POLDISP 1.0c. a software package to estimate Pollen Dispersal. by Juan J. Robledo-Arnuncio, Frédéric Austerlitz and Peter E. Smouse USER'S MANUAL

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1 POLDISP 1.0c a software package to estimate Pollen Dispersal by Juan J. Robledo-Arnuncio, Frédéric Austerlitz and Peter E. Smouse USER'S MANUAL June 009 (see release notes at the end of this document) Available from: Correspondence: poldisp@gmail.com Citation: Robledo-Arnuncio JJ, Austerlitz F, Smouse PE (007) POLDISP: a software package for indirect estimation of contemporary pollen dispersal. Molecular Ecology Notes 7:

2 1. Purpose POLDISP is a software package to make inferences on contemporary pollen dispersal from mapped mother-offspring diploid genotypic data. Unlike paternity-based methods, the indirect procedures implemented in POLDISP do not require mapping and genotyping potential pollen donors within the study area. POLDISP contains two different modules (KINDIST and TWOGENER) that can be used complementarily to estimate the following parameters: - the average of the pollen dispersal distribution - the variance of the pollen dispersal distribution - the kurtosis of the pollen dispersal distribution - the correlated paternity rate within each maternal sibship - the correlated paternity rate between each maternal sibship pair - the effective male population density (d e ). Installation and Program's Use overview POLDISP runs under Windows operating system. The self-extracting file "POLDISP_1-0.zip" containing the program package can be downloaded from You can extract its content into any folder in your computer. There is no particular installation process required; you can use the two extracted programs (KINDIST.exe and TWOGENER.exe) by double-clicking on the corresponding executable file directly. The only requirement is that the input data file must be in the same folder than the used executable file. POLDISP implements both the KINDIST and TWOGENER estimation methods (in separate executable files). KINDIST (Robledo-Arnuncio et al. 006) estimates the distribution of pollen dispersal distances from the spatial coordinates and genotypes of a sample of seed-plants and their respective maternal progenies. It also provides estimates of the correlation of paternity within and among the maternal sibships in the sample. TWOGENER (Austerlitz & Smouse 001, 00) jointly estimates the distribution of pollen dispersal distances and the effective male population density (d e ), from the same input data. KINDIST has proved to yield more accurate estimates of the pollen dispersal distribution than TWOGENER when d e is unknown. Therefore, it is advisable to use KINDIST to estimate the pollen dispersal curve, rather than TWOGENER, unless there is independent information on d e. TWOGENER, on the other hand, can be used to obtain an estimate of d e, which may have biological interest on its own right. The advisable (more accurate) way to estimate d e with POLDISP involves two consecutive actions: (i) to estimate the pollen dispersal distribution using KINDIST; and (ii) to estimate d e with TWOGENER, including as input data the dispersal parameter estimates previously obtained with KINDIST (see Box 1). Note however that the anisotropic normal dispersal model (Austerlitz et al, 007) is only available for the moment in TWOGENER and thus can only performed there using the complete TWOGENER estimation procedure.

3 To estimate the pollen dispersal curve To estimate d e Is d e known? Estimate the dispersal parameters () with KINDIST NO YES Use KINDIST Use either KINDIST or TWOGENER Estimate d e with TWOGENER, using as input information Box 1. Usage of the two methods implemented in POLDISP program package. (d e is the effective male population density) 3. POLDISP input data file The input data file for POLDISP (valid for both KINDIST and TWOGENER programs) must be a tab-delimited text file. You can create this file by using a worksheet program (such Microsoft Excel) and saving the file as a tab-delimited text. The input data file must contain the following information (each piece of information within a line must be separated by a tab): First line: two numbers, indicating the number of mothers and the number of loci, in this order. Next n m lines: labels, genotypes and spatial coordinates of each of the n m mothers (one line for each mother). Each line must start with the mother's label (which can be any integer number), followed by the mother's genotype (alleles must be coded with numbers, excepting zero; no missing data are allowed for the mothers), and finally the mother's spatial (x,y) coordinates. Next n o lines: labels and genotypes of each of the n o sampled offspring (one line for each offspring). Each line must start with the offspring's label, which must be the same one than that of its mother, followed by the offspring's genotype (alleles must be coded with numbers, using 0 for missing data). Notes about input data format: - Each maternal family must have a minimum of offspring. - The mother's (and thus the offspring's) labels must be integer numbers. - Alleles must be coded with numbers, 0 (zero) being reserved for missing data.

4 - Missing data are allowed in the offspring's genotypes (coded as 0 -zero-), but not in the mother's genotypes. - Loci, both for mothers and offspring, must always be listed in the same order. - The lines corresponding to the offspring's information must be grouped by maternal family, and the maternal families must be listed in the same order as the corresponding mother's order. Input data example: The following is an example of a POLDISP input data file with 3 mothers and loci Additionally, a test dataset (named test_data.txt) is included in the self-extracting POLDISP package file. 4. Using KINDIST program 4.1. KINDIST Method Overview KINDIST estimates the distribution of pollen dispersal distances from the spatial coordinates and genotypes of a sample of seed-plants and their respective maternal progenies. It has proved to be the most accurate indirect method so far for estimating the pollen dispersal curve. It also provides estimates of the correlation of paternity within and among maternal sibships. KINDIST is based on the expected decay with distance of a normalized measure of correlated paternity among maternal sibship pairs: (z) (defined as the ratio of correlated paternity between sibship pairs a distance z apart to the average correlation of paternity within sibships in the population; see Robledo-Arnuncio et al. 006). Under the same isolation by distance theoretical scheme developed for the TWOGENER model (Austerlitz & Smouse 001, 00), KINDIST calculates the theoretically expected distribution of (z) as a function of the parameters of the pollen dispersal distribution, and then estimates the values of these parameters that yield the best least-square fit of

5 expected to observed (z)-values. Unlike the TWOGENER model, however, KINDIST is not explicitly dependent on the unknown effective density of males, and thus it does not require a joint estimation of this quantity, yielding more accurate estimates of the dispersal function parameters (Robledo-Arnuncio et al. 006). 4.. Running KINDIST program Once the input data is ready (see section 3) in the same folder as the program file (KINDIST_1-0.exe), you can launch the application by double-clicking in the program file. You should proceed through the following two steps to estimate the pollen dispersal function: I) After entering the requested data file name, the program will calculate the correlation of paternity within and among the maternal families in the sample. These estimates will be saved to an output text file in tab-delimited format, which can be edited on any standard worksheet program (such as Microsoft Excel). The program will pause at this point. Using the information contained in this file, you should inspect the relationship between among-sibship correlated paternity and separation distance. If no decrease in among-sibship correlated paternity is detected with distance, the pollen-pool spatial genetic structure is probably too weak to allow a reliable estimation of the pollen dispersal function; in this case, subsequent estimations using the POLDISP package become inadvisable. By contrast, if among-sibship correlated paternity decreases with distance, you may want to continue with the second step of the analysis. II) Upon inspection of the decrease of among-sibship correlated paternity with separation distance, you will need to enter a reference threshold distance to define unrelated pollen pools. This distance is used by the program to calibrate kinship coefficients. You can set this distance at the approximate observed value above which among-sibship correlated paternity stabilizes (it typically stabilizes at a slightly negative value). Although the estimation algorithm is reasonably insensitive to the precise value of this threshold distance, you may test different values of this parameter (in consecutive runs of the program) to check the stability of the results. Once you have entered a value for this threshold distance, the program will ask you to choose a dispersal distribution to fit, and will then start the estimation of the dispersal function. The dispersal estimates will be saved to a second output file KINDIST program output KINDIST program generates two different output text files: "corr_paternity_dataname.txt" and "disp_estimates_dataname.txt" (where "dataname" is the name of the input data file). The first one contains the estimated pollen pool allele frequencies and the estimated correlated paternity rates, within and among maternal sibships. The second one stores the iterations of the dispersal-function estimation algorithm (saved progressively during run time) and the following parameter estimates of the pollen dispersal distribution (saved when the estimation is finished): a (the scale parameter), b (the shape parameter), average, sq-rooted axial variance and kurtosis. This file also contains the least-square residual of the fitted dispersal distribution.

6 average is the estimated mean pollen dispersal distance; sq-rooted axial variance is the square root of the axial variance of pollen dispersal; kurtosis is the two-dimensional kurtosis, calculated as the ratio between the 4 th central moment and the squared nd central moment of the dispersal distribution (note that this is not the kurtosis excess, i.e., no subtraction of the kurtosis value of the normal is made); least-square residual is the sum of the squared differences between the expected and observed (z)-values, under the assumed dispersal distribution. By comparing the least-square residual yielded by different assumed dispersal distributions in consecutive runs of the program, you can have a hint on which one provides a better fit (i.e., has the minimum residual, for a given number of estimated parameters). Notes on output information: - some of the correlated paternity estimates may be negative. This is a normal outcome of genetic marker based relatedness coefficients, on which the correlated paternity estimates computed in KINDIST are based. A negative correlated paternity value indicates that the corresponding paternal gametes are less related than the average in the sample. - average and sq-rooted axial variance are expressed in the same length units that the spatial coordinates in the input file. - average, sq-rooted axial variance and kurtosis may be infinite for some dispersal distributions (namely, for the geometric and Dt families). See Section the kurtosis estimate is two-dimensional (D). Note, for instance, that the D-kurtosis of the normal is, while the 1D-kurtosis is 3 (for large kurtosis values, the D-kurtosis tends to be half the 1D-kurtosis) KINDIST technical notes Estimation of correlated paternity KINDIST computes correlated paternity estimates following the procedure described in Robledo-Arnuncio et al. (006), which is based on the method developed by Hardy et al. (004). This procedure is based on the computation of pairwise kinship coefficients (F) between the paternal gametic genotypes of offspring pairs. The expectation of F between two offspring is 0.5 if they have the same (non-inbred) father, and 0 if they have different (unrelated) fathers. Consequently, twice the average of F over many maternal-sib pairs yields an estimate of within-sibship correlated paternity, and twice the average of F between offspring from two different mothers provides an estimate of among-sibship correlated paternity. KINDIST computes multilocus kinship coefficients using J. Nason's (Loiselle et al. 005) estimator: F gh nl n a, l ( p p )( p p ) 1/( N 1) n a, l lag la lah la l l 1 a1 a1 p la (1 p la ) n n L a, l l 1 a1 p la (1 p la ) where n L is the number of loci, n a,l is the total number of alleles at locus l, p lag and p lah are the frequencies of allele a at locus l in the paternal gametes of the g-th and h-th

7 offspring, respectively, and pla is the average frequency of allele a at locus l over all N l paternal gametes with non-missing information at locus l Dispersal Distributions KINDIST currently can fit two bivariate, one-parameter (a) dispersal distributions (normal and exponential) and three bivariate, two-parameter (a, b) dispersal distributions (exponential-power, geometric and bivariate Student's t) (Table 1; Austerlitz et al. 004). Table 1. Probability density functions (PDF) and expressions for the n-th central moments of the five dispersal distributions implemented in POLDISP 1.0 package (see Austerlitz et al. 004). PDF n-th central moment Normal 1 x y n p n ( a; x, y) exp a πa a (1 n / ) Exponential 1 x y n pe ( a; x, y) exp a a ( n) π a b Exponentialpower b x y n (( n)/ b) pep ( a, b; x, y) exp a a πa Γ(/ b) ( / b) Geometric b n ( n) ( b n) ( b )( b 1) x y a if b n pg ( a, b; x, y) a a 1 ( b ) π if b n Dt (1 n / ) ( b 1 n / ) p b 1 π a x y Dt ( a, b; x, y) 1 a b a n ( b 1) if b 1 n / if 1 b 1 n / These functions accommodate a wide range of shapes (e.g., kurtosis values). Although the (one-parameter) normal and exponential functions are commonly found in the literature (and will be faster to fit with POLDISP, specially the normal), most experimental evidence suggest that the (fixed) shape of these distributions do not reflect properly the markedly leptokurtic pattern of pollen dispersal in natural plant populations, and may yield unrealistic dispersal estimates. It is thus advisable to try the two-parameter distributions (which accommodate more leptokurtic shapes) and check whether they yield more realistic results. The exponential-power, geometric and Dt functions have proved to fit satisfactorily seed and pollen dispersal data (Clark et al. 1998; Austerlitz et al. 004). Note that the exponential-power includes the Gaussian (b=) and the exponential (b=1) as special cases. Note also that some shape parameter (b) values may result in infinite values for the central moments of the Geometric and Dt distributions. Rather than a reflection of a real pattern, infinite moments are likely to be an artefactual consequence of extrapolating the observed data beyond a limited spatial scale of analysis. 5. Using TWOGENER program 5.1. TWOGENER method overview

8 The TWOGENER method (Smouse et al., 001) is based on the computation of the differentiation parameter ( ft ) between the pollen clouds of a sample of females within a population. The allelic frequencies in these pollen clouds are computed from the motheroffspring genotypic data and the ft parameter is estimated through an AMOVA procedure (Excoffier et al., 001). It allowed initially only to compute a global ft parameter for all sampled females, and thus only to estimate the parameter of a onedimensional dispersal function, assuming that the effective density of adults (d e ) equals the census density (Austerlitz and Smouse, 001). Then, the method was extended by computing the pairwise differentiation between the pollen clouds of all pairs of females within the sample (Austerlitz and Smouse, 00). This allows one to infer jointly the effective density (d e ) and the parameters of the one- or two-parameter curves described in Table 1 (Austerlitz and Smouse, 00, Austerlitz et al., 004). While the method works well in the case of one-parameter curves (Austerlitz et al., 00), it proved to have difficulties to converge when d e is estimated jointly with both the scale parameter (a) and the shape parameter (b) for the two-parameters dispersal curves (Austerlitz et al., 004), when sample sizes are limited. In this package, we propose a new estimation procedure, which takes advantage of the greater accuracy of the KINDIST for estimating the dispersal curve (Robledo- Arnuncio et al., 006). Specifically, we use the dispersal parameters estimated by the KINDIST method and use TWOGENER to estimate only the effective density (d e ), keeping the dispersal parameters at their values estimated with KINDIST (Box 1). Thus, by estimating only one parameter with TWOGENER, we increase the accuracy of the method. For completeness, we also provide in the POLDISP package the classical TWOGENER estimation procedure, which allows one to estimate jointly d e and the dispersal parameters for various dispersal curves. This procedure should be used only when large samples of females covering both short and long distances are available. 5.. TWOGENER input data files The format for the input file containing the mother and offspring genotypes is the same as described above (section 3). If you have genotyped other adults than the mothers, you can put theses genotypes in another tab-delimited text input file that you can name arbitrarily. Except if you want to perform the estimation under the normal anisotropic model (see below), this other file will only be used to better estimate the allelic frequencies in the population, which increases to some extent the accuracy of the TWOGENER extraction, but the method can run without it. On each line of this additional input file, you should put only the genotype of each adult for all loci. Missing data are allowed for these individuals. Do not put a label for these individuals. The number of loci in this input file should obviously be the same as in the mother-offspring genotypes input file. So for instance, if you have three other individuals genotyped for two loci, the file should look like this: You can also find a test dataset of this kind (named test_data_others.txt) in the selfextracting POLDISP file.

9 If you want to perform the estimate under a normal anisotropic model and you want to test the significance of the deviation from anisotropy using the simulation procedure described in Austerlitz et al (007), you have also to put for each of these individual its spatial coordinate on the same line as the genotypes. Note also that for this procedure to work properly, you need the spatial location of as many surrounding males as possible, including the ones for which you do not have any genotypic information. For these individuals, just put a line of zeros for their genotypes and their spatial coordinates. The file will then look like this: You can also find a test dataset of this kind (named test_data_others_coord.txt) in the self-extracting POLDISP file Running TWOGENER program Entering the settings The input file containing the mother-offspring genotypes and the optional additional file containing the other adults genotypes must be in the same folder as the program file (TWOGENER_1-0b.exe). You can launch the application by double-clicking on this program file. Once the program is launched, it will first ask you to enter the name of the input file containing the mother-offspring genotypes. Then it will ask you the name of the input file containing the other adults genotype. If you have not genotyped other adults, just press enter as requested. If you have this file, the program will ask you if it contains the spatial coordinates of the adults. Finally it will ask you whether you want only to estimate the effective density (see 5.3.) or if you want to perform the complete estimation procedure (see 5.3.3).. Then the program will ask you which family of dispersal curves you want to use among the ones given in table 1. As KINDIST does not perform yet the estimation for the anisotropic normal dispersal curve, only the complete TWOGENER estimate is possible for this family of curves. Please note that, except for the normal and anisotropic normal dispersal function, these estimation procedures can take quite a long time, up to several days. Dots will appear regularly on the screen each time an iteration of the algorithm is performed, showing that the estimation procedure is going on. Unfortunately, the number of iteration of the algorithm depends on the data set, so the number of dots provides no indication of the remaining time until the estimation procedure is completed Estimating effective density only In that case, the program will ask you for the dispersal parameters (a and eventually b) that were estimated by KINDIST under the same family of dispersal curve that you have chosen. Then, then it will compute the pairwise ft -values between all pairs of mothers. Finally, it will estimate the effective density by finding the value that minimize

10 the squared error between the observed ft -values and the expected ft -values between all pairs of mothers, with the method described in Austerlitz et al. (004). Note that this effective density will be expressed in a unit that will be consistent with the unit of the spatial coordinates of the mothers. For instance, if these spatial coordinates are in meters, the density will be expressed in individuals per squared meter. In that case, you can obtain a density per hectare by multiplying by 10, Estimating jointly all parameters In that case the program will ask you for the census density of adults in the population, because it is needed for the first step of the estimation process. Again, this has to be in a unit consistent with the unit of the spatial coordinates of the mothers. For instance, if these spatial coordinates are in meters, this density has to be in individuals per squared meter. So in that case, if your density is expressed in individuals per hectare, divide it by 10,000. The program will then provide you the global ft -value as computed in Smouse et al. (001) and several outputs depending on the chosen dispersal function, as detailed below in each case. 1) Normal dispersal function (We have only kept the estimates that perform best under this model) The program will first provide you an estimate of the parameter from the global ft value. It corresponds to the estimate ˆ g that takes the average distance between mothers into account (see eq. (6) in Austerlitz and Smouse 00). The program will also provide you the corresponding average distance of pollen dispersal (). Then it will provide you with an estimate of from the pairwise ft -values, assuming that the effective density equals the census density. It corresponds to the estimate ˆ p 3 obtained by minimization of the squared error criterion Q() given in eq. (11) in Austerlitz and Smouse (00). These estimates assume that density is set at its observed value. Then the program provides you the error for that model, i.e. the value of Q() when = ˆ p 3. Finally the program provides you the joint estimate of the effective adult density (d e ) and of. These estimates, denoted ˆd 1 and ˆ d 1 in Austerlitz and Smouse (00) are obtained by minimizing the least square criterion Q(d, ) simultaneously for d and. Then the program provides the error associated for that model, i.e. the value of Q(d, ) when d = ˆd 1 and = ˆ d 1. ) Exponential dispersal function model: The program will perform the same estimates as for the normal dispersal function, except that the dispersal parameter is denoted a. 3) Two-parameter models (Exponential-power, Geometric, Dt): The program will perform the estimates under these models again minimizing the squared error criterion. (see eq. (1) and details in Austerlitz et al. 004), in three steps.

11 Step 1: The program will first provide you with an estimated of the scale parameter (a) and the shape parameter (b), assuming that the effective density equals the census density Step : The program sets the b parameter at its previously estimated value and estimates jointly the effective density (d e ) and the scale parameter (a). Step 3: The three parameters (d e, a and b) are jointly estimated. These three steps correspond to the three lines given for each two-parameter dispersal functions in tables 4 to 7 in Austerlitz et al. (004). Note that step c does not converge well in many cases (see Austerlitz et al., 004). In that case, the program will either indicate that the estimation procedure did not converge after 5000 steps or provide completely unrealistic values. Our experience based on the comparison with paternitybase method show that it is better in that case to keep the estimates of d e and a provided by step and the estimate of b provided by step 1. 4) Anisotropic normal dispersal The program aims at estimating the three parameters (, max, min ) of the following dispersal curve: p (, min, max 1 ; x, y) max min ( x cos( ) y sin( )) exp min ( x sin( ) y cos( )) where denotes the angle made by the major dispersal axis with the north axis, while max and min are the standard deviations of dispersal distance along the major axis and the minor axis respectively (Austerlitz et al. 007). The program will first ask you the number of simulations you want to perform to assess the significance of the deviation from the isotropic model. Note again that this procedure requests to have the spatial position of as many surrounding males as possible, including the one that you have not genotyped (see Austerlitz et al. 007). If you have not these spatial positions, just enter zero in order not to do this simulation analysis. Otherwise, we advise to perform at least 1000 simulations. Then the program will provide you various results in three steps. Step 1: The program will first assume that the effective density equals the census density and provide you under this hypothesis an estimate of the dispersal parameters (, max, min ). It will also provide you the estimate of the deviation from the isotropic model (R = max / min ) and of the average dispersal distance (). The Jack-knife estimates of the standard error of these quantities are also provided. Step : The same estimations are performed with the effective density jointly estimated. Step 3: The program will perform the simulations (unless the number of simulations was set to zero) and provide you the significance level of R in both cases. To show its progress, the program will count the first simulations from 1 to 10, then it will indicate the number of simulations performed every 100 generations. max

12 5.4. TWOGENER program output Suppose your mother-offspring input file is named datafile.txt. The program generates two output files. - A file named phift_datafile.txt. This tab-delimited text file contains for each pair of mothers their label, their physical distance and their ft. You can open it under excel or any equivalent software to make a graphical representation of the measured pairwise ft s as a function of distance, as for instance in Figure 7A in Dick et al (003). You will get only the points represented on this figure and not the best-fitting theoretical curve. To represent this latter curve requires a computer software that can do numerical integration like Mathematica. A Mathematica notebook to represent it is available upon request to F.A. - A file named Twogener_density_xxx_datafile.txt (if only density was estimated, see paragraph 5.3.) or Twogener_complete_xxx_datafile.txt (if the complete TwoGener procedure was performed, see paragraph 5.3.3), where xxx denotes the assumed dispersal function. This file will contain the different estimates described in the previous sections. These estimates will also appear on screen while the program is running. 6. References Austerlitz F, Dick CW, Dutech C, Klein EK, Oddou-Muratorio S, Smouse PE, Sork VL (004) Using genetic markers to estimate the pollen dispersal curve. Molecular Ecology 13, Austerlitz F, Smouse PE (001) Two-generation analysis of pollen flow across a landscape. II. Relation between ft, pollen dispersal and inter-females distance. Genetics 157, Austerlitz F, Smouse PE (00) Two-generation analysis of pollen flow across a landscape. IV. Estimating the dispersal parameter. Genetics 161, Austerlitz F, Dutech C, Smouse PE, Davis FB, Sork VL (007) Estimating anisotropic pollen dispersal: A case study in Quercus lobata. Heredity (in press). Dick CW, Etchelecu G, Austerlitz F (003) Pollen dispersal of tropical trees (Dinizia excelsa: Fabaceae) by native insects and African honeybees in pristine and fragmented Amazonian rainforest. Molecular Ecology 1, Excoffier L, Smouse PE, Quattro JM (199) Analysis of molecular variance inferred from metric distances among DNA haplotypes: Application to human mitochondrial DNA restriction data. Genetics 131, Robledo-Arnuncio JJ, Austerlitz F, Smouse PE (006) A new method of estimating the pollen dispersal curve independently of effective density. Genetics 173, Smouse PE, Dyer RJ, Westfall RD, Sork VL (001) Two-generation analysis of pollen flow across a landscape. I. Male gamete heterogeneity among females. Evolution 55, Release Notes Version 1.0c (June 009): - TWOGENER (ver 1.0c) implements now the possibility of estimating anisotropous pollen dispersal functions. - KINDIST (ver 1.0c): the precision of the minimization algorithm used to estimated dispersal parameters has been increased to 1e-8 (approximately the square root of double machine precision). Running KINDIST

13 will take longer now (maybe about the same as before, if you are using a faster processor than with previous Poldisp versions!), but dispersal estimates should be more accurate and might be different from those yielded by the previous version in cases where there is a weak spatial genetic structure of the pollen pool. Version 1.0b (March 007):.The package includes a new version of KINDIST (ver 1.0b) with the following changes: - The tolerance of the minimization algorithm used to estimate dispersal parameters in KINDIST has been decreased (the precision increased). Running KINDIST will take longer now, but dispersal estimates should be more accurate and might be significantly different from those yielded by the previous version in cases where there is a weak spatial genetic structure of the pollen pool. - Bug fix: KINDIST ver. 1.0 yielded infinite correlated paternity estimates if two paternal gametes did not share any locus with non-missing information (the estimates were not affected otherwise). - Bug fix: KINDIST ver. 1.0 yielded wrong estimates for the scale parameter (a) of the exponential function: the output file reported a* 0.5, rather than a. All other estimates for this function were fine.