Trematode infections in fish eyes
Diplostomum spp. (Trematoda) are common parasites of freshwater fishes. Here we use two datasets describing the distribution of Diplostomum spp. metacercariae in the eyes of fifty ruffe, Gymnocephalus cernuus.
Infections of the eye excluding lens
The first dataset comprises the numbers of metacercariae found in each eye, excluding the lens (i.e. the choroid, retina and humors). The dataset is formatted as a data.frame with two columns; the first column is the number of parasites found in the left eye and the 2nd column is the number of parasites found in the right eye. Here are the data for the first ten hosts:
head(diplostomum_eyes_excl_lenses, 10)## left right
## 1 86 99
## 2 131 199
## 3 195 133
## 4 128 167
## 5 192 211
## 6 212 278
## 7 451 474
## 8 323 312
## 9 394 305
## 10 222 270The row.names are host IDs. To detect bilateral asymmetry we apply replicated G-tests of goodness-of-fit:
results <- g.test(diplostomum_eyes_excl_lenses)First we inspect the pooled, heterogeneity and total G-statistics:
results$summary## Test df G p
## 1 Pooled 1 8.189282 4.213860e-03
## 2 Heterogeneity 49 437.827514 7.448961e-64
## 3 Total 50 446.016796 5.819378e-65The total G-test statistic is highly significant and so the null hypothesis of overall symmetry can be rejected. The pooled G-test (p-value = 0.004) suggests that the total number of parasites from each side differs from symmetry. Let’s calculate the total number of parasites found on each side:
#total number of parasites on left side
sum(diplostomum_eyes_excl_lenses$left)## [1] 15026#total number of parasites on right side
sum(diplostomum_eyes_excl_lenses$right)## [1] 14534#ratio
sum(diplostomum_eyes_excl_lenses$left) / sum(diplostomum_eyes_excl_lenses$right)## [1] 1.033852However, this slight left bias in the pooled data, doesn’t necessarily mean that we would expect to find a left bias in most hosts. The p-value for the heterogeneity G-test is highly significant, revealing that the proportions of metacercariae in the left and right eyes varies from host to host. The individual G-test identifies which of the hosts have asymmetrical parasite distributions. Here are the results for the first ten hosts:
head(results$hosts, 10)## Host Left Right G p BH Holm
## 1 1 86 99 0.9142668 0.3389848373 0.452402486 1.000000000
## 2 2 131 199 14.1130061 0.0001721493 0.001434578 0.007746721
## 3 3 195 133 11.7903193 0.0005953952 0.003307751 0.025006599
## 4 4 128 167 5.1710572 0.0229662351 0.055002553 0.711953287
## 5 5 192 211 0.8961138 0.3438258892 0.452402486 1.000000000
## 6 6 212 278 8.9168733 0.0028254783 0.008310230 0.096066262
## 7 7 451 474 0.5719508 0.4494847458 0.548152129 1.000000000
## 8 8 323 312 0.1905607 0.6624503507 0.690052449 1.000000000
## 9 9 394 305 11.3627210 0.0007493296 0.003746648 0.030722515
## 10 10 222 270 4.6903841 0.0303318672 0.068936062 0.879624148The seven columns of the above data.frame are:
- Host - host identifier
- Left - count of parasites on left side
- Right - count of parasites on right side
- G - Individual G-statistic
- p - p-value
- BH - p-value adjusted for multiplicity using procedure of Benjamini and Hochberg (1995)
- Holm - p-value adjusted for multiplicity using the procedure of Holm (1979)
The raw p-value has to be adjusted for multiplicity, because an individual G-test is applied to each host. If the null hypothesis of symmetry is true for all 50 hosts, at a significance level of 5% the probability of getting at least one Type I error (false rejection of the null hypothesis) is \(1-(1-0.05)^{50} = 0.923\); the expected number of significant results obtained purely by chance would be \(50 \times 0.05 = 2.5\). Holm’s procedure controls the familywise error rate (FWER), i.e. the probability of a single Type I error. However, guarding against a single Type I error will be unnecessarily conservative for many studies, especially those surveying a large number of hosts. If a small number of false positives among the set of rejected null hypotheses can be tolerated, then it will be preferable to control the false discovery rate (FDR) using the procedure of Benjamini and Hochberg (1995).
At the conventional significance level of 5%, Holm’s procedure finds 11 of the 50 individual G-tests to be significant:
sum(results$hosts$Holm<0.05)## [1] 11Benjamini and Hochberg’s method provides more power to detect asymmetric infections, identifying a total of 19, although one (5% of 19) of these is likely to be a false positive:
sum(results$hosts$BH<0.05)## [1] 19These asymmetric infections identified with the aid of Benjamini and Hochberg’s procedure can be classified according to whether they show left or right bias:
# Number of asymmetric infections showing left bias
sum(results$hosts$BH<0.05 & results$hosts$Left>results$hosts$Right)## [1] 11# Number of asymmetric infections showing right bias
sum(results$hosts$BH<0.05 & results$hosts$Right>results$hosts$Left)## [1] 8We conclude that Diplostomum spp. infections of the eyes (excluding the lenses) show bilateral asymmetry in a large proportion of the sampled ruffe. Moreoever, the bias is inconsistent, with parasites favoring the left eye in some hosts and the right in others.
For visualization the proportion of parasites in each eye can be expressed as a binary log transformed ratio. These ratios can be plotted as a histogram:
plotHistogram(diplostomum_eyes_excl_lenses, nBreaks=20, main="Histogram")
Alternatively, these ratios can be combined with corresponding p-values from individual G-tests in a volcano plot:
plotVolcano(diplostomum_eyes_excl_lenses, test="G", pAdj="BH", sigThresh=0.1,
main="Volcano Plot")
The dashed horizontal line in the volcano plot represents the chosen p-value threshold. Parasite distributions deviating significantly from symmetry are shown as red squares, whereas those not differing significantly from a 1:1 ratio are represented by blue circles.
Infections of the lens of the eye
The second dataset comprises observations on the distribution of Diplostomum spp. metacercariae in the lenses of the eyes of the ruffe. Here are the data for the first ten hosts:
head(diplostomum_lenses, 10)## left right
## 1 0 0
## 2 0 0
## 3 0 0
## 4 0 0
## 5 0 0
## 6 0 0
## 7 0 0
## 8 0 0
## 9 6 5
## 10 0 2Note that the first eight hosts are not infected. The total number of ruffe with Diplostomum spp. infection of the lens is:
sum(diplostomum_lenses$left>0 | diplostomum_lenses$right>0)## [1] 31G-tests involve logarithmic transformation and so cannot be applied to counts of zero, as found in this dataset. Furthermore, G-tests are not recommended when counts are below ten. The g.test function will ignore uninfected hosts in this dataset, but will raise an error, because some hosts only have an infection in one eye. In this situation, binomial exact test should be used instead:
results <- eb.test(diplostomum_lenses)
# pooled test p-value
results$pooled## [1] 0.8439127# results for the first ten infected hosts
head(results$hosts, 10)## Host Left Right p BH Holm
## 9 9 6 5 1.0000000 1.0000000 1.000000
## 10 10 0 2 0.5000000 1.0000000 1.000000
## 28 28 0 3 0.2500000 0.9687500 1.000000
## 46 46 11 8 0.6476059 1.0000000 1.000000
## 15 15 0 1 1.0000000 1.0000000 1.000000
## 16 16 0 7 0.0156250 0.4843750 0.484375
## 17 17 3 2 1.0000000 1.0000000 1.000000
## 18 18 0 5 0.0625000 0.5535714 1.000000
## 19 19 4 7 0.5488281 1.0000000 1.000000
## 20 20 8 2 0.1093750 0.5535714 1.000000# smallest FDR adjusted p-value for an individual host
min(results$hosts$BH)## [1] 0.484375No evidence of asymmetry in lens infections was found by either the pooled test or in the tests of the individual host infections. Note that the eb.test function restricted analysis to the 31 infected hosts:
length(results$hosts[,1])## [1] 31