HIest (version 2.0)

HItest: Compare the likelihood of hybrid classification to MLE estimates of ancestry and heterozygosity.

Description

HItest compares the best fit of six early generation diploid hybrid genotypes (parental, F1, F2, backcross) to the maximum likelihood genotype decribed by ancestry (S) and interclass heterozygosity (H).

Usage

HItest(class, MLE, thresholds = c(2, 1))

Arguments

class

Output from HIclass: a data frame summarizing the fit of each individual to the six genotype classes.

MLE

Output from HIest: a data frame giving the MLE S and H and associated log-likelihood.

thresholds

Criteria for classification. The first criterion (thresholds[1]) is a cutoff for the difference in log-likelihood for the best vs. second best genotype class. The second criterion (thresholds[2]) is a cutoff for the difference in log-likelihood for the best genotype class vs. the MLE.

Value

A data frame with one row per individual. Columns are:

S

The maximim likelihood estimate of the ancestry index from HIest.

H

The maximum likelihood estimate of the interclass heterozygosity from HIest.

Best.class

The class with the highest likelihood of the six from HIclass.

LL.class

The log-likelihood of the data for the best-fit class from HIclass.

LLD.class

The difference in log-likelihood between the best and second-best fit class from HIclass.

LL.max

The maximum log-likelihood from HIest.

dAIC

The difference in AIC between the continuous model MLE (2 estimated parameters) and the best-fit class (1 estimated parameter).

c1

Logical: TRUE if the best-fit class is supported by more than thresholds[1] log-likelihood units over the second best.

c2

Logical: TRUE if the best-fit class is WITHIN thresholds[2] log-likelihood units of the MLE.

Details

As a quick-and-dirty rule of thumb, one might accept a putative classification as credible if the log-likelihood of the best-fit class was over 2 units greater than the log-likelihood of the second best-fit class AND within 2 units of the maximum log-likelihood. The first criterion is based on the approximate equivalence of a 2 x log-likelihood interval to a 95 percent confidence interval for some distributions (Hudson 1971; Hillborn and Mangel 1997). The second is based on the conventional penalty of two log-likelihood units for an additional estimated parameter in model selection (Edwards 1972; Burnaham and Anderson 2004). The classification model can be viewed as having one free parameter (once the best-fit class is set to "chosen", the other five are constrained to "not chosen"), while the continuous model has two (S and H). This approach has the disadvantage of effectively treating the classification as a null model, which is not biologically justified.

A better approach might be to accept the classification only if its AIC is lower than the AIC of the MLE, i.e., if dAIC is negative (Fitzpatrick 2012). Note that dAIC cannot be less than -2 (the case where the MLE is identical to the expectation for a class).

References

Burnham, K. P., and D. R. Anderson. 2004. Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33:261-304.

Edwards, A. W. F. 1972. Likelihood. Cambridge University Press, Cambridge.

Fitzpatrick, B. M. 2008. Hybrid dysfunction: Population genetic and quantitative genetic perspectives. American Naturalist 171:491-198.

Fitzpatrick, B. M. 2012. Estimating ancestry and heterozygosity of hybrids using molecular markers. BMC Evolutionary Biology 12:131. http://www.biomedcentral.com/1471-2148/12/131

Hilborn, R., and M. Mangel. 1997. The ecological detective: Confronting models with data. Princeton University Press, New Jersey.

Hudson, D. J. 1971. Interval estimation from the likelihood function. Journal of the Royal Statistical Society, Series B 33: 256-262.

Lynch, M. 1991. The genetic interpretation of inbreeding depression and outbreeding depression. Evolution 45:622-629.

See Also

HIest for maximum likelihood estimation of S and H, HIsurf for a likelihood surface, HIclass for likelihoods of early generation hybrid classes, HILL for the basic likelihood function.

Examples

	## Not run: 
# data(Bluestone)
# Bluestone <- replace(Bluestone,is.na(Bluestone),-9)
# # parental allele frequencies (assumed diagnostic)
# BS.P <- data.frame(Locus=names(Bluestone),Allele="BTS",P1=1,P2=0)
# 
# # estimate ancestry and heterozygosity
# BS.est <-HIC(Bluestone)
# 
# # calculate likelihoods for early generation hybrid classes
# BS.class <- HIclass(Bluestone,BS.P,type="allele.count")
# 
# # compare classification with maximum likelihood estimates
# BS.test <- HItest(BS.class,BS.est)
# 
# table(BS.test$c1)
# # all 41 are TRUE, meaning the best classification is at least 2 log-likelihood units
# # better than the next best
# 
# table(BS.test$c2)
# # 2 are TRUE, meaning the MLE S and H are within 2 log-likelihood units of the best
# # classification, i.e., the simple classification is rejected in all but 2 cases.
# 
# table(BS.test$Best.class,BS.test$c2)
# # individuals were classified as F2-like (class 3) or backcross to CTS (class 4), but
# # only two of the F2's were credible 
# 
# BS.test[BS.test$c2,]
# # in only one case was the F2 classification a better fit (based on AIC) than the
# # continuous model.
# ## End(Not run)