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More tutorial code.

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[SVN r3122]
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John Maddock
2006-08-10 18:16:39 +00:00
parent 5c645698c3
commit fe21fb46b3
2 changed files with 156 additions and 28 deletions
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165
doc/dist_tutorial.qbk
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@@ -178,7 +178,7 @@ different from this colloquialism. More background information can be found
The formula for the interval can be expressed as:
Y[sub s] +- t[sub (__alpha/2,N-1)] * s / sqrt(N)
[$../equations/dist_tutorial4.png]
Where, ['Y[sub s]] is the sample mean, /s/ is the sample standard deviation,
/N/ is the sample size, /__alpha/ is the desired significance level and
@@ -192,7 +192,7 @@ From the formula it should be clear that:
* The width increases as the confidence level gets smaller (stronger).
The following example code is taken from the example program
students_t_single_sample.cpp.
[@../../example/students_t_single_sample.cpp students_t_single_sample.cpp].
We'll begin by defining a procedure to calculate intervals for
various confidence levels, the procedure will print these out
@@ -333,7 +333,7 @@ on the NIST site].
often a "traditional" method of measurement).
The following example code is taken from the example program
students_t_single_sample.cpp.
[@../../example/students_t_single_sample.cpp students_t_single_sample.cpp].
We'll begin by defining a procedure to determine which of the
possible hypothesis are accepted or rejected at a given confidence level:
@@ -418,9 +418,9 @@ calibration and stability analysis.
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Mean != 5.000 ACCEPTED
Mean < 5.000 REJECTED
Mean > 5.000 ACCEPTED
Mean != 5.000 ACCEPTED
Mean < 5.000 REJECTED
Mean > 5.000 ACCEPTED
You will note the line that says the probability that the difference is
due to chance is zero. From a philosophical point of view, of course,
@@ -454,9 +454,9 @@ atomic absorption.
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Mean != 38.900 REJECTED
Mean < 38.900 REJECTED
Mean > 38.900 REJECTED
Mean != 38.900 REJECTED
Mean < 38.900 REJECTED
Mean > 38.900 REJECTED
As you can see the small number of measurements (3) has led a large uncertainty
in the location of the true mean. So even though there is a clear difference
@@ -482,9 +482,9 @@ we see a different output:
Results for Alternative Hypothesis and alpha = 0.1000
Alternative Hypothesis Conclusion
Mean != 38.900 ACCEPTED
Mean < 38.900 ACCEPTED
Mean > 38.900 REJECTED
Mean != 38.900 ACCEPTED
Mean < 38.900 ACCEPTED
Mean > 38.900 REJECTED
In this case we really have a borderline result, and more data should
be collected.
@@ -501,7 +501,8 @@ result is borderline. At this point one might go off and collect more data,
but first ask the question "How much more?". The parameter estimators of the
students_t_distribution class can provide this information.
This section is based on the example code in students_t_single_sample.cpp
This section is based on the example code in
[@../../example/students_t_single_sample.cpp students_t_single_sample.cpp]
and we begin by defining a procedure that will print out a table of
estimated sample sizes for various confidence levels:
@@ -594,6 +595,9 @@ Car Mileage sample data] from the
[@http://www.itl.nist.gov NIST website]. The data compares
miles per gallon of US cars with miles per gallon of Japanese cars.
The sample code is in
[@../../example/students_t_two_samples.cpp students_t_two_samples.cpp].
There are two ways in which this test can be conducted: we can assume
that the true standard deviations of the two samples are equal or not.
If the standard deviations are assumed to be equal, then the calculation
@@ -693,7 +697,7 @@ skip over that, and take a look at the sample output for alpha=0.05
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Alternative Hypothesis Conclusion
Sample 1 Mean != Sample 2 Mean ACCEPTED
Sample 1 Mean < Sample 2 Mean ACCEPTED
Sample 1 Mean > Sample 2 Mean REJECTED
@@ -717,7 +721,11 @@ And for the combined degress of freedom we have:
[$../equations/dist_tutorial3.png]
Putting these into code that produces:
Note that this is one of the rare situation where the degrees-of-freedom
parameter to the Student's t distribution is a real number, and not an
integer value.
Putting these formulae into code we get:
// Degrees of freedom:
double v = Sd1 * Sd1 / Sn1 + Sd2 * Sd2 / Sn2;
@@ -734,7 +742,7 @@ Putting these into code that produces:
double t_stat = (Sm1 - Sm2) / sqrt(Sd1 * Sd1 / Sn1 + Sd2 * Sd2 / Sn2);
cout << setw(55) << left << "T Statistic" << "= " << t_stat << "\n";
Thereafter the code and the tests are performed the same as before, using
Thereafter the code and the tests are performed the same as before. Using
are car mileage data again, here's what the output looks like:
__________________________________________________
@@ -753,7 +761,7 @@ are car mileage data again, here's what the output looks like:
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Alternative Hypothesis Conclusion
Sample 1 Mean != Sample 2 Mean ACCEPTED
Sample 1 Mean < Sample 2 Mean ACCEPTED
Sample 1 Mean > Sample 2 Mean REJECTED
@@ -766,6 +774,129 @@ than Japanese models.
[endsect]
[section:size2 Estimating how large a sample size would have to become
in order to give a significant Students-t test result with a two sample test]
Imagine that you have compare the means of two samples with a Student's-t test
and that the result is borderline. The question one would like to ask is
"How large would the two samples have to become in order for the observed
difference to be significant?"
The student's t distribution has two parameter-estimators that can be used
for this purpose. However, the problem domain is rather more complex
than it is for the single sample case. Firstly we have two sample sizes
to deal with: this can be handled by assuming either than one of the sample
sizes is fixed (as happens when comparing against historical data), or by
assuming that both sample sizes are always equal. Secondly, the estimators
always assume that the variances of the two samples are equal, without this
assumption it's impossible to relate the sample sizes to the number of degrees
of freedom in any direct way.
In this example, we'll be using the
[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3531.htm
Car Mileage sample data] from the
[@http://www.itl.nist.gov NIST website]. The data compares
miles per gallon of US cars with miles per gallon of Japanese cars.
The sample code is in
[@../../example/students_t_two_samples.cpp students_t_two_samples.cpp].
We'll define a procedure that prints a table of sample size estimates
required to obtain a range of statistical outcomes.
void two_samples_estimate_df(
double m1, // m1 = Sample 1 Mean.
double s1, // s1 = Sample 1 Standard Deviation.
unsigned n1, // n1 = Sample 1 Size.
double m2, // m2 = Sample 2 Mean.
double s2) // s2 = Sample 2 Standard Deviation.
{
using namespace std;
using namespace boost::math;
// Print out general info:
cout <<
"_____________________________________________________________\n"
"Estimated sample sizes required for various confidence levels\n"
"_____________________________________________________________\n\n";
cout << setprecision(5);
cout << setw(40) << left << "Sample 1 Mean" << "= " << m1 << "\n";
cout << setw(40) << left << "Sample 1 Standard Deviation" << "= " << s1 << "\n";
cout << setw(40) << left << "Sample 1 Size" << "= " << n1 << "\n";
cout << setw(40) << left << "Sample 2 Mean" << "= " << m2 << "\n";
cout << setw(40) << left << "Sample 2 Standard Deviation" << "= " << s2 << "\n";
Next we define a table of confidence levels:
double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
Most of the rest of the code is pretty-printing, so let's skip to
calculation of the sample size. For each alpha value, we use
each of the two parameter estimators to obtain the degrees of freedom
required. The arguments are wrapped in a call to `complement(...)`
since the significance levels are the complement of the probability:
// calculate df assuming equal sample sizes:
double df = students_t::estimate_two_equal_degrees_of_freedom(
complement(m1, s1, m2, s2, alpha[i]));
// convert to sample size:
double size = (ceil(df) + 2) / 2;
// Print size:
cout << fixed << setprecision(0) << setw(28) << right << size;
// calculate df with sample 1 size fixed:
df = students_t::estimate_two_unequal_degrees_of_freedom(
complement(m1, s1, n1, m2, s2, alpha[i]));
// convert to sample size:
size = (ceil(df) + 2) - n1;
// Print size:
cout << fixed << setprecision(0) << setw(28) << right << size << endl;
And other than printing the result that's pretty much it. Let's see some
sample output using the fuel efficiency data:
_____________________________________________________________
Estimated sample sizes required for various confidence levels
_____________________________________________________________
Sample 1 Mean = 20.14458
Sample 1 Standard Deviation = 6.41470
Sample 1 Size = 249
Sample 2 Mean = 30.48101
Sample 2 Standard Deviation = 6.10771
_______________________________________________________________________
Confidence Estimated Sample Size Estimated Sample 2 Size
Value (%) (With Two Equal Sizes) (With Fixed Sample 1 Size)
_______________________________________________________________________
50.000 1 0
75.000 2 1
90.000 3 1
95.000 4 2
99.000 6 3
99.900 10 4
99.990 14 6
99.999 18 8
So in order to achieve a 95% confidence level we would only need to
compare 4 American cars with 4 Japanese cars. Alternatively, comparing
just 3 Japanese cars against the data for all 249 American cars would yield
a 99% probability that the Japanese cars were more efficient. However, at
this point a word of caution is in order: comparing just 4 cars from each
country is unlikely to win you friends and admirers. As ever a measure of
common sense, and some analysis of the problem domain is needed when
interpretting such results.
Finally, you will note that the table contains some "nonesence" values
of 0 or 1: these arise if ['there is no solution to the question posed], and
/any/ valid value for the degrees of freedom will cause the null-hypothesis
to fail at the significance level given.
[endsect]
[section:paired_t Comparing two paired samples with the Student's t distribution]
[endsect]
[endsect]
[endsect]

19
example/students_t_two_samples.cpp
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@@ -77,7 +77,7 @@ void two_samples_t_test(
cout << setw(55) << left <<
"Results for Alternative Hypothesis and alpha" << "= "
<< setprecision(4) << fixed << alpha << "\n\n";
cout << "Alternative Hypothesis Conclusion\n";
cout << "Alternative Hypothesis Conclusion\n";
cout << "Sample 1 Mean != Sample 2 Mean " ;
if(q < alpha)
cout << "ACCEPTED\n";
@@ -160,7 +160,7 @@ void two_samples_t_test_equal_sd(
cout << setw(55) << left <<
"Results for Alternative Hypothesis and alpha" << "= "
<< setprecision(4) << fixed << alpha << "\n\n";
cout << "Alternative Hypothesis Conclusion\n";
cout << "Alternative Hypothesis Conclusion\n";
cout << "Sample 1 Mean != Sample 2 Mean " ;
if(q < alpha)
cout << "ACCEPTED\n";
@@ -179,16 +179,13 @@ void two_samples_t_test_equal_sd(
cout << endl << endl;
}
void two_samples_estimate_df(double m1, double s1, unsigned n1, double m2, double s2)
void two_samples_estimate_df(
double m1, // m1 = Sample 1 Mean.
double s1, // s1 = Sample 1 Standard Deviation.
unsigned n1, // n1 = Sample 1 Size.
double m2, // m2 = Sample 2 Mean.
double s2) // s2 = Sample 2 Standard Deviation.
{
//
// m1 = Sample 1 Mean.
// s1 = Sample 1 Standard Deviation.
// n1 = Sample 1 Size.
// m2 = Sample 2 Mean.
// s2 = Sample 2 Standard Deviation.
// alpha = confidence level
//
using namespace std;
using namespace boost::math;