Fisher's F-test
A F-test is
a parametric statistic based on F-distribution. The test assumes that the
underlying distribution of observations is a Fisher’s F distribution under the
null hypothesis. Exact F-tests are specifically applied to least-sqaures
estimations.
The
test was developped by the American mathematical statistician George Waddel
Snedecor and named after the English statistician and biologist Sir Ronald
Aylmer Fisher, who earlier, had developed the statistic as the variance ratio.
Assumptions
F-test assumes an underlying F-distribution
and is applied in severaral cases to test variability of one or multiple samples, related or unrelated.
§
F follows a F-distribution under the null hypothesis
§
Under the null hypothesis, the sums of squares
should be statistically independent, and each should follow a scaled
chi-squared distribution.
§
Hence, the bservations must be independent and
normally distributed with a common variance.
Cases of F-Test
In
parametric cases, F-test may be used to assess the following hypothesis:
General Cases of F-test and
Calculations
Explicit
expressions that can be used to carry out various F-tests are given
in the AroniSmartLytics section explaining the choice of a parametric test. The
description of F-distribution is given in the AroniStat 1.0.1 and the probability distribution module of AroniSmartLytics. The formula used to
estimate the F statistic under the null hypothesis is given in each case.
To
test the significance of each test, one tailed, left or right, or two tailed test may be conducted.
Once
an F value is determined, the p-value can be computed either
from the AroniStat 1.0.1 and AroniSmartLytics distribution modules or
the data analysis module of AroniSmartLytics. If the calculated p-value is below
the statistical significance level chosen by the researcher of
practictioner, then the null hypothesis is rejected in favor of the
alternative hypothesis.
AroniSmartLytics
covers multiple situations where F-test is appropriate, both in parametrics and
non-parametric statistics settings.
In
general, F-tests are derived from the decomposition of the variability in the
samples of observations in terms of sums of squares. The test statistic in an F-test is the ratio of two scaled sums of
squares. The sums of squares
describe the sources of variability. Under the null hypothesis this ratio
of the sums of squares tends to zero, whereas the statistic tends to be larger
when the null hypothesis is not true. In order for the statistic to follow the
F-distribution under the null hypothesis, the sums of squares should be
statistically independent, and each should follow a scaled chi-squared
distribution. A corolary to the chi-squared assumption is that the observations
are independent and normally distributed with a common variance.
F-test and Analysis of Variance
(ANOVA)
F-test is extensively used in assessing the
variability among multiple groups. In this case, it is an extension of
chi-square test of variability. In multiple-comparison settings, the purpose is
to assess whethere the expected values of quantitative observation variables differ
across the groups of interest. The problem is known as Analysis of Variance or ANOVA.
While t-test is useful in comparing two
groups, in several instances researchers and practitioner need to assess the
causes of variations from several potential sources. In this case, Analysis of Variance, usually
designated by its acronym ANOVA,
could be the suitable statistical technique. ANOVA, instead of being one formula, is a collection of statistical
models and procedures. A variable is observed and then the researcher uses
ANOVA techniques to identify and explain the different sources of variation. In
general, multiple dependent variables may be observed and analyzes at the same
time. In that case, ANOVA becomes MANOVA,
or Multiple
Analysis of Variance. In case where the independent or predictor
variables are a mix of categorical (discrete ) and
continuous variables, with at least one categorical and one continuous, and the
independent or outcome variable is continuous, the ANOVA version of Analysis of Covariance (ANCOVA) is
used.
The ANOVA
F-test can be used to assess whether the mean of the observations for a
group, sample or treatment is superior or inferior to the means of the others, with the
null hypothesis that all groups, samples
or treatments have the same mean
response. In some cases, the researecher may be interested in paiwise
comparisons, among all the groups, samples or treatments, hence requiring at M(M − 1)/2
tests for M groups.
The ANOVA F-test is
known as an omnibus test, because
one single test is used to detect several possible differences. In the case of multiple comparisons, the significance
level must be adjusted to account
for increased error and pairwise
comparisons need to be specified a priori.
The omnibus factor of the F-Test constitutes a disadvantage, as without
multiple comparisons, it may not be clear which of the groups, samples, or
tretaments has the greatest or smallest mean.
Several methods exist to hadle multiple comparisons. These methods may
be classified into two groups:
Single-step procedures
§
Tukey-Kramer method, also knownw as Tukey Range test, Tukey
Method, Tukey’honest significance
test, or Tukey’s Honestly
Significant Difference (HSD) test, named after the American statisticians
John Wilder Tukey and
Clyde Kramer.
§
Scheffe Method, developped by Henry Scheffe
Multi-step procedures based
on t- statistic (Studentized range).
§
Duncan’s new multiple range test (MRT), developped by the American
Statistician David B. Duncan
§
Nemenyi test, developped by the American-Hungarian statistician and
Civil Rights activist Peter Nemenyi.
§
Bonferrnoni-Dunn test,
developped by the italian mathematician Carlo Emilio Bonferroni and
§
Student Newman-Keuls test also know as Student Neuman-Keuls Post-Hoc
Analysis Test, developed by D. Newman and M. Keuls .
·
ANOVA
problems
The formula for the one-way ANOVA F-test statistic is based
on the decomposition of the varaiance. The variance may be decomposed as
follows:
Hence
The "explained variance", or "between-group
variability" is:
where denotes the sample mean in the ith group, ni is
the number of observations in the ith group,
and
denotes
the overall mean of the data.
The
"unexplained variance", or "within-group variability" is
where Yij is
the jth observation in the ith out
of m groups and n is the overall sample
size. Under the numm hypothesis, the F-statistic follows the F-distribution
with m-1
and n-m
degrees of freedom under the null hypothesis. The statistic will be large if
the between-group or explained variability is large relative to the within-group or unxplained
variability. When the populations groups means are
equal, the statistic will be very small.
As
a special case, when there are only two groups for the one-way ANOVA F-test, F = t2 where t is
the Student’s t statistic.
·
Regression
problems
For
a regression in which Model i is nested within Model
j. Model i is the Restricted model, and Model j is the Unrestricted one. That is, Model i has fewer paramerers, pi than Model j, pj
. pj > pi,
In
the regression case, under the null hypothesis of no differences among the two
models, Model i and Model j, in getting a better fit, i.e. Model j does not
offer a significant improvement in the fit than Model i.
Under
the null hypothesis that model 2 does not provide a significantly better fit
than model 1, F statistic will have an F-distribution,
with (pj – pi, n – pj) degrees
of freedom.
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