Re: one sample t-test --> nonparamatric equivalent?

On Sat, 15 Nov 2008 05:38:59 EST, Anatol <anatol.sendker@xxxxxxx>
wrote:

Using the one-sample t-test is probably the most
sensible thing.

If you want a test that is strictly non-parametric,
with easy-to-state
assumptions, do a sign test. Compare the count of
those "below
average" (both groups) to the count of the two groups
above average.

If you were to elect to weight the responses
differentially, such as,
2 points for each extreme, and 1 point for the
moderately above/below,
you are pretty much back to the one-sample t-test.

Doing anything with ranks implicitly entails scoring
the groups by
a rank-transform. That will result, most likely, in
uneven intervals
between the groups: which is hardly likely to be an
improvement
on the original scoring.

Also,rank procedures are not very good when there are
too many
ties owing to score categories like these. If you
use the
rank-transforms where there are numerous ties, you
may be apt to
see a variance estimate (for large samples) which is
less accurate
than the computation you would get from doing a
t-test on the
rank-transformed scores. -- If you want to perform
with a rank
transform, then do the transform and use a one-sample
t-test.
But if I were a critic, I would want to see the
simple one-sample
t-test on the original data.

Richard, thank you for your message. It seems anything else than the
one sample t-test will get things just complicated. I take it that the
t-test assumes

1. normally distributed data

2. interval data

Whereas, Ranks assume perfect intervals between discrete
scores.

Those are formal assumptions. In practice, the absence of
any outlier is pretty good evidence for both t-test assumtions,
especially when the manner of generating the numbers gives
no rational reason for expecting log-normal, Poisson, etc.

It is hard for a 5-point scale to have outliers without also
having an absence of face validity for its categories. Anybody
in the social sciences (anyway) is going to generate categories
that *strive* for equal intervals when they make up labels;
they fare fairly well. Frederick Lord argued persuasively with
his "football jersey number" example, c. 1951, that both of
those formal requirements can be excessive.


One formal assumption for rank-transformed tests is that there
are no ties. A 5-point scale fails that one pretty badly,
doesn't it?

Conover demonstated the high similarity for moderate N
between the conventional rank tests and performing simple
ANOVA on the rank-transformed scores. Using ANOVA
could perform better in large samples than using the "non-
parametric test" with an approximation for how the variance
is affected by ties. That tells me that I really, always want
to look the actual "scores" that I get by rank-transforming
my data. Then I as, "Does this distribution look better, and
more interval





as to 1. I ran (with my test data) a Kolmogorov-Smirnov test to
check for normality (to be able to use the t-test) and with K-S sig. =
.104 I would be able to argue for normality, and consequently for the
use of the one sample t-test, right?


Bob Hayden, posting in October, 1994 --
(BH -- Quoting someone else)
If you are t-testing mu1 = mu2 onesided or twosided, you start with testing
H0: var1 = var2 twosided, using F-test. If H0 survives, you continue testing
mu1=mu2 using pooled t-test. (usual requirements: large enough samples, ...
fulfilled)
(BH)
This is generally not considered good practice because the t-test is
so much more robust than the F-test. George Box said this procedure
is "like sending out a rowboat to see if the waters are calm enough
for an ocean liner" (approximate quote).
* * * end of excerpt

I expect that the Shapiro-Wilk test for normality would be better
than the KS in general, and particularly for data where the KS
is distorted by discreteness.

In this case, the sample N has never been mentioned. I'd say,
for 5-point score, you can't have non-normality that matters
unless you have a bimodal distribution, regardless of the N.

For other distributions in general -- You could have
dangerous non-normality with a small sample that has no power
for testing normality; or you could have a significant test in a
large sample where it doesn't matter at all. A report of the
p-level alone is not very informative, whether it is significant
or non.



as to 2. I actually do have a number of sources I can reference on
why Likert data is sometimes treated as interval. It is just that my
supervisor (who will mark my work) suggested to use non-parametric
tests, because that would not have to make the interval data \assumption.


A lot of advisors do want you to learn by considering options,
even the unreasonable ones, without wanting you to fall on your
sword defending them. Unfortunately, a few of advisors can
be pig-headed about their own mistakes on a topic. Are you
sure that you know his intentions?

Show him, explicitly, the spacing that you get with the
Rank-transformation. It's probably about the same.
However, where it is different for a five point scale, it is
apt to be *worse*, by rational consideration.

Example:
If 100 cases are distributed (47, 48, 1, 1, 1) for groups 0-4
The average ranks are (24, 72, 98, 99, 100)
Rescale:
Subtracting 24 gives (0, 48, 74, 75, 76)
Rescaling to 0-4 gives (0, 2.52, 3.89, 3.95, 4),
or, rounding off, (0, 2.5, 4, 4, 4)

Are these, the effective values for the rank-transformed
analyses, a better spacing for the outcomes than (0,1,2,3,4)?

A logistic transformation would make use of the ranks and
create a different, quasi-non-metric scaling, but that is a
more complicated topic.

[snip rest]

--
Rich Ulrich
.

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