Welch's t-test: Difference between revisions

 

 

'''Task Description'''<br>

Your task is to discern whether or not the difference in means between the two sets is statistically significant and worth further investigation. P-values are significance tests to gauge the probability that the difference in means between two data sets is significant, or due to chance. A threshold level, alpha, is usually chosen, 0.01 or 0.05, where p-values below alpha are worth further investigation and p-values above alpha are considered not significant. The p-value is not considered a final test of significance, [http://www.nature.com/news/scientific-method-statistical-errors-1.14700 only whether the given variable should be given further consideration].

 

 

<math>t \quad = \quad {\; \overline{X}_1 - \overline{X}_2 \; \over \sqrt{ \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \quad }} </math>

and

 

 

and

 

 

 

and the degrees of freedom, <math>\nu</math> can be approximated:

<math>\nu \quad \approx \quad

{{\left( \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \; \right)^2 } \over

 

 

<math> p_{2-tail} = I_{\frac{\nu}{t^2+\nu}}\left(\frac{\nu}{2}, \frac{1}{2}\right) </math>

 

 

 

Keeping in mind that

 

 

and

The definite integral can be approximated with [[wp:Simpson's_rule|Simpson's Rule]] but [http://rosettacode.org/wiki/Numerical_integration other methods] are also acceptable.

 

 

=={{header|C}}==

 

Link with <code>-lm</code>

 

This shows how pvalue can be calculated from any two arrays, using Welch's 2-sided t-test, which doesn't assume equal variance.

#include <math.h>

 

return 1.0;

return 1.0;

}

}

}

}

}

}

/

(

);

}

return 1.0;

}

 

int main(void) {

const double d1[] = {27.5,21.0,19.0,23.6,17.0,17.9,16.9,20.1,21.9,22.6,23.1,19.6,19.0,21.7,21.4};

const double d2[] = {27.1,22.0,20.8,23.4,23.4,23.5,25.8,22.0,24.8,20.2,21.9,22.1,22.9,20.5,24.4};

const double x[] = {3.0,4.0,1.0,2.1};

const double y[] = {490.2,340.0,433.9};

 

return 0;

}

{{out}}

<pre>Test sets 1 p-value = 0.021378

Test sets 2 p-value = 0.148842

 

=={{header|J}}==

Implementation:

 

'a b steps'=. 3{.y,128

size=. (b - a)%steps

 

lngamma=: ^.@!@<:`(^.@!@(1 | ]) + +/@:^.@(1 + 1&| + i.@<.)@<:)@.(1&<:)"0

nu=: # - 1:

sampvar=: +/@((- mean) ^ 2:) % nu

hi=. v%(t^2)+v

(F f. simpson integrate lo,hi) % 0.5 B v%2

 

 

 

 

 

 

 

Data for task examples:

d2=: 27.1 22 20.8 23.4 23.4 23.5 25.8 22 24.8 20.2 21.9 22.1 22.9 20.5 24.4

d3=: 17.2 20.9 22.6 18.1 21.7 21.4 23.5 24.2 14.7 21.8

d8=: 29.89 29.93 29.72 29.98 30.02 29.98

d9=: 3 4 1 2.1

 

Task examples:

0.021378

d3 p2_tail d4

0.0907733

d9 p2_tail da

 

=={{header|R}}==

d1 <- c(27.5,21.0,19.0,23.6,17.0,17.9,16.9,20.1,21.9,22.6,23.1,19.6,19.0,21.7,21.4)

d2 <- c(27.1,22.0,20.8,23.4,23.4,23.5,25.8,22.0,24.8,20.2,21.9,22.1,22.9,20.5,24.4)

x <- c(3.0,4.0,1.0,2.1)

y <- c(490.2,340.0,433.9)

results <- t.test(d1,d2, alternative="two.sided", var.equal=FALSE)

results <- t.test(d3,d4, alternative="two.sided", var.equal=FALSE)

results <- t.test(d5,d6, alternative="two.sided", var.equal=FALSE)

results <- t.test(d7,d8, alternative="two.sided", var.equal=FALSE)

results <- t.test(x,y, alternative="two.sided", var.equal=FALSE)

 

{{out}}

</pre>

 

=={{header|Racket}}==

 

(require math/statistics math/special-functions)

 

(p-value (list 3.0 4.0 1.0 2.1)

 

{{out}}

<pre>(0.021378001462867013 0.14884169660532798 0.035972271029796624 0.09077332428567102 0.01075139991904718)</pre>

 

=={{header|Tcl}}==

This is not particularly idiomatic Tcl, but perhaps illustrates some of the language's relationship with the Lisp family.

 

 

package require math::statistics

puts [pValue $left $right]

}

 

{{out}}

0.09077332428458083

0.010751399918798182

</pre>