Psy 612: Data Analysis II

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# Psy 612: Data Analysis II

---

## Welcome back!

**Last term:**

- Probability, sampling, hypothesis testing
- Descriptive statistics
- A little matrix algebra

**This term:**

- Model building

--
- Correlations
- Linear regression
- General linear model
- Multiple regresesion

---

## Website

[UOpsych.github.io/psy612/](uopsych.github.io/psy612/)

Structure of this course:
 - Lectures, Labs, Reading 
 - Weekly quizzes (T/F)
 - Homework assignments (2)
 - Final project (1)
 
If you took PSY 611 in Fall 2018 or earlier, take a look at the materials for last term:
[UOpsych.github.io/psy611/](uopsych.github.io/psy611/)

---
## Relationships

- What is the relationship between IV and DV?

- Measuring relationships depend on type of measurement

- You have primarily been working wtih categorical IVs (*t*-test, chi-square)

---
## Scatter Plot with best fit line

![](1-correlation_files/figure-html/unnamed-chunk-3-1.png)

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## Review of Dispersion

Variation (sum of squares)

`$$SS = {\sum{(x-\bar{x})^2}}$$`
`$$SS = {\sum{(x-\mu)^2}}$$`
---

## Review of Dispersion
Variance

`$$\large s^{2} = {\frac{\sum{(x-\bar{x})^2}}{N-1}}$$`

`$$\large\sigma^{2} = {\frac{\sum{(x-\mu)^2}}{N}}$$`

---

## Review of Dispersion

Standard Deviation

`$$\large s = \sqrt{\frac{\sum{(x-\bar{x})^2}}{N-1}}$$`

`$$\large\sigma = \sqrt{\frac{\sum{(x-\mu)^2}}{N}}$$`

---

class: center

Formula for standard error of the mean?

`$$\sigma_M = \frac{\sigma}{\sqrt{N}}$$`

`$$\sigma_M = \frac{\hat{s}}{\sqrt{N}}$$`

---
## Associations

- i.e., relationships
- to look at continuous variable associations we need to think in terms of how variables relate to one another

---
## Associations

Covariation (cross products)

**Sample:**

`$$\large SS = {\sum{(x-\bar{x})(y-\bar{y})}}$$`

**Population:**

`$$SS = {\sum{{(x-\mu_{x}})(y-\mu_{y})}}$$`

---
## Associations

Covariance

**Sample:**

`$$\large cov_{xy} = {\frac{\sum{(x-\bar{x})(y-\bar{y})}}{N-1}}$$`

**Population:**

`$$\large \sigma_{xy}^{2} = {\frac{\sum{(x-\mu_{x})(y-\mu_{y})}}{N}}$$`

--
>- Covariance matrix is basis for many analyses
>- What are some issues that may arise when comparing covariances?

---
## Associations
Correlations

**Sample:**

`$$\large r_{xy} = {\frac{\sum({z_{x}z_{y})}}{N}}$$`

**Population:**

`$$\large \rho_{xy} = {\frac{cov(X,Y)}{\sigma_{x}\sigma_{y}}}$$`

Many other formulas exist for specific types of data, these were more helpful when we computed everything by hand (more on this later).

---

## Correlations

- How much two variables are linearly related

- -1 to 1

- Invariant to changes in mean or scaling

- Most common (and basic) effect size measure

- Will use to build our regression model

---

## Correlations

![](1-correlation_files/figure-html/unnamed-chunk-4-1.png)

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## Conceptually

Ways to think about a correlation:

* How two vectors of numbers co-relate

* Product of z-scores
  
  + Mathematically, it is
  
* The average squared distance between two vectors in the same space

* The cosine of the angle between Y and the projected Y from X `$(\hat{Y})$`.

---

![](images/cosine.png)

---

## Statistical test

Hypothesis testing

`$$\large H_{0}: \rho_{xy} = 0$$`

`$$\large H_{A}: \rho_{xy} \neq 0$$`

Assumes:
- Observations are independent
- Symmetric bivariate distribution (joint probability distribution)

---

Univariate distributions

![](1-correlation_files/figure-html/unnamed-chunk-5-1.png)

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### Population

![](1-correlation_files/figure-html/unnamed-chunk-6-1.png)

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### Sample

![](1-correlation_files/figure-html/unnamed-chunk-7-1.png)

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### Sampling distribution?

The sampling distribution we use depends on our null hypothesis.

If our null hypothesis is the nil `$(\rho = 0)$` , then we can use a ***t*-distribution** to estimate the statistical significance of a correlation.

---
## Statistical test

Test statistic

`$$\large t = {\frac{r}{SE_{r}}}$$`
--
.pull-left[
`$$\large SE_r = \sqrt{\frac{1-r^2}{N-2}}$$`

`$$\large t = {\frac{r}{\sqrt{\frac{1-r^{2}}{N-2}}}}$$`
]

.pull-right[
`$$\large DF = N-2$$`
]

---
## Power calculations

```r
library(pwr)
pwr.r.test(n = , r = .1, sig.level = .05 , power = .8)
```

```
## 
##      approximate correlation power calculation (arctangh transformation) 
## 
##               n = 781.7516
##               r = 0.1
##       sig.level = 0.05
##           power = 0.8
##     alternative = two.sided
```

```r
pwr.r.test(n = , r = .3, sig.level = .05 , power = .8)
```

```
## 
##      approximate correlation power calculation (arctangh transformation) 
## 
##               n = 84.07364
##               r = 0.3
##       sig.level = 0.05
##           power = 0.8
##     alternative = two.sided
```

---
## Power calculations
- But what is your confidence?

- N = 84 gives you CI[.09, .48]

- Schönbrodt & Perugini (2013) suggest correlations 'stabilize' at 250+ regardless of effect size

- CI[.19, .39]

---
## Fisher's r to z' transformation
.left-column[
If we want to make calculations based on `$\rho \neq 0$` then we will run into a skewed sampling distribution.
]

![](1-correlation_files/figure-html/unnamed-chunk-9-1.png)

---
## Fisher's r to z' transformation

- Skewed sampling distribution will rear its head when:

* `$H_{0}: \rho \neq 0$`

* Calculating confidence intervals

* Testing two correlations against one another

---

![](1-correlation_files/figure-html/unnamed-chunk-10-1.png)

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## Fisher’s r to z’ transformation

- r to z':

`$$\large z^{'} = {\frac{1}{2}}ln{\frac{1+r}{1-r}}$$`

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## Fisher’s r to z’ transformation

![](1-correlation_files/figure-html/unnamed-chunk-11-1.png)

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##  Steps for computing confidence interval

1. Transform r into z'
2. Compute CI as you normally would using z'
3. revert back to r

$$ SE_z = \frac{1}{\sqrt{N-3}}$$

`$$\large r = {\frac{e^{2z'}-1}{e^{2z'}+1}}$$`
---

### Example

In a sample of 42 students, you calculate a correlation of 0.44 between hours spent outside on Saturday and self-rated health. What is the precision of your estimate?

.pull-left[
`$$z = {\frac{1}{2}}ln{\frac{1+.44}{1-.44}} = 0.47$$`
`$$SE_z = \frac{1}{\sqrt{42-3}} = 0.16$$`

`$$CI_{Z_{LB}} = 0.47-(2.021)0.16 = 0.15$$`

`$$CI_{Z_{UB}} = 0.47+(2.021)0.16 = 0.8$$`
]

---

`$$CI_{r_{LB}} = {\frac{e^{2(0.15)}-1}{e^{2(0.15)}+1}} = 0.15$$`

`$$CI_{r_{UB}} = = {\frac{e^{2(0.8)}-1}{e^{2(0.8)}+1}} = 0.66$$`

---

## Comparing two correlations

Again, we use the Fisher’s r to z’ transformation. Here, we're transforming the correlations into z's, then using the difference between z's to calculate the test statistic.

`$$Z = \frac{z_1^{'}- z_2^{'}}{se_{z_1-z_2}}$$`

`$$se_{z_1-z_2} = \sqrt{se_{z_1}+se_{z_2}} = \sqrt{\frac{1}{n_1-3}+\frac{1}{n_2-3}}$$`
---

## Example

Replication of Hill et al. (2012) where they found that the correlation between narcissism and happiness was greater for young adults compared to older adults

.pull-left[
### Young adults
`$$N = 327$$`
`$$r = .402$$`
]

.pull-right[
### Older adults
`$$N = 273$$`
`$$r = .283$$`
]

`$$H_0:\rho_1 = \rho_2$$`
`$$H_1:\rho_1 \neq \rho_2$$`

---

`$$z_1^{'} = {\frac{1}{2}}ln{\frac{1+.402}{1-.402}} = 0.426$$`

`$$z_2^{'} = {\frac{1}{2}}ln{\frac{1+.283}{1-.283}} = 0.291$$`

`$$se_{z_1-z_2} = \sqrt{\frac{1}{327-3}+\frac{1}{273-3}} = 0.082$$`

`$$\text{Test statistic} = \frac{z_1^{'}-z_2^{'}}{se_{z_1-z_2}} = \frac{0.426- 0.291}{0.082} = 1.639$$`

```r
pnorm(abs(zstat), lower.tail = F)*2
```

```
## [1] 0.1011256
```

---
## Effect size

- The strength of relationship between two variables

- `$\eta^2$`, cohen’s d, cohen’s f, hedges g, `$R^2$` , Risk-ratio, etc

- Significance is a function of effect size and sample size

- Statistical significance `$\neq$` practical significance

---
## Effect size
How big is practical?

- Cohen (.1, .3., .5)
- Meyer & Hemphill .3 is average
- Rosenthaul:

Drug TX?  | Alive |  Dead
----------|-------|--------
Treatment |  65   |  35
No Tx     |  35   |  65

---
## What is the size of the correlation?
- Chemotherapy and breast cancer survival?
- Batting ability and hit success on a single at bat?
- Antihistamine use and reduced sneezing/runny nose?
- Combat exposure and PTSD?
- Ibuprofen on pain reduction?
- Gender and weight?
- Therapy and well being?
- Observer ratings of attractiveness?
- Gender and arm strength?
- Nearness to equator and daily temperature for U.S.?

---
## What is the size of the correlation?
- Chemotherapy and breast cancer survival? (.03)
- Batting ability and hit success on a single at bat? (.06)
- Antihistamine use and reduced sneezing/runny nose? (.11)
- Combat exposure and PTSD? (.11)
- Ibuprofen on pain reduction? (.14)
- Gender and weight? (.26)
- Therapy and well being? (.32)
- Observer ratings of attractiveness? (.39)
- Gender and arm strength? (.55)
- Nearness to equator and daily temperature for U.S.? (.60)

---
## Questions to ask yourself:
- What is your N?
- What is the typical effect size in the field?
- Study design?
- What is your DV?
- Importance?
- Same method as IV (method variance)?

---
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## Next time...

More correlations