The distributions of log-likelihood ratios (ΔLL) obtained from fitting ion-channel dwell-time distributions with nested pairs of gating models (Ξ, full model; ΞR, submodel) were studied both theoretically and using simulated data. When Ξ is true, ΔLL is asymptotically normally distributed with predictable mean and variance that increase linearly with data length (n). When ΞR is true and corresponds to a distinct point in full parameter space, ΔLL is Γ-distributed (2ΔLL is χ-square). However, when data generated by an l-component multiexponential distribution are fitted by l+1 components, ΞR corresponds to an infinite set of points in parameter space. The distribution of ΔLL is a mixture of two components, one identically zero, the other approximated by a Γ-distribution. This empirical distribution of ΔLL, assuming ΞR, allows construction of a valid log-likelihood ratio test. The log-likelihood ratio test, the Akaike information criterion, and the Schwarz criterion all produce asymmetrical Type I and II errors and inefficiently recognize Ξ, when true, from short datasets. A new decision strategy, which considers both the parameter estimates and ΔLL, yields more symmetrical errors and a larger discrimination power for small n. These observations are explained by the distributions of ΔLL when Ξ or ΞR is true.
Abbreviations used
LL
log likelihood
ΔLL
log-likelihood ratio
E(X)
expectation value of the random variable X
D2(X)
variance of X
D(X)
standard deviation of X
Ω
parameter space
Ξ
Θtwo different models, defined in terms of both scheme and parameter vector
