If A and B are two independent chi-square variables with m
and n degrees of freedom respectively, the random variable X,
defined as X = (A/m)/(B/n), is said to be F distributed with
parameters m and n, i.e. X is distributed F(m,n).
Clearly m > 0, n > 0 and F >= 0.
If pdf(x) = the probability density function the, cdf(x), or
cumulative distribution function, is the integral of pdf(u)
between u = 0 and u = x.
Upper and lower tail probabilities are defined for an alpha
(0 =< alpha =< 1) and some X-value ( e.g. X = X-critical ) by
lower tail probability = P(X =< X-critical) = 1 - alpha upper tail probability = P(X >= X-critical) = alphawhere P(E) = probability of event E in the sample space. Sometimes percentage points 100(1 - alpha)% or 100*alpha% are preferred.
You input m and n for all subsequent calculations except for options 6 and 7 (which set m and n automatically). Option 1 is selected to change m and/or n.
You input x-values and the program calculates probability density functions, i.e. pdf(x) values.
You input x-values and the program calculates cumulative distribution functions, i.e. cdf(x) values.
You input significance levels alpha and obtain x-critical values such that P(X >= x-critical) = alpha, i.e. inverses of the F distribution.
You input measured values V1, V2, ..., VK and the program
calculates probability integral transforms W1, W2, ...,WK
under the null hypothesis that V is F(m,n) distributed.
Kolmogorov-Smirnov and chi-square tests are performed on
the transforms to test the hypothesis that W is uniformly
distributed on the interval (0,1), which is equivalent to
the null hypothesis.
Use MAKMAT/EDITMT to prepare/edit files (like ftest.tf1).)
The F test for excess variance
Model discrimination can be performed using a variance ratio
VR = [(Q1 - Q2)/(M2 - M1)]/[Q2/(NPTS - M2)], where Q1 = (wtd) sum of squares after fitting model 1, Q2 = (wtd) sum of squares after fitting model 2, M1 = no. of parameters estimated in model 1, M2 = no. of parameters estimated in model 2, and NPTS = number of experimental measurements.Under H0: the experimental error (after weighting) is of the constant variance type and the models are linear and nested, VR should be distributed F(M2 - M1, NPTS - M2). If the error is not of a constant variance type then weights equal to standard deviations of measurements should be used, i.e. the Q values are weighted sums of squares. The test will only be approximate for nonlinear models. The variance ratio test for equality of variance with two independent samples is performed by program TTEST.
Often replicates are used to calculate standard errors for
weighting or for curve fitting to means of replicates which
complicates the F test.
There are three cases.
The 1-, 2- and 3-way Analysis of Variance
For one way analysis of variance, you can type in or prepare a
data matrix file if all columns have equal length. However, if
you want greater versatility, make files of individual columns.
Once you have read in a set of data files (not necessarily of
equal column length) you can edit to suppress/restore selected
columns. You can try variance stabilising transforms such as:-
log(x), sqrt(x), arcsin(sqrt(x)),without altering the original
data. After doing the usual analysis a Kruskall-Wallis one way
nonparametric ANOVA by ranks is always performed.Print out the
file anova1.tf1 for an example of a 1-way ANOVA library file.
For a two way analysis of variance you should prepare a matrix
of data using MAKMAT.After doing the usual analysis a Friedman
nonparametric ANOVA by ranks is always performed. Examine the
file anova2.tf1 to see how to prepare data for a 2-way ANOVA.
For a three way analysis of variance you should make a 2n by n
matrix of data using MAKMAT, with the top n by n submatrix set
up with numbers 1 to n (corresponding to A, B, C,etc.) for the
Latin Square key, as in the 3-way ANOVA data file anova3.tf1.
The full set of AVOVA and MANOVA test files can be viewed from
the main [View] option. Here is a description of some of these.
ftest.tf1
This is a set of F(m,n) pseudo random numbers with parameters
m = 2 and n = 5. First choose option 1 to set the degrees of
freedom equal to 2 and 5. Then read this test file into option
5 to see how the program is used to test if a set of numbers
do come from an F distribution.
anova1.tf1
This is a library file to explore 1 way analysis of variance.
Choose option 7 then read in this test file to see how to use
this program to do 1 way ANOVA interactively, by selecting the
columns to be included and the variance stabilising transform
to employ (dividing by 100 to normalise in this case). There
is a similar version of this data in file anova1.TFL. Note how
nonparametric analysis is done automatically.
anova2.tf1 and anova3.tf1
Use these files to find out how to do 2-way and 3-way ANOVA.