Wrong Statistics, Lasky's "Fixed Ratio vs Spear"

[Mark Johnson <janitor@xxxxxxxxxxxx>]

Regrettably, Paul Lasky has applied the wrong
statistical tool and has therefore come to an
incorrect conclusion.

  Paul1> Perform the usual dependent t test to validate
  Paul1> or invalidate the results.
  
  Paul2> The data show that, overwhelmingly, there
  Paul2> is no statistical difference between Fixed
  Paul2> Ratio, Spear, or Un-equal Fixed Ratio.
  Paul2>
  Paul2>  Fr-Spear     t-test -0.0232  Prob the same 0.9819
  Paul2>  Fr-Unequal   t-test -0.0123  Prob the same 0.99038
  Paul2>  Spear-Unequ  t-test  0.0085  Prob the same 1.0


Unfortunately, the dependent t test is inappropriate
here and it has led Paul to the wrong conclusion.

Here's a little example that shows just how wrong
it is to use this statistical test on trading equity
curves:

    Trader D starts with $80,000 and manages to make
    a profit of $1 per month for ten months.

    Trader E starts with $130,000 and manages to
    lose $10,000 per month for ten months.

    Yet the dependent t test, as Paul Lasky uses it,
    SAYS THESE TWO EQUITY CURVES ARE INDISTINGUISHABLE!
    One has 11 (small) losses in a row, the other has
    11 wins in a row, yet the test says they are
    "the same".  Really!

    The equity curves of the two traders are:

    Month#     D_equity     E_equity
   ------------------------------------
       0        80,000       130,000
       1        80,001       120,000
       2        80,002       110,000
       3        80,003       100,000
       4        80,004        90,000
       5        80,005        80,000
       6        80,006        70,000
       7        80,007        60,000
       8        80,008        50,000
       9        80,009        40,000
      10        80,010        30,000

    Now we do as Paul Lasky did and perform a
    dependent t test on the two equity curves.
    Lucky for us, software exists on the web:

    http://www.physics.csbsju.edu/stats/Paired_t-test_NROW_form.html

    After keying in the two equity curves and
    running the software, we find:

        t_test_statistic = 0.000500
        Probability the same = 1.000


So it is clear and apparent that the dependent t test
is the wrong tool to use -- it produces wrong conclusions
that mislead us badly.

TECHNICAL STUFF

The purpose of the dependent t test is to examine
two POPULATION MEANS via paired comparisons, and to
determine whether the population means are significantly
different.

But when examining equity curves, it is not the
population mean that is of interest.  The dependent
t test, is testing the wrong hypothesis --- in fact
it is testing a hypothesis that we don't care about:

   * Is the mean of equity curve A, significantly
     different from the mean of equity curve B.


Common sense might have prevented this; a cursory
examination of the equity curves would have shown
that the Unequal#FR equity curve was 2.5X higher
than the FR equity curve on 920101.  It's hard
to call this "indistinguishable".
--
   Mark Johnson      Silicon Valley, California    mark@xxxxxxxxxxxx

      "... The world will little note, nor long remember, what we
       say here..."  -- Abraham Lincoln, "The Gettysburg Address"