Correlation

class: center, middle, inverse, title-slide

# Correlation
## Cohen Chapter 9 <br><br> .small[EDUC/PSY 6600]

---

class: center, middle

## "Statistics is not a discipline like physics, chemistry, or biology where we study a subject to solve problems in the same subject. <br> We study statistics with the main aim of solving problems in other disciplines."

### -- C.R. Rao, Ph.D.

---
# Motivating Example

.large[
- Dr. Mortimer is interested in knowing whether people who have a positive view of themselves in one aspect of their lives also tend to have a positive view of themselves in other aspects of their lives. 
- He has 80 men complete a self-concept inventory that contains 5 scales. Four scales involve questions about how competent respondents feel in the areas of intimate relationships, relationships with friends, common sense reasoning and everyday knowledge, and academic reasoning and scholarly knowledge. 
- The 5th scale includes items about how competent a person feels in general. 
- 10 correlations are computed between all possible pairs of variables.
]

---
# Correlation

.pull-left[
.large[
- .dcoral[Interested in **degree** of covariation or co-relation among >1 variables measured on SAME objects/participants]
    - Not interested in group differences, per se
- Variable measurements have:
    - Order: Correlation
    - No order: Association or dependence
]]

.pull-right[.large[
- Level of measurement for each variable determines type of correlation coefficient
- Data can be in raw or standardized format
- Correlation coefficient is .nicegreen[scale-invariant]
- .bluer[Statistical significance] of correlation
    - `$\Large H_0$`: population correlation coefficient = 0
]]

---
background-image: url(figures/fig_spurious.jpeg)
background-position: 50% 50%
background-size: 1200px

.footnote[http://www.tylervigen.com/spurious-correlations]

---
# Always **Visualize** Data First

### Scatterplots

.pull-left[

*Aka: scatterdiagrams, scattergrams*

Notes:
1. Can stratify scatterplots by subgroups
2. Each subject is represented by 1 dot (x and y coordinate)
3. Fit line can indicate nature and degree of relationship (Regression or prediction lines)

```r
library(tidyverse)
df %>%
  ggplot(aes(x, y)) +
    geom_point() +
    geom_smooth(se = FALSE,
                method = "lm")
```
]

.pull-right[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-3-1.png" style="display: block; margin: auto;" />
]

---
# Correlation: Direction

.pull-left[.center[
#### .nicegreen[Positive Association]
.large[
**High values** of one variable tend to occur with **High values** of the other
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-4-1.png" style="display: block; margin: auto;" />
]]]

.pull-right[.center[
#### .bluer[Negative Association]
.large[
**High values** of one variable tend to occur with **Low values** of the other
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-5-1.png" style="display: block; margin: auto;" />
]]]

---
# Correlation: Strength

The strength of the relationship between the two variables can be seen by how much variation, or scatter, there is around the main form.

- With a strong relationship, you can get a pretty good estimate of y if you know x.
- With a weak relationship, for any x you might get a wide range of  y values.

.pull-left[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-7-1.png" style="display: block; margin: auto;" />
]

.pull-right[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-8-1.png" style="display: block; margin: auto;" />
]

---
# Scatterplot Patterns

.pull-left[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-10-1.png" style="display: block; margin: auto;" />
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-11-1.png" style="display: block; margin: auto;" />
]

.pull-right[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-12-1.png" style="display: block; margin: auto;" />
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-13-1.png" style="display: block; margin: auto;" />
]

---
# Predictability

The ability to predict `y` based on `x` is another indication of correlation strength:

.pull-left[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-14-1.png" style="display: block; margin: auto;" />
]

.pull-right[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-15-1.png" style="display: block; margin: auto;" />
]

---
# Scatterplot: Scale

.pull-left[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-16-1.png" style="display: block; margin: auto;" />
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-17-1.png" style="display: block; margin: auto;" />
]

.pull-right[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-18-1.png" style="display: block; margin: auto;" />
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-19-1.png" style="display: block; margin: auto;" />
]

Note: all have the same data! Also, `ggplot2`'s defaults are usually pretty good

???
- Using an inappropriate scale for a scatterplot can give an incorrect impression. 
- Both variables should be given a similar amount of space:
  - Plot roughly square
  - Points should occupy all the plot space (no blank space)

---
# Outliers

.pull-left[.large[
- An .dcoral[outlier] is a data value that has a very low probability of occurrence (i.e., it is unusual or unexpected).

- In a scatterplot, BIVARIATE outliers are points that fall outside of the overall pattern of the relationship.

- *Not all extreme values are outliers.*
]]

.pull-right[

]

---
## Pearson "Product Moment" Correlation Coefficient (r)

### 
.large[
- Used as a measure of: 
    - Magnitude (strength) and direction of relationship between two continuous variables
    - Degree to which coordinates cluster around STRAIGHT regression line

- Test-retest, alternative forms, and split half reliability

- Building block for many other statistical methods
]

.pull-left[.huge[.center[
Population: `$\Huge \rho$`
]]]

.pull-right[.huge[.center[
Sample: `r`
]]]

---
# Pearson "Product Moment" Correlation Coefficient (r)

.large[
- The correlation coefficient is a measure of the .dcoral[direction] and .nicegreen[strength] of a *linear* relationship. 
- It is calculated using the mean and the standard deviation of both the x and y variables. 
- Correlation can only be used to describe quantitative variables. Why?
]

.pull-left[.large[
`r` does not distinguish between x and y

`r` has no units of measurement
]]

.pull-right[.large[
`r` ranges from -1 to +1

Influential points…can change `r` a great deal!
]]

---
# Correlation: Calculating

$$ 
\LARGE r = \frac{1}{n - 1} \sum^n_{i = 1} \LARGE(\normalsize\frac{x_i - \bar{x}}{s_x}\LARGE)(\normalsize \frac{y_i - \bar{y}}{s_y}\LARGE)
$$

### Anyone want to do this by hand??

.large[
Let's use R to do this for us 
]

---
## Correlation: Calculating

.pull-left[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-23-1.png" style="display: block; margin: auto;" />

]

.pull-right[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-24-1.png" style="display: block; margin: auto;" />
]

### Same Plots -- Left is unstandardized, Right is standardized

**Standardization** allows us to compare correlations between data sets where variables are measured in different units or when variables are different. For instance, we might want to compare the correlation between [swim time and pulse], with the correlation between [swim time and breathing rate].

---
# Correlations in R Code

.pull-left[

```r
df %>%
  cor.test(~x + y,
           data = .,
           method = "pearson")
```
<br>

```r
df %>%
  furniture::tableC(x, y)
```
]

.pull-right[

```

Pearson's product-moment correlation

data:  x and y
t = 0.53442, df = 98, p-value = 0.5943
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.1440376  0.2477011
sample estimates:
       cor 
0.05390564 
```
]

```

──────────────────────────
      [1]           [2]  
 [1]x 1.00               
 [2]y 0.054 (0.594) 1.00 
──────────────────────────
```

---
# Relationship Form

.huge[Correlations only describe **linear** relationships]

.pull-left[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-31-1.png" style="display: block; margin: auto;" />
]

.pull-right[
<img src="u03_Ch9_Cor_files/figure-html/unnamed-chunk-32-1.png" style="display: block; margin: auto;" />
]

.large[
Note: You can sometimes *transform* a non-linear association to a linear form, for instance by taking the logarithm.
]

---
# Let's see it in action

## [Correlation App](http://digitalfirst.bfwpub.com/stats_applet/stats_applet_5_correg.html)

.pull-left[
.large[
- Influential Points

- Eye-ball the correlation

- Draw the line of the best fit
]]

.pull-right[
.large[
Why are correlations not resistant to outliers?

When do outliers have more *leverage*?
]]

---
background-image: url(figures/fig_bivariate_normal.png)
background-position: 80% 50%
background-size: 400px

# Assumptions

.pull-left[
.large[
1. Random Sample

2. Relationship is linear (check scatterplot, use transformations)

3. Bivariate normal distribution
    - Each variable should be normally distributed in population
    - Joint distribution should be bivariate normal
    - Curvilinear relationships = violation
    - Less important as N increases
]]

---
# Sampling Distribution of `rho`

.large[
- Normal distribution about 0
- Becomes non-normal as `$\Large \rho$` gets larger and deviates from `$\Large H_0$` value of 0 in the population
    - Negatively skewed with large, positive null hypothesized `$\rho$`
    - Positively skewed with large, negative null hypothesized `$\rho$`
- Leads to 
    - Inaccurate p-values
    - No longer testing `$\Large H_0$` that `$\Large \rho = 0$`
- Fisher's solution: transform sample `r` coefficients to yield normal sampling distribution, regardless of `$\LARGE\rho$`

*We will let the computer worry about the details...*
]

---
background-image: url(figures/fig_t_table2.png)
background-position: 80% 50%
background-size: 400px

# Hypothesis testing for 1-sample `r`

.pull-left[
.large[
$$ \LARGE H_0: \rho = 0$$
`$$\LARGE H_A: \rho \neq 0$$`

.center[`r` is converted to a t-statistic]

$$ 
\LARGE t = \frac{r\sqrt{N - 2}}{\sqrt{1 - r^2}}
$$
- Compare to t-distribution with `$\Large df = N - 2$`
  - Rejection = statistical evidence of relationship
  - Or look up critical values of `r`
]

]

---
# Example

.large[
Researcher wishes to correlate scores from 2 tests: current mood state and verbal recal memory
]

.pull-left[

```
              # A tibble: 7 x 2
                 Mood Recall
                <dbl>  <dbl>
              1    45     48
              2    34     39
              3    41     48
              4    25     27
              5    38     42
              6    20     29
              7    45     30
```

]

.pull-right[

```r
df %>%
  cor.test(~Mood + Recall,
           data = .)
```

```

Pearson's product-moment correlation

data:  Mood and Recall
t = 1.8815, df = 5, p-value = 0.1186
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2120199  0.9407669
sample estimates:
      cor 
0.6438351 
```
]

???
Talk about each piece of the output

---
# Power

.pull-left[.large[
Want to know N necessary to reject `$\Large H_0$` given an effect `$\Large \rho$` (we transform it into a `$\Large d$`)

- Determine effect size needed to detect
- Determine delta ( `$\Large \delta$` ; the value from appendix A.4 that would result in given level of power at `$\Large \alpha = .05$`)
- Solve: 
$$
\Huge(\LARGE \frac{\delta}{d}\Huge)\LARGE^2 + 1 = N
$$
]]

.pull-right[.large[

### Example

.nicegreen[Based on a pilot study, if we had a pearson correlation of .6, how many observations should I plan to study to ensure I have at least 80% power for an `$\Large \alpha = .05$`, two-tailed test?
]]]

---
# Factors Affecting Validity of `r`

.large[
- Range restriction (variance of X and/or Y)
  - r can be inflated or deflated
  - May be related to small N

- Outliers
  - `r` can be heavily influenced

- Use of heterogeneous subsamples
  - Combining data from heterogeneous groups can inflate correlation coefficient or yield spurious results by stretching out data
]

---
# Interpretation and Communcation

.huge[
Correlation `$\Large \neq$` Causation

But, correlation can be causation
]

.large[
- Can infer strength and direction; not form or prediction from r
- Can say that prediction will be better with large r, but cannot predict actual values

- Statistical significance
  - p-value heavily influenced by N
  - Need to interpret size of r-statistic, more than p-value

- APA format: r(df) = -.74, p = .006
]

---
background-image: url(figures/fig_APA_correlations.png)
background-position: 50% 98%
background-size: 1050px

# APA Style of Reporting

---
class: inverse, center, middle

# Let's Apply This to the Cancer Dataset

---
# Read in the Data

```r
library(tidyverse)    # Loads several very helpful 'tidy' packages
library(haven)        # Read in SPSS datasets
library(furniture)    # for tableC()
```

```r
cancer_raw <- haven::read_spss("cancer.sav")
```

### And Clean It

```r
cancer_clean <- cancer_raw %>% 
  dplyr::rename_all(tolower) %>% 
  dplyr::mutate(id = factor(id)) %>% 
  dplyr::mutate(trt = factor(trt,
                             labels = c("Placebo", 
                                        "Aloe Juice"))) %>% 
  dplyr::mutate(stage = factor(stage))
```

---
# R Code: Basic Correlations

.pull-left[

```r
cancer_clean %>%
  cor.test(~ totalcin + totalcw2,
           data = .,
           alternative = "two.sided",
           method = "pearson")
```

]

.pull-right[

```

Pearson's product-moment correlation

data:  totalcin and totalcw2
t = 1.5885, df = 23, p-value = 0.1258
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.09215959  0.63114058
sample estimates:
     cor 
0.314421 
```
]
---
count: false

# R Code: Basic Correlations

.pull-left[

```r
cancer_clean %>%
  cor.test(~ totalcin + totalcw2,
           data = .,
           alternative = "two.sided",
           method = "pearson")
```

```r
cancer_clean %>%
  cor.test(~ totalcin + totalcw2,
           data = .,
           alternative = "less",
           method = "pearson")
```

```r
cancer_clean %>%
  cor.test(~ totalcin + totalcw2,
           data = .,
           alternative = "greater",
           method = "pearson")
```
]

.pull-right[

```

Pearson's product-moment correlation

---
## R Code: Correlation Matrix

.pull-left[

```r
cancer_clean %>%
  furniture::tableC(totalcin, totalcw2, 
                    totalcw4, totalcw6)
```
]

.pull-right[

```r
cancer_clean %>%
  furniture::tableC(totalcin, totalcw2, 
                    totalcw4, totalcw6,
                    na.rm=TRUE)
```
]

```

─────────────────────────────────────────────────────
             [1]           [2]           [3]   [4]  
 [1]totalcin 1.00                                   
 [2]totalcw2 0.314 (0.126) 1.00                     
 [3]totalcw4 0.222 (0.287) 0.337 (0.099) 1.00       
 [4]totalcw6 NA NA         NA NA         NA NA 1.00 
─────────────────────────────────────────────────────
```

```

─────────────────────────────────────────────────────────────
             [1]           [2]           [3]           [4]  
 [1]totalcin 1.00                                           
 [2]totalcw2 0.282 (0.192) 1.00                             
 [3]totalcw4 0.206 (0.346) 0.314 (0.145) 1.00               
 [4]totalcw6 0.098 (0.657) 0.378 (0.075) 0.763 (<.001) 1.00 
─────────────────────────────────────────────────────────────
```

---
# R Code: Scatterplot with Regression Line

```r
cancer_clean %>%
  ggplot(aes(totalcin, totalcw2)) +
    geom_point() +
    geom_smooth(method = "lm")
```

---
# R Code: Scatterplot with Count

```r
cancer_clean %>%
  ggplot(aes(totalcin, totalcw2)) +
    geom_count() +
    geom_smooth(method = "lm")
```

---
class: inverse, center, middle

# Questions?

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class: inverse, center, middle

# Next Topic

### Linear Regression