Line of Best Fit
Model of a linear association between \(x\) and \(y\):
Actual (observed) value: \(y\)
Predicted value: \(\widehat{y} = {\color{#44ee44}b_0} + {\color{#44ee44}b_1} x\)
Prediction error (residual): \(e = y - \widehat{y}\)
Observed value again: \(y = \widehat{y} + e = {\color{#44ee44}b_0} + {\color{#44ee44}b_1} x + e\)
How do we find \(b_0\) and \(b_1\)?
We want the prediction errors or residuals to be as small as possible.
There is one residual for each observation.
Some are positive while others are negative.
Try to make the sum of squares of residuals as small as possible.
What we need:
\[b_1 = \frac{s_y}{s_x}\cdot r\]
\[b_0 = \overline{y} - b_1\cdot\overline{x}\]
\[\widehat{y} - \overline{y} = b_1\cdot \left(x - \overline{x}\right)\]
What we need:
\[ \begin{aligned} b_1 &= \frac{s_y}{s_x}\cdot r\\[1.3ex] &= \frac{5.53}{18.22}\cdot 0.47\\[1.3ex] &= 0.14 \end{aligned} \]
\[ \begin{aligned} b_0 &= \overline{y} - b_1\cdot\overline{x}\\[1.3ex] &= 72.8 - 0.14\cdot 53.7\\[1.3ex] &= 65.28 \end{aligned} \]
\[\widehat{y} = 65.28 + 0.14 x\]
glimpse(sowc_sample)Rows: 20
Columns: 3
$ countries_and_areas <chr> "Guatemala", "Trinidad and Tobago", "Philippines"…
$ percent_urban_2018 <int> 51, 53, 47, 27, 67, 31, 43, 91, 48, 41, 71, 63, 5…
$ life_expectancy_2018 <int> 74, 73, 71, 71, 81, 79, 72, 74, 77, 75, 77, 75, 7…Question: How do we predict life expectancy from percent urban population?
gf_point(life_expectancy_2018 ~ percent_urban_2018, data = sowc_sample)
cor(life_expectancy_2018 ~ percent_urban_2018, data = sowc_sample)[1] 0.4652268
model <- lm(life_expectancy_2018 ~ percent_urban_2018, data = sowc_sample)
model
Call:
lm(formula = life_expectancy_2018 ~ percent_urban_2018, data = sowc_sample)
Coefficients:
(Intercept) percent_urban_2018
65.2169 0.1412 summary(model)
Call:
lm(formula = life_expectancy_2018 ~ percent_urban_2018, data = sowc_sample)
Residuals:
Min 1Q Median 3Q Max
-10.148 -4.305 1.234 2.330 9.405
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 65.21686 3.58200 18.21 4.84e-13 ***
percent_urban_2018 0.14121 0.06333 2.23 0.0387 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.03 on 18 degrees of freedom
Multiple R-squared: 0.2164, Adjusted R-squared: 0.1729
F-statistic: 4.972 on 1 and 18 DF, p-value: 0.03873model$residuals 1 2 3 4 5 6
1.5812752 0.2988491 -0.8538727 1.9703877 6.3218668 9.4055357
7 8 9 10 11 12
0.7109794 -4.0672458 5.0049142 3.9934054 1.7570147 0.8867189
13 14 15 16 17 18
1.7339970 -5.1017723 -5.3120383 -7.8423639 2.3218668 -10.1478076
19 20
2.3563933 -5.0181034
model$fitted.values 1 2 3 4 5 6 7 8
72.41872 72.70115 71.85387 69.02961 74.67813 69.59446 71.28902 78.06725
9 10 11 12 13 14 15 16
71.99509 71.00659 75.24299 74.11328 73.26600 75.10177 69.31204 72.84236
17 18 19 20
74.67813 71.14781 77.64361 70.01810 model$fitted.values + model$residuals 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
74 73 71 71 81 79 72 74 77 75 77 75 75 70 64 65 77 61 80 65
sowc_sample$life_expectancy_2018 [1] 74 73 71 71 81 79 72 74 77 75 77 75 75 70 64 65 77 61 80 65
model <- lm(life_expectancy_2018 ~ percent_urban_2018, data = sowc_sample)
model
Call:
lm(formula = life_expectancy_2018 ~ percent_urban_2018, data = sowc_sample)
Coefficients:
(Intercept) percent_urban_2018
65.2169 0.1412
\(\widehat{\text{life_expectancy_2018}} = 65.217 + 0.141 \times \text{percent_urban_2018}\)
The complete equation:
\[\text{life_expectancy_2018} = {\color{#44ee44} 65.217 + 0.141 \times\text{percent_urban_2018}} {\color{#F9BA02} + e}\]
Association with urbanization (the model)
Other variables unrelated to urbanization (the residuals)
Variation associated with urbanization (the model)
Variation associated with other variables unrelated to urbanization (the residuals)
\[\begin{gather} \text{Variation associated with urbanization}\\ +\\ \text{Variation unrelated to urbanization} \end{gather}\]
Question: What portion of the variation in life expectancy is the variation associated with urbanization?
What portion of the variation in live expectancy is explained by the model?
Question: What fraction of the total variance of life expectancy is explained by the model?
\[\frac{\text{Variance of predicted values}}{\text{Variance of life_expectancy_2018}}\]
Coefficient of Determination \(R^2\)
\[1 - \frac{\text{Variance of residuals}}{\text{Variance of life_expectancy_2018}}\]
summary(model)
Call:
lm(formula = life_expectancy_2018 ~ percent_urban_2018, data = sowc_sample)
Residuals:
Min 1Q Median 3Q Max
-10.148 -4.305 1.234 2.330 9.405
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 65.21686 3.58200 18.21 4.84e-13 ***
percent_urban_2018 0.14121 0.06333 2.23 0.0387 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.03 on 18 degrees of freedom
Multiple R-squared: 0.2164, Adjusted R-squared: 0.1729
F-statistic: 4.972 on 1 and 18 DF, p-value: 0.03873glimpse(sowc_sample)Rows: 20
Columns: 5
$ percent_urban_2018 <int> 51, 53, 47, 27, 67, 31, 43, 91, 48, 41, 71,…
$ pop_urban_growth_rate_2018 <dbl> 2.8, 0.2, 1.8, 2.2, 1.1, -0.2, 2.0, 4.5, 0.…
$ total_pop_2018 <int> 17248, 1390, 106651, 9101, 1189, 287, 98424…
$ life_expectancy_2000 <int> 68, 69, 69, 62, 78, 77, 69, 72, 74, 71, 72,…
$ life_expectancy_2018 <int> 74, 73, 71, 71, 81, 79, 72, 74, 77, 75, 77,…for life_expectancy_2018:
Predicting from percent_urban_2018:
model1 = lm(life_expectancy_2018 ~ percent_urban_2018, data = sowc_sample)Predicting from pop_urban_growth_rate_2018:
model2 = lm(life_expectancy_2018 ~ pop_urban_growth_rate_2018, data = sowc_sample)Predicting from total_pop_2018:
model3 = lm(life_expectancy_2018 ~ total_pop_2018, data = sowc_sample)Predicting from life_expectancy_2000:
model4 = lm(life_expectancy_2018 ~ life_expectancy_2000, data = sowc_sample)summary(model1)
Call:
lm(formula = life_expectancy_2018 ~ percent_urban_2018, data = sowc_sample)
Residuals:
Min 1Q Median 3Q Max
-10.148 -4.305 1.234 2.330 9.405
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 65.21686 3.58200 18.21 4.84e-13 ***
percent_urban_2018 0.14121 0.06333 2.23 0.0387 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.03 on 18 degrees of freedom
Multiple R-squared: 0.2164, Adjusted R-squared: 0.1729
F-statistic: 4.972 on 1 and 18 DF, p-value: 0.03873summary(model2)
Call:
lm(formula = life_expectancy_2018 ~ pop_urban_growth_rate_2018,
data = sowc_sample)
Residuals:
Min 1Q Median 3Q Max
-7.6117 -1.9358 -0.7584 2.1588 6.6903
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 77.0751 1.4092 54.695 < 2e-16 ***
pop_urban_growth_rate_2018 -2.1701 0.5406 -4.015 0.000813 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.127 on 18 degrees of freedom
Multiple R-squared: 0.4724, Adjusted R-squared: 0.4431
F-statistic: 16.12 on 1 and 18 DF, p-value: 0.000813summary(model3)
Call:
lm(formula = life_expectancy_2018 ~ total_pop_2018, data = sowc_sample)
Residuals:
Min 1Q Median 3Q Max
-12.246 -2.135 1.464 3.509 7.505
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.354e+01 1.480e+00 49.701 <2e-16 ***
total_pop_2018 -3.712e-05 4.039e-05 -0.919 0.37
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.554 on 18 degrees of freedom
Multiple R-squared: 0.04481, Adjusted R-squared: -0.008251
F-statistic: 0.8445 on 1 and 18 DF, p-value: 0.3703summary(model4)
Call:
lm(formula = life_expectancy_2018 ~ life_expectancy_2000, data = sowc_sample)
Residuals:
Min 1Q Median 3Q Max
-3.3240 -1.3480 0.2308 1.3781 4.2515
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 25.11782 4.17523 6.016 1.09e-05 ***
life_expectancy_2000 0.69864 0.06083 11.484 1.02e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.969 on 18 degrees of freedom
Multiple R-squared: 0.8799, Adjusted R-squared: 0.8732
F-statistic: 131.9 on 1 and 18 DF, p-value: 1.02e-09
Call:
lm(formula = life_expectancy_2018 ~ percent_urban_2018 + pop_urban_growth_rate_2018,
data = sowc_sample)
Residuals:
Min 1Q Median 3Q Max
-6.0532 -2.0916 0.4063 1.9666 4.8125
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 69.64133 2.52931 27.534 1.53e-15 ***
percent_urban_2018 0.13748 0.04182 3.288 0.004343 **
pop_urban_growth_rate_2018 -2.14419 0.43496 -4.930 0.000127 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.321 on 17 degrees of freedom
Multiple R-squared: 0.6775, Adjusted R-squared: 0.6395
F-statistic: 17.85 on 2 and 17 DF, p-value: 6.649e-05