Univariate Analysis

class: center, middle, inverse, title-slide

.title[
# Univariate Analysis
]
.subtitle[
## Description and Visualization
]
.author[
### Prof. Weldzius
]
.institute[
### Villanova University
]

---

# Agenda

1. Definitions and scope

2. Opening and defining the data

3. Variable classes

4. Univariate description

---

# Definition

- .red[Uni] + .blue[variate]

- .red[One] + .blue[variable]
  
--

- Analysis of one variable
  
---

# Scope

- How to analyze a single variable?

- How to think .blue[scientifically]?

- Typically, scientific theories concern more than one variable
  
--

- I.e., education + wages; gender + voting

- What might be a theory about education in isolation?
  
--

- Is there no point to univariate analysis?

---

# Univariate Analysis is **ESSENTIAL**

- Both from a .red[practical data] perspective...

- Informs how we "wrangle" the data

- ...and from a .blue[scientific theory] perspective

- Generates hypotheses
  
--

---

# Set-up and Load Data

- As always, create your topic folder first

---

# Set-up and Load Data

- Open `R` via `RStudio` and `require(tidyverse)`

``` r
require(tidyverse)
```

- Load data from [github](https://github.com/rweldzius/PSC4175_SUM2025/raw/main/Data/nba_players_2018.Rds)

``` r
nba <- read_rds('https://github.com/rweldzius/PSC4175_SUM2025/raw/main/Data/nba_players_2018.Rds')
```

---

# Introducing the data

- Data on every NBA player active in the 2018-2019 season

| Name               |                          Definition |
|--------------------|------------------------------------:|
| namePlayer         |                         Player name |
| idPlayer           |                    Unique player id |
| slugSeason         |                Season start and end |
| numberPlayerSeason |        Which season for this player |
| isRookie           |        Rookie season, true or false |
| slugTeam           |                     Team short name |
| idTeam             |                      Unique team id |
| gp                 |                        Games Played |
| ...                |                                 ... |

- [`psc4175_hw_4.html`](https://github.com/rweldzius/PSC4175_SUM2025/raw/main/Homeworks/psc4175_hw_4.html) has the full codebook

---

# Thinking like a .blue[scientist]

- What questions do we have? What hypotheses might we want answered?

- Overwhelming? Let's start simpler

- Total points (`pts`)

- What does this measure?
  - What kind of variable is it?

``` r
glimpse(nba %>% select(pts))
```

```
## Rows: 530
## Columns: 1
## $ pts <dbl> 1727, 17, 1108, 165, 729, 37, 211, 32, 108, 7,…
```

---

# Thinking like a .blue[scientist]

- How can we analyze a single variable?

- Want to **summarize** it somehow

- For example, look at the `mean()` and the `median()`

``` r
nba %>%
  summarise(mean_pts = mean(pts,na.rm=T),
            med_pts = median(pts,na.rm=T))
```

```
## # A tibble: 1 × 2
##   mean_pts med_pts
##      <dbl>   <dbl>
## 1     516.     419
```

---

# Thinking like a .blue[scientist]

- Or we could summarise the overall distribution with `summary()`

``` r
summary(nba$pts)
```

```
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     0.0   115.0   419.0   516.2   759.5  2818.0
```

- In English:

- There is at least one player who didn't score at all (`Min.`)
  
  - At least one player scored 2,818 points (`Max.`)
  
  - 25% of players scored less than 115 points (`1st Qu.`)
  
--

- 25% of players scored more than ???
  
--

- What does a decimal mean here?
  
---

# Visualization

- We could try and remember all these statements

- Or we could just visualize the data

``` r
nba %>%
  ggplot(aes(x = pts)) + 
  geom_histogram()
```

---

# Visualization

- Plotting the histogram reveals some things!

- There are **MANY** players who didn't score any points
  - There are **VERY FEW** who scored many
  
--

- We can combine the substantive interpretation with the visualization by plotting vertical lines for the quartiles

- A "quartile" is 25% increments
  
--

- A "decile" is 10% increments, a "quantile" is 20% increments
  
--

- A "percentile" is 1% increments

---

# Visualization

``` r
nba %>%
  ggplot(aes(x = pts)) + 
  geom_histogram() + 
* geom_vline(xintercept = quantile(nba$pts,c(.25,.5,.75)))
```

---

# Visualization

- We can save and update plots using the object assignment operator `<-`

``` r
p <- nba %>%
  ggplot(aes(x = pts)) + 
  geom_histogram()

p <- p + geom_vline(xintercept = quantile(nba$pts,c(.25,.5,.75)),linetype = 'dashed',color = 'red')

p <- p + xlab('Total Points') + ylab('Number of Players')

p <- p + theme(panel.background = element_rect(fill = 'white'))

p <- p + labs(title = 'Points by Players',subtitle = '2018-2019 NBA Season')
```

---

# Visualization

``` r
p
```

---

# Visualization

- Let's add Caitlin Clark to this!

``` r
p <- nba %>%
  ggplot(aes(x = pts)) + 
  geom_histogram()

p <- p + geom_vline(xintercept = quantile(nba$pts,c(.25,.5,.75)),linetype = 'dashed',color = 'red')

p <- p + geom_vline(xintercept = 761,linetype = 'dashed',color = 'blue') +
  annotate("text", x=950, y=90, label= "Caitlin Clark", color='blue')

p <- p + xlab('Total Points') + ylab('Number of Players')

p <- p + theme(panel.background = element_rect(fill = 'white'))

p <- p + labs(title = 'Points by Players',subtitle = '2018-2019 NBA Season')
```

---

# Visualization

``` r
p
```

---

# Visualization informs .blue[science]

- Looking at the data can help generate research questions, theories, and hypotheses

- **Question:** Why do some players not score any points?
  
--

- **Theory:** Players need minutes to score points.
  
--

- **Hypothesis:** The number of points a player scores should be positively correlated with their minutes.

---

# Univariate Description

- Testing this hypothesis comes later

- For now, let's also describe the minutes variable

``` r
summary(nba$minutes)
```

```
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     1.0   322.8  1069.0  1121.6  1836.5  3028.0
```

- At minimum, every player played at least 1 minute
  
--

- Does the distribution of this variable look similar to the points?
  
---

``` r
nba %>%
  ggplot(aes(x = minutes)) + 
  geom_histogram(alpha = .3) + 
  geom_vline(xintercept = quantile(nba$minutes,c(.25,.5,.75))) + 
  labs(title = 'Minutes by Players',subtitle = '2018-2019 NBA Season',x = 'Minutes Played',y = 'Number of Players')
```

---

# Other Variables

- Thus far, `pts` and `minutes` are both `dbl`

``` r
glimpse(nba %>% select(pts,minutes))
```

```
## Rows: 530
## Columns: 2
## $ pts     <dbl> 1727, 17, 1108, 165, 729, 37, 211, 32, 108…
## $ minutes <dbl> 2687, 123, 2669, 588, 1913, 120, 416, 194,…
```

- What about other variable types?

---

# Other Variabes

``` r
glimpse(nba)
```

```
## Rows: 530
## Columns: 37
## $ namePlayer         <chr> "LaMarcus Aldridge", "Quincy Ac…
## $ idPlayer           <dbl> 200746, 203112, 203500, 203518,…
## $ slugSeason         <chr> "2018-19", "2018-19", "2018-19"…
## $ numberPlayerSeason <dbl> 12, 6, 5, 2, 1, 0, 0, 0, 0, 0, …
## $ isRookie           <lgl> FALSE, FALSE, FALSE, FALSE, FAL…
## $ slugTeam           <chr> "SAS", "PHX", "OKC", "OKC", "MI…
## $ idTeam             <dbl> 1610612759, 1610612756, 1610612…
## $ gp                 <dbl> 81, 10, 80, 31, 82, 10, 38, 19,…
## $ gs                 <dbl> 81, 0, 80, 2, 28, 1, 2, 3, 1, 0…
## $ fgm                <dbl> 684, 4, 481, 56, 280, 13, 67, 1…
## $ fga                <dbl> 1319, 18, 809, 157, 486, 39, 17…
## $ pctFG              <dbl> 0.519, 0.222, 0.595, 0.357, 0.5…
## $ fg3m               <dbl> 10, 2, 0, 41, 3, 3, 32, 6, 25, …
## $ fg3a               <dbl> 42, 15, 2, 127, 15, 12, 99, 23,…
## $ pctFG3             <dbl> 0.2380952, 0.1333333, 0.0000000…
## $ pctFT              <dbl> 0.847, 0.700, 0.500, 0.923, 0.7…
## $ fg2m               <dbl> 674, 2, 481, 15, 277, 10, 35, 5…
## $ fg2a               <dbl> 1277, 3, 807, 30, 471, 27, 79, …
## $ pctFG2             <dbl> 0.5277995, 0.6666667, 0.5960347…
## $ agePlayer          <dbl> 33, 28, 25, 25, 21, 21, 23, 22,…
## $ minutes            <dbl> 2687, 123, 2669, 588, 1913, 120…
## $ ftm                <dbl> 349, 7, 146, 12, 166, 8, 45, 4,…
## $ fta                <dbl> 412, 10, 292, 13, 226, 12, 60, …
## $ oreb               <dbl> 251, 3, 391, 5, 165, 11, 3, 3, …
## $ dreb               <dbl> 493, 22, 369, 43, 432, 15, 20, …
## $ treb               <dbl> 744, 25, 760, 48, 597, 26, 23, …
## $ ast                <dbl> 194, 8, 124, 20, 184, 13, 25, 5…
## $ stl                <dbl> 43, 1, 117, 17, 71, 1, 6, 1, 14…
## $ blk                <dbl> 107, 4, 76, 6, 65, 0, 6, 4, 5, …
## $ tov                <dbl> 144, 4, 135, 14, 121, 8, 33, 6,…
## $ pf                 <dbl> 179, 24, 204, 53, 203, 7, 47, 1…
## $ pts                <dbl> 1727, 17, 1108, 165, 729, 37, 2…
## $ urlNBAAPI          <chr> "https://stats.nba.com/stats/pl…
## $ n                  <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
## $ org                <fct> Texas, NA, Other, FC Barcelona …
## $ country            <chr> NA, NA, NA, "Spain", NA, NA, NA…
## $ idConference       <int> 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2…
```

---

# Categorical Variables

- Already introduced you to `dbl`, `fct`, `chr` and `int`

- Taking a step back: Outside `R`, data science uses "categorical" variables

1. Mutually exclusive: observations can only be in one category
  2. Exhaustive: every observation is assigned to a category
  
--

- For example, `isRookie`

1. Mutually exclusive: Players are either in their rookie season in 2018-2019, or are not
  2. Exhaustive: these categories define every player in the data
  
---

# Categorical Variables

- Categorical variables can be divided into the following sub-types

- **Ordered:** There is a sensible order (i.e., education)

- Should be arranged intuitively (i.e., LTHS, HS Degree, Some coll, etc.)
  
--

- To summarize, calculate the proportions for each category.
  
--

- If there are too many categories, use the "mode"

---

# Categorical Variables

- Categorical variables can be divided into the following sub-types

- **Ordered, Binary:** An ordered categorical variable with just two levels

- Should be arranged in intuitive order (i.e., is not a rookie / is a rookie)
  
--

- To summarize, just convert to a [0,1] number and take the mean

---

# Categorical Variables

- Categorical variables can be divided into the following sub-types

- **Unordered**: No sensible order of categories (i.e., major degree)

- Order by most commonly occurring categories
  
--

- As before, use the mode for too many categories

---

# Categorical Variables

- Categorical variables can be divided into the following sub-types

- **Unordered, Binary**: No sensible order and only two levels

---

# Categorical Variables

- Categorical variables are meaningfully different from continuous variables

- Continuous variables are ordered and can theoretically be divided into arbitrarily small measures
  
--

- Technically can be defined as either **interval** or **ratio** variables
  
--

- In practice, we rarely worry about this distinction, but we **DO** care about continuous versus categorical variables

---

# Categorical Variables

- `fct` is a class that is unique to `R`

- Meant for ordered categorical variables
  
--

- `fct` stores the order and assigns a numeric value + a definition
  
--

- Most of the time, better to store as a `chr` (but not always)
  
---

# Variables

- `R` may store categorical variables as `chr`, `fct`, `lgl`, `int`, or even `dbl`

- Continuous variables typically stored as `int` or `dbl`

- Up to the data scientist to look at the data and determine

- Simple **process**

1. Look at a few observations and make a guess about the variable type
  2. Create a plot or table based on that guess
  3. If the result is sensible, proceed. Otherwise go back to #1.
  
---

# In Practice

- Let's look at field goals (`fgm`)

- What type of variable should this be?

- *Technically* not continuous, since it can't be divided into fractions (i.e., what is 35.5 field goals?)
  
--

- But we typically don't care about this distinction
  
--

- We just want to make sure it is not a categorical variable (i.e., less than 20 FGs, 20-40 FGs...etc. would be categorical)
  
--

- To check, follow the process!

---

# The Process: #1 Look

``` r
nba %>% 
  select(namePlayer,slugTeam,fgm) %>%
  arrange(-fgm)
```

```
## # A tibble: 530 × 3
##    namePlayer            slugTeam   fgm
##    <chr>                 <chr>    <dbl>
##  1 James Harden          HOU        843
##  2 Bradley Beal          WAS        764
##  3 Kemba Walker          CHA        731
##  4 Giannis Antetokounmpo MIL        721
##  5 Kevin Durant          GSW        721
##  6 Paul George           OKC        707
##  7 Nikola Vucevic        ORL        701
##  8 LaMarcus Aldridge     SAS        684
##  9 Damian Lillard        POR        681
## 10 Karl-Anthony Towns    MIN        681
## # ℹ 520 more rows
```

---

# The Process: #2 Create

``` r
nba %>%
  ggplot(aes(x = fgm)) + 
  geom_density()
```

---

# The Process: #3 Evaluate

- Looks like a continuous variable to me!

- Summarize it!

``` r
nba %>%
  summarise(mean_fg = mean(fgm,na.rm=T),
            med_fg = median(fgm,na.rm=T))
```

```
## # A tibble: 1 × 2
##   mean_fg med_fg
##     <dbl>  <dbl>
## 1    191.    157
```

- `mean()` is more easily understood, but more sensitive to outliers

- `median()` is harder to explain to a general audience, but more sensible when there are outliers

---

# Other Variables: Use the process!

- What kind of variable is field goal percentage?

- Follow the process!

``` r
# INSERT CODE HERE
```

---

# Another example

- Player age

- What kind of variable do we think this might be?

- Continuous? It is ordered and divisible to arbitrary fractions! (Just ask any 6 and three quarters year old)
  
--

- But is it also useful to think of it as a categorical? In the context of NBA players, there aren't many categories!
  
--

- Time for the **process**!

---

# The Process: #1 Look

``` r
nba %>%
  select(namePlayer,agePlayer) %>%
  arrange(-agePlayer)
```

```
## # A tibble: 530 × 2
##    namePlayer     agePlayer
##    <chr>              <dbl>
##  1 Vince Carter          42
##  2 Dirk Nowitzki         41
##  3 Jamal Crawford        39
##  4 Udonis Haslem         39
##  5 Pau Gasol             38
##  6 Kyle Korver           38
##  7 Jose Calderon         37
##  8 Tony Parker           37
##  9 Dwyane Wade           37
## 10 Channing Frye         36
## # ℹ 520 more rows
```

---

# The Process: #2 Create

``` r
nba %>%
  ggplot(aes(x = agePlayer)) + 
  geom_density()
```

---

# The Process: #2 Create

``` r
nba %>%
  ggplot(aes(x = factor(agePlayer))) + 
  geom_bar(stat = 'count')
```

---

# The Process: #2 Create

---

# The Process: #3 Evaluate

``` r
quantile(nba$agePlayer,c(.1,.25,.5,.75,.9,.95))
```

```
## 10% 25% 50% 75% 90% 95% 
##  21  23  26  29  32  34
```

---

# Some more examples!

- Which of these variables is an unordered categorical variable?

- Follow the process and calculate which category in this variable is the most commonly occurring

``` r
# INSERT CODE HERE
```

---

# Career Prior to NBA (`org`)

- If you chose this as your unordered categorical variable, you probably saw something like the following in step #1 of the process

``` r
nba %>%
  count(org) %>%
  arrange(-n)
```

```
## # A tibble: 68 × 2
##    org                        n
##    <fct>                  <int>
##  1 <NA>                     157
##  2 Other                     85
##  3 Kentucky                  25
##  4 Duke                      17
##  5 California-Los Angeles    15
##  6 Kansas                    11
##  7 Arizona                   10
##  8 Texas                     10
##  9 North Carolina             9
## 10 Michigan                   8
## # ℹ 58 more rows
```

---

# Career Prior to NBA (`org`)

- The most commonly occurring categories are `NA` and `Other`!

- Wrangle some data and re-calculate

``` r
nba %>%
  filter(!is.na(org)) %>%
  filter(org != 'Other') %>%
  count(org) %>%
  arrange(-n)
```

```
## # A tibble: 66 × 2
##    org                        n
##    <fct>                  <int>
##  1 Kentucky                  25
##  2 Duke                      17
##  3 California-Los Angeles    15
##  4 Kansas                    11
##  5 Arizona                   10
##  6 Texas                     10
##  7 North Carolina             9
##  8 Michigan                   8
##  9 Villanova                  7
## 10 Indiana                    6
## # ℹ 56 more rows
```

---

# Categorical: Unordered, Binary

- Which variable is an unordered binary categorical?

- Follow the process and summarize it

``` r
# INSERT CODE HERE
```

---

# Categorical: Unordered, Binary (`idConference`)

- Example of the default variable class (`dbl`) not corresponding to the type of variable (unordered binary)

- Should wrangle into something better

``` r
nba <- nba %>%
  mutate(west_conference = ifelse(idConference == 1,1,0))

nba %>%
  summarise(propWest = mean(west_conference))
```

```
## # A tibble: 1 × 1
##   propWest
##      <dbl>
## 1    0.508
```

---

# A Preview of Multivariate Analysis

- Let's take a "conditional mean"

- I.e., conditional on players going to Kentucky, how many points did NBA players score in the 2018-2019 season?
  
--

- (Simpler is just to say "how many points did NBA players who went to Kentucky score?")
  
- Recall the `group_by()` command

---

# A Preview of Multivariate Analysis

``` r
nba %>%
  filter(!is.na(org)) %>%
  filter(org != 'Other') %>%
  group_by(org) %>%
  summarise(tot_pts = sum(pts,na.rm=T))
```

```
## # A tibble: 66 × 2
##    org                    tot_pts
##    <fct>                    <dbl>
##  1 Anadolu Efes S.K.         1270
##  2 Arizona                   5467
##  3 Baylor                     861
##  4 Boston College            1659
##  5 Butler                    1255
##  6 California                1942
##  7 California-Los Angeles    9061
##  8 Cincinnati                 531
##  9 Colorado                  2367
## 10 Connecticut               3634
## # ℹ 56 more rows
```

---

# A Preview of Multivariate Analysis

- Some non-college organizations snuck in there

- `Anadolu Efes S.K.` is a professional Turkish basketball team
  
---

# A Preview of Multivariate Analysis

``` r
nba %>%
  filter(!is.na(org)) %>%
  filter(org != 'Other') %>%
  filter(!str_detect(org,"CB|KK|rytas|FC|B.C.|S.K.|Madrid")) %>%
  group_by(org) %>%
  summarise(tot_pts = sum(pts,na.rm=T))
```

```
## # A tibble: 57 × 2
##    org                    tot_pts
##    <fct>                    <dbl>
##  1 Arizona                   5467
##  2 Baylor                     861
##  3 Boston College            1659
##  4 Butler                    1255
##  5 California                1942
##  6 California-Los Angeles    9061
##  7 Cincinnati                 531
##  8 Colorado                  2367
##  9 Connecticut               3634
## 10 Creighton                 1230
## # ℹ 47 more rows
```

---

# Another Preview

- Do the same but for free throw percentage (`pctFT`)

- **NB**: should you summarise with `sum()` or `mean()`? Why?

``` r
# INSERT CODE HERE
```