Are genetic drift and inbreeding the same thing?

Does it ever happen to you that the more you try to understand something, the more difficult to understand it turns out to be? Recently, I’ve had such a problem with two of the very basic microevolutionary phenomena – genetic drift and inbreeding.
Genetic drift and inbreeding are associated with changes in allele frequencies and heterozygosity, and are particularly important in small populations. Their causes and effects are so intertwined that I ended up asking “Are genetic drift and inbreeding the same thing?”

The problem is that at small population sizes, the combined effect of genetic drift and inbreeding leads to increased homozygosity and fixation of alleles, including deleterious alleles. Sometimes these processes are described as independent forces operating at the same level, while elsewhere inbreeding tends to be addressed as a result of genetic drift.
I realised that despite using these terms in everyday scientific discussions, writing, and when presenting my research, I wasn’t able to properly address them. Feeling like a very poor PhD student, I turned to the best friend of every desperate [fill in your occupation], i.e. I asked Google.
I’m not sure if it made me feel relieved or even more concerned when I found out that the relationship of genetic drift and inbreeding was something that already (two of) the fathers of the modern evolutionary synthesis couldn’t agree upon.

 
 
 
 
 
 
 
 
 
The turbulent relationship of genetic drift and inbreeding (as well as the relationship of R. A. Fisher and Sewall Wright) has been evolving through decades. It has started in the 1920s and 1930s with Wright’s definition of the inbreeding coefficient (Wright 1922), Wright-Fisher’s model of binomial sampling in a finite population (Fisher 1930, Wright 1931) and introduction of the concept of effective population size (N_e; Wright 1931, 1939).

“[Effective population size] is the size of an idealized population with the same gene frequency drift or inbreeding as the observed population. An idealized population is panmictic with each parent having an equal expectation of progeny.” (Crow 2010).

Aged 94, a year before his death, James F. Crow wrote a perspective article titled “Wright and Fisher on inbreeding and random drift” (Crow 2010). On three pages, Crow managed to explain both of the giants of theoretical population genetics, and provided a simple explanation to the complex problem.

“Fisher did not consider the irregular consanguineous matings that occur, especially in animal pedigrees, and for which Wright’s inbreeding algorithm is especially useful. I doubt, however, that this would have changed Fisher’s opinion. He clearly thought that consanguineous mating within a large population, whether systematic or not, was quite different from increased fixation due to small population size.” (Crow 2010)

On the other hand…

“Wright was particularly pleased that his F statistics could be used to measure random drift as well as consanguineous mating…. Since he could use the same formulas for both inbreeding and random drift, Wright naturally thought of these as two sides of the same coin.” (Crow 2010)

And not surprisingly, Crow found the answer to the problem in the effective population size. He realised that there is not just one way of defining N_e and in association with genetic drift and inbreeding described variance (N_eV) and inbreeding (N_eI) effective sizes, respectively (Crow 1954). These were later even further developed to consider separate sexes and selection (Crow & Denniston 1988, Caballero 1994).


I won’t go into details, but what these formulas clearly show is that N_eI and N_eV are not the same thing, in which case R. A. Fisher is the winner, voilà!
…unless the population size remains constant, and then they can be both reduced to:

and then Wright is the winner. Touché!
In reality, the conditions of constant population size are much less common, so if I were to choose the winner, it would be Fisher. Or maybe Crow.
References
Caballero, A., 1994. Developments in the prediction of effective population size. Heredity 73, 657–679.
Crow JF (2010) Wright and Fisher on Inbreeding and Random Drift. Genetics, 184(3), 609-611. doi:  10.1534/genetics.109.110023
Crow JF & Denniston C (1988) Inbreeding and variance population numbers. Evolution 42, 482–495.
Fisher RA (1930) The Genetical Theory of Natural Selection. Clarendon Press, Oxford. 
Wright S (1922) Coefficients of inbreeding and relationship. Am. Nat. 56, 330–338.
Wright S (1931) Evolution in Mendelian Populations. Genetics 16, 97–159.
Wright S (1939) Statistical Genetics in Relation to Evolution (Exposes de Biometrie et de Statistique Biologique, Vol. 13) Hermann & Cie, Paris.