Week 10 R tutorial
Aaron Shaw
November 10, 2020
This week’s R tutorial focuses on the basics of correlations and linear regression. I’ll work with the mtcars dataset that comes built-in with R.
data(mtcars)Correlations and covariance
Calculating correlation coefficients is straightforward: use the cor() function:
with(mtcars, cor(mpg, hp))## [1] -0.7761684All you prius drivers out there will be shocked to learn that miles-per-gallon is negatively correlated with horsepower.
The cor() function works with two variables or with more—the following generates a correlation matrix for the whole dataset!
cor(mtcars)## mpg cyl disp hp drat wt
## mpg 1.0000000 -0.8521620 -0.8475514 -0.7761684 0.68117191 -0.8676594
## cyl -0.8521620 1.0000000 0.9020329 0.8324475 -0.69993811 0.7824958
## disp -0.8475514 0.9020329 1.0000000 0.7909486 -0.71021393 0.8879799
## hp -0.7761684 0.8324475 0.7909486 1.0000000 -0.44875912 0.6587479
## drat 0.6811719 -0.6999381 -0.7102139 -0.4487591 1.00000000 -0.7124406
## wt -0.8676594 0.7824958 0.8879799 0.6587479 -0.71244065 1.0000000
## qsec 0.4186840 -0.5912421 -0.4336979 -0.7082234 0.09120476 -0.1747159
## vs 0.6640389 -0.8108118 -0.7104159 -0.7230967 0.44027846 -0.5549157
## am 0.5998324 -0.5226070 -0.5912270 -0.2432043 0.71271113 -0.6924953
## gear 0.4802848 -0.4926866 -0.5555692 -0.1257043 0.69961013 -0.5832870
## carb -0.5509251 0.5269883 0.3949769 0.7498125 -0.09078980 0.4276059
## qsec vs am gear carb
## mpg 0.41868403 0.6640389 0.59983243 0.4802848 -0.55092507
## cyl -0.59124207 -0.8108118 -0.52260705 -0.4926866 0.52698829
## disp -0.43369788 -0.7104159 -0.59122704 -0.5555692 0.39497686
## hp -0.70822339 -0.7230967 -0.24320426 -0.1257043 0.74981247
## drat 0.09120476 0.4402785 0.71271113 0.6996101 -0.09078980
## wt -0.17471588 -0.5549157 -0.69249526 -0.5832870 0.42760594
## qsec 1.00000000 0.7445354 -0.22986086 -0.2126822 -0.65624923
## vs 0.74453544 1.0000000 0.16834512 0.2060233 -0.56960714
## am -0.22986086 0.1683451 1.00000000 0.7940588 0.05753435
## gear -0.21268223 0.2060233 0.79405876 1.0000000 0.27407284
## carb -0.65624923 -0.5696071 0.05753435 0.2740728 1.00000000Note that if you are calculating correlations with variables that are not distributed normally you should use cor(method="spearman") because it calculates rank-based correlations (look it up online for more details).
To calculate covariance, you use the cov() function:
with(mtcars, cov(mpg, hp))## [1] -320.7321While OpenIntro does not spend much time on either covariance or correlation, I highly recommend taking the time to review at least the corresponding Wikipedia articles and ideally another statistics textbook) to understand what goes into each one. 1 A key point to notice/understand is that covariance is calculated in such a way that the magnitude may vary depending on the scale of the variables involved. Correlation is not (every correlation coefficient falls between -1 and 1).
Fitting a linear model
Linear models are fit using the lm() command. As with aov(), the lm() function requires a formula as an input and is usually presented with a call to summary(). You can enter the formula directly in the call to lm() or define it separately. For this example, I’ll regress mpg on a single predictor, hp:
model1 <- lm(mpg ~ hp, data=mtcars)
summary(model1)##
## Call:
## lm(formula = mpg ~ hp, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.7121 -2.1122 -0.8854 1.5819 8.2360
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.09886 1.63392 18.421 < 2e-16 ***
## hp -0.06823 0.01012 -6.742 1.79e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.863 on 30 degrees of freedom
## Multiple R-squared: 0.6024, Adjusted R-squared: 0.5892
## F-statistic: 45.46 on 1 and 30 DF, p-value: 1.788e-07Notice how much information the output of summary() gives you for a linear model! You have details about the residuals, the usual information about the coefficients, standard errors, t-values, etc., little stars corresponding to conventional significance levels, \(R^2\) values, degrees of freedom, F-statistics (remember those?) and p-values for the overall model fit.
There’s even more under the hood. Try looking at all the different things in the model object R has created:
names(model1)## [1] "coefficients" "residuals" "effects" "rank"
## [5] "fitted.values" "assign" "qr" "df.residual"
## [9] "xlevels" "call" "terms" "model"You can directly inspect the residuals using model1$residuals. This makes plotting and other diagnostic activities pretty straightforward:
summary(model1$residuals)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -5.7121 -2.1122 -0.8854 0.0000 1.5819 8.2360More on that in a moment. In the meantime, you can also use the items generated by the call to summary() as well:
names(summary(model1))## [1] "call" "terms" "residuals" "coefficients"
## [5] "aliased" "sigma" "df" "r.squared"
## [9] "adj.r.squared" "fstatistic" "cov.unscaled"summary(model1)$coefficients## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.09886054 1.6339210 18.421246 6.642736e-18
## hp -0.06822828 0.0101193 -6.742389 1.787835e-07CIs around coefficients and predicted values
There are also functions to help you do things with the model such as predict the fitted values for new (unobserved) data. For example, if I found some new cars with horsepowers ranging from 90-125, what would this model predict for the corresponding mpg for each car?
new.data <- data.frame(hp=seq(90,125,5))
predict(model1, new.data, type="response")## 1 2 3 4 5 6 7 8
## 23.95832 23.61717 23.27603 22.93489 22.59375 22.25261 21.91147 21.57033A call to predict can also provide standard errors around these predictions (which you could use, for example, to construct a 95% confidence interval around the model-predicted values):
predict(model1, new.data, type="response", se.fit = TRUE)## $fit
## 1 2 3 4 5 6 7 8
## 23.95832 23.61717 23.27603 22.93489 22.59375 22.25261 21.91147 21.57033
##
## $se.fit
## 1 2 3 4 5 6 7 8
## 0.8918453 0.8601743 0.8303804 0.8026728 0.7772744 0.7544185 0.7343427 0.7172804
##
## $df
## [1] 30
##
## $residual.scale
## [1] 3.862962Linear model objects also have a built-in method for generating confidence intervals around the values of \(\beta\):
confint(model1, "hp", level=0.95) # Note that I provide the variable name in quotes## 2.5 % 97.5 %
## hp -0.08889465 -0.0475619Feeling old-fashioned? You can always calculate residuals or confidence intervals (or anything else) “by hand”:
# Residuals
mtcars$mpg - model1$fitted.values## Mazda RX4 Mazda RX4 Wag Datsun 710 Hornet 4 Drive
## -1.59374995 -1.59374995 -0.95363068 -1.19374995
## Hornet Sportabout Valiant Duster 360 Merc 240D
## 0.54108812 -4.83489134 0.91706759 -1.46870730
## Merc 230 Merc 280 Merc 280C Merc 450SE
## -0.81717412 -2.50678234 -3.90678234 -1.41777049
## Merc 450SL Merc 450SLC Cadillac Fleetwood Lincoln Continental
## -0.51777049 -2.61777049 -5.71206353 -5.02978075
## Chrysler Imperial Fiat 128 Honda Civic Toyota Corolla
## 0.29364342 6.80420581 3.84900992 8.23597754
## Toyota Corona Dodge Challenger AMC Javelin Camaro Z28
## -1.98071757 -4.36461883 -4.66461883 -0.08293241
## Pontiac Firebird Fiat X1-9 Porsche 914-2 Lotus Europa
## 1.04108812 1.70420581 2.10991276 8.01093488
## Ford Pantera L Ferrari Dino Maserati Bora Volvo 142E
## 3.71340487 1.54108812 7.75761261 -1.26197823# 95% CI for the coefficient on horsepower
est <- model1$coefficients["hp"]
se <- summary(model1)$coefficients[2,2]
est + 1.96 * c(-1,1) * se## [1] -0.08806211 -0.04839444Plotting residuals
You can generate diagnostic plots of residuals in various ways:
hist(residuals(model1))
hist(model1$residuals)
Plot the residuals against the original predictor variable:
library(ggplot2)
qplot(x=mtcars$hp, y=residuals(model1), geom="point")
Quantile-quantile plots can be done using qqnorm() on the residuals:
qqnorm(residuals(model1))
The easiest way to generate a few generic diagnostic plots in ggplot is documented pretty well on StackExchange and elsewhere:
library(ggfortify)
autoplot(model1)## Warning: `arrange_()` is deprecated as of dplyr 0.7.0.
## Please use `arrange()` instead.
## See vignette('programming') for more help
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_warnings()` to see where this warning was generated.
Adding additional variables (multiple regression—really useful next week)
You can, of course, have models with many variables. This might happen by creating a brand new formula or using a command update.formula() to…well, you probably guessed it:
f1 <- formula(mpg ~ hp)
f2 <- formula(mpg ~ hp + disp + cyl + vs)
f2a <- update.formula(f1, . ~ . + disp + cyl + vs) ## Same as f2 above
model2 <- lm(f2, data=mtcars)
summary(model2)##
## Call:
## lm(formula = f2, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.0190 -2.1712 -0.7994 1.6104 6.9770
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 36.00980 4.46560 8.064 1.15e-08 ***
## hp -0.01594 0.01506 -1.058 0.2992
## disp -0.01825 0.01061 -1.720 0.0969 .
## cyl -1.44613 0.91674 -1.577 0.1263
## vs -0.96916 1.91824 -0.505 0.6175
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.097 on 27 degrees of freedom
## Multiple R-squared: 0.7701, Adjusted R-squared: 0.736
## F-statistic: 22.61 on 4 and 27 DF, p-value: 2.745e-08Estimating linear models with predictor variables that are not continuous (numeric or integers) is no problem. Just go for it:
mtcars$cyl <- factor(mtcars$cyl)
mtcars$vs <- as.logical(mtcars$vs)
## Refit the same model:
model2 <- lm(f2, data=mtcars)
summary(model2)##
## Call:
## lm(formula = f2, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.4430 -1.7856 -0.3927 1.9359 6.0095
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 31.41674 2.43984 12.877 8.65e-13 ***
## hp -0.02143 0.01457 -1.472 0.1532
## disp -0.02575 0.01076 -2.392 0.0243 *
## cyl6 -4.16031 1.85492 -2.243 0.0337 *
## cyl8 -2.74149 3.80548 -0.720 0.4777
## vsTRUE -0.30254 1.85252 -0.163 0.8715
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.941 on 26 degrees of freedom
## Multiple R-squared: 0.8003, Adjusted R-squared: 0.7619
## F-statistic: 20.84 on 5 and 26 DF, p-value: 2.391e-08We’ll talk more about how to interpret these results with categorical predictors next week, but for now you can see that R has no trouble handling multiple types or classes of variables in a regression model.
Producing nice regression tables
Generating regression tables directly from your statistical software is very important for preventing mistakes and typos. There are many ways to do this and a variety of packages that may be helpful (LaTex users: see this StackExchange post for a big list).
One especially easy-to-use package that can output text and html (both eminently paste-able into a variety of typesetting/word-processing systems) is called stargazer. Here I use it to generate an ASCII table summarizing the two models we’ve fit in this tutorial.
library(stargazer)##
## Please cite as:## Hlavac, Marek (2018). stargazer: Well-Formatted Regression and Summary Statistics Tables.## R package version 5.2.2. https://CRAN.R-project.org/package=stargazerstargazer(model1, model2, type="text")##
## =================================================================
## Dependent variable:
## ---------------------------------------------
## mpg
## (1) (2)
## -----------------------------------------------------------------
## hp -0.068*** -0.021
## (0.010) (0.015)
##
## disp -0.026**
## (0.011)
##
## cyl6 -4.160**
## (1.855)
##
## cyl8 -2.741
## (3.805)
##
## vs -0.303
## (1.853)
##
## Constant 30.099*** 31.417***
## (1.634) (2.440)
##
## -----------------------------------------------------------------
## Observations 32 32
## R2 0.602 0.800
## Adjusted R2 0.589 0.762
## Residual Std. Error 3.863 (df = 30) 2.941 (df = 26)
## F Statistic 45.460*** (df = 1; 30) 20.837*** (df = 5; 26)
## =================================================================
## Note: *p<0.1; **p<0.05; ***p<0.01Back to ANOVAs for a moment
You may recall that I mentioned that R actually calls lm() when it estimates an ANOVA. As I said before, I’m not going to walk through the details, but an important thing to note is that the F-statistics and the p-values for those F-statistics are identical when you use aov() and when you use lm(). That means that you already know what hypothesis is being tested there and how to interpret that part of the regression model output.
summary(aov(data=mtcars, mpg ~ factor(cyl)))## Df Sum Sq Mean Sq F value Pr(>F)
## factor(cyl) 2 824.8 412.4 39.7 4.98e-09 ***
## Residuals 29 301.3 10.4
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(lm(data=mtcars, mpg ~ factor(cyl)))##
## Call:
## lm(formula = mpg ~ factor(cyl), data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.2636 -1.8357 0.0286 1.3893 7.2364
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 26.6636 0.9718 27.437 < 2e-16 ***
## factor(cyl)6 -6.9208 1.5583 -4.441 0.000119 ***
## factor(cyl)8 -11.5636 1.2986 -8.905 8.57e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.223 on 29 degrees of freedom
## Multiple R-squared: 0.7325, Adjusted R-squared: 0.714
## F-statistic: 39.7 on 2 and 29 DF, p-value: 4.979e-09Here are those corresponding Wikipedia articles: correlation and covariance.↩︎