Re: [Boost-users] [math/distributions] Problem using inv-chi**2 and quantiles

Subject: Re: [Boost-users] [math/distributions] Problem using inv-chi**2 and quantiles
From: Paul A. Bristow (pbristow_at_[hidden])
Date: 2011-06-15 10:38:45

> -----Original Message-----
> From: boost-users-bounces_at_[hidden] [mailto:boost-users-bounces_at_[hidden]] On Behalf
Of
> Paul Johnson
> Sent: Tuesday, June 14, 2011 1:39 PM
> To: boost-users_at_[hidden]
> Subject: [Boost-users] [math/distributions] Problem using inv-chi**2 and quantiles
> 
> I'm trying to duplicate Excel's CHIINV to calculate confidence limits for an observed number of
events. I
> think I'm nearly there, largely as a result of trial and error, but the boost results still
significantly differ
> from the Excel results, so I've presumably messed up somewhere.
> Any advice will be *greatly* appreciated.

I've checked and for the formula John suggests Boost.Math and Excel agree - see below and attached.

Excel 2007 has new versions claimed to be more accurate - see below, but I don't think this is the
trouble.

I think there is confusion about the inverse of the chi squared - quantile in the new terminology,
and the inverse_chi_squared_distribution - a completely different distribution.

(Boost.Math and R agree in tests for both these distributions).

quantile(boost::math::chi_squared(2*17),  (0.05)/2) / 2 = (34, 0.025) = 9.90313
=CHISQ.INV(0.025,34)/2 =9.90313
and

quantile(boost::math::chi_squared(2*17), 1 - (0.05)/2) / 2 = (34, 0. 925) = 25.9829976
=CHISQ.INV(0.975,34)/2 = 25.9829976

quantile(boost::math::chi_squared(2*(17+1)), 1 - (0.05)/2) / 2 = (36, 0. 925) =
=CHISQ.INV(0.975,36)/2 = 27.2186
=CHISQ.INV(0.975,36)/2 = 27.21864682

I'm also not sure what you really want here, but I suspect it isn't the
inverse_chi_squared_distribution,
but I'm not expert enough to know if it is chi_squared either ;-)

HTH - a bit!

Paul
--
Paul A. Bristow,
Prizet Farmhouse, Kendal LA8 8AB  UK
[phone]  07714330204
pbristow_at_[hidden]
> The Excel formulas I'm using are:
> 
> lo limit = CHIINV((1+$C$3)/2, 2*B6)/2) == chiinv (0.05/2, 2 * 17) = chiinv(0.025, 34) =
19.80625294
> hi limit = CHIINV((1-$C$3)/2, 2*(B6+1))/2)  = chiinv(0.975, 36) = > 
> where $C$3 is the confidence level required (95%), and B6 is the number of events observed. In
Excel,
> 'CHIINV' is documented as:
> 
> > CHIINV(probability,degrees_freedom)
> > Probability   is a probability associated with the chi-squared distribution.
> > Degrees_freedom   is the number of degrees of freedom.
> 
> For 17 events, for example, Excel is producing limits of [9.90, 27.22].
> 
> I've attached my complete boost test program below, which I hope does the same thing, but there
was a
> large amount of guesswork here. This program produces results of [10.67, 25.98] for 17 events.
CHIINV function
Returns the inverse of the right-tailed probability of the chi-squared distribution. If probability
= CHIDIST(x,...), then CHIINV(probability,...) = x. Use this function to compare observed results
with expected ones in order to decide whether your original hypothesis is valid.
 Important   This function has been replaced with one or more new functions that may provide
improved accuracy and whose names better reflect their usage. Although this function is still
available for backward compatibility, you should consider using the new functions from now on,
because this function may not be available in future versions of Excel.
For more information about the new functions, see CHISQ.INV function and CHISQ.INV.RT function.
Syntax
CHIINV(probability,deg_freedom)The CHIINV function syntax has the following arguments (argument: A
value that provides information to an action, an event, a method, a property, a function, or a
procedure.):
Probability     Required. A probability associated with the chi-squared distribution.
Deg_freedom     Required. The number of degrees of freedom.
CHISQ.INV function
Returns the inverse of the left-tailed probability of the chi-squared distribution.
The chi-squared distribution is commonly used to study variation in the percentage of something
across samples, such as the fraction of the day people spend watching television.
Syntax
CHISQ.INV(probability,deg_freedom)The CHISQ.INV function syntax has the following arguments
(argument: A value that provides information to an action, an event, a method, a property, a
function, or a procedure.):
Probability     Required. A probability associated with the chi-squared distribution.
Deg_freedom     Required. The number of degrees of freedom.
CHISQ.INV.RT function
Returns the inverse of the right-tailed probability of the chi-squared distribution.
If probability = CHISQ.DIST.RT(x,...), then CHISQ.INV.RT(probability,...) = x. Use this function to
compare observed results with expected ones in order to decide whether your original hypothesis is
valid.
Syntax
CHISQ.INV.RT(probability,deg_freedom)The CHISQ.INV.RT function syntax has the following arguments
(argument: A value that provides information to an action, an event, a method, a property, a
function, or a procedure.):
Probability     Required. A probability associated with the chi-squared distribution.
Deg_freedom     Required. The number of degrees of freedom.