Linear model with categorical variables in R
I am trying to fit a lineal model with some categorical variables
model <- lm(price ~ carat+cut+color+clarity)
summary(model)
The answer is:
Call:
lm(formula = price ~ carat + cut + color + clarity)
Residuals:
Min 1Q Median 3Q Max
-11495.7 -688.5 -204.1 458.2 9305.3
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -3696.818 47.948 -77.100 < 2e-16 ***
carat 8843.877 40.885 216.311 < 2e-16 ***
cut.L 755.474 68.378 11.049 < 2e-16 ***
cut.Q -349.587 60.432 -5.785 7.74e-09 ***
cut.C 200.008 52.260 3.827 0.000131 ***
cut^4 12.748 42.642 0.299 0.764994
color.L 1905.109 61.050 31.206 < 2e-16 ***
color.Q -675.265 56.056 -12.046 < 2e-16 ***
color.C 197.903 51.932 3.811 0.000140 ***
color^4 71.054 46.940 1.514 0.130165
color^5 2.867 44.586 0.064 0.948729
color^6 50.531 40.771 1.239 0.215268
clarity.L 4045.728 108.363 37.335 < 2e-16 ***
clarity.Q -1545.178 102.668 -15.050 < 2e-16 ***
clarity.C 999.911 88.301 11.324 < 2e-16 ***
clarity^4 -665.130 66.212 -10.045 < 2e-16 ***
clarity^5 920.987 55.012 16.742 < 2e-16 ***
clarity^6 -712.168 52.346 -13.605 < 2e-16 ***
clarity^7 1008.604 45.842 22.002 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1167 on 4639 degrees of freedom
Multiple R-squared: 0.9162, Adjusted R-squared: 0.9159
F-statistic: 2817 on 18 and 4639 DF, p-value: < 2.2e-16
But I don't understand why the answers are with ".L,.Q,.C,^4, ...", something is wrong but I don't know what is wrong, I already tried with the function factor for each variable.
Answer
You are encountering how “ordered” ( ordinal ) factor variables are handled by regression functions and the default set of contrasts are orthogonal polynomial contrasts up to degree n-1, where n is the number of levels for that factor. It's not going to be very easy to interpret that result ... especially if there is no natural order. Even if there is, and there might well be in this case, you might not want the default ordering (which is alphabetical by factor level) and you probably don't want to have more than a few of degrees in the polynomial contrasts.
In the case of ggplot2's diamonds dataset, the factor levels are set up correctly but most newbies when they stumble across ordered factors get ordered levels like "Excellent" <"Fair" < "Good"< "Poor". (Fail)
> levels(diamonds$cut)
[1] "Fair" "Good" "Very Good" "Premium" "Ideal"
> levels(diamonds$clarity)
[1] "I1" "SI2" "SI1" "VS2" "VS1" "VVS2" "VVS1" "IF"
> levels(diamonds$color)
[1] "D" "E" "F" "G" "H" "I" "J"
One methid to use ordered factors when they have been set up correctly is to just wrap them in as.numeric which gives you a linear test of trend.
> contrasts(diamonds$cut) <- contr.treatment(5) # Removes ordering
> model <- lm(price ~ carat+cut+as.numeric(color)+as.numeric(clarity), diamonds)
> summary(model)
Call:
lm(formula = price ~ carat + cut + as.numeric(color) + as.numeric(clarity),
data = diamonds)
Residuals:
Min 1Q Median 3Q Max
-19130.3 -696.1 -176.8 556.9 9599.8
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -5189.460 36.577 -141.88 <2e-16 ***
carat 8791.452 12.659 694.46 <2e-16 ***
cut2 909.433 35.346 25.73 <2e-16 ***
cut3 1129.518 32.772 34.47 <2e-16 ***
cut4 1156.989 32.427 35.68 <2e-16 ***
cut5 1264.128 32.160 39.31 <2e-16 ***
as.numeric(color) -318.518 3.282 -97.05 <2e-16 ***
as.numeric(clarity) 522.198 3.521 148.31 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1227 on 53932 degrees of freedom
Multiple R-squared: 0.9054, Adjusted R-squared: 0.9054
F-statistic: 7.371e+04 on 7 and 53932 DF, p-value: < 2.2e-16