# “ANOVA on a non-random non-Normal sample from a Normal Population”

How can I run ANOVA or tests for statistical significance on a bi-modal sample that came from a normal population?

Context:

I was tasked with running an ANOVA to see if genotypes (treatments / factors) had an effect on phenotypes (responses) in a simple fixed-effects model. The standard is to apply a fixed-effects linear model on each genotype to phenotype, run ANOVA, then a Tukey's HSD. See here for more context.

Unfortunately, out of the 416 samples phenotyped, only 356 were genotyped (~85%). Furthermore, the samples genotyped were non-random; most of the excluded samples came from the peak, shifting the samples from a normal distribution to a bi-normal distribution.

I know ANOVA doesn't apply when we have a non-normal population.

To fix this, would I use bootstrapping or parametrization? How would I set that up?

Mock-data to show extent of sampling bias (in R):

    data_pre_sample <-  structure(list(phenotype = structure(1:9, .Label = c("1", "2", "3", "4", "5", "6", "7", "8", "9"), class = "factor"),
frequency = c(4, 16, 48, 108, 116, 88, 32, 4, 0)),
.Names = c("phenotype", "frequency"), row.names = c(NA, -9L), class = "data.frame")
data_post_sample <- structure(list(phenotype = structure(1:9, .Label = c("1", "2", "3", "4", "5", "6", "7", "8", "9"), class = "factor"),
frequency = c(4, 16, 48, 108, 64, 80, 32, 4, 0)),
.Names = c("phenotype", "frequency"), row.names = c(NA, -9L), class = "data.frame")


Phenotypes:
- 1, 2, 3, 4, 5, 6, 7, 8, 9

Pre-selection phenotype count:
- 4, 16, 48, 108, 116, 88, 32, 4, 0

Post-selection phenotype count:
- 4, 16, 48, 108, 64, 80, 32, 4, 0

Sources Consulted:

I apologize if my question isn't clear; I am relatively new to stats stackexchange and I am just refreshing and expanding my knowledge in statistics.

• While I do have a second experiment, it needs to be curated a LOT still (super noisy clustering, even when using HDBSCAN. Clustering was done to find the genotypes). I was hoping to run this ANOVA as a preliminary test to see whether it was worth curating the other population. This population has sampling bias literally because the geneticists couldn't genotype all the samples and decided to cut off the ones in the middle so they could have more samples on the extremes (another standard in biology apparently). – A Duv Jun 19 '18 at 2:23