H₀ Null Hypothesis

The null hypothesis represents the case in which the sample data does not differ from the population data, i.e., (informally) nothing interesting is going on.

H_a Alternate Hypothesis

The alternate hypothesis represents the hypothesis to be tested; that is, the presence of a trend. It is directly contradictory to the null hypothesis.

It’s important to note that we cannot definitively accept the null hypothesis; instead, we write that we “fail to reject the null hypothesis,” as a lack of evidence does not necessarily indicate that a trend does not exist.

The formula for a Goodness of Fit Test is as follows:

\(χ^2=\sum\frac{(o_i-e_i)^2}{e_i}\), where:

Observed Value (o_i)

The experimental/observed data you wish to test.

Expected Value (e_i)

Expected data from the distribution model you wish to test your sample data against.

Test statistic (χ²)

The value which is compared against a known distribution of critical values to determine if the results are significant.

• If χ² > critical value, the results are insignificant, and we fail to reject the null hypothesis
• If χ² < critical value, the results are significant, and we reject the null hypothesis

Degrees of Freedom (df)

The number of categories that are free to vary.

• Equal to the number of categories (k) minus one.
• e.g. If your potential outputs are heads or tails, your df = (k-1) = (2-1) = 1.

Alpha (α)

The degree of certainty you wish to perform your test with. Alongside degrees of freedom, this statistic will inform which critical value is used to compare the test statistic to.

• e.g., α = 0.01 means that you want to be 99% certain that the sample distribution is not due to chance.

Mendelian inheritance predicts a 9:3:3:1 ratio of offspring phenotypes in a dihybrid cross for two genes expressing complete dominance. If a cross is performed and the offspring express a different phenotypic ratio, these results could be due to chance or due to gene linkage.

Phenotype: A characteristic of an organism determined by the arrangement of alleles in a gene.

Dihybrid Cross: A cross performed between two parents which are both heterozygous (BbEe) for two genes.

Complete Dominance: The presence of the dominant allele (B) will fully mask the effects of the recessive allele (b) on phenotype.

Gene Linkage: Genes closer together on the same chromosome may have a higher likelihood of being inherited together.

Imagine we cross two heterozygous fruit flies (BbEe), where:

• The body color gene, where B codes for brown bodies and b codes for black bodies
• The eye color gene, where E codes for red eyes and e codes for brown eyes.

We would expect to see a cross as follows:

For an experimental cross of heterozygous, Brown-bodied, red-eyed (BbEe) fruit flies, the following offspring distribution was obtained:

We would like to investigate whether the genes for body color and eye color are linked.

Perform Goodness of Fit Test

We can use the R function chisq.test() to test our hypothesis. chisq.test(x, p) takes two parameters for our calculation; a numeric vector X, and a vector of probabilities p, where length(x) = length(p)

probabilities <- c(9/16, 3/16, 3/16, 1/16)
flychi <- chisq.test(flycounts$Observed, p=probabilities)
flychi

## 
##  Chi-squared test for given probabilities
## 
## data:  flycounts$Observed
## X-squared = 23.554, df = 3, p-value = 3.094e-05

Here is a barplot of the observed vs. expected data.

The observed values are visually different from the expected values, but is this significant enough to suggest linkage?

The package gginference contains a function that allows us to plot Chi-Square distribution for our data.

\[23.554 > 7.815\]

Our test statistic is larger than the critical value for a distribution with df = 3, and we are therefore able to reject the null hypothesis; something in this offspring distribution is different from what we would expect from a typical dihybrid distribution, and we are 95% certain this is not due to chance.

The distribution we obtained suggests that the genes for body color and eye color are linked, as well as suggesting potential allelic arrangements on the parental chromosomes.