Linear Regression- Day 2
Heather Harvey
February 22, 2019
We will study linear regression with a database from the
library(yarrr)To estimate the beta weights of a linear model in R we use the lm() function. The function has two key arguments: formula and data.
- What are the names of the variables? And what kind of data available in the data set (integer or decimal etc.)
names(diamonds)## [1] "weight" "clarity" "color" "value"str(diamonds)## 'data.frame': 150 obs. of 4 variables:
## $ weight : num 9.35 11.1 8.65 10.43 10.62 ...
## $ clarity: num 0.88 1.05 0.85 1.15 0.92 0.44 1.09 1.43 0.95 1.05 ...
## $ color : num 4 5 6 5 5 4 6 4 6 5 ...
## $ value : num 182 191 176 195 182 ...# The values are in decimal formatCan you view the data set in HTML document? If not, can you print a few data sample?
Yes, the data can be viewed in HTML.
The linear model will estimate each diamonds value using the following equation: \[ \text{Diamond value =} \beta_(intercept) + \beta_1 * weight + \beta_2* clarity + \beta_3 * color\] Here is the R-command for that formula:
diamonds.lm <- lm(formula = value ~ weight + clarity + color, data=diamonds)- Print the above output and understand the coefficients.
diamonds.lm##
## Call:
## lm(formula = value ~ weight + clarity + color, data = diamonds)
##
## Coefficients:
## (Intercept) weight clarity color
## 148.3354 2.1894 21.6922 -0.4549- Print the summary of linear model
summary(diamonds.lm)##
## Call:
## lm(formula = value ~ weight + clarity + color, data = diamonds)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.4046 -3.5473 -0.1134 3.2552 11.0464
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 148.3354 3.6253 40.917 <2e-16 ***
## weight 2.1894 0.2000 10.948 <2e-16 ***
## clarity 21.6922 2.1429 10.123 <2e-16 ***
## color -0.4549 0.3646 -1.248 0.214
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.672 on 146 degrees of freedom
## Multiple R-squared: 0.6373, Adjusted R-squared: 0.6298
## F-statistic: 85.49 on 3 and 146 DF, p-value: < 2.2e-16- Calculate the fitted values of linear model diamonds.lm
fitted(diamonds.lm)## 1 2 3 4 5 6 7 8
## 186.0758 193.1401 182.9826 193.8424 189.2692 183.0995 193.8593 198.9261
## 9 10 11 12 13 14 15 16
## 192.1142 189.7465 192.1629 188.3562 190.3111 175.8357 183.5867 193.1738
## 17 18 19 20 21 22 23 24
## 182.4543 183.3600 186.0564 172.1894 190.8769 191.2587 191.9566 187.3536
## 25 26 27 28 29 30 31 32
## 195.0996 189.1340 191.6832 188.9047 198.0162 187.7150 197.4707 186.7516
## 33 34 35 36 37 38 39 40
## 194.0486 182.9471 197.6990 181.8749 189.0689 192.3530 181.4667 199.4478
## 41 42 43 44 45 46 47 48
## 192.6166 195.5992 186.2338 188.6963 180.4489 191.4205 188.6512 195.4749
## 49 50 51 52 53 54 55 56
## 186.4460 182.0177 183.9607 182.3691 194.3771 179.6129 198.1373 189.8638
## 57 58 59 60 61 62 63 64
## 189.4808 195.4418 195.2050 187.5684 195.2332 174.1453 188.9927 190.8006
## 65 66 67 68 69 70 71 72
## 197.1481 193.9149 187.2343 188.9602 191.7916 196.3235 202.1371 197.8254
## 73 74 75 76 77 78 79 80
## 194.2708 192.7675 179.0534 182.9247 196.1669 185.3859 193.3181 187.2589
## 81 82 83 84 85 86 87 88
## 201.0018 198.4333 184.3770 192.0118 183.3099 189.2979 190.9494 193.6273
## 89 90 91 92 93 94 95 96
## 196.9222 196.0434 198.0521 186.6527 178.9600 187.5789 190.2013 183.8228
## 97 98 99 100 101 102 103 104
## 181.8627 196.5341 194.6557 182.6076 189.7448 186.4550 203.3765 193.2738
## 105 106 107 108 109 110 111 112
## 187.7032 184.5395 190.0623 183.7670 182.1457 196.8296 186.3046 183.5932
## 113 114 115 116 117 118 119 120
## 196.1479 193.8122 201.6535 189.7461 187.3012 186.4676 189.2750 189.6210
## 121 122 123 124 125 126 127 128
## 190.4658 186.7303 176.4423 188.1299 187.0176 187.1431 187.2087 183.3231
## 129 130 131 132 133 134 135 136
## 196.9590 177.9258 181.6754 180.9373 190.5306 186.5017 198.1243 175.8418
## 137 138 139 140 141 142 143 144
## 195.7068 202.0438 190.5316 186.6816 183.6889 194.2321 182.3883 192.3458
## 145 146 147 148 149 150
## 194.7501 190.5253 189.2768 190.8429 187.7496 186.6139- Add one column to the dataset “diamonds” with a column-name “value.lm” and the entries will be the fitted values.
diamonds$value.lm <- diamonds.lm$fitted.values- Check the new dataset whether you inserted the new column correctly by printing a few rows of data.
diamonds.lm##
## Call:
## lm(formula = value ~ weight + clarity + color, data = diamonds)
##
## Coefficients:
## (Intercept) weight clarity color
## 148.3354 2.1894 21.6922 -0.4549- Plot the relationship between the “true value” (provided in the data) and linear model fitted values.
plot(x=diamonds$value, # True values on x-axis
y=diamonds.lm$fitted.values) # fitted values on y-axis
- Add x-label as true dieamond values, y-label as linear model fitted values and title as Regression fits of diamond values.
plot(x=diamonds$value, y=diamonds.lm$fitted.values, xlab = "True Diamond Values", ylab = "Linear Model Fitted Values", main= "Regression Fits of Diamond values")
- Create a straight line with following command
plot(x=diamonds$value, y=diamonds.lm$fitted.values, xlab = "True Diamond Values", ylab = "Linear Model Fitted Values", main= "Regression Fits of Diamond values")
abline(b=1,a=0, col = "red")
Explain what are these “a” and “b” presenting.
“a” represents where the line will cross the y-axis, and “b” is the slope of the line.
Finally, how do you relate/ interpret the fit-plot versus the straight line?
The straight line goes about directly though the center of most of the data, almost line a line of regression