STT 430/530, Nonparametric Statistics

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STT 430/530, Nonparametric Statistics

• The empirical cumulative distribution function
  (ecdf), F-hat(x), counts the fraction of
  observations less than or equal to x.
• Note that the graph of this ecdf is a step
  function that takes a step at each observed data
  value also notice that if all n data values are
  distinct then the step size is 1/n and whenever
  there are k tied values, the next step is k/n
• F-hat(x) is an estimate of the true c.d.f. - in
  fact,
• E(F-hat(x))F(x) and
• SD(F-hat(x)) sqrt((F-hat(x))(1-F-hat(x))/n) as
  you would expect for a binomial r.v. like
  F-hat(x) n of obs, pP(an obs. lt x)F(x)
• We can use SAS to sketch a plot of the ecdf and
  compare it with several theoretical
  distributions. Of course, we are most interested
  in whether the data is following the Normal
  distribution, so I show you how to check for that
  one
• proc capability
• cdfplot sodium/normal(colorred)
• This statement will do an ecdf and overlay a
  theoretical normal cdf with mean and sd estimated
  from the data.

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• Another important graph for checking normality of
  data is called a normal quantile plot . This
  plots the sorted data values against the
  corresponding normal quantile. That is,
• first, sort the data from smallest to largest
• second, for each data point find the ecdf (i.e.,
  the fraction of the data lt that point)
• third, get the corresponding standard normal
  z-score for that fraction.
• Try this SAS code to check it out (recall that
  the sodium values are already sorted from
  smallest to largest if they weren't, then you'd
  have to use PROC SORT and OUTPUT the sorted data
  to a SAS data set
• fract_n_/40 zprobit(fract)
• probit is the SAS function that returns the
  z-score corresponding to the cumulative
  probability under the standard normal curve
  between 0 and 1.
• PROC UNIVARIATE PLOT will give you a normal
  quantile plot but not a very nice oneTry this
  code to make it better
• proc capability
• qqplot sodium/normal(mu76 sigma2.25)
• This last option tells SAS to put in a reference
  line with mean76 and slope2.25 (I remembered
  these values from PROC UNIVARIATE output)