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# Re: st: sign test output

 From Nick Cox To statalist@hsphsun2.harvard.edu Subject Re: st: sign test output Date Thu, 17 Jan 2013 10:59:43 +0000

```Sorry; I misread radically what your variable is, and it is helpful
that you have now explained it.

My suggestion of a binomial confidence interval still makes sense when
understood in this way: equal numbers of positive and negative
differences imply a fraction of 0.5 for pr(positive) and also
pr(negative).

from somewhere

>  Shapiro-Wilk is used to test normality, when the number of
> observations is less than 30. Otherwise, we should use
> Kolmogorov-Smirnov for large sample (as in my sample).

would never be my two sentences of advice. I would always start out
with -qnorm- and often end with it. Kolmogorov-Smirnov is more
sensitive in the middle than in the tails of a distribution, which is
precisely the wrong way round.

All that said, there is a lot of literature to the effect that the
t-test can work very well even when assumptions are not well
satisfied. See for example Rupert Miller, Beyond ANOVA

http://www.amazon.com/Beyond-ANOVA-Applied-Statistics-Statistical/dp/0412070111

Nick

On Thu, Jan 17, 2013 at 10:21 AM, Nahla Betelmal <nahlaib@gmail.com> wrote:
> Dear Nick,
>
> Thank you for the comments. the variable I am testing is not binary ,
> and the literary of my field is concerned whether the mean (median) of
> this variable is different than zero. So, U is the mean in case the
> variable is normally distributed, or U is the median in case the
> distribution is not normal.
>
> from my readings in statistics , I know that in order to decide
> whether to use parametric or non-parametric tests, the data normality
> distribution should be checked first.
>
>  Shapiro-Wilk is used to test normality, when the number of
> observations is less than 30. Otherwise, we should use
> Kolmogorov-Smirnov for large sample (as in my sample).
>
> So, when the test accepts the null (normality), we should use the
> parametric test (i.e. t-test) which examines the mean. On the other
> hand if the null of normality was reject, we should use the
> non-parametric test ( sign test) instead which examines the median (As
> in my case).
>
> Also,  for the comment about robust, I meant exactly what said (I used
> the robust term loosely)
>
> Thanks for suggesting to read again, sure I will do.
>
> Many thanks again
>
> Nahla
>
> On 17 January 2013 09:49, Nick Cox <njcoxstata@gmail.com> wrote:
>> Your t-test is testing a quite different hypothesis. If the two states
>> 0 and 1 of a binary variable have equal frequencies, then its mean is
>> 0.5, not 0.
>>
>> That aside, the t-test can not be more appropriate for a binary
>> variable than what you have done already, and this is predictable in
>> advance, as a distribution with two distinct states is not a normal
>> distribution. You do not need a Kolmogorov-Smirnov test to tell you
>> that.
>>
>> For the record, what I suggested is best not described as a robust
>> test. It was calculating a confidence interval, and I showed that for
>> your data the result was robust to the method of calculation, meaning
>> merely not sensitive. The word "robust" was used informallly.
>>
>> You never define what you mean by u, so I am not commenting on any
>>
>> I recommend that you read (or re-read) a good introductory text on
>> statistics, as you appear confused on some basic matters.
>>
>> Nick
>>
>> On Thu, Jan 17, 2013 at 7:52 AM, Nahla Betelmal <nahlaib@gmail.com> wrote:
>>
>>> Thank you Maarten and Nick  for the great help.
>>>
>>>  So, in this case I would reject the null in favour of the alternative
>>> u>0 as p value 0.000. However, using t-test on the same sample
>>> provided the opposite (i.e. accept the null).
>>>
>>> ttest DA_T_1 == 0
>>>
>>> One-sample t test
>>> ------------------------------------------------------------------------------
>>> Variable |     Obs        Mean    Std. Err.   Std. Dev.   [95% Conf. Interval]
>>> ---------+--------------------------------------------------------------------
>>>   DA_T_1 |     346    1.564346     1.68628    31.36663   -1.752338     4.88103
>>> ------------------------------------------------------------------------------
>>>     mean = mean(DA_T_1)                                           t =   0.9277
>>> Ho: mean = 0                                     degrees of freedom =      345
>>>
>>>     Ha: mean < 0                 Ha: mean != 0                 Ha: mean > 0
>>>  Pr(T < t) = 0.8229         Pr(|T| > |t|) = 0.3542          Pr(T > t) = 0.1771
>>>
>>>
>>> I think this is due to the distribution of the sample, so I performed
>>> K-S normality test. It shows that data is not normally distributed,
>>> hence I should use the non-parametric sign test instead of t-test. In
>>> other words I would reject the null u=0 in favor of u>0 , right?
>>>
>>>
>>> ksmirnov  DA_T_1 = normal((DA_T_1-DA_T_1_mu)/  DA_T_1_s)
>>>
>>> One-sample Kolmogorov-Smirnov test against theoretical distribution
>>>            normal((DA_T_1-DA_T_1_mu)/  DA_T_1_s)
>>>
>>>  Smaller group       D       P-value  Corrected
>>>  ----------------------------------------------
>>>  DA_T_1:             0.4878    0.000
>>>  Cumulative:        -0.4330    0.000
>>>  Combined K-S:    0.4878    0.000      0.000
>>>
>>>
>>> N.B. Thank you so much Nick for the robust test you mentioned, I will
>>> use that as well)
>>>
>>> Many thanks
>>>
>>> Nahla
>>>
>>> On 16 January 2013 09:33, Nick Cox <njcoxstata@gmail.com> wrote:
>>>> In addition, it could be as or more useful to think in terms of
>>>> confidence intervals. With this sample size and average, 0.5 lies well
>>>> outside 95% intervals for the probability of being positive, and that
>>>> is robust to method of calculation:
>>>>
>>>> . cii 346 221
>>>>
>>>>                                                          -- Binomial Exact --
>>>>     Variable |        Obs        Mean    Std. Err.       [95% Conf. Interval]
>>>> -------------+---------------------------------------------------------------
>>>>              |        346    .6387283    .0258248        .5856497    .6894096
>>>>
>>>> . cii 346 221, jeffreys
>>>>
>>>>                                                          ----- Jeffreys -----
>>>>     Variable |        Obs        Mean    Std. Err.       [95% Conf. Interval]
>>>> -------------+---------------------------------------------------------------
>>>>              |        346    .6387283    .0258248        .5871262    .6880204
>>>>
>>>> . cii 346 221, wilson
>>>>
>>>>                                                          ------ Wilson ------
>>>>     Variable |        Obs        Mean    Std. Err.       [95% Conf. Interval]
>>>> -------------+---------------------------------------------------------------
>>>>              |        346    .6387283    .0258248        .5868449    .6875651
>>>>
>>>> Nick
>>>>
>>>> On Wed, Jan 16, 2013 at 9:13 AM, Maarten Buis <maartenlbuis@gmail.com> wrote:
>>>>> On Wed, Jan 16, 2013 at 9:38 AM, Nahla Betelmal wrote:
>>>>>> I have generated this output using  non-parametric test "one sample
>>>>>> sign test" with null: U=0 , & Ua > 0
>>>>>>
>>>>>> However, I do not understand the output. where is the p-value? is it
>>>>>> 0.5 in all cases or the 0.000 ( as in the first and third cases) and
>>>>>> 1.000 as in the second case?
>>>>>>
>>>>>>. signtest DA_T_1= 0
>>>>>>
>>>>>> Sign test
>>>>>>
>>>>>>         sign |    observed    expected
>>>>>> -------------+------------------------
>>>>>>     positive |         221         173
>>>>>>     negative |         125         173
>>>>>>         zero |           0           0
>>>>>> -------------+------------------------
>>>>>>          all |         346         346
>>>>>>
>>>>>> One-sided tests:
>>>>>>   Ho: median of DA_T_1 = 0 vs.
>>>>>>   Ha: median of DA_T_1 > 0
>>>>>>       Pr(#positive >= 221) =
>>>>>>          Binomial(n = 346, x >= 221, p = 0.5) =  0.0000
>>>>>
>>>>> The p-value is the last number, so in your case 0.0000. The stuff
>>>>> before the p-value tells you how it is computed: it is based on the
>>>>> binomial distribution, and in particular it is the chance of observing
>>>>> 221 successes or more in 346 trials when the chance of success at each
>>>>> trial is .5. For this tests this chance is the p-value, and it is very
>>>>> small, less than 0.00005. If you type in Stata -di binomialtail(346,
>>>>> 221, 0.5)- you will see that this chance is 1.381e-07, i.e.
>>>>> 0.00000001381.
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