Derivatives

For overdispersed Poisson, negative binomial and gamma GLMMs with log link, the observation-level variance can be obtained via the variance of the lognormal distribution (electronic supplementary material, appendix S1). This is the approach that has led to the terms presented above. There are two more alternative methods to obtain the same target: the delta method and the trigamma function. The two alternatives have different advantages and we will therefore discuss them in some detail in the following.
The delta method for variance approximation uses a first-order Taylor series expansion, which is often employed to approximate the standard error (error variance) for transformations (or functions) of a variable x when the (error) variance of x itself is known (see [18 (link)]; for an accessible reference for biologists, [19 (link)]). The delta method for variance approximation can be written as where x is a random variable (typically represented by observations), f represents a function (e.g. log or square-root), var denotes variance and d/dx is a (first) derivative with respect to variable x. Taking derivatives of any function can be easily done using the R environment (examples can be found in the electronic supplementary material, appendices). It is the delta method that Foulley et al. [20 (link)] used to derive the distribution-specific variance for Poisson GLMMs as 1/λ (see also [21 (link)]). Given that in the case of Poisson distributions and , it follows that (note that for Poisson distributions without overdispersion, is equal to because ).
One clear advantage of the delta method is its flexibility. We can easily obtain the observation-level variance for all kinds of distributions/link functions. For example, by using the delta method, it is straightforward to obtain for the Tweedie distribution, which has been used to model non-negative real numbers in ecology (e.g. [22 (link),23 (link)]). For the Tweedie distribution, the variance on the observed scale has the relationship where μ is the mean on the observed scale and φ is the dispersion parameter, comparable to λ and ω in equation (3.1), and p is a positive constant called an index parameter. Therefore, when used with the log-link function, can be approximated by according to equation (4.1). The lognormal approximation is also possible (see the electronic supplementary material, appendix S1; table 1).
The use of the trigamma function is limited to distributions with log link, but it is considered to provide the most accurate estimate of the observation-level variance in those cases. This is because the variance of a gamma-distributed variable on the log scale is equal to where ν is the shape parameter of the gamma distribution [24 (link)] and hence is . At the level of the statistical parameters (table 1; on the ‘expected data’ scale; sensu [25 (link)]; see their fig. 1), both Poisson and negative binomial distributions can be seen as special cases of gamma distributions, and can be obtained using the trigamma function (table 1). For example, for the Poisson distribution is (note that ). As shown in the electronic supplementary material, appendix S2, ln(1 + 1/λ) (lognormal approximation), 1/λ (delta method approximation) and (trigamma function) give similar results when λ is greater than 2. Our recommendation is to use the trigamma function for obtaining whenever this is possible.
The trigamma function has been previously used to obtain observation-level variance in calculations of heritability (which can be seen as a type of ICC although in a strict sense, it is not; see [25 (link)]) using negative binomial GLMMs ([24 (link),26 (link)]; cf. [25 (link)]). Table 1 summarizes observation-level variance for overdispersed Poisson, negative binomial and gamma distributions for commonly used link functions.

We have developed a protocol that builds on the RADseq method [19] (link) but which differs in two principal respects (Figure 2). First, our method eliminates random shearing and end repair of genomic DNA (an advantage shared with a family of partially overlapping protocols such as MSG, CrOPS, and other recent RADseq derivatives [9] , [20] (link), [21] (link)). Instead, we use a double restriction enzyme (RE) digest (i.e., a restriction digest with two enzymes simultaneously) that results in at least five-fold reduction in library production cost–complete ddRADseq libraries cost ∼$5 per sample, while the necessary enzymatic steps following the initial restriction digest and ligation in random shearing RAD libraries alone introduce a cost of ∼$25 per library (NEB, Ipswich, MA). Furthermore, the elimination of several high-DNA-loss steps permits construction of ddRAD libraries from 100 ng or less of starting DNA. Second, we introduced a precise selection for genomic fragments by size, which allows greater fine-scale control of the fraction of regions represented in the final library (see results). By combining precise and repeatable size selection with sequence-specific fragmentation, double digest Restriction-Site Associated DNA sequencing (ddRADseq) produces sequencing libraries consisting of only the subset of genomic restriction digest fragments generated by cuts with both REs (i.e., have one end from each cut) and which fall within the size-selection window (Figure 2B). This combination of requirements can be tuned to generate libraries consisting of fragments derived from hundreds to hundreds of thousands of regions genome-wide.
Precise, repeatable size selection offers two further advantages. First, because only a small fraction of restriction fragments will fall in the target size-selection regime (<5% in conditions described here), the probability of sampling both directions from the same restriction site is low. This reduces “duplicate” (i.e., immediately neighboring) region sampling, which effectively halves the number of reads that are required to reach high-confidence sampling of a SNP associated with a given RE cut site. Second, shared bias in region representation favoring fragments closest to the mean of size selection, in turn, biases independent samples (e.g., from different individuals) towards recovering the same genomic regions (Figure 2B). Because of this correlated recovery, regions are “filled in” with reads in approximately the same order across all individual samples, and samples with read recovery counts below saturation will still share a significant number of well-covered regions (“Experimental ddRADseq results” below; Analysis S1 Supporting Figure 4; Analysis S1 “Region recovery: ddRADseq vs. random shearing”). Both of these properties make the ddRADseq method robust to under-sampling with respect to read counts, which is a commonly observed problem arising from unequal read representation across individual samples in pooled sequencing experiments [9] , [22] (link), [23] .

We added certain amounts of carbazole and anhydrous aluminum chloride to a single-port round-bottom flask and used anhydrous dichloromethane as the solvent. We thoroughly mixed the carbazole and catalyst under anhydrous conditions. Carbazole and anhydrous aluminum chloride were completely mixed, dissolved under agitation, and cooled to 0 °C. The 2-chloro-2-methylpropane was dropped into the mixture and stirred quickly at room temperature to make them completely mixed when the temperature of the mixing system reached 0 °C. The reaction ended, and tert-butylcarbazole derivatives were obtained after 3 h.
The synthesis of the propyl carbazole derivatives was similar to that of the tert-butylcarbazole derivatives, which are not detailed here.

As illustrated in Scheme 1, a variety of 2-morpholino-4-anilinoquinoline derivatives were synthesized starting from 2-morpholinoquinolin-4-ol, 1. Chlorination of 1 with phosphorus oxychloride resulted in 4-chloro-2-morpholinoquinoline, 2. Transformation of 2 to 2-morpholino-4-anilinoquinoline derivatives 3a–3e was carried out by substituting the chlorine atom with the corresponding aniline.