naginterfaces.library.anova.rowcol¶
- naginterfaces.library.anova.rowcol(nrep, nrow, ncol, y, nt, it, tol, irdf)[source]¶
rowcol
computes the analysis of variance for a general row and column design together with the treatment means and standard errors.For full information please refer to the NAG Library document for g04bc
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g04/g04bcf.html
- Parameters
- nrepint
, the number of replicates.
- nrowint
, the number of rows per replicate.
- ncolint
, the number of columns per replicate.
- yfloat, array-like, shape
The observations ordered by columns within rows within replicates. That is contains the observation from the th column of the th row of the th replicate, for , for , for .
- ntint
The number of treatments. If only replicates, rows and columns are required in the analysis then set .
- itint, array-like, shape
Note: the required length for this argument is determined as follows: if : ; otherwise: .
If , indicates which of the treatments unit received, for .
If , is not referenced.
- tolfloat
The tolerance value used to check for zero eigenvalues of the matrix . If a default value of is used.
- irdfint
An adjustment to the degrees of freedom for the residual and total.
The degrees of freedom for the total is set to and the residual degrees of freedom adjusted accordingly.
The total degrees of freedom for the total is set to , as usual.
- Returns
- gmeanfloat
The grand mean, .
- tmeanfloat, ndarray, shape
If , contains the (adjusted) mean for the th treatment, , for , where is the mean of the treatment adjusted observations . Otherwise is not referenced.
- tablfloat, ndarray, shape
The analysis of variance table. Column 1 contains the degrees of freedom, column 2 the sum of squares, and where appropriate, column 3 the mean squares, column 4 the -statistic and column 5 the significance level of the -statistic. Row 1 is for replicates, row 2 for rows, row 3 for columns, row 4 for treatments (if ), row 5 for residual and row 6 for total. Mean squares are computed for all but the total row, -statistics and significance are computed for treatments, replicates, rows and columns. Any unfilled cells are set to zero.
- cfloat, ndarray, shape
The upper triangular part of contains the variance-covariance matrix of the treatment effects, the strictly lower triangular part contains the standard errors of the difference between two treatment effects (means), i.e., contains the covariance of treatment and if and the standard error of the difference between treatment and if , for , for .
- irepint, ndarray, shape
If , the treatment replications, , for . Otherwise is not referenced.
- rpmeanfloat, ndarray, shape
If , contains the mean for the th replicate, , for . Otherwise is not referenced.
- rmeanfloat, ndarray, shape
contains the mean for the th row, , for .
- cmeanfloat, ndarray, shape
contains the mean for the th column, , for .
- rfloat, ndarray, shape
The residuals, , for .
- effloat, ndarray, shape
If , the canonical efficiency factors. Otherwise is not referenced.
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, at least one treatment is not present. Treatment does not appear in .
- (errno )
On entry, , and .
Constraint: .
- (errno )
On entry, the values in are constant.
- (errno )
The computation of the eigenvalues has failed to converge.
- (errno )
A computed standard error is zero.
- Warns
- NagAlgorithmicWarning
- (errno )
The treatments are totally confounded with blocks.
- (errno )
The residual degrees of freedom is zero.
- (errno )
The residual mean square is zero.
- (errno )
The design is disconnected.
- Notes
In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.
In a row and column design the experimental material can be characterised by a two-way classification, nominally called rows and columns. Each experimental unit can be considered as being located in a particular row and column. It is assumed that all rows are of the same length and all columns are of the same length. Sets of equal numbers of rows/columns can be grouped together to form replicates, sometimes known as squares or rectangles, as appropriate.
If for a replicate, the number of rows, the number of columns and the number of treatments are equal and every treatment occurs once in each row and each column then the design is a Latin square. If this is not the case the treatments will be non-orthogonal to rows and columns. For example in the case of a lattice square each treatment occurs only once in each square.
For a row and column design, with treatments in rows and columns and replicates or squares with observations the linear model is:
for , , and , where is the effect of the th replicate, is the effect of the th row, is the effect of the th column and the notation indicates that the th treatment is applied to the unit in row , column of replicate .
To compute the analysis of variance for a row and column design the mean is computed and subtracted from the observations to give, . Since the replicates, rows and columns are orthogonal the estimated effects, ignoring treatment effects, , , , can be computed using the appropriate means of the , and the unadjusted sum of squares computed as the appropriate sum of squared totals for the divided by number of units per total. The observations adjusted for replicates, rows and columns can then be computed by subtracting the estimated effects from to give .
In the case of a Latin square design the treatments are orthogonal to replicates, rows and columns and so the treatment effects, , can be estimated as the treatment means of the adjusted observations, . The treatment sum of squares is computed as the sum of squared treatment totals of the divided by the number of times each treatment is replicated. Finally the residuals, and hence the residual sum of squares, are given by .
For a design which is not orthogonal, for example a lattice square or an incomplete Latin square, the treatment effects adjusted for replicates, rows and columns need to be computed. The adjusted treatment effects are found as the solution to the equations:
where is the vector of the treatment totals of the observations adjusted for replicates, rows and columns, , is a diagonal matrix with equal to the number of times the th treatment is replicated, and is the incidence matrix, with equal to the number of times treatment occurs in replicate , with and being similarly defined for rows and columns. The solution to the equations can be written as:
where, is a generalized inverse of . The solution is found from the eigenvalue decomposition of . The residuals are first calculated by subtracting the estimated adjusted treatment effects from the adjusted observations to give . However, since only the unadjusted replicate, row and column effects have been removed and they are not orthogonal to treatments, the replicate, row and column means of the have to be subtracted to give the correct residuals, and residual sum of squares.
Given the sums of squares, the mean squares are computed as the sums of squares divided by the degrees of freedom. The degrees of freedom for the unadjusted replicates, rows and columns are , and respectively and for the Latin square designs the degrees of freedom for the treatments is . In the general case the degrees of freedom for treatments is the rank of the matrix . The -statistic given by the ratio of the treatment mean square to the residual mean square tests the hypothesis:
The standard errors for the difference in treatment effects, or treatment means, for Latin square designs, are given by:
where is the residual mean square. In the general case the variances of the treatment effects are given by:
from which the appropriate standard errors of the difference between treatment effects or the difference between adjusted means can be calculated.
The analysis of a row-column design can be considered as consisting of different strata: the replicate stratum, the rows within replicate and the columns within replicate strata and the units stratum. In the Latin square design all the information on the treatment effects is given at the units stratum. In other designs there may be a loss of information due to the non-orthogonality of treatments and replicates, rows and columns and information on treatments may be available in higher strata. The efficiency of the estimation at the units stratum is given by the (canonical) efficiency factors, these are the nonzero eigenvalues of the matrix, , divided by the number of replicates in the case of equal replication, or by the mean of the number of replicates in the unequally replicated case, see John (1987). If more than one eigenvalue is zero then the design is said to be disconnected and information on some treatment comparisons can only be obtained from higher strata.
- References
Cochran, W G and Cox, G M, 1957, Experimental Designs, Wiley
Davis, O L, 1978, The Design and Analysis of Industrial Experiments, Longman
John, J A, 1987, Cyclic Designs, Chapman and Hall
John, J A and Quenouille, M H, 1977, Experiments: Design and Analysis, Griffin
Searle, S R, 1971, Linear Models, Wiley