Call:
lm(formula = ReproductiveEffort ~ Foodavailability, data = foodav)
Residuals:
Min 1Q Median 3Q Max
-0.37653 -0.25549 -0.04903 0.23662 0.45750
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.89490 0.07262 12.32 <2e-16 ***
Foodavailability 1.19387 0.01590 75.08 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2652 on 61 degrees of freedom
Multiple R-squared: 0.9893, Adjusted R-squared: 0.9891
F-statistic: 5636 on 1 and 61 DF, p-value: < 2.2e-16Generalized Linear Models
Assumptions
This lecture
You can access it from: https://rpubs.com/amolina/presentation2
Review of linear models
Assumptions of glm’s
Mixed effects models
How to interpret GLM’s outputs?
- Let’s remember that linear models, and glm’s have a “linear” deterministic component
How to interpret GLM’s outputs?
Let’s remember that linear models, and glm’s have a “linear” deterministic component
In linear models: \(\mu_i = \beta_0 + \beta_1x_{1,i} ...\) (which means, the expected value of y )
In generalized linear models: \(f(\mu_i) = \beta_0 + \beta_1x_{1,i} ...\) (which means, a function of the expected value of y)
The deterministic component is linear
Which is a good thing! Each linear (not polynomial… let’s forget about that for a little while) has two components!
A slope
An intercept
The deterministic component is linear
Which is a good thing! Each linear (not polynomial… let’s forget about that for a little while) has two components!
A slope
An intercept
\[ y = mx + b \]
The deterministic component is linear
Which is a good thing! Each linear (not polynomial… let’s forget about that for a little while) has two components!
A slope
An intercept
\[ y = mx + b = b + mx \]
is not any different than:
\[ \begin{align} \mu_i & = \beta_0 + \beta_1x_{1,i} \\ y_i & ~ N(mean = \mu_i, var=\sigma^2) \end{align} \]
Let’s remember what a slope and an intercept is
Let’s remember what a slope and an intercept is
Let’s remember what a slope and an intercept is
Let’s remember what a slope and an intercept is
Slope: For very change in x of 1, y changes 1.5. Intercept: 2
So, if x = 5, then y = 2 + 1.5(5) = 9.5
So, what about linear models?
Each line in a linear model simply has: 1 slope and 1 intercept.
So, what about linear models?
What is the slope? what is the intercept?
So, what about linear models?
\[ \begin{split} y_i & \sim \beta_0 + \beta_1x_i + \epsilon_i \\ \text{where } \epsilon & \sim normal(0,\sigma) \end{split} \] Or, in this specific case:
\[ Reproductive \ effort_i \sim \beta_0 + \beta_1Food \ availability_i + \epsilon_i \\ \]
So, what about linear models?
Intercept: 0.895
Slope: 1.194
\[ \begin{split} y_i & \sim \beta_0 + \beta_1x_i + \epsilon_i \\ \text{where } \epsilon & \sim normal(0,\sigma) \end{split} \]
Results
| Coefficient1 | Estimate | Col3 | Col4 | Col5 |
|---|---|---|---|---|
| Intercept \(\beta_0\) | 0.895 | |||
| Slope \(\beta_1\) | 1.194 |
\[ \begin{split} y_i & \sim \beta_0 + \beta_1x_i + \epsilon_i \\ \text{where } \epsilon & \sim normal(0,\sigma) \end{split} \]
Results
| Coefficient1 | Estimate | Std. Error | t-value (test) | Col5 |
|---|---|---|---|---|
| Intercept \(\beta_0\) | 0.895 | 0.072 | 12.32 | <0.0001 |
| Slope \(\beta_1\) | 1.194 | 0.0159 | 75.08 | <0.0001 |
\[ \begin{split} y_i & \sim \beta_0 + \beta_1x_i + \epsilon_i \\ \text{where } \epsilon & \sim normal(0,\sigma) \end{split} \]
Test:
Ho: Coefficient == 0
Ha Coefficient != 0
How about this one:
How about this one:
Call:
lm(formula = FC ~ IC + Dose, data = drugz)
Residuals:
Min 1Q Median 3Q Max
-2.2610 -0.6360 0.0000 0.6514 2.2876
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.51804 0.38892 1.332 0.1849
IC 0.93384 0.05647 16.537 <2e-16 ***
Dosedose1 -0.44007 0.20011 -2.199 0.0294 *
Dosedose2 -2.09915 0.20162 -10.412 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9893 on 146 degrees of freedom
Multiple R-squared: 0.7636, Adjusted R-squared: 0.7588
F-statistic: 157.2 on 3 and 146 DF, p-value: < 2.2e-16- Additive model means only one slope is estimated!
Back to the plot
Parallel lines
1 slope, 3 intercepts
Each line: \(y = mx + b\)
Finding values
Call:
lm(formula = FC ~ IC + Dose, data = drugz)
Residuals:
Min 1Q Median 3Q Max
-2.2610 -0.6360 0.0000 0.6514 2.2876
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.51804 0.38892 1.332 0.1849
IC 0.93384 0.05647 16.537 <2e-16 ***
Dosedose1 -0.44007 0.20011 -2.199 0.0294 *
Dosedose2 -2.09915 0.20162 -10.412 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9893 on 146 degrees of freedom
Multiple R-squared: 0.7636, Adjusted R-squared: 0.7588
F-statistic: 157.2 on 3 and 146 DF, p-value: < 2.2e-16\[ \begin{align} \beta_0 & = 0.51 \ \text{intercept for control} \\ \beta_1 & = 0.93 \ \text{slope} \\ \beta_2 & = -0.44 \ \text{Difference in intercept between control and dose 1} \\ \beta_2 & = -2.099 \ \text{Difference in intercept between control and dose 2} \\ \end{align} \]
\[ y_i \sim \beta_0 + \beta_1x_{1,i} + \beta_2x_{2,i} + \beta_3x_{3,i} + \epsilon_i \\ \]
| Site | Value of \(x_2\) | Value of \(x_3\) |
|---|---|---|
| Control | 0 | 0 |
| Dose 1 | 1 | 0 |
| Dose 2 | 0 | 1 |
Finding values
Call:
lm(formula = FC ~ IC + Dose, data = drugz)
Residuals:
Min 1Q Median 3Q Max
-2.2610 -0.6360 0.0000 0.6514 2.2876
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.51804 0.38892 1.332 0.1849
IC 0.93384 0.05647 16.537 <2e-16 ***
Dosedose1 -0.44007 0.20011 -2.199 0.0294 *
Dosedose2 -2.09915 0.20162 -10.412 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9893 on 146 degrees of freedom
Multiple R-squared: 0.7636, Adjusted R-squared: 0.7588
F-statistic: 157.2 on 3 and 146 DF, p-value: < 2.2e-16| Measurement | Intercept | Slope |
|---|---|---|
| Control | 0.51 | 0.93 |
| Dose 1 | 0.51 - 0.44 | 0.93 |
| Dose 2 | 0.51 -2.099 | 0.93 |
Statistical inference
anova
emmeans
contrast
Analysis of Variance Table
Response: FC
Df Sum Sq Mean Sq F value Pr(>F)
IC 1 342.36 342.36 349.825 < 2.2e-16 ***
Dose 2 119.30 59.65 60.949 < 2.2e-16 ***
Residuals 146 142.89 0.98
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 contrast estimate SE df t.ratio p.value
control - dose1 0.44 0.200 146 2.199 0.0747
control - dose2 2.10 0.202 146 10.412 <.0001
dose1 - dose2 1.66 0.198 146 8.377 <.0001
P value adjustment: tukey method for comparing a family of 3 estimates statistical inference
anova compares variance. Is the variance between groups higher than within groups?
Interactive
- Each line has its own slope and its own intercept
Interactive
Call:
lm(formula = FC ~ IC * Dose, data = drugz)
Residuals:
Min 1Q Median 3Q Max
-2.3167 -0.6262 -0.0031 0.6443 2.1836
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.38733 0.59401 0.652 0.5154
IC 0.95418 0.08981 10.624 <2e-16 ***
Dosedose1 0.07353 0.85237 0.086 0.9314
Dosedose2 -2.21858 0.86308 -2.571 0.0112 *
IC:Dosedose1 -0.08529 0.13509 -0.631 0.5288
IC:Dosedose2 0.02324 0.13917 0.167 0.8676
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9939 on 144 degrees of freedom
Multiple R-squared: 0.7647, Adjusted R-squared: 0.7565
F-statistic: 93.59 on 5 and 144 DF, p-value: < 2.2e-16Doing inference of interactions
Analysis of Variance Table
Response: FC
Df Sum Sq Mean Sq F value Pr(>F)
IC 1 342.36 342.36 346.5514 <2e-16 ***
Dose 2 119.30 59.65 60.3790 <2e-16 ***
IC:Dose 2 0.63 0.31 0.3168 0.729
Residuals 144 142.26 0.99
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 contrast estimate SE df t.ratio p.value
control - dose1 0.44 0.202 144 2.173 0.0794
control - dose2 2.08 0.204 144 10.175 <.0001
dose1 - dose2 1.64 0.201 144 8.136 <.0001
P value adjustment: tukey method for comparing a family of 3 estimates Interactions
Finding values (predictions)
Assignment question 1 📝
Look at the summary. And before continuing make sure you understand it. If you don’t, now is the time to raise your hand.
You need to calculate the expected probability (we’ll call this \(\pi\)) of an individual being infected for each of the following two cases:
- An individual of length 50 from Area 1, in 1999
- An individual of length 50 from Area 3, in 2001
GLM
Model
'data.frame': 1254 obs. of 11 variables:
$ Sample : int 1 2 3 4 5 6 7 8 9 10 ...
$ Intensity : int 0 0 0 0 0 0 0 0 0 0 ...
$ Prevalence: int 0 0 0 0 0 0 0 0 0 0 ...
$ Year : Factor w/ 3 levels "1999","2000",..: 1 1 1 1 1 1 1 1 1 1 ...
$ Depth : int 220 220 220 220 220 220 220 194 194 194 ...
$ Weight : int 148 144 146 138 40 68 52 3848 2576 1972 ...
$ Length : int 26 26 27 26 17 20 19 77 67 60 ...
$ Sex : Factor w/ 3 levels "0","1","2": 1 1 1 1 1 1 1 1 1 1 ...
$ Stage : Factor w/ 5 levels "0","1","2","3",..: 1 1 1 1 1 1 1 1 1 1 ...
$ Age : int 0 0 0 0 0 0 0 0 0 0 ...
$ Area : Factor w/ 4 levels "1","2","3","4": 2 2 2 2 2 2 2 3 3 3 ... Sample Intensity Prevalence Year Depth Weight Length Sex Stage Age Area
1 1 0 0 1999 220 148 26 0 0 0 2
2 2 0 0 1999 220 144 26 0 0 0 2
3 3 0 0 1999 220 146 27 0 0 0 2
4 4 0 0 1999 220 138 26 0 0 0 2
5 5 0 0 1999 220 40 17 0 0 0 2
6 6 0 0 1999 220 68 20 0 0 0 2
7 7 0 0 1999 220 52 19 0 0 0 2
8 8 0 0 1999 194 3848 77 0 0 0 3
9 9 0 0 1999 194 2576 67 0 0 0 3
10 10 0 0 1999 194 1972 60 0 0 0 3
11 11 0 0 1999 194 2272 62 0 0 0 3
12 12 0 0 1999 194 1046 49 0 0 0 3
13 13 0 0 1999 194 1186 52 0 0 0 3
14 14 0 0 1999 194 776 45 0 0 0 3
15 15 0 0 1999 194 2372 61 0 0 0 3
16 16 0 0 1999 194 1194 54 0 0 0 3
17 17 0 0 1999 194 3574 71 0 0 0 3
18 18 0 0 1999 194 2138 65 0 0 0 3
19 19 0 0 1999 194 2222 59 0 0 0 3
20 20 0 0 1999 194 2482 62 0 0 0 3
21 21 0 0 1999 194 2164 61 0 0 0 3
22 22 0 0 1999 194 1772 57 0 0 0 3
23 23 0 0 1999 194 494 39 0 0 0 3
24 24 0 0 1999 194 3130 67 0 0 0 3
25 25 0 0 1999 194 2708 65 0 0 0 3
26 26 0 0 1999 194 2394 66 0 0 0 3
27 27 0 0 1999 194 2212 64 0 0 0 3
28 28 0 0 1999 194 3110 65 0 0 0 3
29 29 0 0 1999 194 2218 62 0 0 0 3
30 30 0 0 1999 194 2222 60 0 0 0 3
31 31 0 0 1999 194 2398 62 0 0 0 3
32 32 0 0 1999 235 368 35 0 0 0 4
33 33 0 0 1999 235 306 34 0 0 0 4
34 34 0 0 1999 235 318 35 0 0 0 4
35 35 0 0 1999 235 276 32 0 0 0 4
36 36 0 0 1999 278 84 22 0 0 0 4
37 37 0 0 1999 51 1016 47 2 1 3 1
38 38 0 0 1999 51 1584 53 1 1 5 1
39 39 0 0 1999 51 1354 55 1 1 4 1
40 40 0 0 1999 51 1518 52 1 1 4 1
41 41 0 0 1999 51 1382 51 2 1 4 1
42 42 0 0 1999 51 1648 55 1 1 4 1
43 43 0 0 1999 51 1354 51 2 1 5 1
44 44 0 0 1999 51 1870 54 2 1 5 1
45 45 0 0 1999 51 1272 53 2 1 5 1
46 46 0 0 1999 51 1360 50 1 1 4 1
47 47 0 0 1999 51 936 45 2 1 3 1
48 48 0 0 1999 51 1650 55 2 1 5 1
49 49 0 0 1999 51 1504 53 1 1 5 1
50 50 0 0 1999 51 1008 48 2 1 3 1
51 51 0 0 1999 51 896 46 2 1 3 1
52 52 0 0 1999 51 3016 68 1 1 5 1
53 53 0 0 1999 85 470 37 1 1 2 1
54 54 0 0 1999 85 520 38 1 1 2 1
55 55 0 0 1999 85 470 38 1 1 2 1
56 56 0 0 1999 85 558 39 2 1 3 1
57 57 0 0 1999 85 3076 68 2 1 5 1
58 58 0 0 1999 85 1536 57 2 1 4 1
59 59 0 0 1999 85 1916 60 2 1 4 1
60 60 0 0 1999 85 320 32 1 1 2 1
61 61 0 0 1999 85 1152 49 1 1 4 1
62 62 0 0 1999 85 1284 54 1 1 5 1
63 63 0 0 1999 85 4712 80 1 1 7 1
64 64 0 0 1999 85 2182 60 2 1 4 1
65 65 0 0 1999 85 976 49 1 1 3 1
66 66 0 0 1999 85 408 36 1 1 2 1
67 67 0 0 1999 85 242 30 2 1 2 1
68 68 0 0 1999 85 644 41 2 1 3 1
69 69 0 0 1999 85 540 39 2 1 2 1
70 70 0 0 1999 85 326 33 1 1 2 1
71 71 0 0 1999 85 300 33 1 1 2 1
72 72 0 0 1999 85 232 29 2 1 2 1
73 73 0 0 1999 85 822 45 1 1 3 1
74 74 0 0 1999 85 692 43 1 1 3 1
75 75 0 0 1999 85 438 36 2 1 2 1
76 76 0 0 1999 85 406 35 2 1 2 1
77 77 0 0 1999 85 240 30 2 1 2 1
78 78 0 0 1999 85 358 34 1 1 2 1
79 79 0 0 1999 137 476 38 2 1 3 1
80 80 0 0 1999 137 1478 55 1 1 4 1
81 81 0 0 1999 137 1672 57 1 1 5 1
82 82 0 0 1999 137 1582 56 2 1 3 1
83 83 0 0 1999 137 390 35 2 1 2 1
84 84 0 0 1999 150 1816 59 1 1 5 2
85 85 0 0 1999 150 3888 71 1 1 5 2
86 86 0 0 1999 150 1986 58 1 1 4 2
87 87 0 0 1999 150 2524 66 1 1 5 2
88 88 0 0 1999 150 1712 59 2 1 4 2
89 89 0 0 1999 150 1968 58 2 1 4 2
90 90 0 0 1999 150 2764 64 1 1 5 2
91 91 0 0 1999 150 3412 69 1 1 5 2
92 92 0 0 1999 150 2294 63 1 1 5 2
93 93 0 0 1999 150 552 38 1 1 2 2
94 94 0 0 1999 150 1422 54 2 1 4 2
95 95 0 0 1999 150 1028 47 2 1 3 2
96 96 0 0 1999 150 466 38 2 1 2 2
97 97 0 0 1999 150 432 35 1 1 2 2
98 98 0 0 1999 150 300 31 2 1 2 2
99 99 0 0 1999 150 2706 67 1 1 5 2
100 100 0 0 1999 150 1154 52 1 1 3 2
101 101 0 0 1999 150 1146 52 1 1 4 2
102 102 0 0 1999 100 2352 64 1 1 6 2
103 103 0 0 1999 100 2084 60 2 1 4 2
104 104 0 0 1999 100 1008 48 2 1 3 2
105 105 0 0 1999 100 3970 73 2 1 5 2
106 106 0 0 1999 100 1738 55 2 1 5 2
107 107 0 0 1999 100 844 45 2 1 3 2
108 108 0 0 1999 100 3688 74 1 1 6 2
109 109 0 0 1999 100 2848 67 1 1 5 2
110 110 0 0 1999 100 1090 47 2 1 3 2
111 111 0 0 1999 292 826 49 2 1 3 2
112 112 0 0 1999 220 2086 61 1 1 5 2
113 113 0 0 1999 220 1744 58 2 1 5 2
114 114 0 0 1999 220 2118 60 1 1 5 2
115 115 0 0 1999 220 750 45 2 1 3 2
116 116 0 0 1999 220 1570 58 2 1 5 2
117 117 0 0 1999 220 326 33 2 1 2 2
118 118 0 0 1999 220 4292 77 1 1 6 2
119 119 0 0 1999 220 220 30 1 1 2 2
120 120 0 0 1999 220 262 32 2 1 2 2
121 121 0 0 1999 220 184 28 1 1 2 2
122 122 0 0 1999 220 180 28 1 1 2 2
123 123 0 0 1999 220 976 48 1 1 3 2
124 124 0 0 1999 220 1344 52 2 1 4 2
125 125 0 0 1999 220 2926 67 1 1 5 2
126 126 0 0 1999 220 544 41 1 1 3 2
127 127 0 0 1999 220 774 44 2 1 3 2
128 128 0 0 1999 220 452 37 2 1 2 2
129 129 0 0 1999 220 258 32 1 1 2 2
130 130 0 0 1999 220 190 29 2 1 2 2
131 131 0 0 1999 220 266 32 1 1 2 2
132 132 0 0 1999 220 196 29 1 1 0 2
133 133 0 0 1999 99 1846 59 1 1 4 3
134 134 0 0 1999 99 1312 53 2 1 4 3
135 135 0 0 1999 99 3302 73 2 1 5 3
136 136 0 0 1999 99 1258 50 2 1 4 3
137 137 0 0 1999 99 1848 57 1 1 6 3
138 138 0 0 1999 99 1246 51 2 1 4 3
139 139 0 0 1999 99 2038 59 2 1 4 3
140 140 0 0 1999 99 4738 77 1 1 6 3
141 141 0 0 1999 99 1248 51 2 1 4 3
142 142 0 0 1999 99 1072 48 1 1 3 3
143 143 0 0 1999 99 854 46 2 1 3 3
144 144 0 0 1999 99 2168 62 2 1 4 3
145 145 0 0 1999 99 1108 50 2 1 4 3
146 146 0 0 1999 99 1122 50 1 1 4 3
147 147 0 0 1999 99 2050 61 1 1 4 3
148 148 0 0 1999 99 1168 51 2 1 4 3
149 149 0 0 1999 99 1334 53 2 1 4 3
150 150 0 0 1999 99 964 47 2 1 3 3
151 151 0 0 1999 99 678 41 2 1 2 3
152 152 0 0 1999 99 1076 49 2 1 4 3
153 153 0 0 1999 99 1238 52 1 1 4 3
154 154 0 0 1999 99 1824 58 1 1 4 3
155 155 0 0 1999 99 860 44 1 1 3 3
156 156 0 0 1999 99 1260 50 2 1 4 3
157 157 0 0 1999 99 1154 52 1 1 4 3
158 158 0 0 1999 99 936 43 1 1 3 3
159 159 0 0 1999 99 1598 55 2 1 4 3
160 160 0 0 1999 99 2500 65 2 1 5 3
161 161 0 0 1999 99 1942 59 1 1 4 3
162 162 0 0 1999 270 770 47 2 1 4 3
163 163 0 0 1999 270 538 40 2 1 3 3
164 164 0 0 1999 270 3862 78 1 1 6 3
165 165 0 0 1999 270 878 48 2 1 4 3
166 166 0 0 1999 270 1438 55 1 1 4 3
167 167 0 0 1999 270 1592 59 2 1 5 3
168 168 0 0 1999 270 1250 53 2 1 4 3
169 169 0 0 1999 270 1226 51 1 1 4 3
170 170 0 0 1999 270 1326 53 1 1 4 3
171 171 0 0 1999 270 1104 49 2 1 4 3
172 172 0 0 1999 270 1982 63 2 1 5 3
173 173 0 0 1999 270 2480 64 1 1 5 3
174 174 0 0 1999 270 1142 51 2 1 4 3
175 175 0 0 1999 270 488 39 1 1 4 3
176 176 0 0 1999 270 1382 55 1 1 4 3
177 177 0 0 1999 256 1714 59 2 1 4 3
178 178 0 0 1999 256 754 45 1 1 4 3
179 179 0 0 1999 256 1618 61 1 1 5 3
180 180 0 0 1999 256 2122 64 2 1 5 3
181 181 0 0 1999 256 604 38 1 1 3 3
182 182 0 0 1999 256 2780 65 1 1 6 3
183 183 0 0 1999 256 2318 64 1 1 4 3
184 184 0 0 1999 256 1344 53 1 1 4 3
185 185 0 0 1999 256 1674 60 2 1 8 3
186 186 0 0 1999 256 1144 53 1 1 7 3
187 187 0 0 1999 256 1052 51 2 1 3 3
188 188 0 0 1999 172 2034 57 2 1 5 3
189 189 0 0 1999 172 674 42 2 1 3 3
190 190 0 0 1999 172 862 47 1 1 3 3
191 191 0 0 1999 172 612 40 1 1 3 3
192 192 0 0 1999 172 2528 64 2 1 4 3
193 193 0 0 1999 172 1516 55 1 1 4 3
194 194 0 0 1999 172 906 47 1 1 4 3
195 195 0 0 1999 172 434 38 2 1 3 3
196 196 0 0 1999 172 182 29 1 1 2 3
197 197 0 0 1999 172 230 31 1 1 2 3
198 198 0 0 1999 172 200 29 1 1 2 3
199 199 0 0 1999 172 222 29 1 1 2 3
200 200 0 0 1999 172 238 30 2 1 2 3
201 201 0 0 1999 172 176 28 2 1 2 3
202 202 0 0 1999 172 198 30 1 1 2 3
203 203 0 0 1999 172 168 29 1 1 2 3
204 204 0 0 1999 172 170 28 1 1 2 3
205 205 0 0 1999 172 204 30 2 1 2 3
206 206 0 0 1999 235 2496 68 1 1 6 4
207 207 0 0 1999 235 1198 50 2 1 3 4
208 208 0 0 1999 235 3678 75 1 1 5 4
209 209 0 0 1999 235 1036 50 1 1 4 4
210 210 0 0 1999 235 2898 73 1 1 5 4
211 211 0 0 1999 278 582 40 1 1 3 4
212 212 0 0 1999 278 986 48 2 1 4 4
213 213 0 0 1999 278 662 44 2 1 3 4
214 214 0 0 1999 278 270 33 1 1 2 4
215 215 0 0 1999 278 980 49 2 1 3 4
216 216 0 0 1999 278 1330 52 1 1 4 4
217 217 0 0 1999 278 364 36 2 1 2 4
218 218 0 0 1999 110 3190 68 2 1 6 4
219 219 0 0 1999 110 626 42 1 1 3 4
220 220 0 0 1999 110 788 46 1 1 3 4
221 221 0 0 1999 110 968 51 2 1 5 4
222 222 0 0 1999 110 1322 54 1 1 4 4
223 223 0 0 1999 110 1400 54 2 1 4 4
224 224 0 0 1999 110 942 48 2 1 4 4
225 225 0 0 1999 110 1104 53 2 1 4 4
226 226 0 0 1999 110 500 40 1 1 3 4
227 227 0 0 1999 110 296 33 2 1 2 4
228 228 0 0 1999 110 168 29 1 1 2 4
229 229 0 0 1999 110 370 36 1 1 3 4
230 230 0 0 1999 51 1544 53 2 2 4 1
231 231 0 0 1999 51 1520 53 2 2 4 1
232 232 0 0 1999 51 1912 58 1 2 6 1
233 233 0 0 1999 51 1144 49 2 2 4 1
234 234 0 0 1999 51 1472 51 2 2 4 1
235 235 0 0 1999 51 1418 51 2 2 5 1
236 236 0 0 1999 51 1330 49 2 2 3 1
237 237 0 0 1999 85 948 47 2 2 3 1
238 238 0 0 1999 200 2292 65 1 2 6 1
239 239 0 0 1999 200 2562 63 1 2 6 1
240 240 0 0 1999 100 1708 55 2 2 5 2
241 241 0 0 1999 100 1670 55 2 2 5 2
242 242 0 0 1999 100 1740 57 2 2 5 2
243 243 0 0 1999 100 2396 62 2 2 5 2
244 244 0 0 1999 220 3484 68 2 2 6 2
245 245 0 0 1999 99 2896 64 2 2 7 3
246 246 0 0 1999 99 1712 55 2 2 4 3
247 247 0 0 1999 99 1778 57 1 2 5 3
248 248 0 0 1999 99 1800 55 2 2 4 3
249 249 0 0 1999 99 4086 77 1 2 9 3
250 250 0 0 1999 99 1772 57 2 2 5 3
251 251 0 0 1999 99 2140 62 1 2 5 3
252 252 0 0 1999 99 1116 49 2 2 4 3
253 253 0 0 1999 99 1844 56 2 2 7 3
254 254 0 0 1999 99 1344 51 2 2 5 3
255 255 0 0 1999 270 1110 54 1 2 5 3
256 256 0 0 1999 270 2228 63 2 2 6 3
257 257 0 0 1999 270 1864 58 2 2 5 3
258 258 0 0 1999 270 1266 51 2 2 4 3
259 259 0 0 1999 270 1082 49 2 2 4 3
260 260 0 0 1999 256 2958 63 2 2 5 3
261 261 0 0 1999 256 3198 66 1 2 5 3
262 262 0 0 1999 256 2490 61 2 2 5 3
263 263 0 0 1999 256 3618 70 2 2 5 3
264 264 0 0 1999 256 3034 68 2 2 4 3
265 265 0 0 1999 172 2244 62 2 2 7 3
266 266 0 0 1999 172 1754 55 1 2 5 3
267 267 0 0 1999 172 178 28 1 2 2 3
268 268 0 0 1999 235 1578 54 2 2 4 4
269 269 0 0 1999 278 2130 59 2 2 5 4
270 270 0 0 1999 110 1304 50 1 2 4 4
271 271 0 0 1999 110 4098 74 2 2 8 4
272 272 0 0 1999 110 6390 88 1 2 8 4
273 273 0 0 1999 150 7854 90 1 4 7 2
274 274 0 0 1999 150 NA NA 1 4 8 2
275 275 0 0 1999 100 6916 86 1 4 9 2
276 276 0 0 1999 100 2980 65 1 4 7 2
277 277 0 0 1999 220 4018 72 1 4 6 2
278 278 0 0 1999 270 2552 63 1 4 5 3
279 279 0 0 2000 57 1474 51 1 1 6 1
280 280 0 0 2000 57 1608 57 2 1 6 1
281 281 0 0 2000 57 830 47 1 1 3 1
282 282 0 0 2000 50 992 48 2 1 3 1
283 283 0 0 2000 121 552 40 1 1 2 1
284 284 0 0 2000 121 1744 58 1 1 4 1
285 285 0 0 2000 121 668 42 1 1 2 1
286 286 0 0 2000 121 448 36 1 1 2 1
287 287 0 0 2000 121 410 36 1 1 2 1
288 288 0 0 2000 121 1222 51 1 1 3 1
289 289 0 0 2000 102 1538 54 1 1 4 2
290 290 0 0 2000 102 2820 62 1 1 6 2
291 291 0 0 2000 102 1116 49 2 1 3 2
292 292 0 0 2000 102 4058 75 2 1 6 2
293 293 0 0 2000 102 544 40 1 1 2 2
294 294 0 0 2000 102 398 35 2 1 2 2
295 295 0 0 2000 102 514 39 2 1 2 2
296 296 0 0 2000 102 1124 48 2 1 4 2
297 297 0 0 2000 102 1038 48 1 1 3 2
298 298 0 0 2000 102 1470 53 2 1 4 2
299 299 0 0 2000 102 4348 75 1 1 6 2
300 300 0 0 2000 102 440 36 2 1 2 2
301 301 0 0 2000 102 1984 59 2 1 6 2
302 302 0 0 2000 102 1966 61 1 1 4 2
303 303 0 0 2000 102 1094 49 2 1 4 2
304 304 0 0 2000 102 986 46 2 1 3 2
305 305 0 0 2000 293 3370 71 2 1 6 2
306 306 0 0 2000 293 920 46 2 1 3 2
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844 844 18 1 1999 270 492 40 1 1 3 3
845 845 18 1 1999 235 708 43 1 1 3 4
846 846 19 1 1999 172 200 30 1 1 2 3
847 847 19 1 1999 200 1520 53 1 2 5 1
848 848 20 1 1999 200 842 46 1 1 3 1
849 849 20 1 1999 278 562 40 2 1 2 4
850 850 21 1 1999 256 1108 54 2 1 5 3
851 851 21 1 1999 278 198 29 1 1 2 4
852 852 21 1 1999 278 2034 58 2 2 4 4
853 853 22 1 1999 278 198 30 1 1 2 4
854 854 22 1 1999 200 1416 52 2 2 5 1
855 855 23 1 1999 235 1242 53 1 1 5 4
856 856 23 1 1999 235 1682 56 2 1 4 4
857 857 24 1 1999 235 932 49 0 0 0 4
858 858 25 1 1999 235 1126 49 0 0 0 4
859 859 25 1 1999 110 1238 49 2 2 4 4
860 860 26 1 1999 137 1582 56 2 1 5 1
861 861 27 1 1999 137 2566 68 1 1 4 1
862 862 28 1 1999 292 794 46 1 1 3 2
863 863 28 1 1999 235 1396 56 1 1 4 4
864 864 28 1 1999 235 1996 59 2 2 5 4
865 865 32 1 1999 270 1342 53 1 1 4 3
866 866 33 1 1999 278 382 37 1 1 2 4
867 867 35 1 1999 235 174 30 0 0 0 4
868 868 35 1 1999 235 1182 53 2 1 4 4
869 869 36 1 1999 292 680 45 1 1 3 2
870 870 38 1 1999 278 156 28 0 0 0 4
871 871 39 1 1999 235 714 44 0 0 0 4
872 872 41 1 1999 137 456 38 1 1 3 1
873 873 41 1 1999 200 2312 61 1 2 6 1
874 874 45 1 1999 278 154 27 0 0 0 4
875 875 50 1 1999 137 1284 51 2 1 5 1
876 876 51 1 1999 235 386 36 0 0 0 4
877 877 52 1 1999 200 1458 53 2 2 5 1
878 878 56 1 1999 235 664 43 0 0 0 4
879 879 65 1 1999 278 194 30 1 1 2 4
880 880 68 1 1999 235 320 34 0 0 0 4
881 881 73 1 1999 235 206 29 0 0 0 4
882 882 84 1 1999 150 1732 58 2 1 5 2
883 883 86 1 1999 278 178 28 2 1 2 4
884 884 126 1 1999 235 1248 52 1 1 4 4
885 885 160 1 1999 235 658 42 0 0 0 4
886 886 183 1 1999 235 236 30 0 0 0 4
887 887 NA 1 1999 194 3722 77 0 0 0 3
888 888 NA 1 1999 194 3594 70 0 0 0 3
889 889 NA 1 1999 194 1104 52 0 0 0 3
890 890 NA 1 1999 194 1368 54 0 0 0 3
891 891 NA 1 1999 194 1762 61 0 0 0 3
892 892 NA 1 1999 194 1240 52 0 0 0 3
893 893 NA 1 1999 194 668 42 0 0 0 3
894 894 NA 1 1999 194 2914 74 0 0 0 3
895 895 NA 1 1999 194 1756 55 0 0 0 3
896 896 NA 1 1999 194 1828 57 0 0 0 3
897 897 NA 1 1999 194 2720 66 0 0 0 3
898 898 NA 1 1999 194 1368 55 0 0 0 3
899 899 NA 1 1999 194 1834 60 0 0 0 3
900 900 NA 1 1999 194 420 36 0 0 0 3
901 901 NA 1 1999 194 2020 60 0 0 0 3
902 902 NA 1 1999 194 1886 61 0 0 0 3
903 903 NA 1 1999 194 3056 71 0 0 0 3
904 904 NA 1 1999 100 1744 61 2 1 6 2
905 905 NA 1 1999 100 1562 55 2 1 4 2
906 906 NA 1 1999 100 3002 68 1 1 5 2
907 907 NA 1 1999 100 1412 54 2 1 4 2
908 908 NA 1 1999 278 1320 50 1 1 4 4
909 909 NA 1 1999 278 332 36 1 1 2 4
910 910 NA 1 1999 278 1190 54 2 1 4 4
911 911 NA 1 1999 278 854 45 2 1 3 4
912 912 NA 1 1999 278 1140 50 2 1 4 4
913 913 NA 1 1999 278 984 49 2 1 4 4
914 914 NA 1 1999 278 652 42 1 1 4 4
915 915 NA 1 1999 278 580 42 2 1 3 4
916 916 NA 1 1999 278 476 39 1 1 3 4
917 917 NA 1 1999 278 360 35 2 1 2 4
918 918 NA 1 1999 110 1654 60 2 1 5 4
919 919 NA 1 1999 110 1786 61 1 1 5 4
920 920 NA 1 1999 110 912 48 1 1 3 4
921 921 NA 1 1999 110 2430 67 1 1 6 4
922 922 NA 1 1999 110 252 31 2 1 2 4
923 923 NA 1 1999 110 176 29 2 1 2 4
924 924 NA 1 1999 110 2136 62 1 1 5 4
925 925 NA 1 1999 110 916 49 2 1 3 4
926 926 NA 1 1999 110 526 39 2 1 3 4
927 927 NA 1 1999 110 354 36 2 1 3 4
928 928 NA 1 1999 110 418 36 1 1 3 4
929 929 NA 1 1999 110 356 35 1 1 3 4
930 930 NA 1 1999 110 670 42 2 1 3 4
931 931 NA 1 1999 110 816 45 2 1 3 4
932 932 NA 1 1999 110 324 35 1 1 2 4
933 933 NA 1 1999 110 234 32 2 1 2 4
934 934 NA 1 1999 110 206 31 2 1 2 4
935 935 NA 1 1999 278 2556 63 2 2 5 4
936 936 NA 1 1999 278 3172 68 1 2 6 4
937 937 NA 1 1999 278 2426 65 2 2 5 4
938 938 NA 1 1999 110 1380 52 1 2 5 4
939 939 NA 1 1999 110 1692 56 1 2 5 4
940 940 NA 1 1999 110 2160 60 1 2 5 4
941 941 NA 1 1999 110 1236 50 2 2 5 4
942 942 NA 1 1999 110 1608 53 2 2 5 4
943 943 NA 1 1999 110 1972 57 1 2 5 4
944 944 1 1 2000 121 950 49 1 1 3 1
945 945 1 1 2000 121 1672 62 1 1 5 1
946 946 1 1 2000 121 1280 51 1 1 3 1
947 947 1 1 2000 121 1278 52 1 1 4 1
948 948 1 1 2000 121 3836 78 1 1 6 1
949 949 1 1 2000 121 1072 51 1 1 6 1
950 950 1 1 2000 293 1908 62 1 1 5 2
951 951 1 1 2000 293 1034 51 2 1 3 2
952 952 1 1 2000 237 1548 59 2 1 6 2
953 953 1 1 2000 237 420 36 1 1 5 2
954 954 1 1 2000 285 1680 58 2 1 6 3
955 955 1 1 2000 285 1826 59 1 1 6 3
956 956 1 1 2000 285 2020 62 1 1 6 3
957 957 1 1 2000 285 1418 53 2 1 4 3
958 958 1 1 2000 285 1954 60 1 1 5 3
959 959 1 1 2000 285 1344 53 1 1 4 3
960 960 1 1 2000 285 954 49 2 1 3 3
961 961 1 1 2000 270 432 36 1 1 2 4
962 962 1 1 2000 260 1396 56 1 1 5 4
963 963 1 1 2000 121 2094 61 1 2 6 1
964 964 1 1 2000 121 3388 76 2 2 8 1
965 965 1 1 2000 102 2414 63 2 2 8 2
966 966 1 1 2000 293 4240 82 2 2 9 2
967 967 1 1 2000 237 3594 71 2 2 8 2
968 968 1 1 2000 285 2296 60 2 2 5 3
969 969 1 1 2000 285 2390 64 2 2 6 3
970 970 1 1 2000 285 2724 64 1 3 6 3
971 971 2 1 2000 121 1506 56 2 1 4 1
972 972 2 1 2000 121 6020 89 2 1 8 1
973 973 2 1 2000 102 940 45 2 1 3 2
974 974 2 1 2000 102 3892 69 2 1 7 2
975 975 2 1 2000 102 542 40 1 1 4 2
976 976 2 1 2000 285 2150 64 2 1 6 3
977 977 2 1 2000 285 1348 54 1 1 5 3
978 978 2 1 2000 285 1170 52 2 1 4 3
979 979 2 1 2000 270 2908 69 2 1 5 4
980 980 2 1 2000 270 554 40 2 1 3 4
981 981 2 1 2000 146 764 42 1 1 3 3
982 982 2 1 2000 121 2404 61 1 2 6 1
983 983 2 1 2000 121 2576 67 2 2 6 1
984 984 2 1 2000 121 2676 65 1 4 5 1
985 985 2 1 2000 146 2182 60 1 4 6 3
986 986 2 1 2000 146 1984 59 1 4 7 3
987 987 3 1 2000 121 2710 69 1 1 5 1
988 988 3 1 2000 102 1258 53 2 1 4 2
989 989 3 1 2000 285 2148 62 2 1 5 3
990 990 3 1 2000 285 3364 72 2 1 5 3
991 991 3 1 2000 270 3200 69 2 1 5 4
992 992 3 1 2000 260 990 51 1 1 5 4
993 993 3 1 2000 146 3258 73 1 1 7 3
994 994 3 1 2000 121 4310 74 1 2 6 1
995 995 3 1 2000 121 6600 92 1 2 7 1
996 996 3 1 2000 270 2162 60 2 2 7 4
997 997 3 1 2000 121 2582 61 2 3 5 1
998 998 3 1 2000 121 2778 62 1 4 6 1
999 999 3 1 2000 270 3152 69 1 4 6 4
1000 1000 3 1 2000 146 3200 66 2 4 7 3
1001 1001 3 1 2000 146 2236 64 1 4 4 3
1002 1002 4 1 2000 121 2234 63 1 1 5 1
1003 1003 4 1 2000 102 932 48 2 1 4 2
1004 1004 4 1 2000 237 1802 56 2 1 5 2
1005 1005 4 1 2000 237 870 47 2 1 4 2
1006 1006 4 1 2000 237 1660 55 1 1 4 2
1007 1007 4 1 2000 270 1748 58 2 1 4 4
1008 1008 4 1 2000 146 3372 72 2 1 5 3
1009 1009 4 1 2000 121 2160 59 1 2 6 1
1010 1010 4 1 2000 121 2736 62 1 2 5 1
1011 1011 4 1 2000 121 1286 52 2 2 6 1
1012 1012 4 1 2000 121 1652 55 1 4 6 1
1013 1013 4 1 2000 121 2032 59 1 4 6 1
1014 1014 4 1 2000 260 2804 68 1 4 5 4
1015 1015 5 1 2000 121 600 41 1 1 2 1
1016 1016 5 1 2000 285 3708 71 1 1 7 3
1017 1017 5 1 2000 270 472 38 2 1 3 4
1018 1018 5 1 2000 146 3622 78 2 1 8 3
1019 1019 5 1 2000 146 856 45 1 1 4 3
1020 1020 5 1 2000 121 2094 60 2 2 5 1
1021 1021 5 1 2000 121 4824 79 1 4 7 1
1022 1022 5 1 2000 260 3328 71 1 4 5 4
1023 1023 5 1 2000 146 2146 60 1 4 7 3
1024 1024 6 1 2000 102 902 45 2 1 3 2
1025 1025 6 1 2000 285 1052 51 2 1 4 3
1026 1026 6 1 2000 260 492 38 2 1 2 4
1027 1027 7 1 2000 260 828 46 1 1 3 4
1028 1028 7 1 2000 121 3012 66 2 2 6 1
1029 1029 7 1 2000 146 3002 70 1 4 5 3
1030 1030 8 1 2000 121 986 49 2 1 4 1
1031 1031 8 1 2000 121 988 49 1 1 4 1
1032 1032 8 1 2000 285 1952 60 1 1 5 3
1033 1033 8 1 2000 260 1936 58 2 1 4 4
1034 1034 8 1 2000 146 790 46 1 1 3 3
1035 1035 8 1 2000 146 682 43 2 1 3 3
1036 1036 8 1 2000 260 1712 55 2 2 4 4
1037 1037 8 1 2000 121 2762 65 1 4 6 1
1038 1038 9 1 2000 121 3040 70 1 1 6 1
1039 1039 9 1 2000 237 2490 66 1 1 8 2
1040 1040 9 1 2000 270 756 43 2 1 4 4
1041 1041 10 1 2000 121 726 43 1 1 3 1
1042 1042 10 1 2000 270 278 33 1 1 2 4
1043 1043 11 1 2000 260 1262 53 1 1 4 4
1044 1044 11 1 2000 260 442 35 2 1 3 4
1045 1045 12 1 2000 285 506 38 2 1 3 3
1046 1046 12 1 2000 285 686 44 2 1 3 3
1047 1047 12 1 2000 270 190 28 2 1 2 4
1048 1048 12 1 2000 260 444 35 1 1 3 4
1049 1049 13 1 2000 146 186 29 2 1 3 3
1050 1050 14 1 2000 260 2554 60 1 1 5 4
1051 1051 14 1 2000 121 2426 63 1 2 6 1
1052 1052 14 1 2000 121 2570 68 2 2 6 1
1053 1053 15 1 2000 121 1926 58 1 2 6 1
1054 1054 17 1 2000 260 2374 67 2 1 6 4
1055 1055 19 1 2000 121 1946 58 2 2 5 1
1056 1056 20 1 2000 121 1664 58 1 1 4 1
1057 1057 20 1 2000 260 2188 61 2 2 7 4
1058 1058 22 1 2000 270 1344 54 2 1 4 4
1059 1059 26 1 2000 285 1106 53 2 1 6 3
1060 1060 26 1 2000 270 1120 49 1 1 3 4
1061 1061 27 1 2000 270 1672 53 2 2 5 4
1062 1062 28 1 2000 260 2096 63 1 4 5 4
1063 1063 30 1 2000 260 3008 67 1 1 5 4
1064 1064 30 1 2000 260 418 38 2 1 4 4
1065 1065 30 1 2000 146 974 47 1 1 4 3
1066 1066 31 1 2000 146 700 43 2 1 3 3
1067 1067 31 1 2000 260 5658 80 1 2 6 4
1068 1068 35 1 2000 270 1304 51 2 1 4 4
1069 1069 39 1 2000 285 1430 57 2 1 6 3
1070 1070 40 1 2000 270 1432 53 1 1 5 4
1071 1071 43 1 2000 270 292 34 2 1 2 4
1072 1072 43 1 2000 146 6314 93 1 4 9 3
1073 1073 44 1 2000 270 1548 55 1 1 5 4
1074 1074 45 1 2000 285 1768 60 1 1 6 3
1075 1075 45 1 2000 260 1138 51 1 1 4 4
1076 1076 49 1 2000 270 1072 48 2 1 4 4
1077 1077 52 1 2000 270 1096 49 2 1 5 4
1078 1078 56 1 2000 146 198 29 2 1 2 3
1079 1079 56 1 2000 121 1488 54 2 2 5 1
1080 1080 58 1 2000 270 550 41 1 1 3 4
1081 1081 67 1 2000 102 1250 54 2 1 6 2
1082 1082 67 1 2000 260 2088 60 2 1 5 4
1083 1083 71 1 2000 146 234 30 2 1 2 3
1084 1084 81 1 2000 260 380 35 2 1 2 4
1085 1085 186 1 2000 146 92 22 1 1 1 3
1086 1086 210 1 2000 260 444 37 1 1 2 4
1087 1087 223 1 2000 270 434 36 2 1 3 4
1088 1088 1 1 2001 90 3754 72 1 1 5 1
1089 1089 1 1 2001 199 2472 67 2 1 6 2
1090 1090 1 1 2001 199 984 48 2 1 3 2
1091 1091 1 1 2001 199 2410 62 1 1 6 2
1092 1092 1 1 2001 199 3314 77 1 1 6 2
1093 1093 1 1 2001 199 1448 56 2 1 5 2
1094 1094 1 1 2001 199 3578 71 2 1 5 2
1095 1095 1 1 2001 98 1860 57 1 1 3 3
1096 1096 1 1 2001 220 3412 68 1 1 5 4
1097 1097 1 1 2001 125 2348 68 1 1 5 4
1098 1098 1 1 2001 125 3146 72 2 1 6 4
1099 1099 1 1 2001 140 836 46 1 1 3 4
1100 1100 1 1 2001 140 744 44 1 1 3 4
1101 1101 1 1 2001 140 620 43 2 1 3 4
1102 1102 1 1 2001 140 748 45 2 1 3 4
1103 1103 1 1 2001 140 938 51 1 1 3 4
1104 1104 1 1 2001 228 1558 58 1 1 5 4
1105 1105 1 1 2001 228 910 49 2 1 3 4
1106 1106 1 1 2001 228 600 40 1 1 3 4
1107 1107 1 1 2001 228 1104 52 1 1 4 4
1108 1108 1 1 2001 180 1940 59 1 1 5 3
1109 1109 1 1 2001 193 1266 56 2 1 5 3
1110 1110 1 1 2001 193 266 32 2 1 2 3
1111 1111 1 1 2001 193 414 35 2 1 2 3
1112 1112 1 1 2001 193 274 33 1 1 2 3
1113 1113 1 1 2001 122 1238 54 1 1 5 3
1114 1114 1 1 2001 122 1490 59 2 1 5 3
1115 1115 1 1 2001 122 1478 55 1 1 4 3
1116 1116 1 1 2001 122 2222 68 2 1 5 3
1117 1117 1 1 2001 122 2138 65 2 1 6 3
1118 1118 1 1 2001 122 1730 56 1 1 5 3
1119 1119 1 1 2001 256 1772 61 2 1 5 3
1120 1120 1 1 2001 276 1276 53 2 1 4 2
1121 1121 1 1 2001 276 2592 68 1 1 6 2
1122 1122 1 1 2001 256 2284 57 2 2 6 3
1123 1123 1 1 2001 98 1936 58 2 4 5 3
1124 1124 1 1 2001 125 6044 84 2 4 7 4
1125 1125 1 1 2001 125 6172 84 2 4 7 4
1126 1126 1 1 2001 140 5442 85 2 4 8 4
1127 1127 1 1 2001 140 4098 79 1 4 6 4
1128 1128 1 1 2001 140 3990 72 2 4 6 4
1129 1129 1 1 2001 256 1988 59 1 4 5 3
1130 1130 1 1 2001 276 5732 82 1 4 7 2
1131 1131 2 1 2001 112 252 29 2 1 2 1
1132 1132 2 1 2001 112 198 27 2 1 1 1
1133 1133 2 1 2001 199 3660 72 1 1 6 2
1134 1134 2 1 2001 199 1194 52 1 1 4 2
1135 1135 2 1 2001 220 1246 53 1 1 3 4
1136 1136 2 1 2001 125 1760 61 1 1 4 4
1137 1137 2 1 2001 228 570 42 1 1 3 4
1138 1138 2 1 2001 228 970 49 1 1 3 4
1139 1139 2 1 2001 193 1350 54 1 1 4 3
1140 1140 2 1 2001 122 2114 63 1 1 5 3
1141 1141 2 1 2001 122 1520 56 2 1 5 3
1142 1142 2 1 2001 256 1602 56 1 1 4 3
1143 1143 2 1 2001 256 1812 56 1 1 6 3
1144 1144 2 1 2001 260 968 49 1 1 3 2
1145 1145 2 1 2001 110 560 41 1 1 2 2
1146 1146 2 1 2001 110 1074 57 2 1 4 2
1147 1147 2 1 2001 140 2164 59 2 2 5 4
1148 1148 2 1 2001 125 4076 76 2 4 6 4
1149 1149 2 1 2001 140 NA NA 1 4 9 4
1150 1150 2 1 2001 122 2192 60 2 4 7 3
1151 1151 2 1 2001 256 2808 65 1 4 6 3
1152 1152 2 1 2001 276 7888 94 2 4 9 2
1153 1153 3 1 2001 112 398 34 2 1 2 1
1154 1154 3 1 2001 140 748 44 1 1 3 4
1155 1155 3 1 2001 140 302 34 2 1 2 4
1156 1156 3 1 2001 140 366 34 1 1 2 4
1157 1157 3 1 2001 228 768 45 2 1 4 4
1158 1158 3 1 2001 228 1636 58 1 1 5 4
1159 1159 3 1 2001 228 596 41 2 1 3 4
1160 1160 3 1 2001 180 778 44 1 1 3 3
1161 1161 3 1 2001 193 1334 52 2 1 4 3
1162 1162 3 1 2001 283 66 19 2 1 1 2
1163 1163 3 1 2001 110 1738 65 2 1 4 2
1164 1164 3 1 2001 112 4432 80 1 4 7 1
1165 1165 3 1 2001 140 2154 63 1 4 5 4
1166 1166 3 1 2001 140 1994 61 2 4 5 4
1167 1167 3 1 2001 228 4938 81 2 4 7 4
1168 1168 3 1 2001 256 3412 74 1 4 5 3
1169 1169 4 1 2001 220 1094 52 1 1 4 4
1170 1170 4 1 2001 220 1042 50 2 1 5 4
1171 1171 4 1 2001 220 1588 58 1 1 4 4
1172 1172 4 1 2001 140 444 38 1 1 2 4
1173 1173 4 1 2001 140 760 44 1 1 4 4
1174 1174 4 1 2001 228 546 39 2 1 3 4
1175 1175 4 1 2001 180 886 50 2 1 5 3
1176 1176 4 1 2001 122 1060 49 1 1 4 3
1177 1177 4 1 2001 260 1476 55 2 1 4 2
1178 1178 4 1 2001 228 1820 56 2 2 5 4
1179 1179 4 1 2001 110 4494 78 2 2 6 2
1180 1180 4 1 2001 140 6478 92 1 4 9 4
1181 1181 4 1 2001 180 3074 64 2 4 7 3
1182 1182 4 1 2001 122 2286 63 1 4 7 3
1183 1183 5 1 2001 220 2556 60 1 1 5 4
1184 1184 5 1 2001 140 480 39 1 1 3 4
1185 1185 5 1 2001 228 3620 74 1 1 5 4
1186 1186 5 1 2001 228 1718 59 2 1 6 4
1187 1187 5 1 2001 122 636 44 2 1 3 3
1188 1188 5 1 2001 256 1938 64 2 1 6 3
1189 1189 5 1 2001 283 2650 65 1 1 5 2
1190 1190 5 1 2001 140 2306 60 2 2 7 4
1191 1191 5 1 2001 283 3244 71 2 2 6 2
1192 1192 5 1 2001 140 2604 63 1 4 8 4
1193 1193 6 1 2001 125 2056 63 1 1 4 4
1194 1194 6 1 2001 228 556 41 2 1 3 4
1195 1195 6 1 2001 122 1392 53 1 1 4 3
1196 1196 6 1 2001 276 1252 51 1 1 3 2
1197 1197 6 1 2001 276 366 36 2 1 3 2
1198 1198 6 1 2001 260 1594 53 2 2 4 2
1199 1199 6 1 2001 112 4286 78 2 4 6 1
1200 1200 6 1 2001 199 3716 78 2 4 7 2
1201 1201 7 1 2001 140 1014 47 1 1 4 4
1202 1202 7 1 2001 228 388 37 1 1 3 4
1203 1203 7 1 2001 276 2708 65 1 1 6 2
1204 1204 8 1 2001 199 2640 67 1 1 6 2
1205 1205 8 1 2001 228 416 38 1 1 4 4
1206 1206 8 1 2001 122 2754 70 1 1 6 3
1207 1207 8 1 2001 256 1360 54 1 1 4 3
1208 1208 8 1 2001 256 746 42 2 1 3 3
1209 1209 8 1 2001 110 786 45 1 1 3 2
1210 1210 8 1 2001 140 3398 67 1 2 7 4
1211 1211 8 1 2001 140 2952 67 2 2 6 4
1212 1212 9 1 2001 228 1688 62 2 1 5 4
1213 1213 9 1 2001 193 386 37 1 1 3 3
1214 1214 9 1 2001 283 2916 67 2 4 7 2
1215 1215 10 1 2001 140 504 39 2 1 2 4
1216 1216 10 1 2001 228 698 43 2 1 3 4
1217 1217 10 1 2001 228 2310 59 2 1 4 4
1218 1218 10 1 2001 228 798 46 2 1 3 4
1219 1219 10 1 2001 193 446 39 2 1 3 3
1220 1220 11 1 2001 256 902 49 2 1 4 3
1221 1221 13 1 2001 125 1828 64 2 1 6 4
1222 1222 13 1 2001 140 718 43 1 1 3 4
1223 1223 13 1 2001 228 382 37 2 1 3 4
1224 1224 14 1 2001 112 560 39 1 1 2 1
1225 1225 14 1 2001 256 444 36 2 1 3 3
1226 1226 14 1 2001 140 2602 65 1 4 7 4
1227 1227 16 1 2001 260 3566 71 1 4 7 2
1228 1228 17 1 2001 140 1542 55 2 1 4 4
1229 1229 17 1 2001 122 146 26 2 1 2 3
1230 1230 18 1 2001 228 692 42 2 1 3 4
1231 1231 18 1 2001 260 652 43 1 1 3 2
1232 1232 18 1 2001 256 2514 64 1 2 7 3
1233 1233 19 1 2001 260 2246 67 1 1 5 2
1234 1234 20 1 2001 228 244 31 1 1 2 4
1235 1235 20 1 2001 140 5900 84 1 4 9 4
1236 1236 30 1 2001 228 1004 48 2 1 3 4
1237 1237 34 1 2001 140 1408 52 2 2 6 4
1238 1238 36 1 2001 140 2052 60 2 2 6 4
1239 1239 40 1 2001 256 1962 61 2 1 5 3
1240 1240 42 1 2001 228 372 34 1 1 4 4
1241 1241 45 1 2001 256 1472 53 2 4 5 3
1242 1242 46 1 2001 220 660 43 1 1 3 4
1243 1243 46 1 2001 122 256 32 2 1 2 3
1244 1244 49 1 2001 140 1558 55 1 4 5 4
1245 1245 50 1 2001 228 484 38 2 1 3 4
1246 1246 55 1 2001 228 704 45 1 1 4 4
1247 1247 75 1 2001 140 884 46 2 1 3 4
1248 1248 84 1 2001 260 910 48 2 1 4 2
1249 1249 89 1 2001 260 1414 56 1 1 6 2
1250 1250 90 1 2001 228 224 31 1 1 2 4
1251 1251 104 1 2001 140 690 43 2 1 3 4
1252 1252 125 1 2001 140 754 44 2 1 3 4
1253 1253 128 1 2001 140 1270 55 2 4 7 4
1254 1254 257 1 2001 228 370 35 2 1 3 4Model
Call:
glm(formula = Prevalence ~ Length + Year + Area, family = binomial(link = "logit"),
data = cod)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.465947 0.269333 -1.730 0.083629 .
Length 0.009654 0.004468 2.161 0.030705 *
Year2000 0.566536 0.169715 3.338 0.000843 ***
Year2001 -0.680315 0.140175 -4.853 1.21e-06 ***
Area2 -0.626192 0.186617 -3.355 0.000792 ***
Area3 -0.510470 0.163396 -3.124 0.001783 **
Area4 1.233878 0.184652 6.682 2.35e-11 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 1727.8 on 1247 degrees of freedom
Residual deviance: 1537.6 on 1241 degrees of freedom
(6 observations deleted due to missingness)
AIC: 1551.6
Number of Fisher Scoring iterations: 4These are all from a linear model. Need to transform them to get the real value in probabilistic scale
Additive! One slope
How to estimate the slopes?
Call:
glm(formula = Prevalence ~ Length + Year + Area, family = binomial(link = "logit"),
data = cod)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.465947 0.269333 -1.730 0.083629 .
Length 0.009654 0.004468 2.161 0.030705 *
Year2000 0.566536 0.169715 3.338 0.000843 ***
Year2001 -0.680315 0.140175 -4.853 1.21e-06 ***
Area2 -0.626192 0.186617 -3.355 0.000792 ***
Area3 -0.510470 0.163396 -3.124 0.001783 **
Area4 1.233878 0.184652 6.682 2.35e-11 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 1727.8 on 1247 degrees of freedom
Residual deviance: 1537.6 on 1241 degrees of freedom
(6 observations deleted due to missingness)
AIC: 1551.6
Number of Fisher Scoring iterations: 4\[ logistic(\lambda_i) = \beta_0 + \beta_1 x_{1,i}+ \beta_2 x_{2,i} + \beta_3 x_{3,i}+ \beta_4 x_{4,i} + \beta_5 x_{5,i} + \beta_6 x_{6,i} \]
Site
| Value of \(x_2\) | Value of \(x_3\) | Value of \(x_4\) | Value of \(x_5\) | Value of \(x_6\) | Value of \(x_7\) | |
|---|---|---|---|---|---|---|
| 2000 | 2001 | Area 1 | Area 2 | Area 3 | Area 4 | |
| Area-1 1999 | 0 | 0 | 0 | 0 | 0 | 0 |
| Area-1 2000 | 1 | 0 | 0 | 0 | 0 | 0 |
| Area-1 2001 | 0 | 1 | 0 | 0 | 0 | 0 |
| Area-2 1999 | 0 | 0 | 0 | 1 | 0 | 0 |
| Area-2 2000 | 0 | 1 | 0 | 1 | 0 | 0 |
| Area-2 2001 | 0 | 0 | 1 | 1 | 0 | 0 |
| Area 3- 1999 | 0 | 0 | 0 | 0 | 1 | 0 |
| Area 3 -2000 | 1 | 0 | 0 | 0 | 1 | 0 |
| Area 3 - 2001 | 0 | 1 | 0 | 0 | 1 | 0 |
| Area 4 - 1999 | 0 | 0 | 0 | 0 | 0 | 1 |
| Area 4 - 2000 | 0 | 1 | 0 | 0 | 0 | 1 |
| Area 4 2001 | 0 | 0 | 1 | 0 | 0 | 1 |
Categorical
This week’s lab
Create your own assignment
I have been reviewing assignments, and I will meet with some of you to go over some concepts
Next week: repeated measures, pseudoreplication, mixed models
Logistic regression assumptions
Binomial distribution
Binary data (1, 0)
Poisson assumptions
We have talked about count-data
While we usually think Poisson, it has an important assumption: \(Var(y_i) = \lambda\_i\)
How to check for overdispersion?
Package: performance
What to do when there is overdispersion?
I have never encountered non-overdispersed count data!
So, what to do?
Add more covariates
Make a mixed effects model
Zero inflated models!
What to do when there is overdispersion?
If we suspect zero inflation, we have several options for zero inflated models
Negative binomial regression
Hurdle models
Zero-inflated mixture model (mixed-effects)
GLM’s other assumptions
Data is independently distributed
Error is independent
Next classes
Mixed effects models
Repeated measures
zero inflation
Other models
Question 1
Interested in the concentration of mercury in Lake Michigan
We use code to randomly select 4 sites
At each site we set up a net
Return later and test Hg concentration of each fish
Question 1:
What is the experimental unit?
What is the response variable?
Where is sampling coming from or being introduced?
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What do we care about?
Question 2:
We are testing infection prevalence in turkeys 🦃
We trap multiple individuals from a flock, and test them all 🦃
We do this for multiple flocks
What is introducing variance?
Question 3
We are testing the effects of a new drug on mice
You have 100 🐁. And three groups: control, dose 1 and dose 2
You measure their heartbeat pre-trial, 5 weeks post trial, and 10 weeks post trial
Do we care about individual 🐭 or the general effect of the drug?
Question 4
Each color represents a different feeding treatment.
Samples are taken daily
Experiments
- Fish and Hg. Research question: Should we limit consumption of fish over a certain size
- Infection prevalence in turkeys: Research question: You are estimating the probability of an individual being infected to be used as a section on a demographic model?
- Mice. Research question:Is the drug affecting the mice heart rate?
- Salmon: Does energy availability (food), affects the activity of the ATPase enzime?
Experiments : 🎣 🦃 🐁🐟
For each experiment answer:
How would you analyze the data
Are they breaking any assumptions?
What is the individual unit, and what is the experimental unit
What is the response variable
Where is the variance coming from?
How to analyze them
There are many ways:
Split plot or repeated measures
Mixed effects models
Split plot or repeated measures
Very simple to do
Extension of the linear models
Mouse example: we have treatment, time, interaction. Error (among individuals) (within individuals)
HOWEVER: Equations for repeated measures are balanced! What happens if there are mortalities? Or sampling doesn’t allow to have balance
Difficult to add structure
Linear mixed models
Linear mixed models have become the more popular method for data that is autocorrelated (non-independent)
- They can have autocorrelation or not
So far we have looked at linear models (or glms)
\[ \underbrace{E[y_i]}_{\text{expected value}} = \underbrace{\beta_0 + \beta_1x_{1,i} + ... \beta_mx_{m,i}}_{deterministic} \]
\[ y_i \sim \underbrace{N(mean=E[y_i], var=\sigma^2)}_{stochastic} \]
Same as:
\[ y_i = \underbrace{\beta_0 + \beta_1x_{1,i} + ... \beta_mx_{m,i}}_{deterministic} \ + \underbrace{\epsilon}_{stachastic} \]
\[\epsilon \sim nomral(0,\sigma) \]
linear model
\[ \underbrace{E[y_i]}_{\text{expected value}} = \underbrace{\beta_0 + \beta_1x_{1,i} + ... \beta_mx_{m,i}}_{deterministic} \]
\[ y_i \sim \underbrace{N(mean=E[y_i], var=\sigma^2)}_{stochastic} \]
Glm
\[ \underbrace{log(\lambda)}_{\text{link function}} = \underbrace{\beta_0 + \beta_1x_{1,i} + ... \beta_mx_{m,i}}_{deterministic} \]
\[ y_i \sim \underbrace{Poisson(\lambda)}_{stochastic} \]
Mixed model
In a mixed model we introduce stochastic effects to the “deterministic component”.
In this case:
\[ \underbrace{E[y_i]}_{\text{expected value}} = \underbrace{\beta_0 + \beta_1x_{1,i} + ... beta_mx_{m,i}}_{deterministic} \]
\[ y_i \sim \underbrace{N(mean=E[y_i], var=\sigma^2)}_{stochastic} \]
Same as:
\[ y_i \sim \underbrace{\beta_0 + \beta_1x_{1,i} + ... \beta_mx_{m,i}}_{deterministic} \ + \underbrace{\epsilon}_{stachastic} \]
They all are deterministic (are a single value!)
Mixed model
When we talk about the stochastic portion of a model:
- \[ y_i \sim \underbrace{\beta_0 + \beta_1x_{1,i} + ... \beta_mx_{m,i}}_{deterministic} \ + \underbrace{\epsilon}_{stachastic} \]
- What is introducing the variance?
- Is the variance (or observation error) independent?
- In a mixed model, what is introducing the variance?
- 🎣 🦃 🐁🐟
Mixed model
Think… how do we introduce that variance?
\[ y_i \sim \underbrace{\beta_0 + \beta_1x_{1,i} + ... \beta_mx_{m,i}}_{deterministic} \ + \underbrace{\epsilon}_{stachastic} \]
Think for a moment on this specific question: 🐟
\[ y_i \sim \underbrace{\beta_0 + \beta_1x_{1,i}}_{deterministic} + \underbrace{\epsilon}_{stachastic} \]
\[ Hg_i \sim \underbrace{\beta_0 + \beta_1size_{i}}_{deterministic} + \underbrace{\epsilon}_{stachastic} \]
\[ \epsilon \sim N(0,\sigma)\]
The variance can affect the slope, the intercept, or both
What is the problem with this? We can actually do a linear model
- But remember! there might be some effect of the locations where nets were placed
Plotting the data as a linear model
- You might think this is all good… but
So… we do a mixed model! Where we introduce the variance from the site
But… why don’t we simply add site as a covariate?
You will be tasked with answering this on your next assignment!
The variance can affect the slope, the intercept, or both
Random effects introduce variance.
It can introduce varianceto the intercept or to the slope
Mixed model
\[ Hg_i \sim \beta_0 + \beta_1size_{i} + \epsilon \]
i individuals, j sites (4)
\[ Hg_i \sim \underbrace{(\beta_0 +\underbrace{\gamma_j}_{\text{Random intercept}})}_{intercept} + \underbrace{\beta_1size_{i}}_{slope} +\underbrace{\epsilon}_\text{ind var} \]
Variance comes from random “selection” of fish within a net
Variance comes from random “selection” of sites
Mixed model
This is the result of a mixed model:
Mixed effects model output
Family: gaussian ( identity )
Formula: Hg ~ size + (1 | region)
Data: HgDat_df
AIC BIC logLik deviance df.resid
-136.0 -126.8 72.0 -144.0 70
Random effects:
Conditional model:
Groups Name Variance Std.Dev.
region (Intercept) 0.033167 0.18212
Residual 0.006538 0.08086
Number of obs: 74, groups: region, 4
Dispersion estimate for gaussian family (sigma^2): 0.00654
Conditional model:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.2131568 0.0975371 2.185 0.0289 *
size 0.0188170 0.0008715 21.591 <2e-16 ***
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Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Output