Correlations in R
1 Introduction
We will create Tables for Correlations, and graphs for Correlations
in R. As always, we will consistently use the Project Mosaic
ecosystem of packages in R (mosaic, mosaicData
and ggformula).
Note the standard method for all commands from the
mosaic package:
goal( y ~ x | z, data = mydata, …)
1.1 Case Study -1:
Dataset from mosaicData
Let us inspect what datasets are available in the package
mosaicData. Run this command in your Console:
# Run in Console
data(package = "mosaicData")The popup tab shows a lot of datasets we could use. Let us continue
to use the famous Galton dataset and inspect
it:
data("Galton")
inspect(Galton)
#>
#> categorical variables:
#> name class levels n missing
#> 1 family factor 197 898 0
#> 2 sex factor 2 898 0
#> distribution
#> 1 185 (1.7%), 166 (1.2%), 66 (1.2%) ...
#> 2 M (51.8%), F (48.2%)
#>
#> quantitative variables:
#> name class min Q1 median Q3 max mean sd n missing
#> 1 father numeric 62 68 69.0 71.0 78.5 69.232851 2.470256 898 0
#> 2 mother numeric 58 63 64.0 65.5 70.5 64.084410 2.307025 898 0
#> 3 height numeric 56 64 66.5 69.7 79.0 66.760690 3.582918 898 0
#> 4 nkids integer 1 4 6.0 8.0 15.0 6.135857 2.685156 898 0The inspect command already gives us a series of
statistical measures of different variables of interest. As discussed
previously, we can retain the output of inspect and use it
in our reports: (there are ways of dressing up these tables too)
galton_describe <- inspect(Galton)
galton_describe$categoricalname <chr> | class <chr> | levels <int> | n <int> | missing <int> | distribution <chr> |
|---|---|---|---|---|---|
| family | factor | 197 | 898 | 0 | 185 (1.7%), 166 (1.2%), 66 (1.2%) ... |
| sex | factor | 2 | 898 | 0 | M (51.8%), F (48.2%) |
galton_describe$quantitativename <chr> | class <chr> | min <dbl> | Q1 <dbl> | median <dbl> | Q3 <dbl> | max <dbl> | mean <dbl> | sd <dbl> | ||
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | father | numeric | 62 | 68 | 69.0 | 71.0 | 78.5 | 69.232851 | 2.470256 | |
| 2 | mother | numeric | 58 | 63 | 64.0 | 65.5 | 70.5 | 64.084410 | 2.307025 | |
| 3 | height | numeric | 56 | 64 | 66.5 | 69.7 | 79.0 | 66.760690 | 3.582918 | |
| 4 | nkids | integer | 1 | 4 | 6.0 | 8.0 | 15.0 | 6.135857 | 2.685156 |
Try help("Galton") in your Console. The dataset is
described as:
A data frame with 898 observations on the following variables.
-familya factor with levels for each family
-fatherthe father’s height (in inches)
-motherthe mother’s height (in inches)
-sexthe child’s sex: F or M
-heightthe child’s height as an adult (in inches)
-nkidsthe number of adult children in the family, or, at least, the number whose heights Galton recorded.
There is a lot of Description generated by the
mosaic::inspect() command ! What can we say about the
dataset and its variables? How big is the dataset? How many variables?
What types are they, Quant or Qual? If they are Qual, what are the
levels? Are they ordered levels? Discuss!
1.2 Correlations and Plots
What Questions might we have, that we could answer with a Statistical Measure, or Correlation chart?
Q.1 Which are the variables that have significant pair-wise correlations? What polarity are these correlations?
# Pulling out the list of Quant variables from NHANES
galton_quant <- galton_describe$quantitative
galton_quant$name
#> [1] "father" "mother" "height" "nkids"
GGally::ggpairs(
Galton,
columns = c("father", "mother", "height", "nkids"),
diag = list("densityDiag"),
title = "Galton Data Correlations Plot"
)
Can we plot a Correlogram for this dataset?
#library(corrplot)
galton_num_var <- Galton %>% select(father, mother, height, nkids)
galton_cor <- cor(galton_num_var)
galton_correlogram <- galton_cor %>%
corrplot(method = "ellipse",
type = "lower",
main = "Correlogram for Galton dataset")
galton_correlogram
#> $corr
#> father mother height nkids
#> father 1.00000000 0.07366461 0.2753548 -0.16002294
#> mother 0.07366461 1.00000000 0.2016549 -0.02002964
#> height 0.27535483 0.20165489 1.0000000 -0.12691012
#> nkids -0.16002294 -0.02002964 -0.1269101 1.00000000
#>
#> $corrPos
#> xName yName x y corr
#> 1 father father 1 4 1.00000000
#> 2 father mother 1 3 0.07366461
#> 3 father height 1 2 0.27535483
#> 4 father nkids 1 1 -0.16002294
#> 5 mother mother 2 3 1.00000000
#> 6 mother height 2 2 0.20165489
#> 7 mother nkids 2 1 -0.02002964
#> 8 height height 3 2 1.00000000
#> 9 height nkids 3 1 -0.12691012
#> 10 nkids nkids 4 1 1.00000000
#>
#> $arg
#> $arg$type
#> [1] "lower"Clearly height is positively correlated to
father and mother; interestingly,
height is negatively correlated ( slightly) with
nkids.
Q.2: Let us confirm with a test:
height ~ father:
mosaic::cor_test(height ~ father, data = Galton)
#>
#> Pearson's product-moment correlation
#>
#> data: height and father
#> t = 8.5737, df = 896, p-value < 2.2e-16
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#> 0.2137851 0.3347455
#> sample estimates:
#> cor
#> 0.2753548height ~ mother:
mosaic::cor_test(height ~ mother, data = Galton)
#>
#> Pearson's product-moment correlation
#>
#> data: height and mother
#> t = 6.1628, df = 896, p-value = 1.079e-09
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#> 0.1380554 0.2635982
#> sample estimates:
#> cor
#> 0.2016549Q.3 What does this correlation look when split by sex of
Child?
Sons ~ Parents
# For the sons
cor_test(height ~ father, data = Galton %>% filter(sex == "M"))
#>
#> Pearson's product-moment correlation
#>
#> data: height and father
#> t = 9.1498, df = 463, p-value < 2.2e-16
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#> 0.3114667 0.4656805
#> sample estimates:
#> cor
#> 0.3913174
cor_test(height ~ mother, data = Galton %>% filter(sex == "M"))
#>
#> Pearson's product-moment correlation
#>
#> data: height and mother
#> t = 7.628, df = 463, p-value = 1.367e-13
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#> 0.2508178 0.4125305
#> sample estimates:
#> cor
#> 0.3341309Daughters ~ Parents
# For the daughters
cor_test(height ~ father, data = Galton %>% filter(sex == "F"))
#>
#> Pearson's product-moment correlation
#>
#> data: height and father
#> t = 10.719, df = 431, p-value < 2.2e-16
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#> 0.3809944 0.5300812
#> sample estimates:
#> cor
#> 0.4587605
cor_test(height ~ mother, data = Galton %>% filter(sex == "F"))
#>
#> Pearson's product-moment correlation
#>
#> data: height and mother
#> t = 6.8588, df = 431, p-value = 2.421e-11
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#> 0.2261463 0.3962226
#> sample estimates:
#> cor
#> 0.3136984Q.4. How can we show this correlation in a set of Scatter Plots + Regression Lines? Can we recreate Galton’s famous diagram?
# For the sons
gf_point(height ~ father,
data = Galton %>% filter(sex == "M")) %>%
gf_smooth(method = "lm")
gf_point(height ~ mother,
data = Galton %>% filter(sex == "M")) %>%
gf_smooth(method = "lm")
# For the daughters
gf_point(height ~ father,
data = Galton %>% filter(sex == "F")) %>%
gf_smooth(method = "lm")
gf_point(height ~ mother,
data = Galton %>% filter(sex == "F")) %>%
gf_smooth(method = "lm")
An approximation to Galton’s famous plot:
gf_point(height ~ (father + mother)/2, data = Galton) %>%
gf_smooth(method = "lm") %>%
gf_density_2d(n = 8) %>%
gf_abline(slope = 1)
How would you interpret this plot?
1.3 Case Study -2:
Dataset from NHANES
We will “live code” this in class!
2 Conclusion
We have a decent Correlations related workflow in R:
- Load the dataset
inspectthe dataset, identify Quant and Qual variables- Develop Pair-Wise plots + Correlations using
GGally::ggpairs() - Develop Correlogram
corrplot::corrplot - Check everything with a
cor_test - Plot scatter plots using
gf_pointif needed. - Add extra lines using
gf_abline()to compare hypotheses that you may have.
2.1 Reference
The Farnam Street Blog. Regression Toward the Mean: An Introduction with Examples. https://fs.blog/regression-to-the-mean/
---
title: "Correlations in R"
author: "Arvind Venkatadri"
date: "`r Sys.Date()`"
output:
  rmdformats::readthedown:
    highlight: kate
    toc_float: TRUE
    toc_depth: 2
    df_print: paged
    number_sections: TRUE
    code_folding: show
    code_download: TRUE
editor_options: 
  markdown: 
    wrap: 72
---

# Introduction

We will create Tables for Correlations, and graphs for Correlations in
R. As always, we will consistently use the [**Project
Mosaic**](https://mosaic-web.org) ecosystem of packages in R (`mosaic`,
`mosaicData` and `ggformula`).

```{r, echo = FALSE, message = FALSE, include=TRUE}
knitr::opts_chunk$set(collapse = T, comment = "#>", echo = TRUE, fig.align = "center",fig.height = 4, fig.width = 6)
options(tibble.print_min = 4L, tibble.print_max = 4L)
library(mosaic) # package for stats, simulations, and basic plots
library(mosaicData) # package containing datasets
library(ggformula) # package for professional looking plots, that use the formula interface from mosaic
library(NHANES) # survey data collected by the US National Center for Health Statistics (NCHS)
library(corrplot) # For Correlogram plots

```

::: {#note .illustration style="background: beige"}
Note the standard method for all commands from the `mosaic` package:

`goal( y ~ x | z, data = mydata, …)`
:::

## Case Study -1: Dataset from `mosaicData`

Let us inspect what datasets are available in the package `mosaicData`.
Run this command in your Console:

```{r, eval=FALSE}
# Run in Console
data(package = "mosaicData")

```

The popup tab shows a lot of datasets we could use. Let us continue to
use the famous `Galton` dataset and `inspect` it:

```{r}
data("Galton")
inspect(Galton)

```

The `inspect` command already gives us a series of statistical measures
of different variables of interest. As discussed previously, we can
retain the output of `inspect` and use it in our reports: (there are
ways of dressing up these tables too)

```{r Galton-inspect}

galton_describe <- inspect(Galton)

galton_describe$categorical
galton_describe$quantitative

```

Try `help("Galton")` in your Console. The dataset is described as:

> A data frame with 898 observations on the following variables.\
> - `family` a factor with levels for each family\
> - `father` the father's height (in inches)\
> - `mother` the mother's height (in inches)\
> - `sex` the child's sex: F or M\
> - `height` the child's height as an adult (in inches)\
> - `nkids` the number of adult children in the family, or, at least,
> the number whose heights Galton recorded.

There is a lot of Description generated by the `mosaic::inspect()`
command ! What can we say about the dataset and its variables? How big
is the dataset? How many variables? What types are they, Quant or Qual?
If they are Qual, what are the *levels*? Are they *ordered* levels?
Discuss!

## Correlations and Plots

What Questions might we have, that we could answer with a Statistical
Measure, or Correlation chart?

Q.1 Which are the variables that have significant pair-wise
correlations? What polarity are these correlations?

```{r,message=FALSE}


# Pulling out the list of Quant variables from NHANES
galton_quant <- galton_describe$quantitative
galton_quant$name

GGally::ggpairs(
  Galton,
  columns = c("father", "mother", "height", "nkids"),
  diag = list("densityDiag"),
  title = "Galton Data Correlations Plot"
)

```

Can we plot a Correlogram for this dataset?

```{r}
#library(corrplot)

galton_num_var <- Galton %>% select(father, mother, height, nkids)
galton_cor <- cor(galton_num_var)
galton_correlogram <- galton_cor %>%
  corrplot(method = "ellipse",
           type = "lower",
           main = "Correlogram for Galton dataset")
galton_correlogram

```

Clearly `height` is positively correlated to `father` and `mother`;
interestingly, `height` is negatively correlated ( slightly) with
`nkids`.

Q.2: Let us confirm with a test:

`height ~ father`:

```{r}
mosaic::cor_test(height ~ father, data = Galton)
```

`height ~ mother`:

```{r}
mosaic::cor_test(height ~ mother, data = Galton)
```

Q.3 What does this correlation look when split by `sex` of Child?

`Sons ~ Parents`

```{r}

# For the sons
cor_test(height ~ father, data = Galton %>% filter(sex == "M"))
cor_test(height ~ mother, data = Galton %>% filter(sex == "M"))

```

`Daughters ~ Parents`

```{r}
# For the daughters
cor_test(height ~ father, data = Galton %>% filter(sex == "F"))
cor_test(height ~ mother, data = Galton %>% filter(sex == "F"))
         
```

Q.4. How can we show this correlation in a set of Scatter Plots +
Regression Lines? Can we recreate Galton's famous diagram?

```{r}
# For the sons
gf_point(height ~ father, 
         data = Galton %>% filter(sex == "M")) %>% 
  gf_smooth(method = "lm")

gf_point(height ~ mother, 
         data = Galton %>% filter(sex == "M")) %>% 
  gf_smooth(method = "lm")

# For the daughters
gf_point(height ~ father, 
         data = Galton %>% filter(sex == "F")) %>% 
  gf_smooth(method = "lm")

gf_point(height ~ mother, 
         data = Galton %>% filter(sex == "F")) %>% 
  gf_smooth(method = "lm")
```

An approximation to [Galton's famous
plot](https://en.wikipedia.org/wiki/Regression_toward_the_mean):

```{r}
gf_point(height ~ (father + mother)/2, data = Galton) %>% 
  gf_smooth(method = "lm") %>% 
  gf_density_2d(n = 8) %>% 
  gf_abline(slope = 1)
```

How would you interpret this plot?

## Case Study -2: Dataset from `NHANES`

We will "live code" this in class!

# Conclusion

We have a decent Correlations related *workflow* in R:\

-   Load the dataset
-   `inspect` the dataset, identify Quant and Qual variables
-   Develop Pair-Wise plots + Correlations using `GGally::ggpairs()`
-   Develop Correlogram `corrplot::corrplot`
-   Check everything with a `cor_test`
-   Plot scatter plots using `gf_point` if needed.
-   Add extra lines using `gf_abline()` to compare hypotheses that you
    may have.

## Reference

1.  <https://en.wikipedia.org/wiki/Regression_toward_the_mean>

2.  The Farnam Street Blog. *Regression Toward the Mean: An Introduction
    with Examples.* <https://fs.blog/regression-to-the-mean/>
