0 0 0 0.0 0.5
## 154 154 140 2 1 69.37714 m 0 0 1 1.0 2.4
## 155 155 2666 0 2 43.55647 f 0 1 1 0.5 0.6
## 156 156 853 2 2 59.40862 f 0 1 0 0.0 25.5
## 157 157 2835 0 2 48.75838 f 0 0 0 0.0 0.6
## 158 158 2475 1 1 36.49281 f 0 0 0 0.0 3.4
## 159 159 1536 2 2 45.76044 m 0 0 0 0.0 2.5
## 160 160 2772 0 2 57.37166 f 0 0 0 0.0 0.6
## 161 161 2797 0 2 42.74333 f 0 0 0 0.0 2.3
## 162 162 186 2 2 58.81725 f 0 1 1 0.0 3.2
## 163 163 2055 2 1 53.49760 f 0 0 0 0.0 0.3
## 164 164 264 2 2 43.41410 f 0 1 1 0.5 8.5
## 165 165 1077 2 1 53.30595 m 0 1 0 0.0 4.0
## 166 166 2721 0 2 41.35524 f 0 1 0 0.0 5.7
## 167 167 1682 2 1 60.95825 m 0 1 0 0.0 0.9
## 168 168 2713 0 2 47.75359 f 0 1 0 0.0 0.4
## 169 169 1212 2 2 35.49076 f 0 0 0 0.0 1.3
## 170 170 2692 0 1 48.66256 f 0 0 0 0.0 1.2
## 171 171 2574 0 1 52.66804 f 0 0 0 0.0 0.5
## 172 172 2301 0 2 49.86995 f 0 0 1 0.0 1.3
## 173 173 2657 0 1 30.27515 f 0 1 1 0.0 3.0
## 174 174 2644 0 1 55.56742 f 0 0 0 0.0 0.5
## 175 175 2624 0 2 52.15332 f 0 0 0 0.0 0.8
## 176 176 1492 2 1 41.60986 f 0 1 1 0.0 3.2
## 177 177 2609 0 2 55.45243 f 0 0 0 0.0 0.9
## 178 178 2580 0 1 70.00411 f 0 0 0 0.0 0.6
## 179 179 2573 0 2 43.94251 f 0 1 0 0.0 1.8
## 180 180 2563 0 2 42.56810 f 0 0 0 0.0 4.7
## 181 181 2556 0 1 44.56947 f 0 1 1 0.0 1.4
## 182 182 2555 0 1 56.94456 f 0 1 0 0.0 0.6
## 183 183 2241 1 2 40.26010 f 0 0 0 0.0 0.5
## 184 184 974 2 2 37.60712 f 0 1 0 0.0 11.0
## 185 185 2527 0 1 48.36140 f 0 0 0 0.0 0.8
## 186 186 1576 2 1 70.83641 f 0 0 1 0.5 2.0
## 187 187 733 2 2 35.79192 f 0 1 0 0.0 14.0
## 188 188 2332 0 1 62.62286 f 0 1 0 0.0 0.7
## 189 189 2456 0 2 50.64750 f 0 1 0 0.0 1.3
## 190 190 2504 0 1 54.52704 f 0 0 1 0.0 2.3
## 191 191 216 2 2 52.69268 f 1 1 1 0.0 24.5
## 192 192 2443 0 1 52.72005 f 0 1 0 0.0 0.9
## 193 193 797 2 2 56.77207 f 0 0 0 0.0 10.8
## 194 194 2449 0 1 44.39699 f 0 0 0 0.0 1.5
## 195 195 2330 0 1 29.55510 f 0 1 0 0.0 3.7
## 196 196 2363 0 1 57.04038 f 0 1 1 0.0 1.4
## 197 197 2365 0 1 44.62697 f 0 0 0 0.0 0.6
## 198 198 2357 0 2 35.79740 f 0 0 1 0.0 0.7
## 199 199 1592 0 1 40.71732 f 0 0 0 0.0 2.1
## 200 200 2318 0 2 32.23272 f 0 0 1 0.0 4.7
## 201 201 2294 0 2 41.09240 f 0 1 0 0.0 0.6
## 202 202 2272 0 1 61.63997 f 0 0 0 0.0 0.5
## 203 203 2221 0 2 37.05681 f 0 1 0 0.0 0.5
## 204 204 2090 2 2 62.57906 f 0 0 0 0.0 0.7
## 205 205 2081 2 1 48.97741 f 1 0 0 0.0 2.5
## 206 206 2255 0 1 61.99042 f 0 0 0 0.0 0.6
## 207 207 2171 0 1 72.77207 f 0 0 0 0.5 0.6
## 208 208 904 2 1 61.29500 f 0 1 0 0.0 3.9
## 209 209 2216 0 2 52.62423 f 0 1 1 0.0 0.7
## 210 210 2224 0 2 49.76318 m 0 1 0 0.0 0.9
## 211 211 2195 0 2 52.91444 f 0 0 0 0.0 1.3
## 212 212 2176 0 2 47.26352 f 0 0 0 0.0 1.2
## 213 213 2178 0 1 50.20397 f 0 0 1 0.0 0.5
## 214 214 1786 2 2 69.34702 f 0 1 0 0.0 0.9
## 215 215 1080 2 2 41.16906 f 0 0 0 0.0 5.9
## 216 216 2168 0 1 59.16496 f 0 0 0 0.0 0.5
## 217 217 790 2 2 36.07940 f 0 1 0 0.0 11.4
## 218 218 2170 0 1 34.59548 f 0 0 0 0.0 0.5
## 219 219 2157 0 2 42.71321 f 0 0 0 0.0 1.6
## 220 220 1235 2 1 63.63039 f 0 0 1 0.0 3.8
## 221 221 2050 0 2 56.62971 f 0 1 0 0.0 0.9
## 222 222 597 2 2 46.26420 f 0 1 0 0.0 4.5
## 223 223 334 2 1 61.24298 f 1 1 0 1.0 14.1
## 224 224 1945 0 1 38.62012 f 0 0 0 0.0 1.0
## 225 225 2022 0 1 38.77070 f 0 0 0 0.0 0.7
## 226 226 1978 0 2 56.69541 f 0 1 0 0.0 0.5
## 227 227 999 2 1 58.95140 m 0 0 0 0.0 2.3
## 228 228 1967 0 2 36.92266 f 0 0 0 0.0 0.7
## 229 229 348 2 1 62.41478 f 1 1 0 0.5 4.5
## 230 230 1979 0 2 34.60917 f 0 1 1 0.0 3.3
## 231 231 1165 2 2 58.33539 f 0 1 1 0.0 3.4
## 232 232 1951 0 1 50.18207 f 0 1 0 0.0 0.4
## 233 233 1932 0 1 42.68583 f 0 1 1 0.0 0.9
## 234 234 1776 0 2 34.37919 f 0 0 0 0.0 0.9
## 235 235 1882 0 2 33.18275 f 0 1 0 0.0 13.0
## 236 236 1908 0 1 38.38193 f 0 1 1 0.0 1.5
## 237 237 1882 0 1 59.76181 f 0 1 0 0.0 1.6
## 238 238 1874 0 2 66.41205 f 0 0 0 0.5 0.6
## 239 239 694 2 1 46.78987 f 0 1 1 0.0 0.8
## 240 240 1831 0 1 56.07940 f 0 0 0 0.0 0.4
## 241 241 837 1 2 41.37440 f 0 1 1 0.0 4.4
## 242 242 1810 0 1 64.57221 f 0 1 0 0.0 1.9
## 243 243 930 2 2 67.48802 f 0 1 0 0.0 8.0
## 244 244 1690 2 1 44.82957 f 0 0 1 0.0 3.9
## 245 245 1790 0 2 45.77139 f 0 1 0 0.0 0.6
## 246 246 1435 1 1 32.95003 f 0 1 0 0.0 2.1
## 247 247 732 1 1 41.22108 f 0 1 0 0.0 6.1
## 248 248 1785 0 2 55.41684 f 0 1 0 0.0 0.8
## 249 249 1783 0 1 47.98084 f 0 0 1 0.0 1.3
## 250 250 1769 0 2 40.79124 f 0 1 0 0.0 0.6
## 251 251 1457 0 1 56.97467 f 0 0 0 0.0 0.5
## 252 252 1770 0 1 68.46270 f 0 1 1 0.0 1.1
## 253 253 1765 0 1 78.43943 m 1 1 1 0.0 7.1
## 254 254 737 1 1 39.85763 f 0 1 1 0.0 3.1
## 255 255 1735 0 2 35.31006 f 0 1 1 0.0 0.7
## 256 256 1701 0 1 31.44422 f 0 0 0 0.0 1.1
## 257 257 1614 0 1 58.26420 f 0 0 0 0.0 0.5
## 258 258 1702 0 1 51.48802 f 0 0 0 0.0 1.1
## 259 259 1615 0 2 59.96988 f 0 1 0 0.0 3.1
## 260 260 1656 0 2 74.52430 m 0 1 0 0.0 5.6
## 261 261 1677 0 2 52.36413 f 0 1 1 0.0 3.2
## 262 262 1666 0 2 42.78713 f 0 1 0 0.0 2.8
## 263 263 1301 1 2 34.87474 f 0 1 1 0.5 1.1
## 264 264 1542 1 2 44.13963 f 0 1 1 0.0 3.4
## 265 265 1084 1 2 46.38193 f 0 1 0 0.0 3.5
## 266 266 1614 0 1 56.30938 f 0 0 0 0.0 0.5
## 267 267 179 2 1 70.90760 f 1 1 1 1.0 6.6
## 268 268 1191 2 1 55.39493 f 1 1 0 0.5 6.4
## 269 269 1363 0 2 45.08419 f 0 0 0 0.0 3.6
## 270 270 1568 0 1 26.27789 f 0 1 1 0.0 1.0
## 271 271 1569 0 2 50.47228 f 0 1 0 0.0 1.0
## 272 272 1525 0 1 38.39836 f 0 0 0 0.0 0.5
## 273 273 1558 0 2 47.41958 f 0 0 1 0.0 2.2
## 274 274 1447 1 1 47.98084 f 0 0 0 0.0 1.6
## 275 275 1349 0 1 38.31622 f 0 0 0 0.0 2.2
## 276 276 1481 0 1 50.10815 f 0 0 0 0.0 1.0
## 277 277 1434 0 2 35.08830 f 0 0 0 0.5 1.0
## 278 278 1420 0 2 32.50376 f 0 0 0 0.0 5.6
## 279 279 1433 0 2 56.15332 f 0 0 0 0.0 0.5
## 280 280 1412 0 1 46.15469 f 0 0 0 0.0 1.6
## 281 281 41 2 1 65.88364 f 1 0 0 1.0 17.9
## 282 282 1455 0 2 33.94387 f 0 1 0 0.0 1.3
## 283 283 1030 0 2 62.86105 f 0 0 0 0.0 1.1
## 284 284 1418 0 2 48.56400 f 0 0 0 0.0 1.3
## 285 285 1401 0 1 46.34908 f 0 0 0 0.0 0.8
## 286 286 1408 0 1 38.85284 f 0 1 1 0.0 2.0
## 287 287 1234 0 1 58.64750 f 0 0 1 0.0 6.4
## 288 288 1067 1 2 48.93634 f 0 1 0 0.5 8.7
## 289 289 799 2 1 67.57290 m 0 1 0 0.5 4.0
## 290 290 1363 0 1 65.98494 f 0 0 0 0.0 1.4
## 291 291 901 1 1 40.90075 f 0 0 0 0.0 3.2
## 292 292 1329 0 2 50.24504 m 0 1 0 0.0 8.6
## 293 293 1320 0 2 57.19644 f 0 1 1 1.0 8.5
## 294 294 1302 0 1 60.53662 m 0 1 0 0.0 6.6
## 295 295 877 1 1 35.35113 m 0 0 0 0.0 2.4
## 296 296 1321 0 2 31.38125 f 0 0 0 0.0 0.8
## 297 297 533 1 1 55.98631 m 0 1 0 0.0 1.2
## 298 298 1300 0 2 52.72553 f 0 1 0 0.0 1.1
## 299 299 1293 0 1 38.09172 f 0 0 0 0.0 2.4
## 300 300 207 2 2 58.17112 f 0 1 0 0.0 5.2
## 301 301 1295 0 2 45.21013 f 0 0 0 0.0 1.0
## 302 302 1271 0 1 37.79877 f 0 0 0 0.0 0.7
## 303 303 1250 0 2 60.65982 f 0 1 1 0.0 1.0
## 304 304 1230 0 1 35.53457 f 0 0 0 0.0 0.5
## 305 305 1216 0 2 43.06639 f 0 1 1 0.0 2.9
## 306 306 1216 0 2 56.39151 f 0 1 0 0.0 0.6
## 307 307 1149 0 2 30.57358 f 0 0 0 0.0 0.8
## 308 308 1153 0 1 61.18275 f 0 1 0 0.0 0.4
## 309 309 994 0 2 58.29979 f 0 0 0 0.0 0.4
## 310 310 939 0 1 62.33265 f 0 0 0 0.0 1.7
## 311 311 839 0 1 37.99863 f 0 0 0 0.0 2.0
## 312 312 788 0 2 33.15264 f 0 0 1 0.0 6.4
## 313 313 4062 0 NA 60.00000 f NA NA NA 0.0 0.7
## 314 314 3561 2 NA 64.99932 f NA NA NA 0.5 1.4
## 315 315 2844 0 NA 54.00137 f NA NA NA 0.0 0.7
## 316 316 2071 2 NA 75.00068 f NA NA NA 0.5 0.7
## 317 317 3030 0 NA 62.00137 f NA NA NA 0.0 0.8
## 318 318 1680 0 NA 43.00068 f NA NA NA 0.0 0.7
## 319 319 41 2 NA 46.00137 f NA NA NA 0.0 5.0
## 320 320 2403 0 NA 44.00000 f NA NA NA 0.5 0.4
## 321 321 1170 0 NA 60.99932 m NA NA NA 0.5 1.3
## 322 322 2011 2 NA 64.00000 f NA NA NA 0.0 1.1
## 323 323 3523 0 NA 40.00000 f NA NA NA 0.0 0.6
## 324 324 3468 0 NA 63.00068 f NA NA NA 0.0 0.6
## 325 325 4795 0 NA 34.00137 f NA NA NA 0.0 1.8
## 326 326 1236 0 NA 52.00000 f NA NA NA 0.0 1.5
## 327 327 4214 0 NA 48.99932 f NA NA NA 0.0 1.2
## 328 328 2111 2 NA 54.00137 f NA NA NA 0.0 1.0
## 329 329 1462 2 NA 63.00068 f NA NA NA 0.0 0.7
## 330 330 1746 2 NA 54.00137 m NA NA NA 0.0 3.5
## 331 331 94 2 NA 46.00137 f NA NA NA 0.5 3.1
## 332 332 785 2 NA 52.99932 f NA NA NA 0.0 12.6
## 333 333 1518 2 NA 56.00000 f NA NA NA 0.0 2.8
## 334 334 466 2 NA 56.00000 f NA NA NA 0.0 7.1
## 335 335 3527 0 NA 55.00068 f NA NA NA 0.0 0.6
## 336 336 2635 0 NA 64.99932 f NA NA NA 0.0 2.1
## 337 337 2286 2 NA 56.00000 f NA NA NA 0.0 1.8
## 338 338 791 2 NA 47.00068 f NA NA NA 0.0 16.0
## 339 339 3492 0 NA 60.00000 f NA NA NA 0.0 0.6
## 340 340 3495 0 NA 52.99932 f NA NA NA 0.0 5.4
## 341 341 111 2 NA 54.00137 f NA NA NA 0.0 9.0
## 342 342 3231 0 NA 50.00137 f NA NA NA 0.0 0.9
## 343 343 625 2 NA 48.00000 f NA NA NA 0.0 11.1
## 344 344 3157 0 NA 36.00000 f NA NA NA 0.0 8.9
## 345 345 3021 1 NA 48.00000 f NA NA NA 0.0 0.5
## 346 346 559 2 NA 70.00137 f NA NA NA 0.5 0.6
## 347 347 2812 2 NA 51.00068 f NA NA NA 0.0 3.4
## 348 348 2834 0 NA 52.00000 m NA NA NA 0.0 0.9
## 349 349 2855 0 NA 54.00137 f NA NA NA 0.0 1.4
## 350 350 662 2 NA 48.00000 f NA NA NA 0.0 2.1
## 351 351 727 2 NA 66.00137 f NA NA NA 0.0 15.0
## 352 352 2716 0 NA 52.99932 f NA NA NA 0.0 0.6
## 353 353 2698 0 NA 62.00137 f NA NA NA 0.0 1.3
## 354 354 990 2 NA 59.00068 f NA NA NA 0.0 1.3
## 355 355 2338 0 NA 39.00068 f NA NA NA 0.0 1.6
## 356 356 1616 2 NA 67.00068 f NA NA NA 0.5 2.2
## 357 357 2563 0 NA 58.00137 f NA NA NA 0.0 3.0
## 358 358 2537 0 NA 64.00000 f NA NA NA 0.0 0.8
## 359 359 2534 0 NA 46.00137 f NA NA NA 0.0 0.8
## 360 360 778 2 NA 64.00000 f NA NA NA 0.0 1.8
## 361 361 617 1 NA 40.99932 f NA NA NA 0.0 5.5
## 362 362 2267 1 NA 48.99932 f NA NA NA 0.0 18.0
## 363 363 2249 0 NA 44.00000 f NA NA NA 0.0 0.6
## 364 364 359 2 NA 59.00068 f NA NA NA 0.0 2.7
## 365 365 1925 0 NA 63.00068 f NA NA NA 0.0 0.9
## 366 366 249 2 NA 60.99932 f NA NA NA 0.0 1.3
## 367 367 2202 0 NA 64.00000 f NA NA NA 0.0 1.1
## 368 368 43 2 NA 48.99932 f NA NA NA 0.0 13.8
## 369 369 1197 2 NA 42.00137 f NA NA NA 0.0 4.4
## 370 370 1095 2 NA 50.00137 f NA NA NA 0.0 16.0
## 371 371 489 2 NA 51.00068 f NA NA NA 0.5 7.3
## 372 372 2149 0 NA 36.99932 f NA NA NA 0.0 0.6
## 373 373 2103 0 NA 62.00137 f NA NA NA 0.0 0.7
## 374 374 1980 0 NA 51.00068 f NA NA NA 0.0 0.7
## 375 375 1347 1 NA 52.00000 f NA NA NA 0.0 1.7
## 376 376 1478 2 NA 44.00000 m NA NA NA 0.0 9.5
## 377 377 1987 0 NA 32.99932 f NA NA NA 0.0 2.2
## 378 378 1168 2 NA 60.00000 f NA NA NA 0.5 1.8
## 379 379 597 2 NA 63.00068 f NA NA NA 0.5 3.3
## 380 380 1725 1 NA 32.99932 f NA NA NA 0.0 2.9
## 381 381 1899 0 NA 40.99932 m NA NA NA 0.0 1.7
## 382 382 221 2 NA 51.00068 f NA NA NA 0.0 14.0
## 383 383 1022 1 NA 36.99932 f NA NA NA 0.5 0.8
## 384 384 1639 0 NA 59.00068 f NA NA NA 0.0 1.3
## 385 385 1635 0 NA 55.00068 f NA NA NA 0.0 0.7
## 386 386 1654 0 NA 54.00137 m NA NA NA 0.0 1.7
## 387 387 1653 0 NA 48.99932 f NA NA NA 0.5 13.6
## 388 388 1560 0 NA 40.00000 f NA NA NA 0.0 0.9
## 389 389 1581 0 NA 67.00068 f NA NA NA 0.0 0.7
## 390 390 1419 0 NA 68.00000 m NA NA NA 0.0 3.0
## 391 391 1443 0 NA 40.99932 f NA NA NA 0.0 1.2
## 392 392 1368 0 NA 68.99932 f NA NA NA 0.0 0.4
## 393 393 193 2 NA 52.00000 f NA NA NA 0.5 0.7
## 394 394 1367 0 NA 56.99932 f NA NA NA 0.5 2.0
## 395 395 1329 0 NA 36.00000 f NA NA NA 0.0 1.4
## 396 396 1343 0 NA 50.00137 f NA NA NA 0.0 1.6
## 397 397 1328 0 NA 64.00000 f NA NA NA 0.0 0.5
## 398 398 1375 0 NA 62.00137 f NA NA NA 0.0 7.3
## 399 399 1260 0 NA 42.00137 f NA NA NA 0.0 8.1
## 400 400 1223 0 NA 44.00000 f NA NA NA 0.0 0.5
## 401 401 935 2 NA 68.99932 f NA NA NA 0.0 4.2
## 402 402 943 0 NA 52.00000 f NA NA NA 0.0 0.8
## 403 403 1141 0 NA 66.00137 f NA NA NA 0.0 2.5
## 404 404 1092 0 NA 40.00000 f NA NA NA 0.0 4.6
## 405 405 1150 0 NA 52.00000 f NA NA NA 0.0 1.0
## 406 406 703 2 NA 46.00137 f NA NA NA 0.0 4.5
## 407 407 1129 0 NA 54.00137 m NA NA NA 0.0 1.1
## 408 408 1086 0 NA 51.00068 f NA NA NA 0.5 1.9
## 409 409 1067 0 NA 43.00068 f NA NA NA 0.0 0.7
## 410 410 1072 0 NA 39.00068 f NA NA NA 0.0 1.5
## 411 411 1119 0 NA 51.00068 f NA NA NA 0.0 0.6
## 412 412 1097 0 NA 67.00068 f NA NA NA 0.0 1.0
## 413 413 989 0 NA 35.00068 f NA NA NA 0.0 0.7
## 414 414 681 2 NA 67.00068 f NA NA NA 0.0 1.2
## 415 415 1103 0 NA 39.00068 f NA NA NA 0.0 0.9
## 416 416 1055 0 NA 56.99932 f NA NA NA 0.0 1.6
## 417 417 691 0 NA 58.00137 f NA NA NA 0.0 0.8
## 418 418 976 0 NA 52.99932 f NA NA NA 0.0 0.7
## chol albumin copper alk.phos ast trig platelet protime stage
## 1 261 2.60 156 1718.0 137.95 172 190 12.2 4
## 2 302 4.14 54 7394.8 113.52 88 221 10.6 3
## 3 176 3.48 210 516.0 96.10 55 151 12.0 4
## 4 244 2.54 64 6121.8 60.63 92 183 10.3 4
## 5 279 3.53 143 671.0 113.15 72 136 10.9 3
## 6 248 3.98 50 944.0 93.00 63 NA 11.0 3
## 7 322 4.09 52 824.0 60.45 213 204 9.7 3
## 8 280 4.00 52 4651.2 28.38 189 373 11.0 3
## 9 562 3.08 79 2276.0 144.15 88 251 11.0 2
## 10 200 2.74 140 918.0 147.25 143 302 11.5 4
## 11 259 4.16 46 1104.0 79.05 79 258 12.0 4
## 12 236 3.52 94 591.0 82.15 95 71 13.6 4
## 13 281 3.85 40 1181.0 88.35 130 244 10.6 3
## 14 NA 2.27 43 728.0 71.00 NA 156 11.0 4
## 15 231 3.87 173 9009.8 127.71 96 295 11.0 3
## 16 204 3.66 28 685.0 72.85 58 198 10.8 3
## 17 274 3.15 159 1533.0 117.80 128 224 10.5 4
## 18 178 2.80 588 961.0 280.55 200 283 12.4 4
## 19 235 3.56 39 1881.0 93.00 123 209 11.0 3
## 20 374 3.51 140 1919.0 122.45 135 322 13.0 4
## 21 252 3.83 41 843.0 65.10 83 336 11.4 4
## 22 271 3.63 464 1376.0 120.90 55 173 11.6 4
## 23 395 2.94 558 6064.8 227.04 191 214 11.7 4
## 24 456 4.00 124 5719.0 221.88 230 70 9.9 2
## 25 298 4.10 40 661.0 106.95 66 324 11.3 2
## 26 1128 3.68 53 3228.0 165.85 166 421 9.9 3
## 27 175 3.31 221 3697.4 101.91 168 80 12.0 4
## 28 222 3.23 209 1975.0 189.10 195 144 13.0 4
## 29 370 3.78 24 5833.0 73.53 86 390 10.6 2
## 30 260 2.54 172 7277.0 121.26 158 124 11.0 4
## 31 296 3.44 114 9933.2 206.40 101 195 10.3 2
## 32 262 3.34 101 7277.0 82.56 158 286 10.6 4
## 33 210 3.19 82 1592.0 218.55 113 180 12.0 3
## 34 364 3.70 37 1840.0 170.50 64 273 10.5 2
## 35 314 3.20 201 12258.8 72.24 151 431 10.6 3
## 36 172 3.39 18 558.0 71.30 96 311 10.6 2
## 37 334 3.01 150 6931.2 180.60 118 102 12.0 4
## 38 383 3.53 102 1234.0 137.95 87 234 11.0 4
## 39 282 3.00 52 9066.8 72.24 111 563 10.6 4
## 40 NA 3.34 105 11046.6 104.49 NA 358 11.0 4
## 41 NA 3.26 96 1215.0 151.90 NA 226 11.7 4
## 42 NA 3.54 122 8778.0 56.76 NA 344 11.0 4
## 43 361 3.64 36 5430.2 67.08 89 203 10.6 2
## 44 299 3.55 131 1029.0 119.35 50 199 11.7 3
## 45 NA 3.93 19 1826.0 71.30 NA 474 10.9 2
## 46 482 2.84 161 11552.0 136.74 165 518 12.7 3
## 47 316 3.65 68 1716.0 187.55 71 356 9.8 3
## 48 259 3.70 281 10396.8 188.34 178 214 11.0 3
## 49 NA 3.82 58 678.0 97.65 NA 233 11.0 4
## 50 257 3.36 43 1080.0 106.95 73 128 10.6 4
## 51 276 3.60 54 4332.0 99.33 143 273 10.6 2
## 52 614 3.70 158 5084.4 206.40 93 362 10.6 1
## 53 NA 3.10 94 6456.2 56.76 NA 214 11.0 4
## 54 288 3.40 262 5487.2 73.53 125 254 11.0 4
## 55 416 3.94 121 10165.0 79.98 219 213 11.0 3
## 56 498 3.80 88 13862.4 95.46 319 365 10.6 2
## 57 260 3.18 231 11320.2 105.78 94 216 12.4 3
## 58 242 4.08 73 5890.0 56.76 118 NA 10.6 1
## 59 329 3.50 49 7622.8 126.42 124 321 10.6 3
## 60 604 3.40 82 876.0 71.30 58 228 10.3 3
## 61 216 3.94 28 601.0 60.45 188 211 13.0 1
## 62 302 2.75 58 1523.0 43.40 112 329 13.2 4
## 63 932 3.12 95 5396.0 244.90 133 165 11.6 3
## 64 373 3.50 52 1009.0 150.35 188 178 11.0 3
## 65 256 3.60 74 724.0 141.05 108 430 10.0 1
## 66 427 3.70 105 1909.0 182.90 171 123 11.0 3
## 67 466 3.91 84 1787.0 328.60 185 261 10.0 3
## 68 174 4.09 58 642.0 71.30 46 203 10.6 3
## 69 652 3.46 159 3292.0 215.45 184 227 12.4 3
## 70 NA 4.64 20 666.0 54.25 NA 265 10.6 2
## 71 258 3.57 79 2201.0 120.90 76 410 11.5 4
## 72 320 3.54 51 1243.0 122.45 80 225 10.0 3
## 73 132 3.60 17 423.0 49.60 56 265 11.0 1
## 74 558 3.99 280 967.0 89.90 309 278 11.0 4
## 75 674 2.53 207 2078.0 182.90 598 268 11.5 4
## 76 394 3.08 111 2132.0 155.00 243 165 11.6 4
## 77 244 3.41 199 1819.0 170.50 91 132 12.1 3
## 78 436 3.02 75 2176.0 170.50 104 236 10.6 4
## 79 315 4.24 13 1637.0 170.50 70 426 10.9 3
## 80 247 3.72 269 1303.0 176.70 91 360 11.2 4
## 81 448 3.65 34 1218.0 60.45 318 385 11.7 4
## 82 472 4.09 154 1580.0 117.80 272 412 11.1 3
## 83 250 3.50 48 1138.0 71.30 100 81 12.9 4
## 84 263 3.76 29 1345.0 137.95 74 181 11.2 3
## 85 262 3.48 58 2045.0 89.90 84 225 11.5 4
## 86 1600 3.21 75 2656.0 82.15 174 181 10.9 3
## 87 345 4.40 75 1860.0 218.55 72 447 10.7 3
## 88 296 4.06 37 1032.0 80.60 83 442 12.0 3
## 89 408 3.65 50 1083.0 110.05 98 200 11.4 2
## 90 660 4.22 94 1857.0 151.90 155 337 11.0 2
## 91 325 3.47 110 2460.0 246.45 56 430 11.9 4
## 92 206 3.13 36 1626.0 86.80 70 145 12.2 4
## 93 353 3.67 73 2039.0 232.50 68 380 11.1 2
## 94 201 3.11 178 1212.0 159.65 69 188 11.8 4
## 95 NA 2.64 182 559.0 119.35 NA 401 11.7 2
## 96 NA 3.70 33 1258.0 99.20 NA 338 10.4 3
## 97 420 3.26 62 3196.0 77.50 91 344 11.4 3
## 98 239 3.77 77 1877.0 97.65 101 312 10.2 1
## 99 460 3.35 148 1472.0 108.50 118 172 10.2 2
## 100 178 3.00 145 746.0 178.25 122 119 12.0 4
## 101 400 3.60 31 1689.0 164.30 166 327 10.4 3
## 102 248 3.97 172 646.0 62.00 84 128 10.1 1
## 103 188 3.67 57 1273.0 119.35 102 110 11.1 4
## 104 303 3.64 20 2108.0 128.65 53 349 11.1 2
## 105 464 4.20 38 1644.0 151.90 102 348 10.3 3
## 106 NA 3.90 50 1087.0 103.85 NA 137 10.6 2
## 107 212 4.03 10 648.0 71.30 77 316 17.1 1
## 108 127 3.50 14 1062.0 49.60 84 334 10.3 2
## 109 120 3.61 53 804.0 110.05 52 271 10.6 3
## 110 486 3.54 74 1052.0 108.50 109 141 10.9 3
## 111 528 4.18 77 2404.0 172.05 78 467 10.7 3
## 112 267 3.67 89 754.0 196.85 90 136 11.8 4
## 113 374 3.74 103 979.0 128.65 100 266 11.1 4
## 114 259 4.30 208 1040.0 110.05 78 268 11.7 3
## 115 303 4.19 81 1584.0 111.60 156 307 10.3 3
## 116 458 3.63 74 1588.0 106.95 382 438 9.9 3
## 117 950 3.11 111 2374.0 170.50 149 354 11.0 4
## 118 390 3.30 67 878.0 137.95 93 207 10.2 3
## 119 636 3.83 129 944.0 97.65 114 306 9.5 3
## 120 325 3.98 444 766.0 130.20 210 344 10.6 3
## 121 151 3.08 73 1112.0 46.50 49 213 13.2 4
## 122 298 4.13 29 758.0 65.10 85 256 10.7 3
## 123 NA 3.23 18 790.0 179.80 NA 104 13.0 4
## 124 251 3.90 25 681.0 57.35 107 182 10.8 4
## 125 316 3.51 75 1162.0 147.25 137 238 10.0 4
## 126 269 3.12 NA 1441.0 165.85 68 166 11.1 4
## 127 268 4.08 9 1174.0 86.80 95 453 10.0 2
## 128 NA 2.89 42 1828.0 299.15 NA 123 12.6 4
## 129 420 3.87 30 1009.0 57.35 232 NA 9.7 3
## 130 1775 3.43 205 2065.0 165.85 97 418 11.5 3
## 131 242 3.80 74 614.0 136.40 104 121 13.2 4
## 132 448 3.83 60 1052.0 127.10 175 181 9.8 3
## 133 331 3.95 13 577.0 128.65 99 165 10.1 4
## 134 578 3.67 35 1353.0 127.10 105 427 10.7 2
## 135 263 3.57 123 836.0 74.40 121 445 11.0 2
## 136 263 3.35 27 1636.0 116.25 69 206 9.8 2
## 137 399 3.60 79 3472.0 155.00 152 344 10.1 2
## 138 426 3.93 262 2424.0 145.70 218 252 10.5 3
## 139 328 3.31 159 1260.0 94.55 134 142 11.6 4
## 140 290 4.09 38 2120.0 186.00 146 318 10.0 3
## 141 346 3.77 59 794.0 125.55 56 336 10.6 2
## 142 364 3.48 20 720.0 134.85 88 283 9.9 2
## 143 332 3.60 86 1492.0 134.85 103 277 11.0 4
## 144 556 3.26 152 3896.0 198.40 171 335 10.0 3
## 145 309 3.84 96 858.0 41.85 106 253 11.4 3
## 146 NA 3.89 58 1284.0 173.60 NA 239 9.4 3
## 147 288 3.37 32 791.0 57.35 114 213 10.7 2
## 148 1015 3.26 247 3836.0 198.40 280 330 9.8 3
## 149 257 3.79 290 1664.0 102.30 112 140 9.9 4
## 150 NA 3.63 57 1536.0 134.85 NA 233 10.0 1
## 151 460 3.03 57 721.0 85.25 174 301 9.4 2
## 152 586 3.01 243 2276.0 114.70 126 339 10.9 3
## 153 217 3.85 68 453.0 54.25 68 270 11.1 1
## 154 168 2.56 225 1056.0 120.90 75 108 14.1 3
## 155 220 3.35 57 1620.0 153.45 80 311 11.2 4
## 156 358 3.52 219 2468.0 201.50 205 151 11.5 2
## 157 286 3.42 34 1868.0 77.50 206 487 10.0 2
## 158 450 3.37 32 1408.0 116.25 118 313 11.2 2
## 159 317 3.46 217 714.0 130.20 140 207 10.1 3
## 160 217 3.62 13 414.0 75.95 119 224 10.5 3
## 161 502 3.56 4 964.0 120.90 180 269 9.6 2
## 162 260 3.19 91 815.0 127.10 101 160 12.0 4
## 163 233 4.08 20 622.0 66.65 68 358 9.9 3
## 164 NA 3.34 161 1428.0 181.35 NA 88 13.3 4
## 165 196 3.45 80 2496.0 133.30 142 212 11.3 4
## 166 1480 3.26 84 1960.0 457.25 108 213 9.5 2
## 167 376 3.86 200 1015.0 83.70 154 238 10.3 4
## 168 257 3.80 44 842.0 97.65 110 NA 9.2 2
## 169 408 4.22 67 1387.0 142.60 137 295 10.1 3
## 170 390 3.61 32 1509.0 88.35 52 263 9.0 3
## 171 NA 4.52 31 784.0 74.40 NA 361 10.1 3
## 172 205 3.34 65 1031.0 91.45 126 217 9.8 3
## 173 236 3.42 76 1403.0 89.90 86 493 9.8 2
## 174 NA 3.85 63 663.0 79.05 NA 311 9.7 1
## 175 283 3.80 152 718.0 108.50 168 340 10.1 3
## 176 NA 3.56 77 1790.0 139.50 NA 149 10.1 4
## 177 258 4.01 49 559.0 43.40 133 277 10.4 2
## 178 NA 4.08 51 665.0 74.40 NA 325 10.2 4
## 179 396 3.83 39 2148.0 102.30 133 278 9.9 4
## 180 478 4.38 44 1629.0 237.15 76 175 10.4 3
## 181 248 3.58 63 554.0 75.95 106 79 10.3 4
## 182 NA 3.69 161 674.0 26.35 NA 539 9.9 2
## 183 201 3.73 44 1345.0 54.25 145 445 10.1 2
## 184 674 3.55 358 2412.0 167.40 140 471 9.8 3
## 185 256 3.54 42 1132.0 74.40 94 192 10.5 3
## 186 225 3.53 51 933.0 69.75 62 200 12.7 3
## 187 808 3.43 251 2870.0 153.45 137 268 11.5 3
## 188 187 3.48 41 654.0 120.90 98 164 11.0 4
## 189 360 3.63 52 1812.0 97.65 164 256 9.9 3
## 190 NA 3.93 24 1828.0 133.30 NA 327 10.2 2
## 191 1092 3.35 233 3740.0 147.25 432 399 15.2 4
## 192 308 3.69 67 696.0 51.15 101 344 9.8 4
## 193 932 3.19 267 2184.0 161.20 157 382 10.4 4
## 194 293 4.30 50 975.0 125.55 56 336 9.1 2
## 195 347 3.90 76 2544.0 221.65 90 129 11.5 4
## 196 226 3.36 13 810.0 72.85 62 117 11.6 4
## 197 266 3.97 25 1164.0 102.30 102 201 10.1 2
## 198 286 2.90 38 1692.0 141.05 90 381 9.6 2
## 199 392 3.43 52 1395.0 184.45 194 328 10.2 3
## 200 236 3.55 112 1391.0 137.95 114 332 9.9 3
## 201 235 3.20 26 1758.0 106.95 67 228 10.8 4
## 202 223 3.80 15 1044.0 80.60 89 514 10.0 2
## 203 149 4.04 227 598.0 52.70 57 166 9.9 2
## 204 255 3.74 23 1024.0 77.50 58 281 10.2 3
## 205 382 3.55 108 1516.0 238.70 NA 126 10.3 3
## 206 213 4.07 12 5300.0 57.35 68 240 11.0 1
## 207 NA 3.33 14 733.0 85.25 NA 259 10.1 4
## 208 396 3.20 58 1440.0 153.45 131 156 10.0 4
## 209 252 4.01 11 1210.0 72.85 58 309 9.5 2
## 210 346 3.37 81 1098.0 122.45 90 298 10.0 2
## 211 NA 3.76 27 1282.0 100.75 NA 114 10.3 3
## 212 232 3.98 11 1074.0 100.75 99 223 9.9 3
## 213 400 3.40 9 1134.0 96.10 55 356 10.2 3
## 214 404 3.43 34 1866.0 79.05 224 236 9.9 3
## 215 1276 3.85 141 1204.0 203.05 157 216 10.7 3
## 216 NA 3.68 20 856.0 55.80 NA 146 10.4 3
## 217 608 3.31 65 1790.0 151.90 210 298 10.8 4
## 218 NA 3.89 29 897.0 66.65 NA 423 10.1 1
## 219 215 4.17 67 936.0 134.85 85 176 9.6 3
## 220 426 3.22 96 2716.0 210.80 113 228 10.6 2
## 221 360 3.65 72 3186.0 94.55 154 269 9.7 4
## 222 372 3.38 227 2310.0 167.40 135 240 12.4 3
## 223 448 2.43 123 1833.0 134.00 155 210 11.0 4
## 224 309 3.66 67 1214.0 158.10 101 309 9.7 3
## 225 274 3.66 108 1065.0 88.35 135 251 10.1 2
## 226 223 3.70 39 884.0 75.95 104 231 9.6 3
## 227 316 3.35 172 1601.0 179.80 63 394 9.7 2
## 228 215 3.35 41 645.0 93.00 74 165 9.6 3
## 229 191 3.05 200 1020.0 175.15 118 139 11.4 4
## 230 302 3.41 51 310.0 83.70 44 95 11.5 4
## 231 518 1.96 115 2250.0 203.05 90 190 10.7 4
## 232 267 3.02 47 1001.0 133.30 87 265 10.6 3
## 233 514 3.06 412 2622.0 105.40 87 284 9.8 4
## 234 578 3.35 78 976.0 116.25 177 322 11.2 2
## 235 1336 4.16 71 3510.0 209.25 111 338 11.9 3
## 236 253 3.79 67 1006.0 139.50 106 341 9.7 3
## 237 442 2.95 105 820.0 85.25 108 181 10.1 3
## 238 280 3.35 NA 1093.0 128.65 81 295 9.8 2
## 239 300 2.94 231 1794.0 130.20 99 319 11.2 4
## 240 232 3.72 24 369.0 51.15 139 326 10.1 3
## 241 316 3.62 308 1119.0 114.70 322 282 9.8 4
## 242 354 2.97 86 1553.0 196.85 152 277 9.9 3
## 243 468 2.81 139 2009.0 198.40 139 233 10.0 4
## 244 350 3.22 121 1268.0 272.80 231 270 9.6 3
## 245 273 3.65 48 794.0 52.70 214 305 9.6 3
## 246 387 3.77 63 1613.0 150.35 33 185 10.1 4
## 247 1712 2.83 89 3681.0 158.10 139 297 10.0 3
## 248 324 3.51 39 1237.0 66.65 146 371 10.0 3
## 249 242 3.20 35 1556.0 175.15 71 195 10.6 4
## 250 299 3.36 23 2769.0 220.10 85 303 10.9 4
## 251 227 3.61 40 676.0 83.00 120 249 9.9 2
## 252 246 3.35 116 924.0 113.15 90 317 10.0 4
## 253 243 3.03 380 983.0 158.10 154 97 11.2 4
## 254 227 3.75 121 1136.0 110.00 91 264 10.0 3
## 255 193 3.85 35 466.0 53.00 118 156 10.3 3
## 256 336 3.74 48 823.0 84.00 108 242 9.7 3
## 257 280 4.23 36 377.0 56.00 146 227 10.6 2
## 258 414 3.44 80 1003.0 99.00 55 271 9.6 1
## 259 277 2.97 42 1110.0 125.00 126 221 9.8 3
## 260 232 3.59 188 1120.0 98.00 128 248 10.9 4
## 261 375 3.14 129 857.0 89.00 NA 375 9.5 3
## 262 322 3.06 65 2562.0 91.00 209 231 9.5 3
## 263 432 3.57 45 1406.0 190.00 77 248 11.4 4
## 264 356 3.12 188 1911.0 92.00 130 318 11.2 3
## 265 348 3.20 121 938.0 120.00 146 296 10.0 4
## 266 318 3.32 52 613.0 70.00 260 279 10.2 3
## 267 222 2.33 138 620.0 106.00 91 195 12.1 4
## 268 344 2.75 16 834.0 82.00 179 149 11.0 4
## 269 374 3.50 143 1428.0 188.00 44 151 10.1 2
## 270 448 3.74 102 1128.0 71.00 117 228 10.2 3
## 271 321 3.50 94 955.0 111.00 177 289 9.7 3
## 272 226 2.93 22 674.0 58.00 85 153 9.8 1
## 273 328 3.46 75 1677.0 87.00 116 202 9.6 3
## 274 NA 3.07 136 1995.0 128.00 NA 372 9.6 4
## 275 572 3.77 77 2520.0 92.00 114 309 9.5 4
## 276 219 3.85 67 640.0 145.00 108 95 10.7 2
## 277 317 3.56 44 1636.0 84.00 111 394 9.8 3
## 278 338 3.70 130 2139.0 185.00 193 215 9.9 4
## 279 198 3.77 38 911.0 57.00 56 280 9.8 2
## 280 325 3.69 69 2583.0 142.00 140 284 9.6 3
## 281 175 2.10 220 705.0 338.00 229 62 12.9 4
## 282 304 3.52 97 1622.0 71.00 169 255 9.5 4
## 283 412 3.99 103 1293.0 91.00 113 422 9.6 4
## 284 291 3.44 75 1082.0 85.00 195 251 9.5 3
## 285 253 3.48 65 688.0 57.00 80 252 10.0 1
## 286 310 3.36 70 1257.0 122.00 118 143 9.8 3
## 287 373 3.46 155 1768.0 120.00 151 258 10.1 4
## 288 310 3.89 107 637.0 117.00 242 298 9.6 2
## 289 416 3.99 177 960.0 86.00 242 269 9.8 2
## 290 294 3.57 33 722.0 93.00 69 283 9.8 3
## 291 339 3.18 123 3336.0 205.00 84 304 9.9 4
## 292 546 3.73 84 1070.0 127.00 153 291 11.2 3
## 293 194 2.98 196 815.0 163.00 78 122 12.3 4
## 294 1000 3.07 88 3150.0 193.00 133 299 10.9 4
## 295 646 3.83 102 855.0 127.00 194 306 10.3 3
## 296 328 3.31 62 1105.0 137.00 95 293 10.9 4
## 297 275 3.43 100 1142.0 75.00 91 217 11.3 4
## 298 340 3.37 73 289.0 97.00 93 243 10.2 3
## 299 342 3.76 90 1653.0 150.00 127 213 10.8 3
## 300 NA 2.23 234 601.0 135.00 NA 206 12.3 4
## 301 393 3.57 50 1307.0 74.00 103 295 10.5 4
## 302 335 3.95 43 657.0 52.00 104 268 10.6 2
## 303 372 3.25 108 1190.0 140.00 55 248 10.6 4
## 304 219 3.93 22 663.0 45.00 75 246 10.8 3
## 305 426 3.61 73 5184.0 288.00 144 275 10.6 3
## 306 239 3.45 31 1072.0 55.00 64 227 10.7 2
## 307 273 3.56 52 1282.0 130.00 59 344 10.5 2
## 308 246 3.58 24 797.0 91.00 113 288 10.4 2
## 309 260 2.75 41 1166.0 70.00 82 231 10.8 2
## 310 434 3.35 39 1713.0 171.00 100 234 10.2 2
## 311 247 3.16 69 1050.0 117.00 88 335 10.5 2
## 312 576 3.79 186 2115.0 136.00 149 200 10.8 2
## 313 NA 3.65 NA NA NA NA 378 11.0 NA
## 314 NA 3.04 NA NA NA NA 331 12.1 4
## 315 NA 4.03 NA NA NA NA 226 9.8 4
## 316 NA 3.96 NA NA NA NA NA 11.3 4
## 317 NA 2.48 NA NA NA NA 273 10.0 NA
## 318 NA 3.68 NA NA NA NA 306 9.5 2
## 319 NA 2.93 NA NA NA NA 260 10.4 NA
## 320 NA 3.81 NA NA NA NA 226 10.5 3
## 321 NA 3.41 NA NA NA NA 259 10.9 4
## 322 NA 3.69 NA NA NA NA 139 10.5 NA
## 323 NA 4.04 NA NA NA NA 130 11.2 2
## 324 NA 3.94 NA NA NA NA 234 11.5 2
## 325 NA 3.24 NA NA NA NA NA 18.0 2
## 326 NA 3.42 NA NA NA NA 246 10.3 3
## 327 NA 3.99 NA NA NA NA NA 11.2 2
## 328 NA 3.60 NA NA NA NA NA 12.1 2
## 329 NA 3.40 NA NA NA NA 371 10.1 4
## 330 NA 3.63 NA NA NA NA 325 10.3 2
## 331 NA 3.56 NA NA NA NA 142 13.6 4
## 332 NA 2.87 NA NA NA NA 114 11.8 4
## 333 NA 3.92 NA NA NA NA NA 10.6 4
## 334 NA 3.51 NA NA NA NA 721 11.8 NA
## 335 NA 4.15 NA NA NA NA 280 10.1 2
## 336 NA 3.34 NA NA NA NA 155 10.1 4
## 337 NA 3.64 NA NA NA NA 141 10.0 NA
## 338 NA 3.42 NA NA NA NA 475 13.8 2
## 339 NA 4.38 NA NA NA NA 269 10.6 2
## 340 NA 4.19 NA NA NA NA 141 11.2 2
## 341 NA 3.29 NA NA NA NA 286 13.1 4
## 342 NA 4.01 NA NA NA NA 244 10.5 3
## 343 NA 2.84 NA NA NA NA NA 12.2 2
## 344 NA 3.76 NA NA NA NA 209 10.6 3
## 345 NA 3.76 NA NA NA NA 388 10.1 2
## 346 NA 3.81 NA NA NA NA 160 11.0 4
## 347 NA 3.92 NA NA NA NA NA 9.3 2
## 348 NA 3.14 NA NA NA NA 191 12.3 2
## 349 NA 3.82 NA NA NA NA 249 10.3 2
## 350 NA 4.10 NA NA NA NA 200 9.0 3
## 351 NA 3.40 NA NA NA NA 150 11.1 4
## 352 NA 4.19 NA NA NA NA 330 9.9 1
## 353 NA 3.40 NA NA NA NA 167 10.6 4
## 354 NA 3.12 NA NA NA NA 125 9.6 2
## 355 NA 3.75 NA NA NA NA 145 10.4 3
## 356 NA 3.26 NA NA NA NA 171 11.1 4
## 357 NA 3.46 NA NA NA NA 109 10.4 4
## 358 NA 3.49 NA NA NA NA 314 10.3 3
## 359 NA 2.89 NA NA NA NA 419 NA 1
## 360 NA 3.15 NA NA NA NA 183 10.4 4
## 361 NA 2.31 NA NA NA NA 517 10.4 4
## 362 NA 3.04 NA NA NA NA 432 9.7 2
## 363 NA 3.50 NA NA NA NA 150 9.9 3
## 364 NA 3.35 NA NA NA NA 142 11.5 4
## 365 NA 3.58 NA NA NA NA 224 10.0 3
## 366 NA 3.01 NA NA NA NA 223 10.7 3
## 367 NA 3.49 NA NA NA NA 166 9.8 3
## 368 NA 2.77 NA NA NA NA 388 NA 4
## 369 NA 4.52 NA NA NA NA 102 10.8 4
## 370 NA 3.36 NA NA NA NA 384 10.0 3
## 371 NA 3.52 NA NA NA NA 265 11.1 1
## 372 NA 3.55 NA NA NA NA 248 10.3 2
## 373 NA 3.29 NA NA NA NA 190 9.8 2
## 374 NA 3.10 NA NA NA NA 274 10.6 3
## 375 NA 3.24 NA NA NA NA 231 10.5 3
## 376 NA 3.63 NA NA NA NA 292 10.2 3
## 377 NA 3.76 NA NA NA NA 253 9.9 3
## 378 NA 3.62 NA NA NA NA 225 9.9 2
## 379 NA 2.73 NA NA NA NA 224 11.1 4
## 380 NA 4.08 NA NA NA NA 418 10.5 3
## 381 NA 3.66 NA NA NA NA 92 11.0 4
## 382 NA 2.58 NA NA NA NA 190 11.6 4
## 383 NA 3.00 NA NA NA NA 76 10.8 4
## 384 NA 3.40 NA NA NA NA 243 9.7 1
## 385 NA 2.93 NA NA NA NA 209 10.6 3
## 386 NA 2.38 NA NA NA NA 166 9.8 3
## 387 NA 3.00 NA NA NA NA 233 9.9 3
## 388 NA 3.50 NA NA NA NA 117 10.9 4
## 389 NA 3.06 NA NA NA NA 165 10.0 4
## 390 NA 3.15 NA NA NA NA 139 10.0 3
## 391 NA 2.80 NA NA NA NA 120 11.0 2
## 392 NA 3.03 NA NA NA NA 173 10.9 3
## 393 NA 2.96 NA NA NA NA 319 9.9 4
## 394 NA 3.07 NA NA NA NA 80 12.1 4
## 395 NA 3.98 NA NA NA NA 402 11.0 1
## 396 NA 3.48 NA NA NA NA 277 10.2 2
## 397 NA 3.65 NA NA NA NA 425 10.2 4
## 398 NA 3.49 NA NA NA NA 189 10.9 4
## 399 NA 2.82 NA NA NA NA 193 10.4 2
## 400 NA 3.34 NA NA NA NA 258 10.6 2
## 401 NA 3.19 NA NA NA NA 120 11.1 4
## 402 NA 3.01 NA NA NA NA 256 10.6 3
## 403 NA 3.33 NA NA NA NA 256 10.8 4
## 404 NA 3.60 NA NA NA NA 337 10.4 3
## 405 NA 3.64 NA NA NA NA 340 10.6 3
## 406 NA 2.68 NA NA NA NA 219 11.5 4
## 407 NA 3.69 NA NA NA NA 220 10.8 3
## 408 NA 3.17 NA NA NA NA 162 10.7 3
## 409 NA 3.73 NA NA NA NA 214 10.8 3
## 410 NA 3.81 NA NA NA NA 255 10.8 3
## 411 NA 3.57 NA NA NA NA 286 10.6 3
## 412 NA 3.58 NA NA NA NA 244 10.8 3
## 413 NA 3.23 NA NA NA NA 312 10.8 3
## 414 NA 2.96 NA NA NA NA 174 10.9 3
## 415 NA 3.83 NA NA NA NA 180 11.2 4
## 416 NA 3.42 NA NA NA NA 143 9.9 3
## 417 NA 3.75 NA NA NA NA 269 10.4 3
## 418 NA 3.29 NA NA NA NA 350 10.6 4
Consider Trends vs Indicators
library(tidyverse)
pbc <- pbc %>%
filter(!is.na(stage)) %>%
mutate(stage_dummy = as.factor(stage)) %>%
mutate(mean_cent_age= age-mean(age))
Regression plot with trend:
library(ggplot2)
ggplot(pbc, aes(mean_cent_age, platelet, color=stage)) + geom_smooth(method="lm", se=FALSE)
Regression plot with trend:
Regression plot with Indicators:
library(ggplot2)
ggplot(pbc, aes(mean_cent_age, platelet, color=stage_dummy)) + geom_smooth(method="lm")
Regression plot with Indicators:
Regressions: Trend
library(broom)
mod1 <- lm(data=pbc, platelet~mean_cent_age + stage)
tidy(mod1)
## # A tibble: 3 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 333. 16.9 19.7 6.72e-61
## 2 mean_cent_age -1.07 0.449 -2.38 1.78e- 2
## 3 stage -25.4 5.35 -4.75 2.80e- 6
Interpretations
- age: For 2 people with the same disease stage, a person 1 year older has an average platelet count of 1 less than the younger person.
- stage: For 2 people of the same age, a person 1 disease stage higher has an average platelet count 25 less than the person with the lower disease stage.
Regressions: Trend
library(broom)
mod2 <- lm(data=pbc, platelet~age + stage_dummy)
tidy(mod2)
## # A tibble: 5 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 340. 29.6 11.5 1.50e-26
## 2 age -1.03 0.453 -2.28 2.33e- 2
## 3 stage_dummy2 -2.16 22.9 -0.0941 9.25e- 1
## 4 stage_dummy3 -26.8 21.9 -1.22 2.22e- 1
## 5 stage_dummy4 -60.1 22.2 -2.70 7.19e- 3
Interpretations
- age: For 2 people with the same disease stage, a person 1 year older has an average platelet count of 1 less than the younger person.
- stage_dummy 2: For 2 people of the same age, a person in diease stage 2 higher has an average platelet count 2 less than the person with disease stage 1.
- stage_dummy 3: For 2 people of the same age, a person in diease stage 3 higher has an average platelet count 27 less than the person with disease stage 1.
- stage_dummy 4: For 2 people of the same age, a person in diease stage 4 higher has an average platelet count 60 less than the person with disease stage 1.
Is there a difference?
- Yes!!
- If we look between diease stage 1 and 2 the difference is on average 2 in the model with dummy variables.
- In the trend model the difference between any 2 stages is on average 25.
Is this difference Significant?
- We can test for significane with the F-test
anova(mod1,mod2)
## Analysis of Variance Table
##
## Model 1: platelet ~ mean_cent_age + stage
## Model 2: platelet ~ age + stage_dummy
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 398 3383405
## 2 396 3370016 2 13389 0.7866 0.4561
- Based on our test, the trend gives us just as much information.
How about \(R^2\)
library(broom)
glance1 <- glance(mod1)[,c(1:2)]
glance2 <- glance(mod2)[,c(1:2)]
bind_rows(glance1, glance2)
## # A tibble: 2 x 2
## r.squared adj.r.squared
## <dbl> <dbl>
## 1 0.0774 0.0728
## 2 0.0811 0.0718