LECTURE 05: Estimation

.title[
# LECTURE 05: Estimation
]
.subtitle[
## ENVS475: Exp. Design and Analysis
]
.author[
### Spring 2023
]

---

# Outline

#### 1) Descriptive Statistics

#### 2) Inferential Statistics

#### 3) Confidence Intervals

---
# Review: populations vs samples

#### Population  
- A collection of subjects of interest

- Often, a biologically meaningful unit

- Sometimes a process of interest

#### Sample

- A finite subset of the population of interest, i.e. the data we collect

- Samples allow us to draw inferences about the population

- Good samples are:
    + Random  
    + Representative  
    + Sufficiently large

---

# Review: parameters vs statistics

### Parameters 
- Attributes of the population  
  + Mean ( `$\mu$` )  
  + Variance ( `$\sigma^2$` )  
  + Standard deviation ( `$\sigma$` )  
- Parameters are the quantities of interest, and usually unknown

--
### Statistics
- Attributes of the sample  
  + Mean ( `$\bar{y}$` or `$\hat{\mu}$` )  
  + Variance ( `$s^2$` or `$\hat{\sigma}^2$` )  
  + Standard deviation ( `$s$` or `$\hat{\sigma}$` )  
- Often treated as estimates of parameters
---

---
# Descriptive Statistics

### Measures of central tendency

- **Sample mean**

`$$\large \bar{y} = \frac{\sum_{i=1}^n y_i}{n}$$`

--
- **Median**: "middle" of the data

+ 50% of data is below, 50% of data is above median  
  
  + Useful for non-normal, or skewed data  
  
  + If data is truly normal, mean = median

--

- **Mode**: Most frequent observation in the data

---

### Central Tendency Example:

* Weight of Wolves (kg) from Yellowstone:
40, 35, 32, 37, 35, 37, 35, 35

* Mean (rounded): 35.8  
  * `mean()`

* Median (rounded): 35  
  * `median()`

* Mode: 35  
  * no default in function in R  
  * For small data sets, `sort()` vector and count repeats
  * 32, 35, 35, 35, 35, 37, 37, 40

---

### Measures of dispersion

- Sample variance  
  - How far, on average, are the *observations* from the mean?  
  - squared scale - different from data

`$$\large s^2 = \frac{\sum_{i=1}^n (y_i - \bar{y})^2}{n-1}$$`

--
- Sample standard deviation  
  - How far, on average, are the *observations* from the mean?  
  - same scale as data

`$$\large s = \sqrt{s^2}$$`

--
- Range  
  * min and max values of data

---
### Dispersion Example:

* Weight of Wolves (kg) from Yellowstone:
40, 35, 32, 37, 35, 37, 35, 35

* Variance: 5.4  
  * `var()`  
  * Remember, different scale!

* Standard Deviation: 2.3  
  * `sd()`  
  * same scale as data  
  * SD is mostly what we will use

* Range: 32, 40  
  * `range()`  
---
# Describing the sample

* When describing the data, always include the mean and a measure of dispersion

+ be sure to label measure of dispersion (var or SD?)
  
> The weight of Yellowstone wolves is 35.8 `$\pm$` 2.3 kg (mean `$\pm$` SD)

* Alternate:

> The average ( `$\pm$` SD)  weight of wolves in Yellowstone is 35.8 `$\pm$` 2.3 kg

---
# Inferential Statistics

* So far, we've been discussing *descriptive statistics* for our observations

* Analyses often interested in comparing means, or estimates of `$\mu$`.

* This requires *inferential statistics*  
---
# Inferential Statistics

* Weight of Wolves (kg) from Yellowstone:
40, 35, 32, 37, 35, 37, 35, 35
* Mean: 35.8  
* SD: 2.3

### How confident are we in this value? 
* If we measured another 8 wolves, what would our estimates be?

* 2nd Sample: 33, 28, 40, 43, 35, 31, 40, 42
* 2nd Mean: 36.5  
* 2nd SD: 5.5

---

# Inferential Statistics
### Standard error of the mean: SEM

* How far, on average, is the sample mean `$\bar{y}$` from the true mean `$\mu$`?

* We could repeat experiment over and over again to get a *sampling distribution*

* Rarely done due to time, logistics, funding, etc.

* Luckily, we can estimate the SEM from a single sample

---

# Inferential Statistics
### Standard error of the mean: SEM

`$$\Large SEM = \frac{s}{\sqrt{n}}$$`

* Remember, SEM tells us how far, on average, the sample mean `$\bar{y}$` is from the true mean `$\mu$`

* SEM is like a standard deviation *of the mean*

---
### Helpful(?) mnemonic

* SD: dispersion for the data (**D**)

* How far is an individual observation from the sample mean?

* SEM: dispersion for the mean (**M**)

* How far is the sample mean from the true mean?

* Note that SEM is sometimes just abbreviated as SE, so this mnemonic isn't always obvious. 
  
---
### Point estimates

* `$\bar{y}$` is a point estimate of the true mean, `$\mu$`  
* point estimates on their own are of limited value

* Always include a measure of precision when you report a mean

`$$\Large \bar{y} \pm SEM$$`

#### Wolf sample 1:  
* Mean: 35.8
* SEM: 0.8

`$35.8 \pm 0.8$` (mean `$\pm$` SEM)

* Note similarity with sample description previously

* Always label your dispersion estimate!

* You can also present the interval by completing the equation:

* Interval = 35, 36.6

---

---
# Confidence Intervals  
- Confidence Intervals (CI) is a range of estimates for an unknown parameter.  
- CI computed for different confidence levels: most often 95%  
- Assuming a normal distribution, 95% CI is approximately:

`$$\Large 95\%CI =\bar{y} \pm 2*SEM$$`

+ Recall empirical rule: 95% of data within `$\pm 2~SD$`  
  
  + In future classes we will discuss equation for CI from a *t*-distribution, i.e., modify value of 2 in above equation

- If we calculated a 95% CI from `$n$` samples, about 95% of them would contain the true population mean

+ **NOTE** it *does not mean* that we are 95% sure that the true mean is in the CI  
  
---

### Repeated samples from Population  
- 100 repeated samples of wolf weights  
- Increased to 25 wolves per sample

* 97 / 100 CIs contain true mean ~97%  
---
### Approximate 95% CI for Yellowstone Wolves

`$$\Large 95\%CI =\bar{y} \pm 2*SEM$$`
--

$$ \large \bar{y} = 35.8$$ 
--

$$ \large SEM = 0.8 $$
--

`$$\large 2 * SEM = 1.6$$`

--

`$$\Large 95\%CI = 34.2, 37.4$$`

---

### Using Confidence Intervals to test hypotheses

* Previous research has indicated that mean wolf weight in the Arctic is 33 kg.

* Does this differ from the wolves in Yellowstone?

* CI for Yellowstone wolf weight (1st sample)

`$$95\%CI =\bar{y} \pm 2*SEM$$`

`$$35.6 \pm 2 * 0.8 = (34,~37.2)$$`

* Arctic weight of 33 is outside (below) 95% CI for Yellowstone wolves.

* **Interpretation**: The data support the conclusion, with 95% confidence, that the weight of Yellowstone Wolves is between 34 and 37.2 kg, and this is higher than the mean estimated weight for wolves in the Arctic of 33 kg.

---
### Changing the Confidence Interval

* We can approximate a 99% CI by multiplying the SEM by 3

`$$99\%CI =\bar{y} \pm 3*SEM$$`

`$$35.6 \pm 3 * 0.8 = (33.2,~38)$$`

* How does the 99% CI compare with the 95% CI?  
  + *i.e.*, which is wider and narrower?  
  + Why do you think this is?  
  
---

### Using Confidence Intervals to test hypotheses

* Data from British Columbia indicates that the mean weight of wolves there is 34.2 kg.

* Do the weight of wolves in BC differ from those in Yellowstone?

* 34.2 falls within the 95% CI (34, 37.2) for Yellowstone Wolves.

* **Interpretation**: The data support the conclusion that the weight of Yellowstone Wolves is between 34 and 37.2 kg (95% CI), and that this is not different from the mean estimated weight of 34.2 for wolves in British Columbia.

* Note we did not use CI's for Arctic or BC wolves.  
  
  * Using CI's is one form of (rough) hypothesis testing  
  
  * We can more formally test these hypotheses with t-test and general linear models  
  
  * We will get to this later in the course.

---

# Looking ahead

### **Wednesday**: Point estimates and CI Lab

### **Friday**: Hw 05 - For a grade

### **Reading**: Hector Chapter 5