1 R Markdown

This is an R Markdown document. Markdown is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. For more details on using R Markdown see http://rmarkdown.rstudio.com.

When you click the Knit button a document will be generated that includes both content as well as the output of any embedded R code chunks within the document. Consider Table 1.1.

Table 1.1: First six rows of mtcars
mpg cyl disp hp drat wt qsec vs am gear carb
Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1

2 Including Plots

You can also embed plots such as Figure 2.1.

Scatterplot of `pressure` versus `temperature`

Figure 2.1: Scatterplot of pressure versus temperature

Note that the echo = FALSE parameter was added to the code chunk to prevent printing of the R code that generated the plot.

3 Inline R Code

mtcars$cyl <- factor(mtcars$cyl)
library(tidyverse)
SI <- mtcars %>% 
  group_by(cyl) %>% 
  summarize(Mean = mean(qsec), SD = sd(qsec))
SI
# A tibble: 3 x 3
  cyl    Mean    SD
  <fct> <dbl> <dbl>
1 4      19.1  1.68
2 6      18.0  1.71
3 8      16.8  1.20

The mean qsec time for six cylinder cars is 17.98 seconds.

4 Chi-Square Test?

Two different professors teach an introductory statistics course. Table 4.1 shows the distribution of final grades they reported. We wonder whether one of these professors is an “easier” grader.

Table 4.1: Grades for Alpha and Beta
Alpha Beta
A 3 10
B 11 11
C 14 8
D 10 1
F 4 0

\(H_0:\) The distribution of grades is the same for the two professors. \(H_A:\) The distribution of grades is different for the two professors.

chisq.test(tt)$exp
     Professor
Grade     Alpha     Beta
    A  7.583333 5.416667
    B 12.833333 9.166667
    C 12.833333 9.166667
    D  6.416667 4.583333
    F  2.333333 1.666667

4.1 Make tt tidy

ttt <- tt %>% 
  broom::tidy() %>% 
  uncount(n)
ttt
# A tibble: 72 x 2
   Grade Professor
   <chr> <chr>    
 1 A     Alpha    
 2 A     Alpha    
 3 A     Alpha    
 4 B     Alpha    
 5 B     Alpha    
 6 B     Alpha    
 7 B     Alpha    
 8 B     Alpha    
 9 B     Alpha    
10 B     Alpha    
# ... with 62 more rows

4.2 Using infer

T1 <- xtabs(~Grade + Professor, data = ttt)
obs_stat <- chisq.test(T1)$stat
obs_stat
X-squared 
 15.19121 
library(infer)
null <- ttt %>% 
  specify(Grade ~ Professor) %>% 
  hypothesize(null = "independence") %>% 
  generate(reps = 999, type = "permute") %>% 
  calculate(stat = "Chisq")
visualize(null, method = "both")

get_pvalue(null, obs_stat, direction = "right")
# A tibble: 1 x 1
  p_value
    <dbl>
1 0.00601
(pvalue <- (sum(null$stat >= obs_stat) + 1)/(999 + 1))
[1] 0.007

4.3 Using a for() loop

Figure 4.1 shows the theoretical \(\chi^2_4\) distribution in blue, while the permutation distribution from using computation is shown as the pink density.

sims <- 10^3 - 1
x2 <- numeric(sims)
for(i in 1:sims){
  TT <- xtabs(~Grade + sample(Professor), data = ttt)
  x2[i] <- chisq.test(TT)$stat
}
pvalue <- (sum(x2 >= obs_stat) + 1)/(sims + 1)
pvalue
[1] 0.004
DF <- data.frame(x = x2)
ggplot(data = DF, aes(x = x)) + 
  geom_density(fill = "pink", alpha = 0.3) + 
  theme_bw() + 
  stat_function(fun = dchisq, args = list(df = 4), color = "blue")
Theoretical and computation null distribution

Figure 4.1: Theoretical and computation null distribution