Statistics: Difference between revisions
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H<sub>A</sub> : μ<sub>1</sub> ? μ<sub>2</sub> | H<sub>A</sub> : μ<sub>1</sub> ? μ<sub>2</sub> | ||
== Familywise Error Rate == | === Chi-Squared === | ||
Given: | |||
{| class="wikitable" | |||
| | |||
|DF1 | |||
|DF2 | |||
|... | |||
|DFn | |||
|- | |||
|Observed | |||
|O<sub>1</sub> | |||
|O<sub>2</sub> | |||
|... | |||
|O<sub>n</sub> | |||
|- | |||
|Excpected | |||
|E<sub>1</sub> | |||
|E<sub>2</sub> | |||
|... | |||
|E<sub>n</sub> | |||
|} | |||
x<sup>2</sup><sub>c</sub> = Σ (O<sub>i</sub> - E<sub>i</sub>)<sup>2</sup> / E<sub>i</sub> | |||
x<sup>2</sup><sub>c</sub> is then used against a reference value in the Critical values table. Look up a comparison value by the degrees of freedom and probability threshold | |||
{| | |||
| DF | |||
| >0.9 | |||
| >0.95 | |||
| >0.975 | |||
| >0.99 | |||
| >0.999 | |||
|- | |||
| 1 | |||
| 2.706 | |||
| 3.841 | |||
| 5.024 | |||
| 6.635 | |||
| 10.828 | |||
|- | |||
| 2 | |||
| 4.605 | |||
| 5.991 | |||
| 7.378 | |||
| 9.210 | |||
| 13.816 | |||
|- | |||
| 3 | |||
| 6.251 | |||
| 7.815 | |||
| 9.348 | |||
| 11.345 | |||
| 16.266 | |||
|- | |||
| 4 | |||
| 7.779 | |||
| 9.488 | |||
| 11.143 | |||
| 13.277 | |||
| 18.467 | |||
|- | |||
| 5 | |||
| 9.236 | |||
| 11.070 | |||
| 12.833 | |||
| 15.086 | |||
| 20.515 | |||
|- | |||
| 6 | |||
| 10.645 | |||
| 12.592 | |||
| 14.449 | |||
| 16.812 | |||
| 22.458 | |||
|} | |||
=== Familywise Error Rate === | |||
The probability of make one or more false discoveries (Type I errors) among all hypothesis when performing multiple hypothesis tests. | The probability of make one or more false discoveries (Type I errors) among all hypothesis when performing multiple hypothesis tests. | ||
== Fisher's Exact Test == | === Fisher's Exact Test === | ||
Smaller data sets than Chi-squared. | Smaller data sets than Chi-squared. | ||
Revision as of 14:28, 8 April 2013
- mean
- x_bar = 1/n Σ xi
- permutation
- nPk = n! / (n-k)!
- n: size of the set
- k: elements drawn
Normal Distribution
Z distribution is a normal distribution with a mean (μ) of 0 and a standard deviation (δ) of 1. Any normal distribution is described by its μ and δ value.
To convert a normal distribution to a Z distribution
Z = (x - μ) / δ
Standard deviation is the root of the sum of the square errors:
δ = ( Σ (i - μ)2 ).5
Binomial Distribution
Similar to normal distribution, but for values that can only be discrete.
Probability of an event happening k times given n trials, where p is the probability of n:
nCk * pk(1-p)n-k nCk = n! / ( k! * (n-k)! )
Example: The probability of two heads in six coin flips:
6C2 = 6! / ( 2! * (6-2)! ) = 6! / (2!4!) = 720 / (24 * 2) = 15 15 * .52(1-.5)6-2 = 15 * .25 * .54 = 15 * 0.015625 = 0.234375
Hypothesis Testing
Null hypothesis h0. Alternate hypothesis HA or H1
Single Tail Test
H0 : μ1 = μ2 HA : μ1 > μ2
Two Tailed Test
H0 : μ1 = μ2 HA : μ1 ? μ2
Chi-Squared
Given:
| DF1 | DF2 | ... | DFn | |
| Observed | O1 | O2 | ... | On |
| Excpected | E1 | E2 | ... | En |
x2c = Σ (Oi - Ei)2 / Ei
x2c is then used against a reference value in the Critical values table. Look up a comparison value by the degrees of freedom and probability threshold
| DF | >0.9 | >0.95 | >0.975 | >0.99 | >0.999 |
| 1 | 2.706 | 3.841 | 5.024 | 6.635 | 10.828 |
| 2 | 4.605 | 5.991 | 7.378 | 9.210 | 13.816 |
| 3 | 6.251 | 7.815 | 9.348 | 11.345 | 16.266 |
| 4 | 7.779 | 9.488 | 11.143 | 13.277 | 18.467 |
| 5 | 9.236 | 11.070 | 12.833 | 15.086 | 20.515 |
| 6 | 10.645 | 12.592 | 14.449 | 16.812 | 22.458 |
Familywise Error Rate
The probability of make one or more false discoveries (Type I errors) among all hypothesis when performing multiple hypothesis tests.
Fisher's Exact Test
Smaller data sets than Chi-squared.
Given:
| a | b |
| c | d |
p = ((a+b)!(c+d)!(a+c)!(b+d)!) / (n!a!b!c!d!)