Download Chi-Squared Analysis

Chi-Squared Analysis
Stickrath
Chi-Squared Analysis
• Suppose I bet you $1,000 that I can
predict whether heads or tails will turn up
each time you flip a coin.
• The first time I say, “heads” you flip the
coin and it is heads.
• I got lucky
• The second time I say, “heads” you flip the
coin and it is heads
Chi-Squared Analysis
• The third time, fourth time, fifth time, sixth time,
seven time, eighth time, and so on I predict
heads. Each time you flip heads.
• At what point do you suspect that I am using a
two-headed coin?
• When do you stop chalking it up to chance and
accuse me of using a two-headed coin?
• You can use statistics to back up your
accusations and save yourself $1,000
Chi-Squared Analysis
• Start with the assumption (null-hypothesis)
that the results of the coin flip are due to
chance
• It is easier to disprove something than to
prove it
• You will attempt to disprove your nullhypothesis
• By showing that it is NOT due to chance
you can accuse me of cheating
Chi-Square Test
• Comparison of observed results and expected results
• Null-hypothesis: It is purely due to chance
Categories
Observed Expected
(O-E)
(O-E)2
(O-E)2/E
Heads
20
10
10
100
10
Tails
0
10
-10
100
10
• X2 value = Sum of (Observed – Expected)2
Expected
• X2 value = 20
What if we do a second experiment
with a new coin and obtain the
results below
• Null-hypothesis: It is purely due to chance
Categories
Observed Expected
(O-E)
(O-E)2
(O-E)2/E
Heads
11
10
1
1
0.1
Tails
9
10
-1
1
0.1
• X2 value = Sum of (Observed – Expected)2
Expected
• X2 value = 0.2
What conclusion would you make
from the data for the two coins?
• Which data is legitimately due to chance, and
which data is not due to chance?
• In the case of the first coin (two-headed) the chisquared (X2) value is 20
• In the case of the second coin (regular) the chisquared (X2) value is 0.2
• So, the higher the (X2) value…the _______
likely the results are due to chance
• The lower the (X2) value…the _______ likely
the results are due to chance
How low is low enough?
• The null-hypothesis is that your results are due
to chance
• You are attempting to disprove the nullhypothesis
• It is easier to disprove something than to prove it
• How can chi-squared (X2) analysis be used to
disprove the null-hypothesis
• There’s an app for that (actually a chart)
• To follow the chart you must know two things
– Degrees of Freedom
– p-value
Degrees of Freedom
• The number of values in the final calculation of a
statistic that are free to change
• Let’s say I give you 4 numbers and tell you that
they must add up to 100. In addition, I tell you
that one of the numbers is 50.
• The three remaining numbers could be a variety
of values as long as the overall total is 100
Choice 1
Choice 2
Choice 3
Number 1 = 50
Number 2 = 30
Number 3 = 10
Number 4 = 10
Number 1 = 50
Number 2 = 5
Number 3 = 25
Number 4 = 20
Number 1 = 50
Number 2 = ?
Number 3 = ?
Number 4 = ?.
• There are many more choices that fulfill the conditions
Degrees of Freedom
• In the example above you have 4 options, one of
which is a fixed value (50)
– 3 numbers are free to change
• 3 degrees of freedom
• What if I said you have 5 options, one of which is
a fixed value (50)
– 4 numbers are free to change
• 4 degrees of freedom
• The more options you have, the more degrees
of freedom you have
• Generally, in biology
degrees of freedom = # categories -1
p-value
• The null-hypothesis is that your results are due to
chance
• p-value: probability that the null-hypothesis is valid (true)
• High p-value means null-hypothesis is true
• Low p-value means that the null-hypothesis is untrue
• How low is low enough?
• The significant p-value is 0.05 (5%)
– A p-value less than 0.05 means that it is less than 5% likely that
the results are due to chance
– A p-value greater than 0.05 means that it is more than 5% likely
that the results are due to chance
Two-headed coin
Big X2 = Small p-value = Not due to chance = Statistically Significant Data
• The X2 value for our two-headed coin was 20
• The number of options were 2 (heads or tails) = 1 degree of freedom
• The significant p-value is always 0.05 or less
• Critical value for 1 degree of freedom is 3.84
• 20 is greater than 3.84 so p-value is less than 0.05
Big X2 = Small p-value = Not due to chance = Statistically Significant Data
20 = 0.000001 = Not due to chance = Statistically significant
Regular coin
Small X2 = Large p-value = Due to chance = Statistically Insignificant Data
• The X2 value for our two-headed coin was 0.2
• The number of options were 2 (heads or tails) = 1 degree of freedom
• The significant p-value is always 0.05 or less
• Critical value for 1 degree of freedom is 3.84
• 0.2 is lower than 3.84 so p-value is more than 0.05
Small X2 = Large p-value = Due to chance = Statistically Insignificant Data
0.2 = 0.65 = Due to chance = Statistically insignificant
Simple vs. Complex
• In the case of the two-headed coin, you
have simple expectations 50:50 heads to
tails
• What about more complex problems?
Teaching Example
• 100 students took my exam
Categories Observed Expected
A
B
C
D
F
20
22
35
23
0
15
25
40
15
5
(O-E)
(O-E)2
(O-E)2/E
5
-3
-5
8
-5
25
9
25
64
25
1.67
0.36
0.63
4.27
5
• X2 =11.93
• Degrees of Freedom = # categories -1 = 5-1 = 4
Did my student meet my
expectations?
• X2 =11.93
• Degrees of Freedom = # categories -1 = 5-1 = 4