Download Chi Square Pg 302 Why Chi - Squared Biologists and other

Chi Square
Pg 302
Why Chi - Squared
▪ Biologists and other scientists use relationships they have discovered
in the lab to predict events that might happen under more real-life
circumstances.
▪ These predictions are, at some point, supported by real data.
– More often than not, these 2 sets of data, do not always match exactly.
▪ The change in the data, or fluctuations, between lab data and real life
data are due to a large variety of variables that can’t be accounted for
in the lab, or something else that a scientists may have missed.
▪ How do we know??
– The most common test used is called the Chi-Squared.
Are two genes linked or unlinked?
▪ As we have talked about, not all patterns of inheritance follow
Mendel’s law of independent assortment.
▪ Genes that are found in close proximity on the same
chromosome tend to be inherited together, which makes them
linked genes.
▪ But how do we tell?
– You guessed it Chi-Squared!!!
How it’s done.
▪ If genes are unlinked and assorting independently, then you would
expect a phenotypic ratio of 1:1:1:1.
▪ If the genes are linked, however this ratio will be more like 1:1:0: 0,
where you would expect the offspring to look like either parent.
– This means that the gens are NOT assorting independently.
▪ The Chi-Square will measure the “goodness of fit”, by comparing
observed data to the expected.
▪ It should also be noted that the size of the experimental set is
important. With small data sets, even if the genes are linked,
discrepancies might be small by chance, if the linkage is weak.
– *For simplicity we ignore the sample size.
The formula
𝟐
▪𝑿
=
(𝒐−𝒆)𝟐
𝜮
𝒆
▪ Where o = observed, and e =
expected.
Model 1- Calculating Chi-Squared (Let’s Try one)
Hypothesis- there is an equal chance of flipping heads or tails on a coin.
Coin A
Observed Data
(o)
Expected
(e)
Heads
108
100
Tails
92
100
Total
200
200
(o-e)
(o-e)^2
(𝑜 − 𝑒)^2
𝑒
𝟐
(𝒐
−
𝒆)
𝑿𝟐 = 𝜮
𝒆
Hypothesis- there is an equal chance of flipping heads or tails on a coin.
Coin B
Observed Data
(o)
Expected
(e)
Heads
120
100
Tails
80
100
Total
200
200
(o-e)
(o-e)^2
(𝑜 − 𝑒)^2
𝑒
𝟐
(𝒐
−
𝒆)
𝑿𝟐 = 𝜮
𝒆
Now let’s answer a couple questions…
▪ 1. What is the hypothesis that is being tested in Model 1?
▪ 2. Describe the experiment being performed in Model 1.
▪ 3. How many flips of the coin will be conducted in each trail for the
experiment in Model 1?
▪ 4. If you were told that one of the coins used in the experiments in
Model 1 was a “trick” coin, which coin would you predict was rigged?
Explain your reasoning.
▪ Choose the correct phrase to complete the sentence is:
– A larger chi-square value means the observed data is very different/ very similar
to the expected data.
The experiments in model one didn’t give us an
expected out come…. Why?
▪ Was it due to random chance?
▪ Or does the coin being flipped favor head for
some reason?
▪ Using our Chi-Square data, and knowing that in
statistics a “significant” difference means there
is less than 5% chance that the variation in that
data is due to random events.
– So! The variation would mostly likely be due to an
environmental factor.
Degrees of Freedom
▪ To determine if the chi-square value is large
enough to be “significant” we used degrees of
freedom.
▪ If an experiment has five possible outcomes, then
in reality there is one result and 4 other
possibilities(total of five).
▪ Degrees of freedom = number of possible
outcomes - 1
Degrees of freedom
▪ How many outcomes were possible in the coin flip experiments?
▪ How many degrees of freedom were there in the coin flip
experiment?
Applying it to genetics!
▪ If 100 heterozygous (Bb) males mate with
100 heterozygous females (Bb).
– Predict the number of offspring from the 100
mating pairs that will be each genotype.
▪ BB
▪ Bb
▪ bb
Observed vs. Expected
Observed
(o)
BB
28
Bb
56
bb
16
Expected
(e)
(o-e)
(o-e)^2
The sum of:
(𝒐 − 𝒆)^𝟐
𝒆
Back to degree’s of freedom
▪ How many genotype outcomes were possible?
▪ How many degrees of freedom are there?
What do we do with all this information???
Use a Chi-Squared Distribution Table! Converts Chi-Square
values to probability that the differences in the data are due to
chance.
Accept
hypothesis
Reject
hypothesis
So… back to our problem
▪ We had a X2 value of 4.32
▪ With 2 degrees of freedom
▪ Where does that fall on our table?
▪ Significant or Not Significant?
▪ We have a P value that is 0.2 < p value < 0.1
– NOT SIGNIFICATNT!
▪ But what does that mean????
– It means there is a 20% to 10% change that variability in the data is only
due to change. This is not significant.
Now, turn to page 302 in your book…
▪ Work out that problem for homework.