7. Product and Process Comparisons
7.2. Comparisons based on data from one process
7.2.1. Do the observations come from a particular distribution?
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Data are often assumed to come from a particular distribution.
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Goodness-of-fit tests indicate whether or not it is reasonable to assume that a random sample comes from a specific distribution. Statistical techniques often rely on observations being of a specific form (e.g., normal, lognormal, Poisson, etc.). Standard control charts for instance, often assume that the data (or at least the plotted sample means) come from an approximately normal distribution. In reliability applications, accurate lifetime modeling generally requires specifying the correct distributional model. There may
be historical or theoretical reasons to assume a sample comes from a particular
population, as well. Past data may have consistently fit a known distribution,
for example, or theory may predict that the underlying population should
be of a specific form.
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Hypothesis Test model for Goodness-of-fit
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Goodness-of-fit tests are a form of hypothesis testing where the
null and alternative hypotheses are
H0: Sample data come from the stated
distribution.
HA: Sample data do not come
from the stated distribution. |
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Parameters may be assumed or estimated from the data |
One needs to consider whether a simple or composite hypothesis is being tested. For a simple hypothesis, values of the distribution's parameters are specified prior
to drawing the sample. For a composite hypothesis, one or more of the
parameters is unknown. Often, these parameters are estimated using the sample
observations.
A simple hypothesis would be:
H0: Data are from a normal distribution,  = 0 and  = 1.
A composite hypotheses would be:
H0: Data are from a normal distribution, unknown
and .
Composite hypotheses are more common because they allow us to decide whether a sample comes from any distribution of a specific type. In this situation, the form of the distribution is of interest, regardless of the values the parameters. Unfortunately, composite hypotheses
are more difficult to work with because the critical values are often hard to compute.
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Problems with censored data |
A second issue that affects a test is whether the data are censored.
When data are censored, sample values are in some way restricted. Censoring
occurs if the range of potential values are limited such that values from
one or both tails of the distribution are unavailable (e.g., right and/or
left censoring - where high and/or low values are missing). Censoring frequently
occurs in reliability testing, when either
the testing time or the number of failures to be observed is fixed in advance.
A thorough treatment of goodness-of-fit testing under censoring is beyond
the scope of this document. See D'Agostino
& Stephens (1986) for more details. |
| Three types of tests will be covered |
Three goodness-of-fit tests are examined in
detail:
- Chi-square test for continuous and discrete distributions;
- Kolmogorov-Smirnov test for continuous distributions based on the empirical distribution
function (EDF);
- Anderson-Darling test for continuous distributions.
A more extensive treatment of goodness of fit techniques is presented
in D'Agostino
& Stephens (1986). Along with the tests mentioned above, other
general and specific tests are examined, including tests based on regression
and graphical techniques. |
