# TTEST: help and advice

Consult the reference manual for further details and worked examples.
W.G.Bardsley, University of Manchester, U.K.

The t distribution

If Z is a standard normal random variable (mu = 0, sigma = 1) and C is an independent chi-square variable with N degrees of freedom, the random variable

         T = Z/[sqrt(C/N)]

is t distributed with N degrees of freedom, i.e. T is distributed t(N).

Also T^2 is distributed F(1,N).

If the probability density function is pdf(t), the cdf(t) or cumulative distribution function is the integral of cdf(u) from u = minus infinity to u = t.

Upper and lower tail probabilities are defined for an alpha 0 =< alpha =< 1, and some t-value (say t = t-critical) by

         lower tail probability = P(t =< t-critical) = 1 - alpha,
         upper tail probability = P(t >= t-critical) = alpha,

where P(E) = probability of event E in the sample space. Sometimes percentage points 100(1 - alpha)% or 100*alpha% are preferred.

Option 1
You input N for all of the subsequent calculations except for 1-sample, paired and unpaired t tests (Options 6,7,8). Option 1 is selected to assign the degrees of freedom.

Option 2
You input t-values and the program calculates probability density functions, i.e. pdf(t) values.

Option 3
You input t-values and the program calculates cumulative distribution functions, i.e. cdf(t) values.

Option 4
You input significance levels alpha and obtain t-critical values such that P(t >= t-critical) = alpha, i.e. inverses of the t distribution.

Option 5
You input measured values V1, V2, ..., VM and the program calculates probability integral transforms W1, W2, ...,WM under the null hypothesis that V is distributed t(N). 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 the V input vector.

Options 6, 7 and 8: The t tests
Suppose you have observed j x-values x(1), x(2), ..., x(j) and k y-values y(1), y(2), ..., y(k), and you suspect that X and Y are normally distributed but with the same variance. Then you can use an unpaired t test for equality of means. If j = k and the x- and y-values are connected in pairs you can also use a paired t test for equality of means. This program reads in vectors (columns) of x and y values, performs a variance ratio test for equality of variances and then an unpaired t test.
If j = k it will also do a paired t test.

Advice

You can input x, y values from the console if you want but it is better to use MAKMAT to write and EDITMT to edit files (without opting to put data in increasing/decreasing order). Observe if the data are consistent with normal distributions with the same variances before accepting t test results. Be sure data are correctly ordered if a paired test is used and note results from the variance ratio test.

Test files

ttest.tf1
First choose option 1 to set the degrees of freedom equal to 10 then choose option 5 and test if these data are consistent with a t distribution

normal.tf1
Use for a 1-sample t test to test if the mean is zero

ttest.tf2 and ttest.tf3
These are a matched pair of data sets to be used in a paired t test.

ttest.tf4 and ttest.tf5
These are an alternative pair of column vectors to use in the unpaired t test.

Note that program SIMSTAT will do all possible pairwise t and nonparametric tests on a library file of data vectors.