# 6 Exploratory statistics

This chapter showcases an exploratory analysis of the distribution of people aged 20 to 24 in Leicester, using the u011 variable from the 2011 Output Area Classification (2011OAC) already used in previous chapters. Before continuing, open the project created for the previous chapter. Create new RMarkdown document within that project to replicate the analysis in this document. Once the document is set up, start by adding an R code snipped including the code below, which is loads the 2011OAC dataset and the libraries used for this chapter.

library(tidyverse)
library(knitr)

leicester_2011OAC <- read_csv("2011_OAC_Raw_uVariables_Leicester.csv")

The graphics above provide preliminary evidence that the distribution of people aged 20 to 24 might, in fact, be different in different 2011 supergroups. In the remainder of the chapter, we are going to explore that hypothesis further. First, load the necessary statistical libraries.

library(pastecs)
library(car)

The code below calculates the percentage of people aged 20 to 24 (i.e., u011) over total population per OA, but it also recodes (see recode) the names of the 2011OAC supergroups to a shorter 2-letter version, which is useful for the tables presented further below. Note that the code below uses the library name as a prefix dplyr:: in front of the function name recode (thus, dplyr::recode) to make sure the function recode of the package car (thus, car::recode) is not used instead.

Only the OA code, the recoded 2011OAC supergroup name, and the newly created perc_age_20_to_24 are retained in the new table leic_2011OAC_20to24. Such a step is sometimes useful as stepping stone for further analysis and can make the code easier to read further down the line. Sometimes it is also a necessary step when interacting with certain libraries, which are not fully compatible with Tidyverse libraries, such as leveneTest (more on that function below).

leic_2011OAC_20to24 <- leicester_2011OAC %>%
mutate(
perc_age_20_to_24 = (u011 / Total_Population) * 100,
supgrpname =
dplyr::recode(
supgrpname,
Suburbanites = "SU",
Cosmopolitans = "CP",
Multicultural Metropolitans = "MM",
Ethnicity Central = "EC",
Constrained City Dwellers = "CD",
Hard-Pressed Living = "HP",
Urbanites = "UR"
)
) %>%
select(OA11CD, supgrpname, perc_age_20_to_24)

leic_2011OAC_20to24 %>%
kable()
OA11CD supgrpname perc_age_20_to_24
E00069517 SU 4.153355
E00069514 CP 30.650155
E00169516 MM 12.316716
E00169048 MM 6.956522
E00169044 MM 6.211180

## 6.1 Descriptive statistics

The first step of any statistical analysis or modelling should be to explore the “shape” of the data involved, by looking at the descriptive statistics of all variables involved. The function stat.desc of the pastecs library provides three series of descriptive statistics.

• base:
• nbr.val: overall number of values in the dataset;
• nbr.null: number of NULL values – NULL is often returned by expressions and functions whose values are undefined;
• nbr.na: number of NAs – missing value indicator;
• desc:
• min (see also min function): minimum value in the dataset;
• max (see also max function): minimum value in the dataset;
• range: difference between min and max (different from range());
• sum (see also sum function): sum of the values in the dataset;
• median (see also median function): median, that is the value separating the higher half from the lower half the values
• mean (see also mean function): arithmetic mean, that is sum over the number of values not NA;
• SE.mean: standard error of the mean – estimation of the variability of the mean calculated on different samples of the data (see also central limit theorem);
• CI.mean.0.95: 95% confidence interval of the mean – indicates that there is a 95% probability that the actual mean is within that distance from the sample mean;
• var: variance ($$\sigma^2$$), it quantifies the amount of variation as the average of squared distances from the mean;
• std.dev: standard deviation ($$\sigma$$), it quantifies the amount of variation as the square root of the variance;
• coef.var: variation coefficient it quantifies the amount of variation as the standard deviation divided by the mean;
• norm (default is FALSE, use norm = TRUE to include it in the output):
• skewness: skewness value indicates
• positive: the distribution is skewed towards the left;
• negative: the distribution is skewed towards the right;
• kurtosis: kurtosis value indicates:
• positive: heavy-tailed distribution;
• negative: flat distribution;
• skew.2SE and kurt.2SE: skewness and kurtosis divided by 2 standard errors. If greater than 1, the respective statistics is significant (p < .05);
• normtest.W: test statistics for the Shapiro–Wilk test for normality;
• normtest.p: significance for the Shapiro–Wilk test for normality.

The Shapiro–Wilk test compares the distribution of a variable with a normal distribution having the same mean and standard deviation. The null hypothesis of the Shapiro–Wilk test is that the sample is normally distributed, thus if normtest.p is lower than 0.01 (i.e., p < .01), the test indicates that the distribution is most probably not normal. The threshold to accept or reject a hypothesis is arbitrary and based on conventions, where p < .01 is the most commonly accepted threshold, or p < .05 for relatively small data sample (e.g., 30 cases).

The next step is thus to apply the stat.desc to the variable we are currently exploring (i.e., perc_age_20_to_24), including the norm section.

leic_2011OAC_20to24_stat_desc <- leic_2011OAC_20to24 %>%
select(perc_age_20_to_24) %>%
stat.desc(norm = TRUE)

leic_2011OAC_20to24_stat_desc %>%
kable(digits = 3)
perc_age_20_to_24
nbr.val 969.000
nbr.null 0.000
nbr.na 0.000
min 1.064
max 60.751
range 59.687
sum 10238.502
median 7.514
mean 10.566
SE.mean 0.304
CI.mean.0.95 0.596
var 89.386
std.dev 9.454
coef.var 0.895
skewness 2.710
skew.2SE 17.249
kurtosis 7.707
kurt.2SE 24.549
normtest.W 0.645
normtest.p 0.000

The table above tells us that all 969 OA in Leicester have a valid value for the variable perc_age_20_to_24, as no NULL nor NA value have been found. The values vary from about 1% to almost 61%, with an average value of 11% of the population in an OA aged between 20 and 24.

The short paragraph above is reporting on the values on the table, taking advantage of two features of RMarkdown. First, the output of the stat.desc function in the snippet further above is stored in the variable leic_2011OAC_20to24_stat_desc, which is then a valid variable for the rest of the document. Second, RMarkdown allows for in-line R snippets, that can also refer to variables defined in any snippet above the text. As such, the source of the paragraph above reads as below, with the in-line R snipped opened by a single grave accent (i.e., ) followed by a lowercase r and closed by another single grave accent.

Having included all the code above into an RMarkdown document, copy the text below verbatim into the same RMarkdown document and make sure that you understand how the code in the in-line R snippets works.

The table above tells us that all r leic_2011OAC_20to24_stat_desc["nbr.val",
"perc_age_20_to_24"] %>% round(digits = 0) OA in Leicester have a valid
value for the variable perc_age_20_to_24, as no r NULL nor r
NA value have been found.The values vary from about r
leic_2011OAC_20to24_stat_desc["min", "perc_age_20_to_24"] %>% round(digits =
0)% to almost r leic_2011OAC_20to24_stat_desc["max",
"perc_age_20_to_24"] %>% round(digits = 0)%, with an average value of
r leic_2011OAC_20to24_stat_desc["mean", "perc_age_20_to_24"] %>%
round(digits = 0)% of the population in an OA aged between 20 and 24. 

If the data described by statistics presented in the table above was a random sample of a population, the 95% confidence interval CI.mean.0.95 would indicate that we can be 95% confident that the actual mean of the distribution is somewhere between 10.566 - 0.596 = 9.97% and 10.566 + 0.596 = 11.162%.

However, this is not a sample. Thus the statistical interpretation is not valid, in the same way that the sum values doesn’t make sense, as it is the sum of a series of percentages.

Both skew.2SE and kurt.2SE are greater than 1, which indicate that the skewness and kurtosis values are significant (p < .05). The skewness is positive, which indicates that the distribution is skewed towards the left (low values). The kurtosis is positive, which indicates that the distribution is heavy-tailed.

The function skim of the library skimr can also be used to generate a quick summary of the content of a dataset.

library(skimr)

leic_2011OAC_20to24 %>%
skim()
 Name Piped data Number of rows 969 Number of columns 3 _______________________ Column type frequency: character 2 numeric 1 ________________________ Group variables None

Variable type: character

skim_variable n_missing complete_rate min max empty n_unique whitespace
OA11CD 0 1 9 9 0 969 0
supgrpname 0 1 2 2 0 7 0

Variable type: numeric

skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
perc_age_20_to_24 0 1 10.57 9.45 1.06 5.8 7.51 10.13 60.75 ▇▁▁▁▁

## 6.2 Significance

Most statistical tests are based on the idea of hypothesis testing:

1. a null hypothesis is set;
2. the data are fit into a statistical model;
3. the model is assessed with a test statistic;
4. the significance is the probability of obtaining that test statistic value by chance.

The threshold to accept or reject a hypothesis is arbitrary and based on conventions. A 0.05 threshold (p < .05) is quite common with relatively small samples (e.g., dozens of cases), while a more strict 0.01 threshold (p < .01) is commonly advised for large samples (e.g., hundreds of cases). However, Mingfeng Lin, Henry C. Lucas, and Galit Shmueli18 advise that such thresholds might not be suitable when working with big data (e.g., tens of thousands of cases).

## 6.3 Shapiro–Wilk test

As perc_age_20_to_24 is a heavy-tailed distribution, skewed towards low values, it is not surprising that the normtest.p value indicates that the Shapiro–Wilk test is significant, which indicates that the distribution is not normal.

The code below present the output of the shapiro.test function, which only present the outcome of a Shapiro–Wilk test on the values provided as input. The output values are the same as the values reported by the norm section of stat.desc. Note that the shapiro.test function require the argument to be a numeric vector. Thus the pull function must be used to extract the perc_age_20_to_24 column from leic_2011OAC_20to24 as a vector, whereas using select with a single column name as the argument would produce as output a table with a single column.

leic_2011OAC_20to24 %>%
pull(perc_age_20_to_24) %>%
shapiro.test()
##
##  Shapiro-Wilk normality test
##
## data:  .
## W = 0.64491, p-value < 2.2e-16

As the null hypothesis of the Shapiro–Wilk test is that the sample is normally distributed, a p value lower than a 0.01 threshold (p < .01) indicates that the probability of that being true is very low. So, the flipper length of penguins in the Palmer Station dataset is not normally distributed.

The two code snippets below can be used to visualise a density-based histogram including the shape of a normal distribution having the same mean and standard deviation, and a Q-Q plot, to visually confirm the fact that perc_age_20_to_24 is not normally distributed.

leic_2011OAC_20to24 %>%
ggplot(
aes(
x = perc_age_20_to_24
)
) +
geom_histogram(
aes(
y =..density..
),
binwidth = 5
) +
stat_function(
fun = dnorm,
args = list(
mean = leic_2011OAC_20to24 %>% pull(perc_age_20_to_24) %>% mean(),
sd = leic_2011OAC_20to24 %>% pull(perc_age_20_to_24) %>% sd()
),
colour = "red", size = 1
)

A Q-Q plot in R can be created using a variety of functions. In the example below, the plot is created using the stat_qq and stat_qq_line functions of the ggplot2 library. Note that the perc_age_20_to_24 variable is mapped to a particular option of aes that is sample.

If perc_age_20_to_24 had been normally distributed, the dots in the Q-Q plot would be distributed straight on the line included in the plot.

leic_2011OAC_20to24 %>%
ggplot(
aes(
sample = perc_age_20_to_24
)
) +
stat_qq() +
stat_qq_line()

## 6.4 Exercise 202.1

Create a new RMarkdown document, and add the code necessary to recreate the table leic_2011OAC_20to24 used in the example above. Use the code below to re-shape the table leic_2011OAC_20to24 by pivoting the perc_age_20_to_24 column wider into multiple columns using supgrpname as new column names.

leic_2011OAC_20to24_supgrp <- leic_2011OAC_20to24 %>%
pivot_wider(
names_from = supgrpname,
values_from = perc_age_20_to_24
)

That manipulation creates one column per supergroup, containing the perc_age_20_to_24 if the OA is part of that supergroup, or an NA value if the OA is not part of the supergroup. The transformation is illustrated in the two tables below. The first shows an extract from the original leic_2011OAC_20to24 dataset, followed by the wide version leic_2011OAC_20to24_supgrp.

leic_2011OAC_20to24 %>%
slice_min(OA11CD, n = 10) %>%
kable(digits = 3)
OA11CD supgrpname perc_age_20_to_24
E00068657 HP 6.053
E00068658 MM 6.964
E00068659 MM 8.383
E00068660 MM 4.643
E00068661 MM 10.625
E00068662 MM 8.284
E00068663 MM 8.357
E00068664 MM 3.597
E00068665 MM 7.068
E00068666 MM 5.864
leic_2011OAC_20to24_supgrp %>%
slice_min(OA11CD, n = 10) %>%
kable(digits = 3)
OA11CD SU CP MM EC CD HP UR
E00068657 NA NA NA NA NA 6.053 NA
E00068658 NA NA 6.964 NA NA NA NA
E00068659 NA NA 8.383 NA NA NA NA
E00068660 NA NA 4.643 NA NA NA NA
E00068661 NA NA 10.625 NA NA NA NA
E00068662 NA NA 8.284 NA NA NA NA
E00068663 NA NA 8.357 NA NA NA NA
E00068664 NA NA 3.597 NA NA NA NA
E00068665 NA NA 7.068 NA NA NA NA
E00068666 NA NA 5.864 NA NA NA NA

Question 202.1.1: The code below uses the newly created leic_2011OAC_20to24_supgrp table to calculate the descriptive statistics calculated for the variable leic_2011OAC_20to24 for each supergroup. Is leic_2011OAC_20to24 normally distributed in any of the subgroups? If yes, which supergroups and based on which values do you justify that claim? (Write up to 200 words)

leic_2011OAC_20to24_supgrp %>%
select(-OA11CD) %>%
stat.desc(norm = TRUE) %>%
kable(digits = 3)
SU CP MM EC CD HP UR
nbr.val 54.000 83.000 573.000 57.000 36.000 101.000 65.000
nbr.null 0.000 0.000 0.000 0.000 0.000 0.000 0.000
nbr.na 915.000 886.000 396.000 912.000 933.000 868.000 904.000
min 1.462 3.141 2.490 2.066 1.064 1.515 2.256
max 9.562 60.751 52.507 36.299 12.963 11.261 13.505
range 8.100 57.609 50.018 34.233 11.899 9.746 11.249
sum 295.867 2646.551 5214.286 838.415 252.108 619.266 372.010
median 5.476 30.457 7.880 10.881 6.854 6.053 5.380
mean 5.479 31.886 9.100 14.709 7.003 6.131 5.723
SE.mean 0.233 1.574 0.230 1.373 0.471 0.172 0.264
CI.mean.0.95 0.467 3.131 0.452 2.751 0.956 0.341 0.528
var 2.929 205.556 30.285 107.523 7.983 2.980 4.545
std.dev 1.712 14.337 5.503 10.369 2.825 1.726 2.132
coef.var 0.312 0.450 0.605 0.705 0.403 0.282 0.372
skewness 0.005 0.067 3.320 0.633 0.322 0.124 1.042
skew.2SE 0.008 0.127 16.266 1.001 0.410 0.258 1.753
kurtosis -0.391 -0.825 15.143 -1.009 -0.142 0.220 1.441
kurt.2SE -0.306 -0.789 37.156 -0.810 -0.093 0.231 1.229
normtest.W 0.991 0.980 0.684 0.889 0.965 0.993 0.937
normtest.p 0.954 0.239 0.000 0.000 0.310 0.886 0.002

Question 202.1.2: Write the code necessary to test again the normality of leic_2011OAC_20to24 for the supergroups where the analysis conducted for Question 202.1.1 indicated they are normal, using the function shapiro.test, and draw the respective Q-Q plot.

Question 202.1.3: Observe the output of the Levene’s test executed below. What does the result tell you about the variance of perc_age_20_to_24 in supergroups?

Note that the leveneTest was not designed to work with a Tidyverse approach. As such, the code below uses the . argument placeholder to specify that the input table leic_2011OAC_20to24 which is coming down from the pipe should be used as argument for the data parameter.

leic_2011OAC_20to24 %>%
leveneTest(
perc_age_20_to_24 ~ supgrpname,
data = .
)
## Levene's Test for Homogeneity of Variance (center = median)
##        Df F value    Pr(>F)
## group   6  62.011 < 2.2e-16 ***
##       962
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1`

by Stefano De Sabbata – text licensed under the CC BY-SA 4.0, contains public sector information licensed under the Open Government Licence v3.0, code licensed under the GNU GPL v3.0.