Transcript Non-Mendelian Genetics

Cancer and Chi-Square
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How can counting onion root tip cells
help us understand cancer?
Lab: How can fungal pathogens
cause cancer?
Pre-lab questions
• What is cancer?
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• What is lectin or a
“lectin-like protein”?
A whole bunch of
lectin proteins
Does the data fit?
• Problem: How can
you statistically
determine if increased
levels of mitosis are
enough to be
considered cancer?
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•Each tissue has a normal mitotic rate calculated from
Historical counts. We are going to use 85% for onion roots
Cancerous??
as well as for your tissue.
•Example: What if the results are 80 to 20? 80 to 10?
84 to 16?
Does this sample have abnormal
mitotic rates? Use a chi-square
analysis to figure it out!
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The Chi Square Test
• A “Goodness of fit” test
• A statistical way to “Reject the Null
Hypothesis”
Null Hypothesis
A way to state expected outcome
• Example 1: Healthy tissue biopsies will show +85%
of cells in interphase
• Example 2: Flipping a coin should land on heads
50% of the time.
**Chi-square is used to REJECT a hypothesis not to
PROVE it.
Coin Data
• If this coin is fair…then we would expect
the chi-square analysis to show us the
results are due only to chance and not due to
some other hypothesis.
• Let’s look how coin data either:
– Rejects a hypothesis that a coin is fair or
– Suggests that the hypothesis that a coin is fair is
likely.
The Formula
2
(
obs

exp)
2  
exp
Example
• Expected results of a coin toss (the Null
Hypothesis): 50 heads , 50 tails
• Observed results: 55 heads, 45 tails
• Now it's just a matter of plugging into the formula:
2 = (55 - 50)2 / 50 + (45 - 502 / 50
= (5)2 /50 + (-5)2 / 100
= 25 / 50 + 25 / 50
= 0.50 + 0.50
= 1.0
• This is our chi-square value: now we need to see
what it means and how to use it.
Chi-Square Table
Using the Table
• In our example of 55 heads to 45 tails, we
calculated a chi-square value of 1.0, with 1 degree
of freedom.
• Looking at the table, 1 d.f. is the first row, and p =
0.05 is the sixth column. Here we find the critical
chi-square value, 3.841.
• Since our calculated chi-square, 1.0, is less than
the critical value, 3.841, we “fail to reject” the null
hypothesis. Thus, an observed ratio of 55 heads to
45 tails is a good fit to 50:50 ratio.
Degrees of Freedom
• A critical factor in using the chi-square test is the
“degrees of freedom”, which is essentially the
number of independent random variables
involved.
• Degrees of freedom is simply the number of
classes of offspring minus 1.
• For our example, there are 2 classes of offspring:
heads and tails. Thus, degrees of freedom (d.f.) =
2 -1 = 1.
Critical Chi-Square
• Critical values for chi-square are found on tables,
sorted by degrees of freedom and probability
levels. Science uses p = 0.05.
• If your calculated chi-square value is greater than
the critical value from the table, you “reject the
null hypothesis”.
• If your chi-square value is less than the critical
value, you “fail to reject” the null hypothesis (that
is, you accept that your genetic theory about the
expected ratio is correct).