Overview

This document is long showing my thoughts as I dive into each aspect heading. For brevity, I’ve outlined the major points here.

1. Linear extension values are not skewed by their window of observation within realistic bounds. I keep data within +/- 1 month of the target i.e. 5-7 months for 6 month growth rates and 11-13 months for 12-month growth rates. Rates are calculated by daily productivity values within the observation window and scaled to the target to ensure comparability.

2. Scores are standardized using z-scores to calculate mean standard deviation about the mean for each trait and genotype. Positive values indicate above average performance, 0 denotes average performance, and negative values indicate below average performance.

3. Mean standardized scores are calculated for linear extension (6-mo and 12-mo TLE and colony volumetric growth) and calcification (buoyant weight, light calcification, and dark calcification), and then the mean of the calcification and linear extension scores is calculated for the composite growth index.

4. Outliers are identified and removed using modified z-scores. The modified z-scores use median values instead of mean values since medians are more resistant to being skewed by outlier observations. Modified z-scores greater than 3.5 or less than -3.5 are filtered out, and normal z-scores are then calculated. This prevents the inclusion of data which are greater than or less than the central tendency of all other data as is the case of buoyant weight data collected from one study which is >> all other and a few linear extension growth rates of corals that were on average smaller yet had high proportionate growth (~20x initial size).

Previous Studies

There exist 16 studies that have investigated the growth forms of Acropora cervicornis at the genotype level which have been incorporated into our database. These studies began with Diego Lirman who analyzed genotype-specific differences in annual proportionate growth (termed productivity in Lirman et al. 2014) across genotypes within Diego’s Biscayne National Park nurseries and his work in the Dominican Republic. Katie Lohr built upon this and analyzed 10 genotypes for differences in total linear extension and calcification rate over a 13 month study of CRF genotypes (Lohr and Patterson 2017). At the same time, Ford Drury analyzed the genotype niches in a genotype by environment study (Dury, Manzello, Lirman 2017). Other studies did not solely focus on differences among genotypes but their data analyzing growth under OA, temperature stress, or a combination of the two shed light on genotype-specific patterns. There has consistently been genotypic patterns of growth throughout all of these studies. Through the connection of each individual dataset, we can elucidate new, overarching genotype-specific patterns of growth. Below is a table summarizing the aforementioned and complementary studies that have analyzed growth of genotyped A. cervicornis. The remaining tables delve into the combined data of all these studies to begin to draw patterns.

Traits Table

• Note: Proportionate growth indicates multiplicative growth over the time scale. If a coral was 10cm and it’s 6-month linear growth is 2, it will have grown 20cm in the 6-months and it’s new size is 30cm.

Linear Extension Analysis

Here, we see how TLE is changing over time, specifically looking at the 6-month and 12-month time points. For this overview, the colors in the graphs denote the largest windows of +/- 60 days around the time-point targets; 6-month growth is selected between 120 and 240 days (4-8 months) and for 12-month growth is between 10 months and 14 months.

The formula for calculating productivity is:

\[ \frac{TLE_f - TLE_i}{TLEi}/days \times (182 \space|\space 365) \] Thus, we calculate the daily prod over the time scale observed, and scale to its target. For instance a coral viewed over 175 days would have 7 days linearly extrapolated and a coral viewed over 185 days would have 3 days linearly subtracted. The amount of extrapolation or subtraction is proportionate to the length of the window chosen. As noted above, the units are in proportionate growth per time scale, \(yr^{-1}\) or \(6mo^{-1}\)

We can narrow our windows around the 6-month and 12-month growth rates. Here I investigate windows of +/- 15, 30, 45, and 60 days. A window of 15 denotes a single month around the midpoint, 30 days denotes 2 months, 45 days 3 months, and 60 days 4 months.

Number of Days +/- the target and the resulting amount of metrics computed. Targets: annual = 365; 6-month = 182

+/- days	annual	6-month
15	1079	2083
30	1271	2370
45	1274	2633
60	1333	2666

For some reason, the 12-month has lots of outliers that bring it all the way up to ~60 cm/cm/yr, which just doesn’t seem right. Let’s dive in.

Lirman 2014 Metrics vs. Everything Else

datafile_name	average initial TLE	average annual prod
2014_data	7.64	7.37
everything else	12.03	3.64

So, the outliers are explained by a dataset which had a handful of small colonies <5cm which had significant growth relative to its initial size. This is only 31 colonies that have prod >20 and initial TLE <5. Many of these values are filtered out with the adopted outlier removal scheme outlined below. Anyway, back to the problem at hand: does increasing the window cause any scaling problems?

Slopes and R2 for all plots are weak and show mixed patterns. My takeaway is that since we are scaling within fairly limited windows, we are not seeing any extrapolation or subtraction problems. Not quite sure what statistics I should run on this or even if they are needed. But visually, it seems as if increasing the window does not have an effect on the derived productivity value. My final takeaway is that +/-30 days seems like a nice balance of including more data points and not straying too far from the target. We can say, “data within one month of the target metric was included in calculations”. This would include data between 5 months and 7 months and data from 11 months to 13 months.

Connecting Across Datasets

We can then look at the mean value of each metric for each genotype. The columns have been ordered by their amount of data completeness. Columns to the left of the vertical line describe linear extension, while columns to the right of the line describe total calcification. Linear extension includes TLE and volumetric methods for a total of 4 metrics, and calcification metrics include buoyant weight and total alkalinity anomaly methodologies for a total of 3 metrics. The large amounts of missing data prohibit a truly robust composite assessment. However, creating the groundwork for a composite index which will become more robust over time will certainly be useful. One of the goals of this tool is for practitioners and researchers to easily complete their genotype’s row using easy to use metrics. CRANG (or manual incubations), buoyant weight, and a ruler is all that is needed for these 7 metrics. I have eliminated metrics that share data with one another or are overtly complex for practitioners (interstitial metric) to measure.

Standardizing Scores

Because each of these metrics describe a different component of coral growth and are measured on different scales, they cannot be directly aggregated. Instead, we must first calculate standardized scores for each metric prior to aligning disparate measurements. I chose to standardize by calculating z-scores where the difference of each value and the column mean is divided by the column’s standard deviation. Thus, the column mean is now 0 and the standard deviation is 1. Each resulting value indicates the amount of standard deviations about the mean for that metric.

The table below shows the resulting standardized data.

Composite Growth Index

Now that we have standardized scores for 7 different metrics, we can compare and align metrics into a single, composite index to understand relative genotype performance.

Robustness Testing

Programmatically Removing Outliers

Rather than removing the Lohr (2017) calcification data explicitly, we can explore removing outliers programmatically in the calculation of a composite score. The even elimination of outliers in the calculation of the composite index score will help improve the robustness of the composite score. This has precedent in other composite index scores, and I will follow their recommendations (Handbook on constructing composite indicators, methodology and user guide, 2008).

Here, I will identify outliers by using modified z-scores where the median value for a trait is subtracted from each value, and then the value is scaled (divided by) the median of the absolute deviations about the median.

\[ Z_\text{modified} = \frac{(x_i-\widetilde{x})}{\text{median}\{\left|x_i - \widetilde{x}\right|\}\times 1.4826} \] where \(x_i\) is the value to be converted to a modified z-score, \(\widetilde{x}\) is the median value of the trait, and 1.4826 is the constant needed to multiply the median of the absolute deviations about the median to approximate one standard deviation for large values (>80) of n observations for a particular trait as we have in our dataset.

When the modified z score is less than or greater than 3.5, the value is removed and the normal z-scores are then calculated on the filtered data. The threshold of 3.5 was chosen based on recommendations of best practices (Iglewicz & Hoaglin, 1993).

This can be achieved in R quite simply:

data %>%
  group_by(trait) %>%
  mutate(`modified z-score` = scale(value,
                                center = median(value, na.rm = T),
                          #mad function contains the constant outlined above
                                scale = mad(value, na.rm=T))) %>%
  filter(abs(`modified z-score`)<=3.5)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  score by type
## Kruskal-Wallis chi-squared = 4.4062, df = 2, p-value = 0.1105

## 
##  Pairwise comparisons using Wilcoxon rank sum exact test 
## 
## data:  t$score and t$type 
## 
##          Both (19) G (5)
## G (5)    0.11      -    
## LE (106) 0.11      0.37 
## 
## P value adjustment method: BH

The differences among groups are now statistically not significant (p>0.05). The p-value here is certainly not as great as above when we explicitly remove Lohr (2017) data [p-value = 0.9713], but the methodology of removing outliers here is not selective as it was above. While we can visually see differences of the interquartile range, the unevenness of genotypes in each grouping (# in parentheses) prevents us from drawing any conclusions.

Final Rankings

After all is said and done, here are the final genotype rankings:

Final ranking of genotypes

genotype	G	LE	score	type	rank	metrics	datasets
U78	3.039	2.023	2.531	Both	1	3	1
K2	3.523	1.296	2.410	Both	2	4	2
U73	2.756	2.014	2.385	Both	3	2	1
U77	3.160	1.326	2.243	Both	4	4	2
U47	-	2.231	2.231	LE	5	2	1
K25	2.976	1.315	2.145	Both	6	3	1
Sunny Isles-A	-	2.096	2.096	LE	7	1	1
MB-A	-	2.066	2.066	LE	8	1	1
MB-C	-	2.009	2.009	LE	9	1	1
U44	2.692	1.087	1.890	Both	10	4	2
B	-	1.736	1.736	LE	11	2	1
Cheetos-C	-	1.726	1.726	LE	12	1	1
K3	-	1.663	1.663	LE	13	3	2
BC-8B	-	1.594	1.594	LE	14	1	1
Stag-A	-	1.414	1.414	LE	15	1	1
Sunny Isles-C	-	1.323	1.323	LE	16	1	1
Tuna Jelly-B	-	1.292	1.292	LE	17	1	1
Cheetos-D	-	1.267	1.267	LE	18	1	1
MB-D	-	1.225	1.225	LE	19	1	1
BC-11	-	1.194	1.194	LE	20	1	1
Stag-B	-	1.182	1.182	LE	21	1	1
Sunny Isles-B	-	1.177	1.177	LE	22	1	1
MB-B	-	1.155	1.155	LE	23	1	1
BC-8A	-	1.147	1.147	LE	24	1	1
K1	0.180	2.113	1.146	Both	25	3	2
FM17	-	0.894	0.894	LE	26	1	1
Tuna Jelly-A	-	0.861	0.861	LE	27	1	1
Yung’s-B	-	0.847	0.847	LE	28	1	1
Sunny Isles-D	-	0.829	0.829	LE	29	1	1
Yung’s-A	-	0.766	0.766	LE	30	1	1
Kelsey-2	-	0.741	0.741	LE	31	1	1
Cheetos-A	-	0.670	0.670	LE	32	1	1
ML48	-	0.660	0.660	LE	33	1	1
FM20	-	0.623	0.623	LE	34	2	1
Kelsey-1	-	0.566	0.566	LE	35	1	1
U41	-	0.547	0.547	LE	36	3	2
THIN	-	0.477	0.477	LE	37	2	1
BC-1	-	0.426	0.426	LE	38	1	1
DR-A	-	0.395	0.395	LE	39	2	1
ML47	0.391	-	0.391	G	40	3	1
ML5	-	0.375	0.375	LE	41	2	3
BC-G	-	0.309	0.309	LE	42	2	1
Stagreef	-	0.235	0.235	LE	43	2	1
Acerv 1 (A-AC)	-	0.224	0.224	LE	44	2	2
Acerv 5 (E-AC)	-	0.202	0.202	LE	45	2	2
G16	-	0.197	0.197	LE	46	2	1
Site 406	-	0.193	0.193	LE	47	2	2
ML34	-0.176	0.561	0.192	Both	48	4	2
Marker 9	-	0.159	0.159	LE	49	3	3
North Midchannel-A	-	0.141	0.141	LE	50	2	1
Site 211	-	0.135	0.135	LE	51	1	1
ML56	-	0.126	0.126	LE	52	2	3
Acerv 3 (C-AC)	-	0.118	0.118	LE	53	2	3
Dom’s Reef	-	0.114	0.114	LE	54	2	2
MB	-	0.104	0.104	LE	55	1	1
Britt’s Reef	-	0.093	0.093	LE	56	2	2
G17	-	0.080	0.080	LE	57	2	1
Govt Cut	-	0.078	0.078	LE	58	1	1
Steph’s	-	0.060	0.060	LE	59	2	3
Acerv 2 (B-AC)	-	0.043	0.043	LE	60	2	2
ML31	0.302	-0.235	0.034	Both	61	7	3
G1	-	0.027	0.027	LE	62	2	1
ML41	0.165	-0.124	0.020	Both	63	7	4
ML54	-	-0.001	-0.001	LE	64	2	3
Struggle Bus	-	-0.020	-0.020	LE	65	2	3
ELK	-	-0.026	-0.026	LE	66	2	3
Trans 2	-	-0.033	-0.033	LE	67	2	1
ML44	0.096	-0.163	-0.034	Both	68	7	3
Ford’s Reef	-	-0.042	-0.042	LE	69	2	2
ML7	0.123	-0.275	-0.076	Both	70	7	4
Sunny Isles-F	-	-0.076	-0.076	LE	71	2	1
Acerv 4 (D-AC)	-	-0.077	-0.077	LE	72	2	2
M10	-0.078	-	-0.078	G	73	1	1
Trav’s Reef	-	-0.112	-0.112	LE	74	2	2
ML62	-0.342	0.106	-0.118	Both	75	7	3
ML14	-	-0.121	-0.121	LE	76	1	1
FM16	-	-0.131	-0.131	LE	77	1	1
ML3	-0.132	-0.167	-0.149	Both	78	7	2
BC-H	-	-0.153	-0.153	LE	79	2	1
M6	-0.530	0.207	-0.161	Both	80	4	3
TT (Thicket 3)	-	-0.176	-0.176	LE	81	2	2
Grounding	-	-0.186	-0.186	LE	82	2	3
Cheetos-B	-	-0.188	-0.188	LE	83	1	2
Jons	-	-0.200	-0.200	LE	84	1	1
ML36	-	-0.203	-0.203	LE	85	4	1
ML50	-0.267	-0.147	-0.207	Both	86	7	3
Almost Done	-	-0.210	-0.210	LE	87	2	2
Alina-A	-	-0.230	-0.230	LE	88	2	1
ML4	-	-0.231	-0.231	LE	89	2	2
ML13	-0.115	-0.496	-0.306	Both	90	7	3
ML63	-0.312	-	-0.312	G	91	3	1
U24	-	-0.314	-0.314	LE	92	2	1
Cooper’s	-	-0.317	-0.317	LE	93	2	4
ML1	-0.392	-0.256	-0.324	Both	94	7	5
Alina-B	-	-0.331	-0.331	LE	95	2	2
ML20	-0.449	-0.247	-0.348	Both	96	2	2
G11	-	-0.356	-0.356	LE	97	2	1
Inshore	-	-0.361	-0.361	LE	98	2	2
G15	-	-0.367	-0.367	LE	99	2	1
B8	-0.387	-	-0.387	G	100	1	1
G6	-	-0.396	-0.396	LE	101	2	1
G2	-	-0.453	-0.453	LE	102	2	1
Big Bertha	-	-0.457	-0.457	LE	103	2	2
FM5	-	-0.461	-0.461	LE	104	2	1
FM1	-	-0.462	-0.462	LE	105	2	1
THICK	-	-0.468	-0.468	LE	106	2	1
Fowey	-	-0.474	-0.474	LE	107	2	2
U25	-	-0.489	-0.489	LE	108	2	1
MUTANT182	-	-0.537	-0.537	LE	109	2	1
ML38	-	-0.542	-0.542	LE	110	1	1
Trans 1	-	-0.563	-0.563	LE	111	2	1
FM3	-	-0.605	-0.605	LE	112	1	1
FM11	-	-0.611	-0.611	LE	113	1	1
M1	-0.633	-	-0.633	G	114	1	1
W/G	-	-0.658	-0.658	LE	115	2	1
FM7	-	-0.664	-0.664	LE	116	2	1
FM18	-	-0.672	-0.672	LE	117	2	1
FM10	-	-0.698	-0.698	LE	118	2	1
FM13	-	-0.769	-0.769	LE	119	2	1
B/B	-	-0.772	-0.772	LE	120	2	1
FM12	-	-0.816	-0.816	LE	121	2	1
ML43	-	-0.839	-0.839	LE	122	1	1
Navassa	-	-0.850	-0.850	LE	123	2	1
W/B	-	-0.855	-0.855	LE	124	2	1
FM15	-	-0.859	-0.859	LE	125	1	1
FM9	-	-0.881	-0.881	LE	126	2	1
FM19	-	-0.897	-0.897	LE	127	1	1
W/Y	-	-0.944	-0.944	LE	128	2	1
G18	-	-0.983	-0.983	LE	129	2	1
ML11	-	-1.276	-1.276	LE	130	1	1

Concluding Remarks

In conclusion, I believe the composite growth index can align the datasets and metrics within AcDC to best describe relative performance. While the index is not without its flaws, the robustness will only increase over time with the inclusion of more datasets and data types, pushing all genotypes towards having calcification and linear extension data. By using the temporal, experimental, and location filters, one can further increase the robustness of the index. For instance, one can remove lab or field data using the filters to achieve the filtering of the Lohr (2017) calcification data.