require(nycflights13)
require(tidyverse)

trees <- read.csv("tree_census.csv")
wb <- read.csv("world_bank.csv")
sensing <- read.csv("remote_sensing.csv", fileEncoding="UTF-8-BOM")

knitr::opts_chunk$set(
  cache = TRUE,
  message = FALSE
)

1 Working with R

1.1 Vectors

purrr provides is_* functions for testing type (for logical, integer, double, numeric, character, atomic, list, vector).

Vectors can be named on creation, like c(x = 1, y = 2, z = 3), or after the fact using purrr::set_names(1:3, c("a", "b", "c")).

1.1.1 Subsetting vectors

When subsetting vectors, negative values drop elements at a specified position:

x <- c(10, 11, 12, 13, 14, 15)
x[-2]
## [1] 10 12 13 14 15
x[c(-2, -3)]
## [1] 10 13 14 15
# x[c(2, -3)] # can't be combined, returns error

1.1.2 Attributes

Can be set using attr():

attr(x, "greeting")
## NULL
attr(x, "greeting") <- "Hi!"
attributes(x)
## $greeting
## [1] "Hi!"

1.2 Probability distributions

In R, every distribution has four functions, all of which have the syntax is letter + name of distribution

  • d: finds the density for a certain value x (the density function)
  • p: finds the cumulative probability given a quantile q (i.e., the CDF)
  • q: finds the quantile given a cumulative probability p (i.e. the inverse CDF)
  • r: generates n random values

The first argument of each function is either x, q, p, or n, and remaining arguments parametrize the distribution:

  • runif(n, min = 0, max = 1)
  • rbinom(n, size, p) where \(\text{Binom}(n, p)\) corresponds to \(\text{Binom}(\text{size}, p)\)
  • rt(n, df, ncp) where ncp is an optional non-centrality parameter
  • rnorm(n, mean = 0, sd = 1)
  • rexp(n, rate = 1)

and so forth for Beta (rbeta), Cauchy (rcauchy), Chi-Square (rchisq), F (rf), Gamma (rgamma), Geometric (rgeom), Hypergeometric (rhyper), Logistic (rlogis), Negative Binomial (rnbinom), Poisson (rpois), Studentized t (rtukey), Weibull (rweibull), and some others.

1.3 Iteration

1.3.1 General tips on iteration

1.3.1.1 Modifying dataframes

Iterate along seq_along(df) or 1:ncol(df) or seq(ncol(df))(all are identical):

rand <- data.frame(matrix(rnorm(100, 0, 1), ncol = 10))

seq_along(rand)
##  [1]  1  2  3  4  5  6  7  8  9 10
seq(ncol(rand))
##  [1]  1  2  3  4  5  6  7  8  9 10
1:ncol(rand)
##  [1]  1  2  3  4  5  6  7  8  9 10
for(i in seq_along(rand)) {
  rand[[i]] <- rand[[i]] * 10
}

1.3.1.2 Storing loop-generated values

Always create the destination vector (or list) first and then filling up the vector, rather then expanding the vector (or dataframe, or string) on each loop iteration. Examples:

  • To create a long string, create a list of strings, then combine at the end using one paste() call
  • To generate a big dataframe row-wise, create a list of rows, then using dplyr::bind_rows() instead of rbind() in each loop iteration

1.3.2 Apply family of functions

1.3.3 Map family of functions from purrr

The map functions are similar to the apply functions, but provide a more consistent interface to iterate across a vector or list:

  • map() returns a list, and is essentially identical to lapply()
  • map_lgl() returns a logical (vector)
  • map_int() returns an integer (vector)
  • map_dbl() returns a double (vector)
  • map_chr() returns a character (vector)
require(purrr)
map_dbl(rand, mean)
##         X1         X2         X3         X4         X5         X6 
##  1.9180649 -1.8027837  0.3534937 -6.8938235  2.8662889  3.1523431 
##         X7         X8         X9        X10 
## -3.3233261 -2.4402544 -1.1798192 -1.1415138
map_dbl(rand, mean, trim = 0.5) # can pass additional arguments
##        X1        X2        X3        X4        X5        X6        X7 
##  2.181209 -1.739250 -1.287261 -6.801780  3.120486  0.923228 -2.721336 
##        X8        X9       X10 
## -4.101704 -2.229284 -1.264049
rand %>% map_dbl(sd) # can use with pipe
##        X1        X2        X3        X4        X5        X6        X7 
## 10.767827 13.404025 13.173023 12.256388 13.546651 11.994249  8.388493 
##        X8        X9       X10 
##  8.292800  9.676282  7.484229

Tools for working with several sub-categories, in order to fit models:

# verbose method
models <- mtcars %>%
  split(.$cyl) %>% # equivalent to split(mtcars$cyl)
  map(function(df) lm(mpg ~ wt, data = df)) # using anonymous function

# using purrr shortcuts
models <- mtcars %>%
  split(.$cyl) %>%
  map(~lm(mpg ~ wt, data = .))

# now use map functinos
models %>%
  map(summary) %>%
  map_dbl("r.squared") # equivalent to map_dbl(.$r.squared)
##         4         6         8 
## 0.5086326 0.4645102 0.4229655

To iterate along two or more vectors, use map2() and pmap():

mu <- c(1, 5, 10)
sigma <- c(1, 0.1, 0.01)
n <- c(5, 6, 7)

map2(mu, sigma, rnorm, n = 5) # apply rnorm, first using different values of mu, then sigma
## [[1]]
## [1] 0.7869046 1.5298229 1.8065202 0.9393476 0.4519518
## 
## [[2]]
## [1] 5.015145 5.000019 5.128891 4.993876 4.931637
## 
## [[3]]
## [1]  9.992750 10.027761  9.993880  9.997804 10.008609
map2(mu, sigma, rnorm, n = 5) %>% map(sum) # map2() returned a list, so we could use map() again
## [[1]]
## [1] 4.637258
## 
## [[2]]
## [1] 24.96279
## 
## [[3]]
## [1] 49.99408
args1 <- list(mu, sigma, n)
args2 <- list(mean = mu, sd = sigma, n = n)
args3 <- data.frame(args2)

# by default, pmap does positional matching for function parameters
# here, we get a bad result because mu is listed first in args1, but rnorm takes n first
args1 %>% pmap(rnorm)
## [[1]]
## [1] -0.9153081
## 
## [[2]]
## [1] 10.842136  1.558008  8.593264 11.190593 -3.875410
## 
## [[3]]
##  [1]   4.42207583   1.50713219   4.70768691 -13.63213417   8.13799988
##  [6]   1.50883349   0.03286155 -10.36769957  -1.31905532   0.66189064
# instead, pass a named list
args2 %>% pmap(rnorm)
## [[1]]
## [1]  1.5394571  0.7347620  1.0581277  2.9806145 -0.1988132
## 
## [[2]]
## [1] 4.968141 4.993620 4.907933 5.002487 4.993384 4.828735
## 
## [[3]]
## [1]  9.990790 10.016362  9.969783 10.008348 10.004564 10.000355 10.013753
# pmap can also take a dataframe
args3 %>% pmap(rnorm) # the same thing
## [[1]]
## [1] 0.8751826 2.3357094 2.3427401 0.7718666 1.2370848
## 
## [[2]]
## [1] 5.082546 4.962414 4.980114 4.953733 5.065688 4.829931
## 
## [[3]]
## [1] 10.008524 10.010003  9.996549 10.012196 10.003213 10.000935  9.998316

1.4 Logical operators

1.4.1 Comparison operators

  • || and && are the standard logical operators
  • | and & are vectorized comparison operators
  • == is both a logical operator and vectorized
  • To test for equality with NA, use is.na(x) rather than x == NA

1.4.2 Testing equality

Since R uses floating-point arithmetic, using == to test for equality might not always produce the desired results. dplyr::near(x, y) includes a built-in tolerance to safely compare values.

1.4.3 De Morgan’s law

As in any language:

  • !(x & y) is equivalent to !x | !y
  • !(x | y) is equivalent to !x & !y

1.5 Handling factors with forcats

1.6 The magrittr pipe

The pipe (%>%) feeds each previous output as the first argument of the next function.

require(dplyr)

trees %>%
  group_by(borough) %>%
  summarize(n = n(),
            avg = mean(tree_dbh))
## # A tibble: 5 x 3
##   borough           n   avg
##   <fct>         <int> <dbl>
## 1 Bronx          1280  9.46
## 2 Brooklyn       2465 11.9 
## 3 Manhattan       989  8.50
## 4 Queens         3679 12.7 
## 5 Staten Island  1587 10.7
# SAME AS THIS (intermediates): gets tedious
# trees_grouped <- group_by(trees, borough)
# trees_summarized <- summarize(trees_grouped, n = n(), avg = mean(tree_dbh))

# SAME AS THIS (nesting): gets hard to read with other options and arguments
# trees_summarized <- summarize(group_by(trees, borough),
#                               n = n(), avg = mean(tree_dbh))

1.6.1 Variants of the pipe operator

  • The tee pipe (%T>%) essentially “cuts” off a pipeline, returning the left hand side and simplying carrying out the actions on the right side
  • The exposition pipe (%$%) explodes out (exposes) the variables of a dataframe to use in non-dataframe-based functions (such as functions in base-R)
  • The assignment pipe (%<>%) returns the entire pipeline and assigns it to the first variable (but use with caution, because can be unclear)

1.6.2 The attribute placeholder

Using . tells R to use the previous passed value in a position that’s not the first position, unless . is used in a way where it’s nested, in which case it will still apply the first-argument rule:

  • f(y, z = x) is equivalent to x %>% f(y, z = .)
  • f(x, y = nrow(x), z = ncol(x)) is equivalent to x %>% f(y = nrow(.), z = ncol(.)
  • f(y = nrow(x), z = ncol(x) is equivalent to x %>% {f(y = nrow(.), z = ncol(.))}

1.7 R Markdown

1.7.1 knitr chunk options

Option Run code Show code Output Plots Messages Warnings
eval = FALSE - - - - -
include = FALSE - - - - -
echo = FALSE -
results = "hide" -
fig.show = "hide" -
message = FALSE -
warning = FALSE -

Syntax for setting global otions:

knitr::opts_chunk$set(
  message = FALSE,
  fig.height = 4
)

1.7.2 kable and kableExtra

knitr::kable() prints a prettier table:

require(knitr)
require(kableExtra)

seq <- data.frame(matrix(c(1, 2, 3, 4, 5, 6, 7, 8, 9), nrow = 3))
kable(seq, caption = "Sequential numbers") # works fine for PDF knitting, not pretty for HTML
Sequential numbers
X1 X2 X3
1 4 7
2 5 8
3 6 9
seq %>%
  knitr::kable(caption = "Sequential numbers") %>%
  kableExtra::kable_styling() # can be piped
Sequential numbers
X1 X2 X3
1 4 7
2 5 8
3 6 9

1.8 Bits and bobs

1.8.1 do.call

2 Data exploration and manipulation

2.1 Reading data using readr

Compared to reading data with base R, readr:

  • Is faster
  • Doesn’t convert strings to factors
  • Handles dates better
  • Shows a progress bar

General syntax:

read_csv(file,
         col_names = TRUE, # set FALSE to treat first row as data
                           # set to vector to use custom colnames
         col_types = NULL, # see below
         skip = 0, # skip the first n lines
         n_max = Inf, # only read up to n lines
         na = c("", "NA")) # replace with NA

Other useful functions: read_lines(file) reads each line into a string

2.1.1 Datatype parsers in readr

Since readr doesn’t automatically convert characters to factors, we can set default datatypes using col_types:

read_csv("cleaned_subway_stations.csv", 
         col_types = cols(
             .default = col_guess(),
             CensusTract = col_factor()
         ))

Common parsers include:

  • col_guess() is the default
  • col_character(), col_double(), col_integer(), col_logical()
  • col_datetime(format = ""), col_date(format = ""), col_time(format = "")
  • col_factor(levels, ordered = FALSE), col_skip()

2.1.2 Reading other file types using haven

haven (also comes with the tidyverse) contains functions to read and write to other common formats. Specifically:

  • Stata: read_dta(), which reads files up to version 15
  • SAS: read_sas() reads .sas7bdat and .sas7bcat
  • SPSS: read_sav() reads .sav

See full vignette here.

2.2 Aggregate statistics

  • base::summary() provides the five-number summary, plus mean, in a named vector
  • base::fivenum() provides the five-number summary in a non-named vector

2.2.1 Using by()

#s: by(values_to_aggregate, a_factor, function)
#r: always returns formatted output
by(trees$tree_dbh, trees$status, mean)
## trees$status: Alive
## [1] 11.78739
## -------------------------------------------------------- 
## trees$status: Dead
## [1] 5.36
## -------------------------------------------------------- 
## trees$status: Stump
## [1] 0
by(trees[, c("tree_dbh", "latitude")], trees$status, summary)
## trees$status: Alive
##     tree_dbh        latitude    
##  Min.   : 0.00   Min.   :40.50  
##  1st Qu.: 5.00   1st Qu.:40.63  
##  Median :10.00   Median :40.70  
##  Mean   :11.79   Mean   :40.70  
##  3rd Qu.:17.00   3rd Qu.:40.76  
##  Max.   :79.00   Max.   :40.91  
## -------------------------------------------------------- 
## trees$status: Dead
##     tree_dbh        latitude    
##  Min.   : 1.00   Min.   :40.51  
##  1st Qu.: 2.00   1st Qu.:40.66  
##  Median : 4.00   Median :40.72  
##  Mean   : 5.36   Mean   :40.72  
##  3rd Qu.: 6.00   3rd Qu.:40.81  
##  Max.   :26.00   Max.   :40.89  
## -------------------------------------------------------- 
## trees$status: Stump
##     tree_dbh    latitude    
##  Min.   :0   Min.   :40.50  
##  1st Qu.:0   1st Qu.:40.64  
##  Median :0   Median :40.70  
##  Mean   :0   Mean   :40.70  
##  3rd Qu.:0   3rd Qu.:40.76  
##  Max.   :0   Max.   :40.90

2.2.2 Using tapply()

#s: tapply(values_to_aggregate, a_factor, function)
#r: if function returns one value per group, tapply() returns array
#r: else, tapply() returns list
tapply(trees$tree_dbh, trees$status, mean)
##    Alive     Dead    Stump 
## 11.78739  5.36000  0.00000

2.2.3 Using dplyr::summarize()

trees %>% #s: dataset
  group_by(status) %>% #s: grouping variable
  summarize(AvgDiameter = mean(tree_dbh)) #s: function and values to target
## # A tibble: 3 x 2
##   status AvgDiameter
##   <fct>        <dbl>
## 1 Alive        11.8 
## 2 Dead          5.36
## 3 Stump         0
#e: multiple variables, one function
trees %>%
  group_by(status) %>%
  summarize_at(c("tree_dbh", "latitude", "longitude"), mean)
## # A tibble: 3 x 4
##   status tree_dbh latitude longitude
##   <fct>     <dbl>    <dbl>     <dbl>
## 1 Alive     11.8      40.7     -73.9
## 2 Dead       5.36     40.7     -73.9
## 3 Stump      0        40.7     -73.9

2.2.4 Using psych::describe()

Provides summary statistics in dataframe format:

require(psych)

#e: for one variable: pass in vector
psych::describe(trees$tree_dbh)
##    vars     n  mean   sd median trimmed  mad min max range skew kurtosis
## X1    1 10000 11.36 8.57      9   10.29 7.41   0  79    79 1.13     1.32
##      se
## X1 0.09
#e: for many variables: pass in dataframe
#s: set FAST = FALSE for fewer statistics (no kurtosis, skew, etc.)
psych::describe(trees[, c("tree_dbh", "latitude", "longitude")], fast = TRUE)
##           vars     n   mean   sd    min    max range   se
## tree_dbh     1 10000  11.36 8.57   0.00  79.00 79.00 0.09
## latitude     2 10000  40.70 0.09  40.50  40.91  0.41 0.00
## longitude    3 10000 -73.92 0.12 -74.25 -73.70  0.55 0.00

2.3 Cleaning data with tidyr

2.4 Handling missing values

For a dataframe, na.omit() takes care of the entire dataframe, or complete.cases() gets indices:

na.omit(wb) # returns only complete cases

comp_cases <- complete.cases(wb) # returns indices of complete cases
wb[comp_cases, ] # gets only rows for those indices

dplyr::drop_na(wb) # does same thing as above
dplyr::drop_na(wb, Imports) # only drops rows where Imports is missing

2.5 Manipulating data using dplyr

All of the main verbs in dplyr share the same syntax: verb(data, arguments, ...).

Grouping performs all operations within each group, and ungroup(), well, ungroups.

# filter()
filter(flights, month == 1, day == 1, dep_delay < 10) # can add as many as desired
flights %>% filter(month == 1, day == 1, dep_delay < 10) # equivalent, using pipe operator

# arrange()
flights %>% arrange(year, month, day) # ascending order
flights %>% arrange(desc(dep_delay)) # descending order: surround a variable with desc()

# select()
flights %>% select(year, month, day) # selects only those columns
flights %>% select(year:day) # all columns between year and day
flights %>% select(-day) # all columns except
flights %>% select(starts_with("abc"), ends_with("abc"), contains("rst"))
flights %>% select(delay = dep_delay) # rename variables (but keeps only that variable)
flights %>% rename(delay = dep_delay) # rename only that variable, keeping everything else
flights %>% select(year, month, day, everything()) # move a few variables to the front, keep all else

# mutate()
flights %>% mutate(gain = dep_delay - arr_delay) # adds new columns
flights %>% transmute(gain = dep_delay - arr_delay) # keeps only the new columns

# summarize() and group_by()
flights %>% 
  group_by(origin, month) %>%
  summarize(n = n(),
            avgDelay = mean(dep_delay, na.rm = TRUE))

2.5.1 Helper functions for mutate()

  • +, -, *, /, ^, %/% (integer division), %% (remainder)
  • lead() and lag() offsets a vector by 1
    • Calculate running differences: x - lag(x)
    • Find when values change between adjacent indices: x != lag(x)
  • Cumulative and rolling aggregates: cumsum(), cumprod(), cummin(), cummax(), cummean()
    • For rolling aggregates, see the RcppRoll package
  • min_rank() assigns ranks as we would expect, properly handling ties
    • Otherwise, see row_number(), dense_rank(), percent_rank(), cume_dist(), ntile()

For more (TODO), read vignette("window-functions")

2.5.2 Helper functions for summary()

  • mean(), median(), sd(), IQR(), mad() (median absolute deviation, which is more robust)
  • quantile(x, 0.25) for finding a quantile
  • n(), n_distinct(), sum(!is.na(x))

2.6 Handling strings with stringr

2.6.1 Regular expressions

  • ^ or $ in a regex indicates that the pattern must occur at the beginning or end of the string
  • [] indicates that we’re searching for anything within the brackets
  • \\s indicates space character (\s is a space, and another \ escapes it)
  • . is any character
  • * is any number of times

2.7 Handling dates and times with lubridate

2.8 Web scraping with rvest

3 Visualization

3.1 Base-R plots

The basic function is the plot() function:

plot(trees$latitude, trees$longitude) # explicit vectors
plot(Exports ~ Imports, data = wb) # formula notation
plot(Exports ~ Imports, data = wb,
     lwd = 2,
     col = "red",
     xlab = "Imports",
     ylab = "Exports")

Mosaic plots using Base-R (note that the function does the aggregation automatically):

mosaicplot(borough ~ health, data = trees)

3.2 ggplot2

require(ggplot2)

3.2.1 Basics: aesthetics, facets, geoms

We can set the data and mapping arguments either in the global ggplot() call or inside each geom:

  • Setting it in the global call uses those arguments for the entire plot
  • Setting it within a geom uses those arguments just for that geom, overriding the global options
# Set color by class
# Just set an aesthetic to name of a variable
ggplot(data = mpg) +
  geom_point(mapping = aes(x = displ, y = hwy, color = class)) +
  labs(
    title = "Mileage versus engine size",
    subtitle = "by class of car",
    caption = "Courtesy of Hadley Wickham"
  )

# Set an aesthetic property
# Use the property outside aes() but inside the geom
ggplot(data = mpg) +
  geom_point(mapping = aes(x = displ, y = hwy), color = "red")

To facet, use facet_wrap() or facet_grid():

# Facet by one variable (ggplot will wrap automatically)
ggplot(data = mpg) +
  geom_point(mapping = aes(x = displ, y = hwy)) +
  facet_wrap(~ class, nrow = 2) # note specification of formula

# Facet by two variable
ggplot(data = mpg) +
  geom_point(mapping = aes(x = displ, y = hwy)) +
  facet_grid(drv ~ cyl) # note specification of formula

Jittering is done with geom_jitter() or with geom_point(position = "jitter").

Set format of bar charts using the position argument in geom_bar():

base <- ggplot(diamonds, aes(x = cut, fill = clarity))
# note that plots can be assigned to variables and constructed additively

base + geom_bar(position = "stack") # this is the default

base + geom_bar(position = "identity") # bars are overlapping

base + geom_bar(position = "dodge") # side by side barplot

base + geom_bar(position = "fill") # proportion within each group

To flip what’s plotted on the x-axis versus y-axis, use coord_flip().

3.2.2 Quick plots with qplot

qplot provides a wrapper for creating ggplots that:

  • Is more convenient to use quickly
  • Has most (but not all) features in day-to-day use
  • Creates a ggplot object, so can be manipulated (i.e. additively) in the same way

Note that unlike plot() from base-R, qplot does not accept formula notation!

qplot(displ, hwy, data = mpg, main = "Title here", xlab = "x-axis label") # basic structure
qplot(displ, hwy, data = mpg, facets = ~ class) # faceting: use either a one- or two-sided formula

qplot(displ, hwy, data = mpg, color = class) # same graph as before, much less code!
qplot(displ, hwy, data = mpg, color = I("red")) # surround in I() to set aesthetic

qplot(displ, hwy, data = mpg, geom = c("point", "smooth", "jitter")) # multiple geoms (with jitter)

qplot(hwy, data = mpg) # only specify x creates histogram
qplot(y = hwy, data = mpg, geom = "boxplot") # only specify x creates histogram

3.3 Interactive plots using plotly

3.4 Interactive plots using shiny

4 Statistics

4.1 Normality

Generally, use the normal quantile plot fom car:

require(car)
car::qqPlot(trees$tree_dbh)

## [1]  871 5016

4.2 Transformations

4.2.1 Box-Cox transformation

Use either one of these functions, both in the car package:

  • car::boxCox generates a plot on creation, even when assigning to an object
  • car::powerTransform finds per-column Box-Cox powers (and includes several transformation options. car::bcPower raises each column of data to each specified value of lambda.
boxcox <- car::boxCox(name_of_model) # generates plot
power <- boxcox$x[which.max(boxcox$y)]

# For multiple columns at once
bc_param <- car::powerTransform(data)
df_new <- bcPower(data, bc_param$lambda)

# bc_param <- car::powerTransform(data, family = "bcPower") -- default
# bc_param <- car::powerTransform(data, family = "bcnPower") -- allows negatives

4.2.2 Logit transformation

4.3 Heteroskedasticity

4.3.1 Bartlett’s test for equality of variances

Bartlett’s test assumes data come from a normal distribution within each group

  • In general, a very sensitive test, so use with a grain of salt
  • Very sensitive to departures from non-normality, so if data is non-normal, Barlett’s test might just be testing for that instead
  • Note that the null hypothesis is \(H_0: \text{variances are equal}\), so we actually want to fail to reject the null
    • We’ve technically only concluded that we don’t know that the variances aren’t different, but in practice we ignore this distinction
bartlett.test(trees$tree_dbh, trees$borough)
## 
##  Bartlett test of homogeneity of variances
## 
## data:  trees$tree_dbh and trees$borough
## Bartlett's K-squared = 441.91, df = 4, p-value < 2.2e-16
bartlett.test(tree_dbh ~ borough, data = trees) # same thing
## 
##  Bartlett test of homogeneity of variances
## 
## data:  tree_dbh by borough
## Bartlett's K-squared = 441.91, df = 4, p-value < 2.2e-16

4.3.2 Levene’s test for equality of variances

Levene’s test is similar, but doesn’t assume normality within each group (but has lower power):

car::leveneTest(trees$tree_dbh, trees$borough)
## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value    Pr(>F)    
## group    4  90.561 < 2.2e-16 ***
##       9995                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
car::leveneTest(tree_dbh ~ borough, data = trees) # same thing
## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value    Pr(>F)    
## group    4  90.561 < 2.2e-16 ***
##       9995                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

4.3.3 Equality of covariances for multivariate tests (Box’s M)

Compute covariance matrices for each group, or use boxM() from biotools.

for(name in levels(sensing$Crop)) { # loop through categories of factor variable
  print(cov(sensing[sensing$Crop == name, -1]))
}

#s: boxM(predictors, response)
biotools::boxM(data[, -1], data$Crop)

4.4 Basic hypothesis tests

4.4.1 Simple t-test for means

# Generate data
n <- 100
rand <- data.frame(id = 1:n,
                   group = sample(1:2, 100, replace = TRUE),
                   var1 = runif(n, -10, 10),
                   var2 = runif(n, -8, 12))
head(rand)
##   id group       var1       var2
## 1  1     1 -5.2025253  4.0520755
## 2  2     1 -0.3507774 -4.6487709
## 3  3     2  4.5811925  9.7107064
## 4  4     2 -0.4222238  9.4746105
## 5  5     1 -2.8241554 10.4507917
## 6  6     1  0.9294305 -0.9094015
# One-sample t-test
t.test(rand$var1) # tests if equal to 0
## 
##  One Sample t-test
## 
## data:  rand$var1
## t = -0.36117, df = 99, p-value = 0.7187
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -1.3157912  0.9105532
## sample estimates:
## mean of x 
## -0.202619
t.test(rand$var1, mu = 2, conf.level = 0.99) # tests if equal to 2
## 
##  One Sample t-test
## 
## data:  rand$var1
## t = -3.9261, df = 99, p-value = 0.0001596
## alternative hypothesis: true mean is not equal to 2
## 99 percent confidence interval:
##  -1.676068  1.270830
## sample estimates:
## mean of x 
## -0.202619
# Two-sample t-test, paired
t.test(rand$var1, rand$var2, paired = TRUE)
## 
##  Paired t-test
## 
## data:  rand$var1 and rand$var2
## t = -2.1521, df = 99, p-value = 0.03382
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.4609316 -0.1404602
## sample estimates:
## mean of the differences 
##               -1.800696
# Two-sample t-test, unpaired
t.test(var1 ~ group, data = rand)
## 
##  Welch Two Sample t-test
## 
## data:  var1 by group
## t = 1.6455, df = 96.646, p-value = 0.1031
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.3764253  4.0268413
## sample estimates:
## mean in group 1 mean in group 2 
##       0.7282371      -1.0969709

4.5 Bootstrap

4.6 Permutation tests

4.7 Cross-validation

Base R sample() function syntax: sample(from, n, replace = FALSE)

5 Generalized linear models

5.1 Multiple regression

5.1.1 Creating the model

# Create basic model using lm
wb_lm <- lm(Exports ~ Imports, data = wb)

# Plot with regression line
par(mar = c(5.1, 4.1, 4.1, 2.1)) # default R settings
plot(Exports ~ Imports, data = wb, col = "red", pch = 19)
abline(wb_lm$coef, col = "blue", lwd = 3)

# Create confidence intervals
confint(wb_lm) # create for all estimates
##                  2.5 %     97.5 %
## (Intercept) -2.4905683 11.4080643
## Imports      0.6779293  0.9149728
confint(wb_lm, "Imports", level = 0.95) # specify which estimates or a conf level
##             2.5 %    97.5 %
## Imports 0.6779293 0.9149728

5.1.2 Model diagnostics

By default, plot() creates four plots when applied to an lm object:

plot(wb_lm) # press 'return' to go through them
plot(wb_lm, which = 1) # print only the residuals vs fits plot
plot(wb_lm, which = 2) # normal quantile plot only

# Better normal quantile plot in car package
car::qqPlot(wb_lm$residuals)

5.1.3 Robust standard error

After creating a linear model using standard base R lm, use:

lmtest::coeftest(lm, vcov = sandwich::vcovHC(lm, type = "HC1"))

Note: default for type is “HC3”, but must set to “HC1” to obtain same results as Stata

TODO: look more into the rlm() functions in MASS

5.1.4 Joint hypothesis tests

After creating a linear model, use car::linearHypothesis():

car::linearHypothesis(lm, c("meduc = 0", "feduc = 0")) # homoskedastic only
car::linearHypothesis(lm, c("meduc = 0", "feduc = 0"), white.adjust = "hc1") # robust

TODO: look more into options in car::linearHypothesis()

5.2 Analysis of variance

5.2.1 ANOVA-regression equivalence

Since ANOVA is essentially the same as linear regression with categorical predictors, many of the methods are similar:

trees_anova <- aov(tree_dbh ~ borough, data = trees)
summary(trees_anova)
##               Df Sum Sq Mean Sq F value Pr(>F)    
## borough        4  20771    5193   72.72 <2e-16 ***
## Residuals   9995 713742      71                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary.lm(trees_anova)
## 
## Call:
## aov(formula = tree_dbh ~ borough, data = trees)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -12.709  -6.495  -1.709   4.513  67.103 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            9.4617     0.2362  40.059  < 2e-16 ***
## boroughBrooklyn        2.4348     0.2911   8.363  < 2e-16 ***
## boroughManhattan      -0.9663     0.3578  -2.701  0.00693 ** 
## boroughQueens          3.2474     0.2742  11.842  < 2e-16 ***
## boroughStaten Island   1.2648     0.3175   3.984 6.82e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.45 on 9995 degrees of freedom
## Multiple R-squared:  0.02828,    Adjusted R-squared:  0.02789 
## F-statistic: 72.72 on 4 and 9995 DF,  p-value: < 2.2e-16
trees_lm <- lm(tree_dbh ~ borough, data = trees)
summary(trees_lm)
## 
## Call:
## lm(formula = tree_dbh ~ borough, data = trees)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -12.709  -6.495  -1.709   4.513  67.103 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            9.4617     0.2362  40.059  < 2e-16 ***
## boroughBrooklyn        2.4348     0.2911   8.363  < 2e-16 ***
## boroughManhattan      -0.9663     0.3578  -2.701  0.00693 ** 
## boroughQueens          3.2474     0.2742  11.842  < 2e-16 ***
## boroughStaten Island   1.2648     0.3175   3.984 6.82e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.45 on 9995 degrees of freedom
## Multiple R-squared:  0.02828,    Adjusted R-squared:  0.02789 
## F-statistic: 72.72 on 4 and 9995 DF,  p-value: < 2.2e-16
summary.aov(trees_lm) 
##               Df Sum Sq Mean Sq F value Pr(>F)    
## borough        4  20771    5193   72.72 <2e-16 ***
## Residuals   9995 713742      71                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  • By default, calling summary() on an aov object calls summary.aov() (which prints the usual ANOVA-style sum of squares output), but we can also force R to print regression-like output for an ANOVA model by calling summary.lm() on an aov object
  • Can do the same for lm by calling summary.aov() on an lm object

5.2.2 Confidence intervals for group means

We can actually tell R to fit a regression model where the coefficient estimates are the group means by fitting a model without an intercept, which we do by adding -1 to the regression specification:

  • We can do this when we have only one categorical predictor since the base level essentially “replaces” the intercept, using up the degree of freedom that the intercept previously used
  • Then, the coef() function returns the means, and confint() returns confidence intervals
trees_lm2 <- lm(tree_dbh ~ borough - 1, data = trees)
coef(trees_lm2) # same as tapply(trees$tree_dbh, trees$borough, mean)
##         boroughBronx      boroughBrooklyn     boroughManhattan 
##             9.461719            11.896552             8.495450 
##        boroughQueens boroughStaten Island 
##            12.709160            10.726528
round(confint(trees_lm2), 2) # 95% CI for individual group means
##                      2.5 % 97.5 %
## boroughBronx          9.00   9.92
## boroughBrooklyn      11.56  12.23
## boroughManhattan      7.97   9.02
## boroughQueens        12.44  12.98
## boroughStaten Island 10.31  11.14

plotCI() from plotrix can help in plotting confidence intervals:

trees_means <- coef(trees_lm2)
trees_ci <- confint(trees_lm2)

par(mar = c(5, 8, 4, 2))
plotrix::plotCI(x = trees_means, # x-values (coef estimates)
                y = 1:length(trees_means), # y-values (just 1 through 5)
                ui = trees_ci[, 2], # upper bound of each error bar
                li = trees_ci[, 1], # lower bound of each error bar
                err = "x", # direction of bars (x is horizontal)
                xlab = "tree dbh", ylab = "", main = "Means and CIs for tree height",
                axes = FALSE, lwd = 2, col = "blue")
axis(side = 1)
axis(side = 2, at = 1:(length(trees_means)), label = levels(trees$borough), las = 2)

5.2.3 Multiple comparisons

Best way to perform multiple comparisons is using Tukey HSD:

trees_tukey <- TukeyHSD(trees_anova, conf.level = 0.95)

par(mar = c(5, 11, 4, 1))
plot(trees_tukey, las = 1)