Download Chi-square goodness of fit tests

Chi-square goodness of fit tests
Chi-square goodness of fit
Chi-square goodness of fit
• Core issue in statistics:
When are you viewing just
random noise and when is there a real trend?
Chi-square goodness of fit
• Core issue in statistics:
When are you viewing just
random noise and when is there a real trend?
Chi-square goodness of fit
• Core issue in statistics:
When are you viewing just
random noise and when is there a real trend?
Chi-square goodness of fit
• Core issue in statistics:
When are you viewing just
random noise and when is there a real trend?
When to use a chi-square test
• Your response variable is count data
• Your response falls into different categories
• You have a hypothesis for the responses you
expect
• You want to know if the difference between
the responses you observe and the responses
you expect is significant or not
How to interpret a chi-square test
If the test tells you your data is not significantly
different from what you expect, you support
the hypothesis
If the test tells you your data is significantly
different from what you expect, you reject the
hypothesis
No statistical test ever proves a hypothesis.
What is chi-square?
x2 = Chi-square value Σ = Sum of
x2 = Σ (Observed Value – Expected Value)2
Expected Value
Observed Expected Obs-Exp
(Obs-Exp)2
Category
1
Category
2
…..
X2 total
Degrees of Freedom
(Obs-Exp)2
Exp
Example problem #1
Chi-square probability table
0.99
0.95
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.05
0.01
1
0.0001 0.003 0.015 0.064 0.148 0.275 0.455 0.708 1.07
1.64
2.71
3.84
6.63
2
0.020 0.103 0.211 0.446 0.713 1.02
1.39
1.83
2.41
3.22
4.61
5.99
9.21
3
0.115 0.352 0.584 1.00
1.42
1.87
2.37
2.95
3.67
4.64
6.25
7.81
11.3
4
0.297 0.711 1.06
1.65
2.19
2.75
3.36
4.04
4.88
5.99
7.78
9.49
13.3
5
0.554
1.15
1.61
2.34
3.00
3.66
4.35
5.13
6.06
7.29
9.24
11.1
15.1
6
0.872
1.64
2.20
3.07
3.83
4.57
5.35
6.21
7.23
8.56
10.6
12.6
16.8
7
1.24
2.17
2.83
3.82
4.67
5.49
6.35
7.28
8.38
9.80
12.0
14.1
18.5
8
1.65
2.73
3.49
4.59
5.53
6.42
7.34
8.35
9.52
11.0
13.4
15.5
20.1
df
Chi-square probability table
0.95 0.9
0.8
0.7 that
0.6 the
0.5 difference
0.4
0.3
0.2
0.1
Only
5% chance
between your obs and exp values comes
0.0001 0.003 0.015 0.064 0.148 0.275 0.455 0.708 1.07 1.64 2.71
from chance alone
0.99
df
1
2
0.020 0.103 0.211 0.446 0.713 1.02
1.39
1.83
2.41
3
0.115 0.352 0.584 1.00
4
0.297 0.711 1.06
1.65
2.19
2.75
3.36
4.04
4.88
5
0.554
1.15
1.61
2.34
3.00
3.66
4.35
5.13
6
0.872
1.64
2.20
3.07
3.83
4.57
5.35
7
1.24
2.17
2.83
3.82
4.67
5.49
8
1.65
2.73
3.49
4.59
5.53
6.42
0.05
0.01
3.84
6.63
3.22
4.61
5.99
9.21
Your hypothesis
can be rejected
1.42 1.87 2.37 2.95 3.67 4.64
6.25
7.81
11.3
5.99
7.78
9.49
13.3
6.06
7.29
9.24
11.1
15.1
6.21
7.23
8.56
10.6
12.6
16.8
6.35
7.28
8.38
9.80
12.0
14.1
18.5
7.34
8.35
9.52
11.0
13.4
15.5
20.1
Chi-square probability table
50-60% chance differences are from random chance
0.99
0.95
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.05
0.01
1
0.0001 0.003 0.015 0.064 0.148 0.275 0.455 0.708 1.07
1.64
2.71
3.84
6.63
2
0.020 0.103 0.211 0.446 0.713 1.02
1.39
1.83
2.41
3.22
4.61
5.99
9.21
3
0.115 0.352 0.584 1.00
1.42
1.87
2.37
2.95
3.67
4.64
6.25
7.81
11.3
4
0.297 0.711 1.06
1.65
2.19
2.75
3.36
4.04
4.88
5.99
7.78
9.49
13.3
5
0.554
1.15
1.61
2.34
3.00
3.66
4.35
5.13
6.06
7.29
9.24
11.1
15.1
6
0.872
1.64
2.20
3.07
3.83
4.57
5.35
6.21
7.23
8.56
10.6
12.6
16.8
7
1.24
2.17
2.83
3.82
4.67
5.49
6.35
7.28
8.38
9.80
12.0
14.1
18.5
8
1.65
2.73
3.49
4.59
5.53
6.42
7.34
8.35
9.52
11.0
13.4
15.5
20.1
df
Example problem 2
Female Song Worksheet
Bird song: It’s not just for males anymore
Male house wrens sing to attract mates
and defend territories
Females sing a lot too!
Why?
Why do females sing?
H1: To help coordinate their behavior with their male
Prediction
Why do females sing?
H2: To solicit copulations from the male
Prediction
Why do females sing?
H3: To defend their nests against other house wrens
Prediction
Why do females sing?
Null hypothesis: Female song occurs randomly
Prediction