Hypothesis & Estimation

class: center, middle, inverse, title-slide

.title[
# Hypothesis & Estimation
]
.subtitle[
## EDP 619 Week 11
]
.author[
### Dr. Abhik Roy
]

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---

class: highlight-last-item
layout: true

---

# Welcome!

Before moving on, please make note of the following

+ This serves as a toolkit of sorts and is meant for those who have taken a first semester course in descriptive & inferential statistics

+ We will address some of the basic testing approaches and estimation statistics that you likely covered before

+ A logical step after reviewing this content is to explore the idea and need for conducting a power analysis for any statistically driven study

.footnote[As of this writing, some equations may not show up properly in Firefox. Other browsers such as Chrome and Safari do appear to render them correctly.]

---

## Essental Terms

.center2[
**Statistic** - Mathematical expression that describes some aspects of a set of scores for a sample
 
 
**Parameter** - Describes some aspect of a set of scores for a population
]

---

# First a Brief Intro to Hypothesis Testing

.pull-left[

Formally 
Testing an assumption about a population parameter 
&nbsp;

]

.pull-right[

Conversationally 
An assumption about a particular situation of the world that is testable

]

---

# Parts of a Hypothesis

.pull-left[

The Null Hypothesis 

<ul>
 <li>what is expected to happen</li> 
 <li>must be a piece of information that is known</li>
</ul> 
]

.pull-right[

Notation 

<hr style="height:5px; visibility:hidden;" />
`$$H_0$$`
]

.pull-left[

Alternative Hypothesis 

<ul>
 <li>what else could happen</li> 
 <li>may or may not be known</li>
</ul> 
]

.pull-right[

Notation 

<hr style="height:5px; visibility:hidden;" />
`$$H_1 \,\,\textrm{or}\,\, H_A$$`
]

---

# Tests of Statistical Significance

.pull-left[

Formally 
Determination if either `H_0` or `H_1` can be rejected 
&nbsp;

]

.pull-right[

Conversationally 
A test to figure out whether you can reasonably say if your initial assumption won't happen

]

<center>
 

Interpretation 
If results from a study goes the way that was expected, then nothing new was discovered1

 
</center>

.footnote[1 Notice that the term *unimportant* is not included within the Interpretation. Non results are important!]

---

## Essential Term

.center2[
A **(statistical) estimation ** is a sample statistic is used to estimate the value of an unknown population parameter
]

---

# Positive and Negative Outcomes

<center>
 

Assumption 
We assume nothing out of the ordinary is going to happen - aka `H_0` is expected

 
</center>

.pull-left[

If `H_0` happens 
then we have a negative outcome because what you expected to happen happened

]

.pull-right[

If `H_1` happens 
then we have a positive outcome because something that was expected to happen didn't happen

]

---

## Example

<center>
 

Experiment 
Over the span of one year, a group of participants with ADHD in a drug study receives a daily experimental pill that is intended to help them focus for a longer timeframe than their current medication

 
</center>

.pull-left[

If `H_0` happens 
The group of people did not report being focused for a longer timeframe than their current medication resulting in a negative outcome because that was an expected outcome

]

.pull-right[

If `H_1` happens 
The group of people did report being focused for a longer timeframe than their current medication resulting in a positive outcome because that was an unexpected outcome

]

---

# Think Big

.center2[No matter what you have heard or may be told in the future, both Type I Errors and Type II Errors are not represented by a single figure, rather they each contain a different range of probabilities]

---

# Formal Table of Statistical Error Types

.center2[

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
 <tr>
 <th style="text-align:left;background-color: #212121 !important;"> Decision </th>
 <th style="text-align:left;background-color: #212121 !important;"> Null is True </th>
 <th style="text-align:left;background-color: #212121 !important;"> Null is False </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;width: 10em; background-color: #212121 !important;"> Reject Null </td>
 <td style="text-align:left;width: 10em; background-color: #212121 !important;"> False Positive Type I Error 
</td>
 <td style="text-align:left;width: 10em; background-color: #212121 !important;"> Correct Outcome True Positive </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; background-color: #212121 !important;"> Fail to Reject Null </td>
 <td style="text-align:left;width: 10em; background-color: #212121 !important;"> Correct Outcome True Negative 
</td>
 <td style="text-align:left;width: 10em; background-color: #212121 !important;"> False Negative Type II Error </td>
 </tr>
</tbody>
</table>
]

---

# Nutshell Table of Statistical Error Types

<table class="table table-condensed" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
 <tr>
 <th style="text-align:left;background-color: #212121 !important;vertical-align: middle !important;"> </th>
 <th style="text-align:left;background-color: #212121 !important;vertical-align: middle !important;"> </th>
 <th style="text-align:left;background-color: #212121 !important;vertical-align: middle !important;"> </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;width: 12em; background-color: #212121 !important;vertical-align: middle !important;"> </td>
 <td style="text-align:left;width: 15em; background-color: #212121 !important;vertical-align: middle !important;"> </td>
 <td style="text-align:left;width: 15em; background-color: #212121 !important;vertical-align: middle !important;"> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 12em; background-color: #212121 !important;vertical-align: middle !important;"> </td>
 <td style="text-align:left;width: 15em; background-color: #212121 !important;vertical-align: middle !important;"> </td>
 <td style="text-align:left;width: 15em; background-color: #212121 !important;vertical-align: middle !important;"> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 12em; "> You changed your mind </td>
 <td style="text-align:left;width: 15em; "> but it was likely the wrong decision 
 False Positive Type I Error 
</td>
 <td style="text-align:left;width: 15em; "> and it was likely the right decision 
 Correct Outcome True Positive </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 12em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 12em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 12em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 12em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 12em; "> You didn't change your mind </td>
 <td style="text-align:left;width: 15em; "> and it was likely the right decision 
 Correct Outcome True Negative 
</td>
 <td style="text-align:left;width: 15em; "> but it was likely the wrong decision 
 False Negative Type II Error </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 12em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 12em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 12em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 12em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 <td style="text-align:left;width: 15em; "> </td>
 </tr>
</tbody>
</table>

---

## Example

---

## Term

.pull-left[

Alpha 

<ul>
 <li>rejecting `H_0` when it is true</li> 
 <li>the probability of making a Type I Error</li> 
 <li>the chance of making a wrong decision when what was initially expected to happen actually happened</li>
</ul> 
]

.pull-right[

Notation 

<hr style="height:35px; visibility:hidden;" />
`$$\alpha$$`
]

---

## Example

<center>
If an airplane
</center>

.pull-left[
<center>
looks like this
 
 
<img src="img/metal-plane.png" width="220px"/>
 
 
then a low risk of failure - small `\alpha` level - is probably acceptable
</center>
]

.pull-right[
<center>
looks like this
 
 
<img src="img/paper-plane.png" width="200px"/>
 
 
then a higher risk of failure - large `\alpha` level - is probably acceptable
</center>
]

---

## Term

.pull-left[

Beta 

<ul>
 <li>the probability of not rejecting `H_0` when it is false</li> 
 <li>the chance associated with making a Type II Error</li> 
 <li>the possibility of making a wrong decision when something unexpected happened</li>
</ul> 
]

.pull-right[

Notation 

<hr style="height:35px; visibility:hidden;" />
`$$\beta$$`
]

.pull-left[

Statistical Power 

<ul>
 <li>the probability of not rejecting `H_0` when it is false</li> 
 <li>the chance associated with NOT making a Type II Error</li> 
 <li>the possibility of making the right decision when something unexpected happened</li>
</ul> 
]

.pull-right[

Notation 

<hr style="height:35px; visibility:hidden;" />
`$$1 - \beta$$`
]

---
  
# Decision Making

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
 <tr>
 <th style="text-align:left;border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Reality </th>
 <th style="text-align:center;border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Rejected `H_0` </th>
 <th style="text-align:center;border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Did Not Reject `H_0` </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> Type I Error </td>
 <td style="text-align:center;width: 10em; "> Correct Decision </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> `H_0` is true </td>
 <td style="text-align:center;width: 10em; "> `alpha` </td>
 <td style="text-align:center;width: 10em; "> `1-alpha` </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> Level of Significance </td>
 <td style="text-align:center;width: 10em; "> Level of Confidence </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> Correct Outcome </td>
 <td style="text-align:center;width: 10em; "> Type II Error </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> `H_0` is false </td>
 <td style="text-align:center;width: 10em; "> `1-beta` </td>
 <td style="text-align:center;width: 10em; "> `beta` </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> Statistical Power! </td>
 <td style="text-align:center;width: 10em; "> Rate of a Type II Error </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 <td style="text-align:center;width: 10em; "> </td>
 </tr>
</tbody>
</table>

---

# Decision Making

| | 
--------|---------|---------
Null | `$H_0 =$` | Forecast says its NOT going to rain
Alternative | `$H_1 =$` | Something else will happen
 | |

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;border-bottom: 0;">
 <thead>
 <tr>
 <th style="text-align:left;border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Reality </th>
 <th style="text-align:left;border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Rejected forecast </th>
 <th style="text-align:left;border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Did not reject the forecast </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;width: 10em; "> Forecast was right </td>
 <td style="text-align:left;width: 10em; "> Took an umbrella AND you're dry but may look silly or possibly fancy </td>
 <td style="text-align:left;width: 10em; "> Did not take an umbrella AND you're dry </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Forecast was wrong </td>
 <td style="text-align:left;width: 10em; border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Took an umbrella AND you're dry </td>
 <td style="text-align:left;width: 10em; border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Did not take an umbrella AND you're wet </td>
 </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
Note: You could have also gotten wet from snow, a flood, etc. so again the alternative hypothesis generally does not imply the opposite!
</td></tr></tfoot>
</table>

---

# Estimation

.center2[
**(Statistical) Estimation** - a sample statistic is used to estimate the value of an unknown population parameter
]
  
---

## Selecting a Sample Mean

---

# Updating Estimation for Sample Means

.center2[
**Point estimation** - use of sample data to calculate a single *mean* value

**Interval estimation** - use of sample data to calculate a possible range of *mean* values
]

---

# The Characteristic of Hypothesis Testing and Estimation

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
 <tr>
 <th style="text-align:left;border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Question </th>
 <th style="text-align:left;border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Hypothesis Testing </th>
 <th style="text-align:left;border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Point/Interval Estimation </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;width: 10em; "> Do we know the population mean? </td>
 <td style="text-align:left;width: 10em; "> Yes its the Null hypothesis </td>
 <td style="text-align:left;width: 10em; "> No we're trying to estimate it </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> What is the process use dto determine? </td>
 <td style="text-align:left;width: 10em; "> The chance of obtaining a sample mean </td>
 <td style="text-align:left;width: 10em; "> The value of a population mean </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> What is learned? </td>
 <td style="text-align:left;width: 10em; "> Whether the population mean is likely correct </td>
 <td style="text-align:left;width: 10em; "> The range of values within which the population mean is probably contained </td>
 </tr>
 <tr>
 <td style="text-align:left;width: 10em; "> What is our decision? </td>
 <td style="text-align:left;width: 10em; "> To retain or reject the null hypothesis </td>
 <td style="text-align:left;width: 10em; "> No actual decison </td>
 </tr>
</tbody>
</table>

---

# Confidence

.center2[
**Confidence Interval** - an interval that contains an unknown parameter (e.g. `$\mu$`) with certain degree of confidence
 
 
**Level of Confidence** - probability or likelihood that an interval estimate will contain an unknown population parameter
]

---

# Determining the Confidence Interval

1. Calculate the standard error of the mean `$$\sigma_{\overline{Y}} =\dfrac{\sigma}{\sqrt{N}}$$`

2. Decide on a level of confidence

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
 <tr>
 <th style="text-align:center;border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> Probability </th>
 <th style="text-align:center;border-bottom: solid;
 border-bottom-width:1px;
 border-bottom-color: #666666;"> `z`-score </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:center;"> 0.90 </td>
 <td style="text-align:center;"> 1.645 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 0.95 </td>
 <td style="text-align:center;"> 1.96 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 0.99 </td>
 <td style="text-align:center;"> 2.576 </td>
 </tr>
</tbody>
</table>

<center>
Again its typical to have a 95% level of confidence thereby making \[\alpha = 0.05\]
</center>

---

# Determining the Confidence Interval (continued)

<ol start=3>
<li> Calculate the confidence interval \[CI = \overline{Y} \pm z\cdot\sigma_{\overline{Y}}\]
</ol>

<ol start=4>
<li> Interpret the results
</ol>

---

# Example

IQ scores in the general healthy population are approximately normally distributed with `$100 ± 15$`. In a sample of 100 students a sample mean IQ of 103. Find the 90% confidence interval for this data.

Firstly we have `$N = 100$`, `$\mu=100$`, `$\sigma = 15$`, and `$\overline{Y} = 103$`.

1. `$$\sigma_{\overline{Y}} = \dfrac{\sigma}{\sqrt{N}} =\dfrac{15}{\sqrt{100}} = 1.50$$`

<ol start=2>
<li> 
We choose a 90% level of confidence
</ol>

`$$z\cdot \sigma_{\overline{Y}} = 1.645\cdot 1.50 = 2.47$$`

<ol start=4>
<li> <center>So we are 90% confident that the overall mean IQ is between `100.53` and `105.47`</center>
</ol>

---

## Thats it!

If you have any questions, please reach out

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