7: Allometry

class: center, middle, inverse, title-slide

.title[
# 7: Allometry
]
.subtitle[
## The Size-Shape Relationship
]
.author[
### 
]

---

### What is Allometry?

+ From greek:  `$\small\acute\alpha\lambda\lambda\omicron$` (álo) = other, different  + `$\small\mu\epsilon\tau\rho\acute\omega$` (metró) = to measure
+ Differences in body proportions of an organism
+ A departure from geometric similarity
+ The relationship between size and shape
---

### Why Is There Allometry?

+ Allometry occurs due to several reasons:
  + Physical (e.g. surface area/volume ratio)
  + Physiological
  + Biomechanical/functional
  + Selection (e.g. sexual selection and male ornaments)

<img src="LectureData/07.allometry/AllomIntro.png" width="70%" style="display: block; margin: auto;" />
---

### Allometry: Basic Concepts

.pull-left[
+ What is allometry, and compared to what?
  + **Isometry**: no change in proportions
  + **Allometry**: change in proportions

+ Easily envisioned via single trait relative to size:

`$$\small{log(Y)}=log(\beta)+\alpha{log(x)}$$`

]

.pull-right[    
![](07-Allometry_files/figure-html/unnamed-chunk-2-1.png)
]

---

### Allometry: Basic Concepts

.pull-left[
+ **Note: concepts of positive and negative allometry are only applicable to univariate traits relative to another trait! (For multivariate traits, allometry is a vector in multi-dimensional space where 'positive' and 'negative' as directions are not defined and have no meaning).**

+ **Also, it is mathematically impossible for all body parts to grow simultaneously faster (positive) or slower (negative) than total body size.**
]

.pull-right[    
![](07-Allometry_files/figure-html/unnamed-chunk-3-1.png)
]

---

### Types of Allometry

+ Allometric patterns may be observed at several levels of biological organization:
  + **Ontogenetic allometry**: Variation across individuals at differing stages or ages (e.g., growth allometry)
  + **Static allometry** : Variation across individuals in a single ontogenetic stage within a population or species
  + **Evolutionary allometry**: Variation across species

<img src="LectureData/07.allometry/AllometryTypes.png" width="70%" style="display: block; margin: auto;" />
---

### Allometry in GM Data

+ Procrustes shape variables are independent of CS when no allometry is present
  + CS: mathematically independent of shape; biological allometric effects are still there!

.pull-left[
+ Here is isometry...

<img src="LectureData/07.allometry/IsometryPlot.png" width="70%" style="display: block; margin: auto;" />
]

.pull-right[
+ and here is allometry

<img src="LectureData/07.allometry/AllometryPlot.png" width="70%" style="display: block; margin: auto;" />
]

+ How do we assess such patterns?
+ We use linear models!

---

### Multivariate Linear Regression

+ To characterize allometric patterns, we fit a model of `shape~size` as:

`$$\small\mathbf{Z}=\mathbf{X}\mathbf{\beta } +\mathbf{E}$$`

+ We evaluate the allometry model `$\small\mathbf{X}_{F}$` by comparing it to an intercept model `$\small\mathbf{X}_{R}$`:

`$\tiny\mathbf{X}_R = \begin{bmatrix} 1\\ 1\\ 1\\ 1\\ 1\\ 1 \end{bmatrix}$`  & `$\tiny\mathbf{X}_F = \begin{bmatrix} 1 & 0.3 \\ 1 & 0.5 \\ 1 & 0.2 \\ 1 & 1.2 \\ 1 & 0.7 \\ 1 & 1.1 \end{bmatrix}$`

Estimate | `$\small\mathbf{X}_{R}$`| `$\small\mathbf{X}_{F}$`
:----------- | :------------------- | :--------------------------
Coefficients |  `$\tiny\hat{\mathbf{\beta_R}}=\left ( \mathbf{X}_R^{T} \mathbf{X}_R\right )^{-1}\left ( \mathbf{X}_R^{T} \mathbf{Z}\right )$` | `$\tiny\hat{\mathbf{\beta_F}}=\left ( \mathbf{X}_F^{T} \mathbf{X}_F\right )^{-1}\left ( \mathbf{X}_F^{T} \mathbf{Z}\right )$` 
Predicted Values | `$\small\hat{\mathbf{Z}}_R=\mathbf{X}_R\hat{\mathbf{\beta}}_R$` | `$\small\hat{\mathbf{Z}}_F=\mathbf{X}_F\hat{\mathbf{\beta}}_F$`
Model Residuals | `$\small\hat{\mathbf{E}}_R=\mathbf{Z}-\hat{\mathbf{Z}}_R$` | `$\small\hat{\mathbf{E}}_F=\mathbf{Z}-\hat{\mathbf{Z}}_F$`
Model Residual Error    `$(\small{SSE})$` | `$\small\mathbf{S}_R=\hat{\mathbf{E}}_R^T\hat{\mathbf{E}}_R$` | `$\small\mathbf{S}_F=\hat{\mathbf{E}}_F^T\hat{\mathbf{E}}_F$`
---

### Assessing Significance: RRPP `$^1$`

.pull-left[
1: Fit `$\small\mathbf{X}_{R}$` for each `$\small\mathbf{X}_{F}$`; Estimate `$\small\hat{\mathbf{Z}}_{R}$` and `$\small\mathbf{E}_{R}$`

2: Permute, `$\small\mathbf{E}_{R}$`: obtain pseudo-values as: `$\small\mathbf{\mathcal{Z}} = \mathbf{\hat{Z}}_{R} + \mathbf{E}_{R}$`

3: Fit `$\small\mathbf{X}_{F}$` using `$\small\mathbf{\mathcal{Z}}$`: obtain coefficients and summary statistics

4: Calculate `$\small{F}$`-value in every random permutation (observed case counts as one permutation)
]
.pull-right[
5: For `$\small{n}$` permutations, `$\small{P} = \frac{n(F_{random} \geq F_{obs})}{n}$`

6: Calculate *effect size* as a standard deviate of the observed value in a normalized distribution of random values (helps for comparing effects within and between models); i.e.:
`$$\small{z} = \frac{
\log\left( F\right) - \mu_{\log\left(F\right)}
} {
 \sigma_{\log\left(F\right)}
}$$`
where `$\small\mu_{\log\left(F\right)}$` and `$\sigma_{\log\left(F\right)}$` are the expected value and standard deviation from the sampling distribution, respectively.
]

.footnote[
1: Collyer et al. *Heredity.* (2015); Adams & Collyer. *Evolution.* (2016); Adams & Collyer. *Evolution.* (2018)

2: For details see Shape Statistics II
]
---

### Regression: Example

.pull-left[
Does body shape covary with size in *Pecos* pupfish?

This is a hypothesis of *Allometry* (shape~size covariation)
]
.pull-right[

``` r
data(pupfish)
pupfish$logSize <- log(pupfish$CS)
pupfish$Group <- interaction(pupfish$Pop, pupfish$Sex)

fit <- procD.lm(coords ~ logSize, data = pupfish, print.progress = FALSE) 
round(anova(fit)$table, 4)
```

```
##           Df     SS     MS    Rsq      F      Z Pr(>F)    
## logSize    1 0.0140 0.0140 0.2489 17.229 4.3462  0.001 ***
## Residuals 52 0.0423 0.0008 0.7511                         
## Total     53 0.0563                                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

Great, but how does one visualize this multivariate allometry??
]
---

### Visualizing Multivariate Allometry

.pull-left[
+ need to summarize size-related variation in shape
+ Use some rotation/projection method

<img src="LectureData/07.allometry/MultiAllomPlot.png" width="80%" style="display: block; margin: auto;" />
]

.pull-right[
+ Several (complementary) solutions:

+ Regression scores (Drake and Klingenberg 2008): projection of shape data on the allometric line
`$$\small\mathbf{s}=\mathbf{Z\beta^T(\beta^T\beta)^{-1/2}}$$`

+ PC1 of predicted values (Adams and Nistri 2010): PC rotation of Ŷ to summarize variation across the shape-size relationship

`$$\small\mathbf{P_1}=svd(\mathbf{\hat{Z}})=svd(\mathbf{X(\tilde{X}^T\tilde{X})^{-1}(\tilde{X}^T\tilde{Z})})$$`
]
---

### Visualizing Multivariate Allometry (Cont.)

+ And now some shape predictions

.pull-left[

``` r
M <- mshape(pupfish$coords)
preds <- shape.predictor(fit$GM$fitted, x= pupfish$logSize, Intercept = TRUE, 
                         predmin = min(pupfish$logSize), 
                         predmax = max(pupfish$logSize)) 
```

]
.pull-right[

``` r
par(mfrow=c(1,2))
plotRefToTarget(M, preds$predmin, mag=1)
mtext("Regression Min")
plotRefToTarget(M, preds$predmax, mag=1)
mtext("Regression Max")
```

``` r
par(mfrow=c(1,1))
```
]

---

### Allometry Example: *Podarcis*

+ Effects of human-mediated disturbance (urbanization) on *Podarcis* lizards
  + 5 urban and 5 rural populations
	+ 359 individuals
	+ 28 bilateral landmarks

+ How does environmental disturbance influence development?
+ Are size-shape relationships modified in urban populations?

<img src="LectureData/07.allometry/LizardLatImage.png" width="60%" style="display: block; margin: auto;" />
---

### Allometry Example: *Podarcis*

.pull-left[

```
## 
## No curves detected; all points appear to be fixed landmarks.
```

<img src="07-Allometry_files/figure-html/unnamed-chunk-13-1.png" width="80%" style="display: block; margin: auto;" />
]
.pull-right[

``` r
fit <- procD.lm(Y.gpa$coords~CS*type, data = gdf, print.progress = FALSE)
anova(fit)$table
```

```
##            Df      SS       MS     Rsq       F      Z Pr(>F)    
## CS          1 0.05814 0.058139 0.09503 40.0581 7.2274  0.001 ***
## type        1 0.03235 0.032350 0.05287 22.2892 4.6569  0.001 ***
## CS:type     1 0.00610 0.006105 0.00998  4.2061 3.0990  0.001 ***
## Residuals 355 0.51524 0.001451 0.84212                          
## Total     358 0.61183                                           
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]
---

### Allometry Example: *Podarcis* (Cont.)

<img src="07-Allometry_files/figure-html/unnamed-chunk-15-1.png" width="50%" style="display: block; margin: auto;" />
---

### *Podarcis* Allometry: Visualization 
<img src="07-Allometry_files/figure-html/unnamed-chunk-16-1.png" width="50%" style="display: block; margin: auto;" />
---

### *Podarcis* Allometry: Visualization (Cont.)

.pull-left[

``` r
shape.gps <- coords.subset(fit$GM$fitted,type)
CS.R <- gdf$CS[which(type=="R")]
CS.U <-   gdf$CS[which(type=="U")]
mn <- Map(function(x) mshape(x),shape.gps)

predsR <- shape.predictor(shape.gps$R,x = CS.R, Intercept = TRUE,
                          predmin = min(CS.R),
                          predmax = max(CS.R))
predsU <- shape.predictor(shape.gps$U,x = CS.U, Intercept = TRUE,
                          predmin = min(CS.U),
                          predmax = max(CS.U))
M <- mshape(Y.gpa$coords)
```

]
.pull-right[
<img src="07-Allometry_files/figure-html/unnamed-chunk-18-1.png" width="80%" style="display: block; margin: auto;" />
]
---

### Post-Hoc Tests

+ Use randomization methods to test custom hypotheses
+ Do urban populations show higher deviations from their group’s trajectory? – a sign of developmental instability

<img src="LectureData/07.allometry/LizUrbRur.png" width="40%" style="display: block; margin: auto;" />
---

### Size-Shape Space

+ Size effects often dominate shape variation, especially in ontogenetic series
+ However, when other factors are present, PC1 does not necessarily relate directly to size-dependent variation (at least not to all of it)
+ Usual statistical methods do not capture subtle ontogenetic modifications in trajectories

+ Examine allometric variation in **size-shape space**
---

### Size-Shape Space

+ Derived from the PCA of shape variables + `$\small{ln}(Csize)$`
  + 1: Calculate shape variables and `$\small{ln}(Csize)$`, and join them in a single data matrix
  + 2: Calculate PCA of this matrix ( `$\small{SSPC1}$`)
  + 3: Generate the **common allometric component** (CAC): the common size-shape regression coefficient, after mean-centering shapes by group `$\small\mathbf{Z}_{c}$`
  + 4: Calculate the **residual shape component**: SVD of data orthogonal to CAC
  + 5: Visualize allometry as (CAC vs. SSPC1)   & (CAC vs. RSC1)

`$$\small\alpha=(\mathbf{Z}_{c}^T\mathbf{CS})/\mathbf{(CS^TCS)}$$`
`$$\small{CAC}=\mathbf{Z}_{c}\alpha$$`

.center[where]

`$$\small\alpha=\alpha\sqrt{(\alpha^T\alpha)}$$`
---

### Size-Shape Space: Example

<img src="LectureData/07.allometry/Mitteroecker-04.png" width="70%" style="display: block; margin: auto;" />
---

### note: CAC Scores vs. Regression Scores `$^1$`

+ `$\small{CAC}$`: regression under a common slope after group mean-centering
+ Regression scores (**s**): shape data projected on the regression line with size
+ When a single group is considered, `$\small{CAC}$` and **s** are **identical**!
+ Both are projections of mean-centered data on size/shape regression (and are equivalent to PLS1 scores of a PLS of shape on size)

`$$\small\alpha=(\mathbf{Z}_{c}^T\mathbf{CS})/\mathbf{(CS^TCS)}$$`
`$$\small{CAC}=\mathbf{Z}_{c}\alpha$$`     `$$\small\alpha=\alpha\sqrt{}(\alpha^T\alpha)$$`
`$$\small\mathbf{s}=\mathbf{Z\beta^T(\beta^T\beta)^{-1/2}}$$`

.footnote[See Adams et al. 2013. *Hystrix* 24: 7-14.] 
---

### Allometry: Other Tests `$^1$`

+ Many biological hypotheses are related to the comparison of ontogenetic allometries

+ Allometric trajectories are not only about their slope, or variation around it

.footnote[1: See Shape Statistics II lecture for discussion of PTA as a general method for comparing all types of trajectories, including ontogenetic/allometric trajectories (i.e. Collyer and Adams 2007; Adams and Collyer 2009; Collyer and Adams 2013)]

+ Do they have similar starting points or ending points?

+ Are ontogenetic trajectories similar? (parallelism)

+ Do ontogenetic trajectories converge?

+ Do ontogenetic trajectories diverge?

+ We need quantitative methods to test such hypotheses

+ Take advantage of the toolkit of resampling methods

---

### Allometric Convergence and Divergence `$^1$`

+ Convergence: shape is more similar at the end of the trajectory, than at the beginning

+ A simple approach: test angular difference between trajectories

.footnote[1: Adams and Nistri. (2010) *BMC. Evol. Biol.*]
--

+ 1: Obtain allometric vectors `$(\small\mathbf{\hat{Z}}_{start} - \small\mathbf{\hat{Z}}_{end})$`  for groups
  + 2: normalize vectors (divide by their length so they are unit length) and find the cross-product between pairs of vectors; i.e, `$r_{ij} = \mathbf{\beta}_i^T \mathbf{\beta}_j$`
  +  3: Obtain test measure: `$\theta = \cos^{-1}(r_{ij})$`
  + 4: RRPP with appropriate null model to test `$\theta$`

+ **Convergence** when `$\theta$` is significant and vectors tend toward a similar shape
+ **Divergence** when `$\theta$` is significant and vectors tend toward a different shape
+ **Parallelism** when `$\theta$` is not significant and/or effect size is small
  + Can also test absolute difference in vector lengths to determine if one group has greater ontogenetic shape change than another
    
---

### Evolutionary Convergence: *Hydromantes* Example

+ Ontogenetic series for 8 species of *Hydromantes*
+ These salamanders live in caves and they climb extensively on the walls as adults
+ Foot shape is a trait with functional relevance
+ How does foot shape change across ontogeny?

<img src="LectureData/07.allometry/HydromantesEx.png" width="80%" style="display: block; margin: auto;" />
---

### Evolutionary Convergence: *Hydromantes* Example (Cont.)

<img src="LectureData/07.allometry/HydroRes1.png" width="75%" style="display: block; margin: auto;" />
---

### Evolutionary Convergence: *Hydromantes* Example (Cont.)

<img src="LectureData/07.allometry/HydroRes2.png" width="75%" style="display: block; margin: auto;" />
---

### Comparing Allometric Trajectories: Pupfish Example

+ **Question**: Do your groups have parallel allometries?
+ **Question**: Do your groups have unique allometries?
+ **Question**: Which groups have different allometries, and how do they differ?

``` r
data(pupfish)
plotAllSpecimens(pupfish$coords)
```

<img src="07-Allometry_files/figure-html/unnamed-chunk-25-1.png" width="30%" style="display: block; margin: auto;" />
---

### Comparing Allometric Trajectories: Pupfish Example (Cont.)

.pull-left[

``` r
fit.common <- procD.lm(coords ~ logSize + Group, 
                     data = pupfish, print.progress = FALSE) 
fit.unique <- procD.lm(coords ~ logSize * Group, 
                     data = pupfish, print.progress = FALSE)  
```

+ now let's do a homogeneity of slopes test
]

.med[.pull-right[

``` r
anova(fit.common, fit.unique, print.progress = FALSE)
```

```
## 
## Analysis of Variance, using Residual Randomization
## Permutation procedure: Randomization of null model residuals 
## Number of permutations: 1000 
## Estimation method: Ordinary Least Squares 
## Effect sizes (Z) based on F distributions
## 
##                                 ResDf Df      RSS        SS         MS      Rsq
## coords ~ logSize + Group (Null)    49  1 0.024084                      0.000000
## coords ~ logSize * Group           46  3 0.022079 0.0020045 0.00066816 0.035583
## Total                              53    0.056333                              
##                                      F      Z     P Pr(>F)
## coords ~ logSize + Group (Null)                           
## coords ~ logSize * Group        1.3921 1.1451 0.136       
## Total
```
+ Suggests allometric slopes are parallel 
]]

---

### Comparing Allometric Trajectories: Pupfish Example (Cont.)

+ Allometric plots 
  + We can see by plotting the prediction lines for both that a unique-allometries model is not practical

<img src="07-Allometry_files/figure-html/unnamed-chunk-29-1.png" width="50%" style="display: block; margin: auto;" />
---

### Comparing Allometric Trajectories: Pupfish Example (Cont.)

+ Hypothesis tests: vector lengths

.scrollable[

``` r
PW.u <- pairwise(fit.unique, groups = pupfish$Group,
               covariate = log(pupfish$CS), print.progress = FALSE)
summary(PW.u, test = "dist")
```

```
## 
## Pairwise comparisons
## 
## Groups: Marsh.F Sinkhole.F Marsh.M Sinkhole.M 
## 
## RRPP: 1000 permutations
## 
## Slopes (vectors of variate change per one unit of covariate 
##         change, by group):
## Vectors hidden (use show.vectors = TRUE to view)
## 
## Pairwise distances between slope vector 
##             (end-points), plus statistics
##                                d UCL (95%)          Z Pr > d
## Marsh.F:Sinkhole.F    0.07848121 0.1621176 -0.9763904  0.838
## Marsh.F:Marsh.M       0.08279940 0.1640452 -0.9689342  0.829
## Marsh.F:Sinkhole.M    0.12322729 0.1610484  0.7139190  0.236
## Sinkhole.F:Marsh.M    0.06418279 0.1095190 -0.3981426  0.652
## Sinkhole.F:Sinkhole.M 0.11706696 0.1001323  2.2066562  0.015
## Marsh.M:Sinkhole.M    0.10863856 0.1132783  1.5445168  0.063
```
]
---

### Comparing Allometric Trajectories: Pupfish Example (Cont.)

+ Hypothesis tests: angles between vectors

.scrollable[

``` r
summary(PW.u, test = "VC", angle.type = "deg") 
```

```
## 
## Pairwise comparisons
## 
## Groups: Marsh.F Sinkhole.F Marsh.M Sinkhole.M 
## 
## RRPP: 1000 permutations
## 
## Slopes (vectors of variate change per one unit of covariate 
##         change, by group):
## Vectors hidden (use show.vectors = TRUE to view)
## 
## Pairwise statistics based on slopes vector correlations (r) 
##           and angles, acos(r)
## The null hypothesis is that r = 1 (parallel vectors).
## This null hypothesis is better treated as the angle 
##           between vectors = 0
##                               r    angle UCL (95%)           Z Pr > angle
## Marsh.F:Sinkhole.F    0.6318267 50.81498  83.57546 -0.07310658      0.527
## Marsh.F:Marsh.M       0.6139591 52.12367  83.58445 -0.13196270      0.549
## Marsh.F:Sinkhole.M    0.4461018 63.50614  81.89039  0.66791084      0.273
## Sinkhole.F:Marsh.M    0.7175129 44.15048  63.60905  0.24275646      0.391
## Sinkhole.F:Sinkhole.M 0.4627734 62.43378  60.22153  1.77232308      0.037
## Marsh.M:Sinkhole.M    0.5629975 55.73665  65.19951  1.04510764      0.154
```
]
---

### Comparing Allometric Trajectories: Pupfish Example (cont.)

+ Conclusion: allometries do not significantly diverge.  But concluding they are parallel allows one to compare means, accounting for common allometric variation

.scrollable[

``` r
PW.c <- pairwise(fit.common, groups = pupfish$Group,
               print.progress = FALSE)
summary(PW.c, test = "dist") 
```

```
## 
## Pairwise comparisons
## 
## Groups: Marsh.F Sinkhole.F Marsh.M Sinkhole.M 
## 
## RRPP: 1000 permutations
## 
## LS means:
## Vectors hidden (use show.vectors = TRUE to view)
## 
## Pairwise distances between means, plus statistics
##                                d  UCL (95%)        Z Pr > d
## Marsh.F:Sinkhole.F    0.03639758 0.01759231 3.805424  0.001
## Marsh.F:Marsh.M       0.03579214 0.01841414 3.996733  0.001
## Marsh.F:Sinkhole.M    0.03809715 0.01528439 4.529552  0.001
## Sinkhole.F:Marsh.M    0.03681731 0.02381545 2.952476  0.002
## Sinkhole.F:Sinkhole.M 0.01939739 0.01857347 1.833261  0.031
## Marsh.M:Sinkhole.M    0.02694896 0.01911014 2.798642  0.001
```
]
---

### Empirical Example: Cranial Ontogeny in Crocodilians

+ Phylogenetic relationships have long been under debate
+ What information can we extract from cranial ontogeny?

<img src="LectureData/07.allometry/CrocRes1.png" width="75%" style="display: block; margin: auto;" />
---

### Empirical Example: Cranial Ontogeny in Crocodilians (Cont.)

+ Some groups follow different ontogenetic paths
+ There is ontogenetic convergence, where adults are more similar than juveniles

<img src="LectureData/07.allometry/CrocRes2.png" width="75%" style="display: block; margin: auto;" />
---

### A Comment on Heterochrony
+ Evolutionary changes in the rate or timing of developmental sequences
  + i.e. changes in ontogenetic trajectories
+ Can be evaluated via allometric/ontogenetic investigations

<img src="LectureData/07.allometry/Heterochrony.png" width="70%" style="display: block; margin: auto;" />
---

### Allometry: Summary

+ Shape usually changes with size – allometry is a common pattern in biological data
+ We need to account for it when testing hypotheses about shape
+ Many biological hypotheses related directly to allometric variation
+ **Visualization**: 
  + RegScore (**s**) vs. Csize (Drake and Klingenberg 2008)
  + PredLine `$\small{P_1}$` vs. Csize (Adams and Nistri 2010)
  + CAC, RSC, Csize (Mitteroecker et al. 2004)

+ **Comparison of ontogenies**: - MANCOVA
  + Convergence & divergence (Adams and Nistri 2010)
  + Orientation (Piras et al. 2010)

+ Heterochrony and related topics also of interest