Five core activities of data analysis

1. Stating and refining the question
2. Exploring the data
3. Building formal statistical models
4. Interpreting the results
5. Communicating the results

Stating and refining the question

Six types of questions

1. Descriptive: summarize a characteristic of a set of data
2. Exploratory: analyse to see if there are patterns, trends, or relationships between variables (hypothesis generating)
3. Inferential: analyse patterns, trends, or relationships in representative data from a population
4. Predictive: make predictions for individuals or groups of individuals
5. Causal: whether changing one factor will change another factor, on average, in a population
6. Mechanistic: explore "how" as opposed to whether

Eg: COVID-19 and Vitamin D

1. Descriptive: frequency of hospitalisations due to COVID-19 in a set of data collected from a group of individuals

2. Exploratory: examine relationships between a range of dietary factors and COVID-19 hospitalisations

3. Inferential: examine whether any relationship between taking Vitamin D supplements and COVID-19 hospitalisations found in the sample hold for the population at large

4. Predictive: what types of people will take Vitamin D supplements during the next year

5. Causal: whether people with COVID-19 who were randomly assigned to take Vitamin D supplements or those who were not are hospitalised

6. Mechanistic: how increased vitamin D intake leads to a reduction in the number of viral illnesses

Questions to data science problems

• Do you have appropriate data to answer your question?

• Do you have information on confounding variables?
  • Other variables that may explain the relationship seen between the two key variables.

• Was the data you're working with collected in a way that introduces bias?
  • Attempting to make conclusions about a wider group when the data is collected from a typically non-representative cohort.

Is my data biased?

Is my data biased?

• Target: average number of children in households in Edinburgh
• Data: an elementary school
• Potential biases:
  • Do all households have a child at elementary school?
  • Will AN elementary school be representative for across Edinburgh?
  • Could there be double counting? "...how many children, including themselves, ..."

Exploratory data analysis

Checklist:

• Formulate your question
• Read in your data
• Check the dimensions
• Look at the data:
  • The top few rows
  • The bottom few rows
  • Some random rows in the middle
• Validate with at least one external data source
• Make a plot
• Try the easy solution first

Formulate your question/hypothesis

• Examples of questions/hypothesis:
  • Are air pollution levels higher in Scotland compared to elsewhere in the United Kingdom?
  • Are daily average temperatures in Edinburgh higher than what they are in Glasgow? Is this relationship true throughout the year?
  • Are people's opinion about political statement X the same across the country or does it differ between between different regions or perhaps different social-economic status?
• Most importantly: "Do I have the right data to answer this question?"
  • What is the format of the data
  • How do I import it into RStudio
  • Does it need cleaning? (Most certainly Yes!)

Final advice

When doing the data analysis:

• Find some data, any data!
• Look at the data!
• Do some simple exploratory data analysis (data visualisations/summary statistics)
• Formulate your investigation question/hypothesis
• Iterate, iterate and iterate
• Let the data speak for itself!

When communicating with your audience:

• Avoid jargon, uninterpreted results and lengthy output
• Pay attention to organization, presentation and flow
• Don't forget about coding best practices (make it readable)
• Be open to suggestions and feedback
• Write as you go, don't leave it for the end!

Scientific studies

Case study: Breakfast cereal keeps girls slim

Explanatory and response variables

• Response variable
  • The key variable of interest
  • Example: BMI of the participant

• Explanatory variable(s)
  • The variable(s) that may help to describe or explain the response variable
  • Example: Whether the participant ate breakfast or not

Three possible explanations

1. Eating breakfast causes girls to be slimmer
   
2. Being slim causes girls to eat breakfast
   
3. A third variable is responsible for both -- a confounding variable: an extraneous variable that affects both the explanatory and the response variable, and that makes it seem like there is a relationship between them

Correlation != Causation

Studies and conclusions

The survey

A July 2019 YouGov survey asked 1633 GB and 1333 USA randomly selected adults which of the following statements about the global environment best describes their view:

• The climate is changing and human activity is mainly responsible
• The climate is changing and human activity is partly responsible, together with other factors
• The climate is changing but human activity is not responsible at all
• The climate is not changing

Investigation question

Probability 101

• What is the probability that a respondent says "The climate is changing and human activity is mainly responsible"?

  • Unconditional probability, $P(A)$

• What is the probability that a respondent says "The climate is changing and human activity is mainly responsible", given that the respondent is from Great Briton?

  • Conditional probability, $P(A|B)$

• If knowing that event $B$ happened, then does it change the probability of event $A$ happening?

  • If yes, then events $A$ and $B$ are said to be dependant
  • If not, then the two events are said to be independent, $P(A | B) = P(A)$

Simple calcualtions

all <- 1340 / 2966
all

## [1] 0.4517869

gb <- 833 / 1633
gb

## [1] 0.5101041

us <- 507 / 1333
us

## [1] 0.3803451

Berkeley admission data

• Study carried out by the Graduate Division of the University of California, Berkeley in the early 70’s to evaluate whether there was a gender bias in graduate admissions.
• The data come from six departments (labelled A--F).
• We have information on whether the applicant was male or female and whether they were admitted or rejected.
• First, we will evaluate whether the percentage of males admitted is indeed higher than females, overall. Next, we will calculate the same percentage for each department.

Data

## # A tibble: 4,526 × 3
##    admit    gender dept 
##    <fct>    <fct>  <ord>
##  1 Admitted Male   A    
##  2 Admitted Male   A    
##  3 Admitted Male   A    
##  4 Admitted Male   A    
##  5 Admitted Male   A    
##  6 Admitted Male   A    
##  7 Admitted Male   A    
##  8 Admitted Male   A    
##  9 Admitted Male   A    
## 10 Admitted Male   A    
## 11 Admitted Male   A    
## 12 Admitted Male   A    
## 13 Admitted Male   A    
## 14 Admitted Male   A    
## 15 Admitted Male   A    
## # ℹ 4,511 more rows

## # A tibble: 2 × 2
##   gender     n
##   <fct>  <int>
## 1 Female  1835
## 2 Male    2691

## # A tibble: 6 × 2
##   dept      n
##   <ord> <int>
## 1 A       933
## 2 B       585
## 3 C       918
## 4 D       792
## 5 E       584
## 6 F       714

## # A tibble: 2 × 2
##   admit        n
##   <fct>    <int>
## 1 Rejected  2771
## 2 Admitted  1755

Overall gender distribution

ggplot(ucbadmit, aes(y = gender, 
                     fill = admit)) +
  geom_bar(position = "fill") + 
  labs(title = "Admit proportion by gender",
       x = "Proportion", 
       y = NULL,
       fill = "Admission")

Gender distribution, by department

ggplot(ucbadmit, aes(y = gender, fill = admit)) +
  geom_bar(position = "fill") +
  facet_wrap(. ~ dept, nrow = 1) +
  labs(title = "Admissions by gender and department",
       x = "Proportion", 
       y = NULL,
       fill = "Admission")

Case for gender discrimination?

Overall:

• $P(\text{Admit} | \text{Female}) = 0.304$ and $P(\text{Admit} | \text{Male}) = 0.445$
• The proportion of applicants begin admitted is higher given that the applicant is male compared to the applicant being female.

Within departments:

• Department A:
  • $P(\text{Admit} | \text{A}, \text{Female}) = 0.824$ and $P(\text{Admit} | \text{A}, \text{Male}) = 0.621$
• Department E:
  • $P(\text{Admit} | \text{E}, \text{Female}) = 0.239$ and $P(\text{Admit} | \text{E}, \text{Male}) = 0.277$
• 4 out of 6 departments have higher admission proportions given the applicant is female.

Simpson's paradox

• Not considering an important variable when studying a relationship can result in Simpson's paradox
• Simpson's paradox illustrates the effect that omission of an explanatory variable can have on the measure of association between another explanatory variable and a response variable
• The inclusion of a third variable in the analysis can change the apparent relationship between the other two variables

Relationship between two variables

df

## # A tibble: 8 × 3
##       x     y z    
##   <dbl> <dbl> <chr>
## 1     2   4.1 A    
## 2     3   2.9 A    
## 3     4   2.1 A    
## 4     5   0.9 A    
## 5     6  10.9 B    
## 6     7  10.1 B    
## 7     8   8.9 B    
## 8     9   8.1 B

Considering a third variable

df %>% 
  summarise(cor_x_y = cor(x, y))

## # A tibble: 1 × 1
##   cor_x_y
##     <dbl>
## 1   0.683

df %>% 
  group_by(z) %>%
  summarise(cor_x_y = cor(x, y))

## # A tibble: 2 × 2
##   z     cor_x_y
##   <chr>   <dbl>
## 1 A      -0.997
## 2 B      -0.997