Multivariate Analysis

class: center, middle, inverse, title-slide

.title[
# Multivariate Analysis
]
.subtitle[
## Part 1: Conditional Relationships
]
.author[
### Prof. Weldzius
]
.institute[
### Villanova University
]

---

# Agenda

1. Mutivariate

2. What is "conditional"?

3. Understanding Trump support

---

# Definition

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

- .red[Many] + .blue[variables]
  
--

- Analysis of multiple variables
  
--

- When we analyze **multiple** variables, we are in the world of "conditional analysis"

---

# What is .blue[conditional]?

- Put simply: "conditional" means "depending on"

- I.e., How does a variable of interest vary *depending on* some other variable?
  
--

- "Variable of interest": the **outcome** (or **dependent** variable `$Y$`)
  
--

- "Some other variable": the **explanation** or **predictor** (or **independent** variable `$X$`)
  
--

- "Vary depending on": the **relationship**
  
--

- Mapping concepts into .red[data] .blue[science]

- The .blue[relationship] between the .red[outcome] and the .red[predictor]
  
---

# What is .blue[conditional]?

- "Depending on" suggests a **causal** interpretation

- High wages "depend on" education &rarr; education **causes** high wages 
  
--

<center><img src="figs/income_unemp.png" width="90%"></center>
  
---

# What is .blue[conditional]?

- "Depending on" suggests a **causal** interpretation

- High wages "depend on" education &rarr; education **causes** high wages

- In .blue[theory], this is reasonable: students acquire skills in school which are valued by the labor market.
  
--

- But the positive correlation between education and wages might also be **"spurious"**
  
--

- Higher education *AND* higher wages are outcomes of some **true cause** (i.e., upbringing, SES, etc.)

**NOTE: The logic for why a relationship might be spurious is itself CAUSAL.**

---

# (Re-)Introducing the Data

- Using the Michigan exit poll data

- Download pre-wrangled data from [GitHub](https://github.com/rweldzius/PSC4175_SUM2025/raw/main/Data/MI2020_ExitPoll_small.rds) and save to your `data` folder.

- `require(tidyverse)` and `readRDS()` the data to `mi_ep` object

``` r
require(tidyverse)

mi_ep <- read_rds('https://github.com/rweldzius/PSC4175/raw/main/static/data/MI2020_ExitPoll_small.rds')
```

---

# Some Light .red[Data] .blue[Science]

- The "gender gap" in Trump support

- .blue[Theory]: Trump has expressed sexist views against women. Therefore, women should be less likely to support him.

- **NOTE** the causal assumptions in this theory! 
  
--

- .red[Analysis]: compare support for Trump among men and women

- But first, some quick data wrangling

``` r
MI_final_small <- mi_ep %>%
  filter(preschoice=="Donald Trump, the Republican" | preschoice=="Joe Biden, the Democrat") %>%
  mutate(BidenVoter=ifelse(preschoice=="Joe Biden, the Democrat",1,0),
         TrumpVoter=ifelse(BidenVoter==1,0,1),
         AGE10=ifelse(AGE10==99,NA,AGE10))
```

---

# Conditional Means

``` r
MI_final_small %>%
  count(preschoice,SEX) %>%
  mutate(PctSupport = n/sum(n),
         PctSupport = round(PctSupport, digits=2))
```

```
## # A tibble: 4 × 4
##   preschoice                     SEX     n PctSupport
##   <chr>                        <dbl> <int>      <dbl>
## 1 Donald Trump, the Republican     1   247       0.21
## 2 Donald Trump, the Republican     2   212       0.18
## 3 Joe Biden, the Democrat          1   304       0.26
## 4 Joe Biden, the Democrat          2   419       0.35
```

- .red[Results] are **consistent** with the .blue[theory]

- NB: .red[results] do not **prove** the .blue[theory]
  
---

# Conditional Means

- However, note that these proportions are out of *all* voters.

- This isn't directly addressing the .blue[theory]

- We want to know the proportion **of women** who supported Trump
  
--

``` r
MI_final_small %>%
  count(preschoice,SEX) %>%
* group_by(SEX) %>%
  mutate(totGender = sum(n)) %>%
  mutate(pctSupport = n / totGender)
```

```
## # A tibble: 4 × 5
## # Groups:   SEX [2]
##   preschoice                  SEX     n totGender pctSupport
##   <chr>                     <dbl> <int>     <int>      <dbl>
## 1 Donald Trump, the Republ…     1   247       551      0.448
## 2 Donald Trump, the Republ…     2   212       631      0.336
## 3 Joe Biden, the Democrat       1   304       551      0.552
## 4 Joe Biden, the Democrat       2   419       631      0.664
```
  
---

# Additional .blue[Theorizing]

- The strength of the theorized relationship might vary by age

- Younger women might be more offended by Trump's casual sexism
  
--

- Older women might be more inured to Trump's casual sexism
  
--

- .blue[Theory]: the "gender gap" will be larger among younger voters

- (But also recognize that younger Americans are generally more progressive...meaning that **both** younger men and women are more offended by Trump's casual sexism!)
  
---

# Two-Way Conditional Means

- We could just subset with `filter()`

``` r
MI_final_small %>%
* filter(AGE10==1) %>%
  group_by(SEX) %>%
  count(preschoice) %>%
  mutate(PctSupport = n/sum(n),
         PctSupport = round(PctSupport, digits=2))
```

```
## # A tibble: 4 × 4
## # Groups:   SEX [2]
##     SEX preschoice                       n PctSupport
##   <dbl> <chr>                        <int>      <dbl>
## 1     1 Donald Trump, the Republican     7       0.44
## 2     1 Joe Biden, the Democrat          9       0.56
## 3     2 Donald Trump, the Republican     1       0.06
## 4     2 Joe Biden, the Democrat         15       0.94
```

---

# Two-Way Conditional Means

- Or we could add `AGE10` to the `group_by`

``` r
MI_final_small %>%
  group_by(SEX, AGE10) %>%
  summarize(PctTrump = mean(TrumpVoter),.groups = 'drop') %>%
  mutate(PctTrump = round(PctTrump, digits =2))
```

```
## # A tibble: 22 × 3
##      SEX AGE10 PctTrump
##    <dbl> <dbl>    <dbl>
##  1     1     1     0.44
##  2     1     2     0.42
##  3     1     3     0.42
##  4     1     4     0.24
##  5     1     5     0.42
##  6     1     6     0.58
##  7     1     7     0.54
##  8     1     8     0.44
##  9     1     9     0.39
## 10     1    10     0.43
## # ℹ 12 more rows
```

---

# Two-Way Conditional Means

- A little hard to make comparisons; so let's use `spread()`

``` r
MI_final_small %>%
  group_by(SEX, AGE10) %>%
  summarize(PctTrump = mean(TrumpVoter),.groups = 'drop') %>%
  spread(SEX,PctTrump) %>% rename(Male = `1`,Female = `2`)
```

```
## # A tibble: 11 × 3
##    AGE10  Male Female
##    <dbl> <dbl>  <dbl>
##  1     1 0.438 0.0625
##  2     2 0.417 0.0714
##  3     3 0.423 0.308 
##  4     4 0.241 0.294 
##  5     5 0.419 0.484 
##  6     6 0.583 0.4   
##  7     7 0.537 0.367 
##  8     8 0.443 0.263 
##  9     9 0.395 0.311 
## 10    10 0.425 0.387 
## 11    NA 0.667 0.571
```

---

# Introducing `spread()` & `gather()`

- Data in `R` is either "long" or "wide"

- **Long**: One column for a categorical label and multiple rows
  
  - I.e., For each age group, we have one **row** for men and one **row** for women
  
--

- **Wide**: Multiple columns for each categorical label and a single row

- I.e., For each age group, we have one **column** for men and one **column** for women
  
--

- In `R`, we can switch between **wide** and **long** with two functions:

1. `spread()` (or `pivot_wider()`): converts from long to wide
  
  2. `gather()` (or `pivot_longer()`): converts from wide to long

---

# `spread()` and `gather()`

- `spread([key],[value])`

- `key`: variable containing categories to make into columns labels
  
  - `value`: variable containing values put into these new columns

---

# `spread()` and `gather()`

- `gather([key],[value],[columns])`

- `key`: name of **new column** that contains categories
  
  - `value`: values you want to put into this new column

---

# `pivot_wider()`

- `pivot_wider([names_from],[values_from])`

- `names_from`: variable containing categories to make into column labels
  
  - `values_from`: variable containing values put into these new columns
  
--

---

# OR `pivot_longer()`

- `pivot_longer([names_from],[values_from])`

- `names_from`: variable containing categories to make into column labels
  
  - `values_from`: variable containing values put into these new columns

---

# `spread()`

``` r
MI_final_small %>%
  group_by(SEX, AGE10) %>%
  summarize(PctTrump = mean(TrumpVoter),.groups = 'drop') %>%
* spread(key = SEX,value = PctTrump,fill = NA) %>%
  rename(Male = `1`,Female = `2`)
```

---

# `gather()`

``` r
MI_final_small %>%
  group_by(SEX, AGE10) %>%
  summarize(PctTrump = mean(TrumpVoter),.groups = 'drop') %>%
  spread(key = SEX,value = PctTrump,fill = NA) %>%
  rename(Male = `1`,Female = `2`) %>%
* gather(SEX,PctTrump,-AGE10)
```

```
## # A tibble: 22 × 3
##    AGE10 SEX   PctTrump
##    <dbl> <chr>    <dbl>
##  1     1 Male     0.438
##  2     2 Male     0.417
##  3     3 Male     0.423
##  4     4 Male     0.241
##  5     5 Male     0.419
##  6     6 Male     0.583
##  7     7 Male     0.537
##  8     8 Male     0.443
##  9     9 Male     0.395
## 10    10 Male     0.425
## # ℹ 12 more rows
```

---

# Save Summary for Later Use

``` r
SexAge <- MI_final_small %>%
  group_by(SEX, AGE10) %>%
  summarize(PctTrump = mean(TrumpVoter),.groups = 'drop')

SexAge %>% filter(SEX == 2)
```

```
## # A tibble: 11 × 3
##      SEX AGE10 PctTrump
##    <dbl> <dbl>    <dbl>
##  1     2     1   0.0625
##  2     2     2   0.0714
##  3     2     3   0.308 
##  4     2     4   0.294 
##  5     2     5   0.484 
##  6     2     6   0.4   
##  7     2     7   0.367 
##  8     2     8   0.263 
##  9     2     9   0.311 
## 10     2    10   0.387 
## 11     2    NA   0.571
```

---

# Conditional Categorical Analysis

- Want to know **reason** for voting for candidate by **vote choice**

- `Quality`: 4 category unordered
  
--

- `preschoice`: 2 category unordered
  
--

- Some light data wrangling

``` r
toplot <- mi_ep %>% 
    select(Quality,preschoice,SEX) %>%
    filter(grepl('Biden|Trump',preschoice)) %>%
    drop_na() %>%
    filter(Quality != "[DON'T READ] Don’t know/refused") 
```

---

# Conditional Categorical Analysis

``` r
(pReasonOverall <- toplot %>%
  ggplot(aes(x = Quality)) + 
  labs(y = "Number of Voters",x = "",
         title = "Reasons for voting for a candidate",
       subtitle = "Michigan 2020 Exit Poll") +
    geom_bar(color="black",alpha = .4))
```

---

# Conditional Categorical Analysis

- Can swap axes with `coord_flip()`

``` r
pReasonOverall + coord_flip()
```

---

# Conditional Categorical Analysis

- `fill` and `position = "dodge"` for **conditional** analysis

``` r
pReasonChoice <- toplot %>%
  ggplot(aes(x = Quality,fill = preschoice)) + 
  labs(y = "Number of Voters",x = "",
         title = "Reasons for voting for a candidate",
       subtitle = "Michigan 2020 Exit Poll",
       fill = 'Self-Reported Vote') +
    geom_bar(color="black",position = "dodge") + 
   coord_flip()
```

---

# Conditional Categorical Analysis

``` r
pReasonChoice
```

---

# Conditional Categorical Analysis

- What about if we do this by `SEX`?

``` r
toplot %>% 
    ggplot(aes(x= preschoice, fill = SEX)) +     
    labs(y = "Number of Respondents",x = "",
         title = "Vote by Respondent Sex",fill = "Sex") +
    geom_bar(position="dodge") + coord_flip()
```

---

# Be Attentive to `class()`

- How is `SEX` stored in the data?

``` r
class(mi_ep$SEX)
```

```
## [1] "numeric"
```

- Need to convert it to a `character` or `factor`

``` r
pVoteSex <- toplot %>% 
    ggplot(aes(x= preschoice, fill = factor(SEX))) +     
    labs(y = "Number of Respondents",x = "",
         title = "Vote by Respondent Sex",fill = "Sex") +
    geom_bar(position="dodge") + coord_flip()
```

---

# Be Attentive to `class()`

``` r
pVoteSex
```

- Why is this a bad visualization? **Poorly labeled legend!**