Data Set 7: Bioerosion by Parrotfish Background volume of bites The question:

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1 Data Set 7: Bioerosion by Parrotfish Background Bioerosion of coral reefs results from animals taking bites out of the calcium-carbonate skeleton of the reef. Parrotfishes are major bioerosion agents, and excrete ingested coral or rock as sand. Ling Ong, a doctoral student in the Zoology department, has estimated the total rate of such bioerosion and sand production in Hanauma Bay by two species of parrotfishes: The Spectacled parrotfish Chlorurus perspicillatus. This species is endemic to Hawai i. It has two color phases: the initial phase (top at right), typically female and called ulu 'ahu'ula and the terminal phase (second at right), always male and called uhu uli uli. The Redlip parrotfish Scarus rubroviolaceus ( palukaluka ). It also has initial (third at right) and terminal (bottom right) color phases. A nice summary of Ling s research is at To estimate the amount of bioerosion she has: estimated the number of fish in the bay, by species, phase and size class; observed fish to estimate the rate of bites on (dead) coral, by species, phase, size class, time of day, and season; and measured length, width, and depth of scars from bites by observed fish, again by fish species, phase, and size class. This handout will use data from the last of these studies to compare the volume of bites by the two species. The Spectacled parrotfish has what is called an excavator jaw and muscle morphology, which is more suited to removing substrate than the scraper morphology of the Redlip parrotfish, so Ling expected the former to have larger bite volumes. The question: Do the two species of parrotfish differ in the volume of material removed per bite? In particular, are Spectacled parrotfish bites larger than Redlip parrotfish bites?

2 The data Procedure Individual fish were followed by a scuba diver, and the first visible bite it made was marked. The species and phase of the fish was recorded and its length was estimated in cm size classes. The diver then returned to the marked bite and used a caliper to measure the length, width, and depth, in mm. From these measurements the volume, in mm 3, of each bite was calculated. Data were obtained for 71 fish of each species. The variable Bite volume, not surprisingly, is positively correlated with the size of the fish, and there was a wide range of fish sizes in the data set. The distribution of bite sizes was skewed, with a longer right tail, and the variability of the bite sizes was greater for larger fish (and larger bites). The variable used in the following analysis therefore is the natural log of the bite volume, adjusted to remove the effect of fish size. * The data Spectacled parrotfish: Redlip parrotfish: *. The size adjustment was based on a regression of log-transformed bite volume against fish size, with a common slope for the two species but separate intercepts. This regression model suited the data well, according to the usual residual plots and other diagnostics. The slope of this regression then was used to adjust all bite volumes as if all fish were the same size (the overall mean): to each observation was added the product of this slope times the difference between the mean size and the individual fish s size. Data Set 7: Bioerosion by Parrotfish (rev. October 19, 200) 2

3 Data exploration Displays of the distributions Histograms and boxplots Frequency anel variable: Species Redlip Spectacled log of bite volume, adjusted for fish size log of bite volume, adjusted Redlip Spectacled The distributions are similar between the two species. Both are skewed, more fish having large bites and smaller numbers of much smaller bites creating long left tails. The smallest bites are considered outliers in the boxplots; in the histograms the smallest Redlip value does not look outlying while the two smallest Spectacled values do. The centers of the two distributions appear to be very similar. The lower/left end of the Spectacled distribution is slightly more stretched out, but otherwise there seems to be little if any difference between the species. NQQ plots Percent Spectacled Redlip Data 4 6 The superimposed NQQ plots show that the distributions are very similar, except that the Redlip distribution is smoother than the Spectacled distribution. Both clearly are skewed. Data Set 7: Bioerosion by Parrotfish (rev. October 19, 200) 3

4 Statistical summary Species Mean StDev Minimum Q1 Median Q3 Maximum Redlip Spectacled These statistics suggest that bites by Spectacled parrotfish are slightly smaller than those by Redlip parrotfish: the Spectacled maximum, third quartile and mean all are about 0.2 units smaller than the corresponding Redlip statistics, and the first quartile and minimum are 0.4 and 0.9 units smaller. Interestingly, the Spectacled median is very slightly larger than the Redlip median. The distances between quantiles get smaller going from minimum Q1 up through Q3 maximum, reflecting the skew of the distributions; the differences between means and medians also show this. The spreads of the distributions also are similar, but slightly smaller for Redlips (IQR: 1.31 vs ; standard deviation: vs. 1.11). Data Set 7: Bioerosion by Parrotfish (rev. October 19, 200) 4

5 Inference The purpose of the study was to compare two populations, so the appropriate inference is two-sample hypothesis tests. Ling did have an a priori expectation that Spectacled parrotfish, with their excavator jaws, would have larger bites than the Redlip parrotfish, with scaper jaws. I feel, though, that it would be desirable to detect a difference regardless of which direction it was in. The tests therefore will be two-sided. They will be supplemented by estimates of the difference, with confidence intervals. Scope of inference The fish used in this study were selected haphazardly by the diver. Because they spanned a wide range of sizes this selection probably was effectively independent of their (size-adjusted) bite volumes. In addition, there were many of them over the full range of sizes present, and of both phases. I therefore expect that it is safe to generalize the results of this analysis to the two species populations in Hanauma Bay, which is what Ling intends to do. Fish sizes, abundances, and behavior, coral abundance and composition, and many other relevant factors are likely to be different in Hanauma Bay, which has been protected from fishing for many years, than elsewhere, so extending the conclusions beyond the Bay would be unwise. t procedures The test is of H 0 : µ R µ S = 0 vs. H a : µ R µ S 0. The results, from Minitab, are: Difference = mu (Red) - mu (Spec) Estimate for difference: % CI for difference: ( , ) T-Test of difference = 0 (vs not =): T-Value = 1.08 P-Value = DF = 138 This test does not give evidence to reject the null hypothesis of no difference. Although the Redlip sample mean is 0.2 units larger than the Spectacled sample mean, the 9% confidence interval for the difference between the population means extends from about units to about 0.4 units: the margin of error is nearly twice the difference between the sample means. A retrospective power analysis can help interpret this non-significant result. Assuming parameters matching those observed in this data set (n R = n S = 71, σ R = σ S = 1.1, µ R µ S = 0.2, and α = 0.0), the power of the test is only To get the power up to 0., with the same within-species variability, would require the true means to differ by units, nearly twice the observed difference. Alternatively, with the same within-species variability and true means differing by 0.2 units, it would take 234 observations per species to achieve power of 0.. What these analyses indicate is that despite the large sample size 1420 bites marked and measured the test does not have much power for detecting small differences between the species. This is a result of the large within-species variability, relative to the between-species difference. Data Set 7: Bioerosion by Parrotfish (rev. October 19, 200)

6 Nonparametric procedures These procedures test H 0 : M R M S = 0 vs. H a : M R M S 0 and estimate M R - M S (where M is the population median, which Minitab calls ETA). Rank-sum: Point estimate for ETA1-ETA2 is Percent CI for ETA1-ETA2 is ( ,0.4924) W = Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at Mood s median test: Chi-Square = 0.03 DF = 1 P = Species N<= N> Median Q3-Q1 Red Spec Overall median = A 9.0% CI for median(red) - median(spec): (-0.392,0.340) These results are similar to those from the t procedures above. The P-values, especially for the median test (as expected due to its generally lower power), are even larger than for the t test. The confidence intervals for the difference in population medians are about as wide as the t CI for the difference in means, though they are shifted somewhat down. Resampling procedures The tests are of H 0 : µ R µ S = 0 vs. H a : µ R µ S 0. Minitab, bootstrap, unpooled: P = randomization (pooled): P = S-Plus, randomization (pooled): P = These tests all give results similar to those from the t test: P-values around The 9% confidence intervals for the difference in the population means, from S-Plus, are: Percentiles: BCa: Tilting: T using Bootstrap SE: These confidence intervals are slightly narrower than the standard t CI, but they really are quite like each other and the t CI. The resampling distributions produced by S-Plus (next page) are quite close to normal, for both the bootstrap estimation of the CIs (top pair of plots) and the randomization test (bottom pair of plots). The bootstrap distribution shows little bias (its mean is very close to the mean of the actual sample), while the randomization distribution shows that the observed value is somewhat off the center of the distribution but a large fraction of randomization producing statistics more extreme than the observed one. Data Set 7: Bioerosion by Parrotfish (rev. October 19, 200) 6

7 bootstrap : parrotbites2$adjl... : mean : Red - Spec Density Observed Mean mean mean Quantiles of Standard Normal permutation : parrotbites2$adjl... : mean : Red - Spec Density Observed Mean Var Var Quantiles of Standard Normal Which procedure to use? The sample distributions clearly are skewed, but not terribly strongly. With sample sizes of 71, the Central Limit Theorem ensures that the t procedures will be valid. The rank-sum procedure, and the resampling procedures also should be valid. The median test is valid, but lacking in power. Given the preceding appeal to the CLT, supported by the near-normality of the resampling distributions, I feel the standard t test and CI are appropriate for these data. Conclusions Do the two species of parrotfish differ in the volume of material removed per bite? In particular, are Spectacled parrotfish bites larger than Redlip parrotfish bites? There is little if any difference between the species in mean bite size (adjusted for fish size). This study does not provide evidence of a difference at any reasonable level of statistical significance. Furthermore, the difference in observed means is opposite to that expected: Redlip bites were slightly larger than Spectacled bites. Other factors such as abundance and size distribution probably have much greater effects on the relative contributions of the two species to bioerosion in Hanauma Bay than does bite size. Data Set 7: Bioerosion by Parrotfish (rev. October 19, 200) 7