The routine may be called by the names g04daf or nagf_anova_contrasts.
In the analysis of designed experiments the first stage is to compute the basic analysis of variance table, the estimate of the error variance (the residual or error mean square), , and the (variance ratio) -statistic for the treatments. If this -test is significant then the second stage of the analysis is to explore which treatments are significantly different.
If there is a structure to the treatments then this may lead to hypotheses that can be defined before the analysis and tested using linear contrasts. For example, if the treatments were three different fixed temperatures, say , and , and an uncontrolled temperature (denoted by ) then the following contrasts might be of interest.
The first represents the average difference between the controlled temperatures and the uncontrolled temperature. The second represents the linear effect of an increasing fixed temperature.
For a randomized complete block design or a completely randomized design, let the treatment means be , , and let the th contrast be defined by , , then the estimate of the contrast is simply:
and the sum of squares for the contrast is:
where is the number of observations for the th treatment. Such a contrast has one degree of freedom so that the appropriate -statistic is .
The two contrasts and are orthogonal if and the contrast is orthogonal to the overall mean if . In practice these sums will be tested against a small quantity, . If each of a set of contrasts is orthogonal to the mean and they are all mutually orthogonal then the contrasts provide a partition of the treatment sum of squares into independent components. Hence the resulting -tests are independent.
If the treatments come from a design in which treatments are not orthogonal to blocks then the sum of squares for a contrast is given by:
with , for , being adjusted treatment means computed by first eliminating blocks then computing the treatment means from the block adjusted observations without taking into account the non-orthogonality between treatments and blocks. For further details see John (1987).
Cochran W G and Cox G M (1957) Experimental Designs Wiley
John J A (1987) Cyclic Designs Chapman and Hall
Winer B J (1970) Statistical Principles in Experimental Design McGraw–Hill
1: – IntegerInput
On entry: , the number of treatment means.
2: – Real (Kind=nag_wp) arrayInput
On entry: the treatment means,
, for .
3: – Integer arrayInput
On entry: the replication for each treatment mean,
, for .
4: – Real (Kind=nag_wp)Input
On entry: the residual mean square, .
5: – Real (Kind=nag_wp)Input
On entry: the residual degrees of freedom.
6: – IntegerInput
On entry: the number of contrasts.
7: – Real (Kind=nag_wp) arrayInput
On entry: the columns of ct must contain the nc contrasts, that is
must contain , for and .
8: – IntegerInput
On entry: the first dimension of the array ct as declared in the (sub)program from which g04daf is called.
9: – Real (Kind=nag_wp) arrayOutput
On exit: the estimates of the contrast,
, for .
10: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array tabl
must be at least
On entry: the elements of tabl that are not referenced as described below remain unchanged.
On exit: the rows of the analysis of variance table for the contrasts. For each row column 1 contains the degrees of freedom, column 2 contains the sum of squares, column 3 contains the mean square, column 4 the -statistic and column 5 the significance level for the contrast. Note that the degrees of freedom are always one and so the mean square equals the sum of squares.
11: – IntegerInput
On entry: the first dimension of the array tabl as declared in the (sub)program from which g04daf is called.
12: – Real (Kind=nag_wp)Input
On entry: the tolerance, used to check if the contrasts are orthogonal and if they are orthogonal to the mean. If the value machine precision is used.
13: – LogicalInput
On entry: if the means are provided in tx and the formula (2) is used instead of formula (1).
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended since useful values can be provided in some output arguments even when on exit. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases g04daf may return useful information.
On entry, and .
On entry, and .
On entry, .
On entry, .
On entry, .
On entry, .
The and contrasts are not orthogonal. Full results are returned but they should be interpreted with care.
The contrast is not orthogonal to the mean. Full results are returned but they should be interpreted with care.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
The computations are stable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g04daf is not threaded in any implementation.
If the treatments have a factorial structure g04caf should be used and if the treatments have no structure the means can be compared using g04dbf.
The data is from a completely randomized experiment on potato scab with seven treatments representing amounts of sulphur applied, whether the application was in spring or autumn and a control treatment. The one-way anova is computed using g02bbf. Two contrasts are analysed, one comparing the control with use of sulphur, the other comparing spring with autumn application.