35-Relationships-Two-Quant.Rmd

# (PART) Connection RQs: regression and correlation {-}


# Relationships between two quantitative variables {#TwoQuant}


<!-- Introductions; easier to separate by format -->
```{r, child = if (knitr::is_html_output())  {'./introductions/35-Relationships-Two-Quant-HTML.Rmd'} else {'./introductions/35-Relationships-Two-Quant-LaTeX.Rmd'}}
```


## Introduction: red deer {#Chap33-Intro}

So far, RQs about single variables and RQs for comparing two groups have been studied.
Comparing the mean value of a quantitative variable in two groups was studied in Sects. \@ref(CITwoMeans) and \@ref(TestTwoMeans).
Comparing the percentage of times an outcome of interest appears in two groups was studied in Sects. \@ref(OddsRatiosCI) and \@ref(TestsOddsRatio).
In this Chapter (and the next two), the relationship between two quantitative variables is studied.

Our main example in the next three chapters is a study [@data:Holgate1965:StraightLine] that examined the relationship between the age of $n = 78$ male red deer and the weight of their
`r if( knitr::is_latex_output() ) {
  'molars (Table \\@ref(tab:deerData)).'
} else {
  'molars; the data are shown below.'
}
`
The data comprises two quantitative variables.


```{r deerDataDT, fig.cap="The male red deer data", fig.align="center"}
data(RedDeer)

if( knitr::is_html_output() ) {
  DT::datatable(RedDeer,
              list(scrollX = TRUE, 
	            scrollY = TRUE, 
		          ordering = FALSE),
              caption = "The male red deer data")
}
```



\begin{figure}
\begin{minipage}{0.32\textwidth}
\captionof{table}{The first five and last five observations of the male red deer data ($n = 78$)\label{tab:deerData}}
\fontsize{10}{12}\selectfont
```{r deerData}
RDhead <- array( dim = c(11, 2) )

RDhead[1:5, ] <- as.matrix( head(RedDeer, 5) )
RDhead[7:11, ] <- as.matrix( tail(RedDeer, 5) )

RDhead[, 1] <- format( RDhead[, 1], 
                       nsmall = 1)
RDhead[, 2] <- format( RDhead[, 2], 
                       digits = 2,
                       nsmall = 2)

RDhead[6, ] <- c("$\\vdots$", "$\\vdots$")

knitr::kable( RDhead,
         format = "latex",
         booktabs = TRUE,
         longtable = FALSE,
         table.env = "@empty",
         escape = FALSE,
         digits = c(1, 2),
         align = "c",
         linesep = "", # With the \vdots, spacing doesn't seem necessary
         col.names = c( "Age (in years)", 
	                      "Molar weight (in g)")) %>%
   row_spec(0, bold = TRUE) %>%
   #kable_styling(font_size = 10) %>% # CANNOT USE THIS IN THE MINIPAGE
   column_spec(1, width = "13mm") %>%
   column_spec(2, width = "22mm")
```
\end{minipage}
\hspace{0.05\textwidth}
\begin{minipage}{0.60\textwidth}%
\centering
```{r RedDeerScatter, fig.width=5.5, out.width='95%'}
plot(Weight ~ Age, 
     data = RedDeer,
     las = 1,
     pch = 19, 
     ylim = c(0, 5.5),
     xlim = c(4, 15),
     cex = 0.8,
     main = "Molar weight and age\nof male red deer",
     xlab = "Age (in years)",
     ylab = "Molar weight (in g)")
```
\caption{Molar weight verses age for the red deer data}\label{fig:RedDeerScatter}
\end{minipage}
\end{figure}


## Two quantitative variables: graphical summaries {#RedDeerData}

<div style="float:right; width: 222x; border: 1px; padding:10px">
<img src="Illustrations/diana-parkhouse-Qf4eZXC3t2Y-unsplash.jpg" width="200px"/>
</div>


For the red deer data, both variables are *quantitative*, so the appropriate graphical summary (Sect. \@ref(Scatterplots)) is a *scatterplot* 
`r if( knitr::is_latex_output() ) {
  '(Fig. \\@ref(fig:RedDeerScatter)).'
} else {
  '(Fig. \\@ref(fig:RedDeerScatterHTML)).'
}
`
The *response* variable is graphed on the *vertical* axis, and denoted $y$; the *explanatory* variable is graphed on the *horizontal* axis,
and denoted $x$.
In some cases, when only a relationship is being explored, which variable is $x$ and which is $y$ is not important (for example, see Example \@ref(exm:CorTestDogs).)

Since the *explanatory* variable (potentially) *influences* the response variable, in this example:

* The *explanatory variable* ($x$) is the age of the deer (in years), and
* The *response variable* ($y$) is the weight of molars (in grams).

In other words, the age of the deer may influence the weight of the molars.
(Supposing that the weight of the molars may influence the age of the deer is silly.)

Each row in the dataset (and each point on the scatterplot) corresponds to a single deer (the units of analysis); two quantitative variables (age; molar weight) are measured on each deer.


<!-- The figure for LaTeX is in the minipage (combined with data table), so only need show it for the HTML -->
`r if (knitr::is_latex_output()) '<!--'`
```{r RedDeerScatterHTML, fig.cap="Molar weight verses age for the red deer data", fig.align="center", fig.width=4.5, fig.height=3.5}
plot(Weight ~ Age,
     data = RedDeer,
     las = 1,
     pch = 19,
     ylim = c(0, 5.5),
     xlim = c(4, 15),
     cex = 1,
     main = "Molar weight in male red deer\nplotted against the age of the deer",
     xlab = "Age (in years)",
     ylab = "Molar weight (in g)") 
```
`r if (knitr::is_latex_output()) '-->'`


## Understanding scatterplots {#UnderstandingScatterplots}

The purpose of a graph is to help *understand* the data (Sect. \@ref(GraphsIntro)).
For a scatterplot, the *form*, *direction*, and *variation in the relationship* (or the *strength of the relationship*) are described:

1. *Form*: The overall *form* or structure of the relationship (e.g., linear; curved upwards; etc.).
2. *Direction*: The *direction* of the relationship (sometimes not relevant if the relationship is non-linear):
    - The variables are *positively* associated if *high* values of one variable accompany *high* values of the other variable, in general.
    - The variables are *negatively* associated if *high* values of one variable accompany *low* values of the other variable, in general.
3. *Variation*: The amount of *variation* in the relationship.
   A small amount of variation in the response variable for given values of the explanatory variable means the relationship is strong; a lot of variation in the response variable for given values of the explanatory variable means the relationship is less strong.

Anything unusual or noteworthy should also be discussed.
These three features explain the *type* of relationship (*form*; *direction*), and the *strength* of that relationship (*variation*).
Examples are shown in
`r if (knitr::is_html_output()) {
  'the carousel below (click to move through the scatterplots).'
} else {
  'Fig. \\@ref(fig:ScatterplotDescriptionExamples). (The online version has extra examples.)'
}`


```{r, child = if (knitr::is_html_output())  './children/ScatterExampleCarousel.Rmd'}
```

```{r, child = if (knitr::is_latex_output())   './children/ScatterExampleImages.Rmd'}
```


::: {.example #DescribeScatterplots name="Describing scatterplots"}
A study [@data:Tager:FEV; @BIB:data:FEV] measured the lung capacity of children in Boston (using the forced expiratory volume, FEV).
The scatterplot (Fig. \@ref(fig:FEVscatter)) is curved (*form*), where older children have larger FEVs, in general (*direction*).
The *variation* gets larger for taller youth. 
:::


```{r FEVscatter, fig.cap="FEV plotted against height for children in Boston", fig.align="center", fig.width=8, fig.height=3.25, out.width='90%'}
data(LungCap)

par(mfrow = c(1, 2))

plot(FEV ~ Ht, 
     data = LungCap,
     las = 1,
     ylim = c(0, 6),
     cex = 0.7,
     main = "FEV and height",
     xlab = "Height (inches)",
     ylab = "FEV (litres)",
     pch = 19)

scatter.smooth( LungCap$Ht, LungCap$FEV,
     las = 1,
     ylim = c(0, 6),
     cex = 0.7,
     col = "grey",
     lwd = 2,
     main = "FEV and height",
     xlab = "Height (inches)",
     ylab = "FEV (litres)",
     pch = 19)

arrows(x0 = 50,
       x1 = 50,
       y0 = 0.8,
       y1 = 2.2,
       angle = 15,
       length = 0.11,
       lwd = 2,
       code = 3)
text(x = 50,
     y = 2.2,
     pos = 3,
     cex = 0.85,
     labels = "Smaller\nvariation")

arrows(x0 = 70,
       x1 = 70,
       y0 = 2,
       y1 = 5.8,
       angle = 15,
       length = 0.11,
       lwd = 2,
       code = 3)
text(x = 70,
     y = 1.8,
     pos = 1,
     cex = 0.85,
     labels = "Larger\nvariation")
```


::: {.thinkBox .think data-latex="{iconmonstr-light-bulb-2-240.png}"}
Describe the scatterplot of the diameter against the age of $385$ small-leaved lime trees (@data:ForestBiomass2017), shown in Fig. \@ref(fig:LimeTreesScatter)).\label{thinkBox:DescribeLimeTreesScatter}


`r if (knitr::is_latex_output()) '<!--'`
`r webexercises::hide()`
* **Form**: may start off straight-ish, but then seems hard to assess.
* **Direction**: biomass increases as age increases (on average). 
* **Variation**: small-ish for small ages; large-ish for older trees (after about 60 years old).
`r webexercises::unhide()`
`r if (knitr::is_latex_output()) '-->'`
:::


```{r LimeTreesScatter, fig.cap="The age and foliage biomass of small-leaved lime trees grown in Russia ($n = 385$)", fig.align="center", fig.show="hold", fig.width=8, fig.height=3.25, out.width='90%'}
data(Lime)

par( mfrow = c(1, 2))

plot(Foliage ~ Age,
	las = 1,
  main = "Foliage biomass of small-leaved\nlime trees against age",
	xlab = "Age (in years)",
	ylab = "Foliage biomass (in kg)", 
  cex = 0.8,
	pch = 19,
	xlim = c(0, 140),
	ylim = c(0, 14),
	data = Lime)

###


plot(Foliage ~ Age,
	las = 1,
  main = "Foliage biomass of small-leaved\nlime trees against age",
	xlab = "Age (in years)",
	ylab = "Foliage biomass (in kg)", 
  cex = 0.8,
	pch = 19,
	col = "grey",
	xlim = c(0, 140),
	ylim = c(0, 14),
	data = Lime)
abline( coef(lm(Foliage ~ Age, 
                data = Lime)),
        lwd = 2)
#m2 <- lm(log(DBH) ~ log(Age), 
#         data = lime)
#
#newAge <- seq( min(lime$Age), 
#               max(lime$Age),
#               length = 100)
#newDBH <- predict(m2, 
#                  newdata = data.frame(Age = newAge))
#lines( newDBH ~ newAge)
```


::: {.example #DescribeScatterplotsDeer name="Scatterplots"}
For the red deer data 
`r if (knitr::is_latex_output()) {
   '(Fig. \\@ref(fig:RedDeerScatter)),'
} else {
   '(Fig. \\@ref(fig:RedDeerScatterHTML)),'
}`
the relationship is approximately linear (form) with a negative direction (*older* deer generally have *lighter* teeth); the *variation* is... perhaps moderate.
:::


<iframe src="https://learningapps.org/watch?v=pdxmdaf1522" style="border:0px;width:100%;height:500px" allowfullscreen="true" webkitallowfullscreen="true" mozallowfullscreen="true"></iframe>



## Summary {#Chap33-Summary}

A *scatterplot* displays the relationship between two quantitative variables (the response denoted $y$; the explanatory denoted $x$).
The relationship is described by the *form* (linear, or otherwise), the *direction* of the relationship (sometimes not relevant if the graph is not linear), and the *variation* in the relationship (or the *strength* of the relationship).


## Quick review questions {#Chap33-QuickReview}

::: {.webex-check .webex-box}
A study of onion growth [@meadplant] produced the scatterplot shown in Fig. \@ref(fig:MeanScatter).


```{r MeanScatter, fig.cap="Onion yield plotted against planting density", fig.align="center", fig.width=5, fig.height=3.5}
data(YieldDen)

plot(Yield ~ Dens,
	las = 1,
	main = "Planting density and yield per plant\nfor onions",
	xlab = "Planting density (plants/sq. ft)",
	ylab = "Yield/plant (g/plant)", 
  pch = 19,
	ylim = c(0, 140),
	data = YieldDen)
```

1. The $x$-variable is \tightlist
`r if( knitr::is_html_output() ) {longmcq( c(
  answer = "Planting density",
  "Yield per plant",
  "It doesn't matter"
) )} else {"________________."}`

1. The *form* is best described as
`r if( knitr::is_html_output() ) {longmcq( c(
  "Linear",
  answer = "Curved"
) )} else {"________________."}`
1. The *direction* is best described as
`r if( knitr::is_html_output() ) {longmcq( c(
  answer = "Negative",
  "Positive"
) )} else {"________________."}`
1. The *variation* is best described as
`r if( knitr::is_html_output() ) {longmcq( c(
  answer = "Small",
  "Large",
  "Strong"
) )} else {"________________."}`
:::




## Exercises {#TwoQuantExercises}

Selected answers are available in Sect. \@ref(TwoQuantAnswer).


<!-- ::: {.exercise #TwoQuantExercisesSheepFood} -->
<!-- A study evaluated various food mixtures for sheep [@data:Moir1961:RuminantDiet]. -->
<!-- Describe the scatterplot (Fig. \@ref(fig:SheepScatter)) in terms of the *form*, *direction* and *variation* in the relationship. -->
<!-- ::: -->


<!-- ```{r SheepScatter, fig.cap="Scatterplots for the sheep-food data", fig.align="center", fig.width=5, fig.height=3.5} -->
<!-- data(ruminant) -->

<!-- plot(Energy ~ DryMatterDigest,  -->
<!--      data = ruminant, -->
<!--      las = 1, -->
<!--      xlim = c(30, 80), -->
<!--      ylim = c(1, 3.5), -->
<!--      main = "Sheep feed data", -->
<!--      xlab = "Dry matter digestibility (%)", -->
<!--      ylab = "Digestible energy (in Cal/gram)", -->
<!--      pch = 19) -->
<!-- abline( lm( Energy ~ DryMatterDigest,  -->
<!--             data = ruminant),  -->
<!--         lwd = 2,  -->
<!-- 	col = "grey") -->
<!-- ``` -->



::: {.exercise #TwoQuantExercisesSoftdrink}
A study examined the time taken to deliver soft drinks to vending machines [@others:Montgomery:regressionanalysis].
Describe the relationship (Fig. \@ref(fig:MandibleGestationPlot), left panel).
:::



::: {.exercise #TwoQuantExercisesMandible}
A study examined the mandible length and gestational age for 167 foetuses from the 12th week of gestation onward [@data:royston:mandible].
Describe the relationship (Fig. \@ref(fig:MandibleGestationPlot), right panel).
:::



```{r MandibleGestationPlot, fig.cap="Two scatterplots. Left: The time taken to deliver soft drinks to vending machines. Right: The relationship between gestational age and mandible length.", fig.align="center", fig.width=9, fig.height=3.5, out.width='100%'}
par(mfrow = c(1, 2), 
    mar = c(5.1, 5.1, 4.1, 2.1))

data(SDrink)

plot(Time ~ Cases, 
     data = SDrink,
     main = "Time to service vending machine\nand number of products stocked",
     xlab = "Number of cases of product stocked",
     ylab = "Time to service\nmachine (mins)",
     xlim = c(0, 30),
     ylim = c(0, 80),
     las = 1,
     pch = 19)
abline(coef(lm(Time ~ Cases, 
               data = SDrink)), 
       col = "grey")


### 

data(Mandible)

plot(Length ~ Age, 
     data = Mandible,
     main = "Gestational age and\nmandible length",
     xlab = "Gestational age (in weeks)",
     ylab = "Mandible length (in mm)",
     xlim = c(10, 40),
     ylim = c(0, 50),
     las = 1,
     pch = 19)
abline(coef(lm(Length ~ Age, 
            data = Mandible)), 
       col = "grey")
###


```


::: {.exercise #TwoQuantExercisesGorillas}
A study [@wright2021chest] 25 gorillas are recorded information about their chest beating and their size (measured by the breadth of the gorillas' backs).
Describe the relationship (Fig. \@ref(fig:GorillaWindmillPlot), left panel).
:::


```{r GorillaWindmillPlot, fig.cap="Two scatterplots. Left: Chest beating in gorillas; right: The relationship between DC output and wind speed.", fig.align="center", fig.width=9, fig.height=3.5, out.width='100%'}
par(mfrow = c(1, 2), 
    mar = c(5.1, 4.1, 4.1, 2.1))

data(Gorillas)

plot(NoChestBeats ~ BackBreadth, 
     data = Gorillas,
     main = "Chest beating in gorillas",
     xlab = "Back breadth (in cm)",
     ylab = "Number chest beats per minute",
     xlim = c(54, 66),
     ylim = c(0, 51),
     las = 1,
     pch = 19)


###

data(Windmill)

plot(DC ~ Wind, 
     data = Windmill,
     main = "DC output and wind speed",
     xlab = "Wind speed (in miles per hour)",
     ylab = "DC output",
     las = 1,
     xlim = c(2, 10),
     ylim = c(0, 2.5),
     pch = 19)
```




::: {.exercise #TwoQuantExercisesWindmill}
A study examined the relationship between direct current generated by a windmill and wind speed [@data:hand:handbook; @data:joglekar:lackoffit].
Describe the relationship (Fig. \@ref(fig:GorillaWindmillPlot), right panel).
:::





<!-- QUICK REVIEW ANSWERS -->
`r if (knitr::is_html_output()) '<!--'`
::: {.EOCanswerBox .EOCanswer data-latex="{iconmonstr-check-mark-14-240.png}"}
**Answers to in-chapter questions:**

- Sect. \ref{thinkBox:DescribeLimeTreesScatter}: 
  **Form**: may start off straight-ish, but then seems hard to assess.
  **Direction**: biomass increases as age increases (on average). 
  **Variation**: small-ish for small ages; large-ish for older trees (after about 60 years old).

- \textbf{Answers to \textit{Quick Revision} questions:}
**1.** Planting density.
**2.** Curved.
**3.** Negative.
**4.** Small.
:::
`r if (knitr::is_html_output()) '-->'`