Bivariate Tables

class: center, middle, inverse, title-slide

.title[
# Bivariate Tables
]
.subtitle[
## EDP 613
]
.author[
### Week 9
]

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# <span style='color:#bff4ee;'>A Note About The Slides</span>

Currently the equations may not show up properly in Firefox. Other browsers such as Chrome and Safari do appear to render them correctly.

---

# Terms

**bivariate** - Doing something with two variables

**bivariate analysis**

>- *Formally*: A statistical method to detect and describe the relationship between two nominal or ordinal variables (typically independent and dependent variables)

>- *Nutshell*: Finding out if and how two variables are related to each other

**cross-tabulation**

>- *Formally*: A tool for analyzing the relationship between two or more nominal or ordinal variables

>- *Nutshell*: A data table to compare the values between two variables

>- *Note*: A good approach when establishing "control" variables

---

# Bivariate Tables

.footnote[*totals* are also known as *marginals*]

---

# Creating a Cross-Tabulation Using Raw Data

>- Column totals: Add across columns

>- Row totals: Add across rows

---

# Example of Cross-Tabulation Using Raw Data

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption><b>Views on Candy Corn</b></caption>
 <thead>
<tr><th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; padding-right: 4px; padding-left: 4px; background-color: #212121 !important;" colspan="4"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Sentiment</div></th></tr>
  <tr>
   <th style="text-align:center;background-color: #212121 !important;">  </th>
   <th style="text-align:center;background-color: #212121 !important;"> Delicious </th>
   <th style="text-align:center;background-color: #212121 !important;"> Disgusting </th>
   <th style="text-align:center;background-color: #212121 !important;">  </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;"> Yes </td>
   <td style="text-align:center;width: 10em; "> 4 </td>
   <td style="text-align:center;width: 10em; "> 7 </td>
   <td style="text-align:center;width: 10em; "> 11 </td>
  </tr>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;background-color: #212121 !important;"> No </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 6 </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 9 </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 15 </td>
  </tr>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;">  </td>
   <td style="text-align:center;width: 10em; "> 10 </td>
   <td style="text-align:center;width: 10em; "> 16 </td>
   <td style="text-align:center;width: 10em; "> 26 </td>
  </tr>
</tbody>
</table>
---

# Creating a Cross-Tabulation Using Percents

.pull-left[
<span style='color:#e5a8be;'><i>Column</i></span> <i>percentages</i> :<br><br> Use column totals as a denominator of the row values.
]

.pull-right[
<span style='color:#61b2e3;'><i>Row</i></span> <i>percentages</i> :<br><br> Use column totals as a denominator of the row values.
]

<br>
<center>
<i>Note</i>: Percentages are typically given for the independent variable.
</center>

---

# Example of Cross-Tabulation Using Percents

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption><b>Views on Candy Corn</b></caption>
 <thead>
<tr><th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; padding-right: 4px; padding-left: 4px; background-color: #212121 !important;" colspan="4"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Sentiment</div></th></tr>
  <tr>
   <th style="text-align:center;background-color: #212121 !important;">  </th>
   <th style="text-align:center;background-color: #212121 !important;"> Delicious </th>
   <th style="text-align:center;background-color: #212121 !important;"> Disgusting </th>
   <th style="text-align:center;background-color: #212121 !important;">  </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;"> Yes </td>
   <td style="text-align:center;width: 10em; "> 40.00% (4) </td>
   <td style="text-align:center;width: 10em; "> 43.75% (7) </td>
   <td style="text-align:center;width: 10em; "> 42.30% (11) </td>
  </tr>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;background-color: #212121 !important;"> No </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 60.00% (6) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 56.25% (9) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 57.69% (15) </td>
  </tr>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;"> <i>N</i> </td>
   <td style="text-align:center;width: 10em; "> (10) </td>
   <td style="text-align:center;width: 10em; "> (16) </td>
   <td style="text-align:center;width: 10em; "> (26) </td>
  </tr>
</tbody>
</table>

That is a <b>contingency table</b>

</center>
--

<br>
<center>
Specifically a <b>2 x 2 contingency table</b>
</center>

---

# Why Do We Care?

Well we use them if we want to

- *partition* the dependent and independent variables

- detect if a relationship *exists* between the dependent and independent variables

- measure how *strong* a relationship may be (known as a *measure of association*)

- determine the *direction* of a relationship

---

# This Way or That Way

The direction of a relationship can be

.pull-left[
**positive** if the dependent and independent both go in the same direction up or down
]
  
--

.pull-right[ 
**negative** if the dependent and independent go in opposite directions
]

---

# Example of a Positive Relationship

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption><b>Health Condition by SES</b></caption>
 <thead>
<tr><th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; padding-right: 4px; padding-left: 4px; background-color: #212121 !important;" colspan="4"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Sentiment</div></th></tr>
  <tr>
   <th style="text-align:center;background-color: #212121 !important;">  </th>
   <th style="text-align:center;background-color: #212121 !important;"> Low </th>
   <th style="text-align:center;background-color: #212121 !important;"> Middle </th>
   <th style="text-align:center;background-color: #212121 !important;"> High </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;background-color: #212121 !important;"> Poor </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 39% (15) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 12% (32) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 9% (18) </td>
  </tr>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;background-color: #212121 !important;"> Fair </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 36% (14) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 45% (114) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 28% (57) </td>
  </tr>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;background-color: #212121 !important;"> Good </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 25% (10) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 43% (109) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 63% (127) </td>
  </tr>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;background-color: #212121 !important;"> <i>N</i> </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> (39) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> (254) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> (202) </td>
  </tr>
</tbody>
</table>

*Source: General Social Survey: 1987-1992*

---

# Example of a Negative Relationship

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption><b>Frequency of Trauma by SES</b></caption>
 <thead>
<tr><th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; padding-right: 4px; padding-left: 4px; background-color: #212121 !important;" colspan="4"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Sentiment</div></th></tr>
  <tr>
   <th style="text-align:center;background-color: #212121 !important;">  </th>
   <th style="text-align:center;background-color: #212121 !important;"> Low </th>
   <th style="text-align:center;background-color: #212121 !important;"> Middle </th>
   <th style="text-align:center;background-color: #212121 !important;"> High </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;background-color: #212121 !important;"> Poor </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 31% (15) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 41% (90) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 48% (86) </td>
  </tr>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;background-color: #212121 !important;"> Fair </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 22% (10) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 42% (92) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 20% (36) </td>
  </tr>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;background-color: #212121 !important;"> Good </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 47% (23) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 17% (38) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> 32% (58) </td>
  </tr>
  <tr>
   <td style="text-align:center;width: 10em; font-weight: bold;background-color: #212121 !important;"> <i>N</i> </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> (48) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> (220) </td>
   <td style="text-align:center;width: 10em; background-color: #212121 !important;"> (180) </td>
  </tr>
</tbody>
</table>

*Source: General Social Survey: 1987-1992*

---

# Other Explanations

<center>hours studying & grades</center>

<br>
<center>partying & assessments</center>

<br>
<center>sleep & performance</center>

<br>
<center>Color of your car & how well you do in EDP 613</center>

---

# Elaborate

- A **control variable** is a special type of variable that doesn't change. We can use it to compare the possible effects of a treatment.

- **Elaboration** is a specific type of bivariate relationship where control variables are introduced.

---

#  Testing for an intervening relationship

- **Intervening variable** - A control variable that follows an independent variable but precedes the dependent variable in a causal sequence

- **Intervening relationships** - The control variable intervenes between the independent and dependent variables

---

# Example: Examining two variables before considering a third one

- <span style='color:#78c4a9;'>independent</span> variable: Attending weekday parties

- <span style='color:#e5a8be;'>dependent</span> variable: Grades

- <span style='color:#b5caa8;'>intervening</span></span> variable (maybe): Hours studying

---

# Example

- <span style='color:#78c4a9;'>independent</span> variable: Sale of ice cream

- <span style='color:#e5a8be;'>dependent</span> variable: Number of outdoor crimes

- <span style='color:#b5caa8;'>intervening</span></span> variable (maybe): Outdoor temperature

---

###   Testing for a spurious relationship

- **Spurious relationships** - Both the independent variable and the dependent variable are NOT

1. not causally linked
  
--

2. influenced by some third variable 
  
--

3. explained by a control variable

- **Nonspurious relationships** - Both the independent variable and the dependent variable
  - cannot by explained by a control variable

---

# Example

- <span style='color:#78c4a9;'>independent</span> variable: Number of firefighters at the scene of a crime

- <span style='color:#e5a8be;'>dependent</span> variable: Property damage

- Possible cause prior to the control variable: Size of the fire

---

# Elaborate

- A **control variable** is a special type of variable that doesn't change. We can use it to compare the possible effects of a treatment.

- **Elaboration** is a specific type of bivariate relationship where control variables are introduced.

---

# Testing

Elaboration tests

- are useless on relationships that have been determined like

- *causal* : At least one variable is found to directly effect another
  
--

- include relationships that are

- *spurious* : Both an independent and dependent variable are influenced by some third party variable. If the third variable is unknown, it may appear that there is a causal link when there actually isn't one.
  
--

- *intervening* : A control variable that comes after an independent variable but is before the dependent variable in a causal chain
  
--

- *conditional* : An independent variable’s effect on the dependent variable depends something within a control variable

---

#  Testing for a control relationship

<center>
<b>control relationship</b> - An independent variable’s effect on the dependent variable depends on, or is conditioned by, a category of a control variable
</center>

<br>
<center>
<i>Note</i>: The relationship between the independent and dependent variables will change according to the different conditions (or categories) of the control variable
</center>

---

# Example: Examining two variables before considering a control

- <span style='color:#78c4a9;'>independent</span> variable: Number of toys owned

- <span style='color:#e5a8be;'>dependent</span> variable: Hours spent playing with toys

- <span style='color:#BDA8CA;'>conditional</span></span> variable (maybe): SES

---

# Goals of Elaboration

1. *to* test for spurious relationships

2. *to* clear up the causal sequence of bivariate relationships by finding possible intervening variables

3. *to* specify the different conditions under which the original bivariate relationship might hold

---

## That's it. Take a break before our R session!

.footnote[See more ridiculous correlations at [spurious correlations](https://www.tylervigen.com/spurious-correlations)]