The routine may be called by the names g04caf or nagf_anova_factorial.
3Description
An experiment consists of a collection of units, or plots, to which a number of treatments are applied. In a factorial experiment the effects of several different sets of conditions are compared, e.g., three different temperatures, ${T}_{1}$, ${T}_{2}$ and ${T}_{3}$, and two different pressures, ${P}_{1}$ and ${P}_{2}$. The conditions are known as factors and the different values the conditions take are known as levels. In a factorial experiment the experimental treatments are the combinations of all the different levels of all factors, e.g.,
The effect of a factor averaged over all other factors is known as a main effect, and the effect of a combination of some of the factors averaged over all other factors is known as an interaction. This can be represented by a linear model. In the above example if the response was ${y}_{ijk}$ for the $k$th replicate of the $i$th level of $T$ and the $j$th level of $P$ the linear model would be
where $\mu $ is the overall mean, ${t}_{i}$ is the main effect of $T$, ${p}_{j}$ is the main effect of $P$, ${\gamma}_{ij}$ is the $T\times P$ interaction and ${e}_{ijk}$ is the random error term. In order to find unique estimates constraints are placed on the parameters estimates. For the example here these are:
If there is variation in the experimental conditions (e.g., in an experiment on the production of a material different batches of raw material may be used, or the experiment may be carried out on different days), then plots that are similar are grouped together into blocks. For a balanced complete factorial experiment all the treatment combinations occur the same number of times in each block.
g04caf computes the analysis of variance (ANOVA) table by sequentially computing the totals and means for an effect from the residuals computed when previous effects have been removed. The effect sum of squares is the sum of squared totals divided by the number of observations per total. The means are then subtracted from the residuals to compute a new set of residuals. At the same time the means for the original data are computed. When all effects are removed the residual sum of squares is computed from the residuals. Given the sums of squares an ANOVA table is then computed along with standard errors for the difference in treatment means.
The data for g04caf has to be in standard order given by the order of the factors. Let there be $k$ factors, ${f}_{1},{f}_{2},\dots ,{f}_{k}$ in that order with levels ${l}_{1},{l}_{2},\dots ,{l}_{k}$ respectively. Standard order requires the levels of factor ${f}_{1}$ are in order $1,2,\dots ,{l}_{1}$ and within each level of ${f}_{1}$ the levels of ${f}_{2}$ are in order $1,2,\dots ,{l}_{2}$ and so on.
For an experiment with blocks the data is for block $1$ then for block $2$, etc. Within each block the data must be arranged so that the levels of factor ${f}_{1}$ are in order $1,2,\dots ,{l}_{1}$ and within each level of ${f}_{1}$ the levels of ${f}_{2}$ are in order $1,2,\dots ,{l}_{2}$ and so on. Any within block replication of treatment combinations must occur within the levels of ${f}_{k}$.
The ANOVA table is given in the following order. For a complete factorial experiment the first row is for blocks, if present, then the main effects of the factors in their order, e.g., ${f}_{1}$ followed by ${f}_{2}$, etc. These are then followed by all the two factor interactions then all the three factor interactions, etc., the last two rows being for the residual and total sums of squares. The interactions are arranged in lexical order for the given factor order. For example, for the three factor interactions for a five factor experiment the $10$ interactions would be in the following order:
On entry: ${\mathbf{lfac}}\left(\mathit{i}\right)$ must contain the number of levels for the $\mathit{i}$th factor, for $\mathit{i}=1,2,\dots ,k$.
Constraint:
${\mathbf{lfac}}\left(\mathit{i}\right)\ge 2$, for $\mathit{i}=1,2,\dots ,k$.
5: $\mathbf{nblock}$ – IntegerInput
On entry: the number of blocks. If there are no blocks, set ${\mathbf{nblock}}=0$ or $1$.
Constraints:
${\mathbf{nblock}}\ge 0$;
if ${\mathbf{nblock}}\ge 2$, ${\mathbf{n}}/{\mathbf{nblock}}$ must be a multiple of the number of treatment combinations, that is a multiple of $\prod _{i=1}^{k}}{\mathbf{lfac}}\left(i\right)$.
6: $\mathbf{inter}$ – IntegerInput
On entry: the maximum number of factors in an interaction term. If no interaction terms are to be computed, set ${\mathbf{inter}}=0$ or $1$.
On entry: the adjustment to the residual and total degrees of freedom. The total degrees of freedom are set to ${\mathbf{n}}-{\mathbf{irdf}}$ and the residual degrees of freedom adjusted accordingly. For examples of the use of irdf see Section 9.
Constraint:
${\mathbf{irdf}}\ge 0$.
8: $\mathbf{mterm}$ – IntegerInput
On entry: the maximum number of terms in the analysis of variance table, see Section 9.
Constraint:
${\mathbf{mterm}}$ must be large enough to contain the terms specified by nfac, inter and nblock. If the routine exits with ${\mathbf{ifail}}\ge {\mathbf{2}}$, the required minimum value of mterm is returned in itotal.
9: $\mathbf{table}({\mathbf{mterm}},5)$ – Real (Kind=nag_wp) arrayOutput
On exit: the first itotal rows of table contain the analysis of variance table. The first column contains the degrees of freedom, the second column contains the sum of squares, the third column (except for the row corresponding to the total sum of squares) contains the mean squares, i.e., the sums of squares divided by the degrees of freedom, and the fourth and fifth columns contain the $F$ ratio and significance level, respectively (except for rows corresponding to the total sum of squares, and the residual sum of squares). All other cells of the table are set to zero.
The first row corresponds to the blocks and is set to zero if there are no blocks. The itotalth row corresponds to the total sum of squares for y and the $({\mathbf{itotal}}-1)$th row corresponds to the residual sum of squares. The central rows of the table correspond to the main effects followed by the interaction if specified by inter. The main effects are in the order specified by lfac and the interactions are in lexical order, see Section 3.
10: $\mathbf{itotal}$ – IntegerOutput
On exit: the row in table corresponding to the total sum of squares. The number of treatment effects is ${\mathbf{itotal}}-3$.
11: $\mathbf{tmean}\left({\mathbf{maxt}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the treatment means. The position of the means for an effect is given by the index in imean. For a given effect the means are in standard order, see Section 3.
12: $\mathbf{maxt}$ – IntegerInput
On entry: the maximum number of treatment means to be computed, see Section 9. If the value of maxt is too small for the required analysis then the minimum number is returned in ${\mathbf{imean}}\left(1\right)$.
Constraint:
${\mathbf{maxt}}$ must be large enough for the number of means specified by lfac and inter; if ${\mathbf{inter}}={\mathbf{nfac}}$ then ${\mathbf{maxt}}\ge {\displaystyle \prod _{i=1}^{k}}({\mathbf{lfac}}\left(i\right)+1)-1$.
13: $\mathbf{e}\left({\mathbf{maxt}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the estimated effects in the same order as for the means in tmean.
On exit: indicates the position of the effect means in tmean. The effect means corresponding to the first treatment effect in the ANOVA table are stored in ${\mathbf{tmean}}\left(1\right)$ up to ${\mathbf{tmean}}\left({\mathbf{imean}}\left(1\right)\right)$. Other effect means corresponding to the $i$th treatment effect, $i=1,2,\dots ,{\mathbf{itotal}}-3$, are stored in ${\mathbf{tmean}}\left({\mathbf{imean}}\left(i-1\right)+1\right)$ up to ${\mathbf{tmean}}\left({\mathbf{imean}}\left(i\right)\right)$.
15: $\mathbf{semean}\left({\mathbf{mterm}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the standard error of the difference between means corresponding to the $i$th treatment effect in the ANOVA table.
16: $\mathbf{bmean}\left({\mathbf{nblock}}+1\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{bmean}}\left(1\right)$ contains the grand mean, if ${\mathbf{nblock}}>1$, ${\mathbf{bmean}}\left(2\right)$ up to ${\mathbf{bmean}}\left({\mathbf{nblock}}+1\right)$ contain the block means.
17: $\mathbf{r}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{inter}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{inter}}\ge 0$.
On entry, ${\mathbf{inter}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nfac}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{inter}}\le {\mathbf{nfac}}$.
On entry, ${\mathbf{irdf}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{irdf}}\ge 0$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 4$.
On entry, ${\mathbf{nblock}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nblock}}\ge 0$.
On entry, ${\mathbf{nfac}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nfac}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, $i=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{lfac}}\left(i\right)=\u27e8\mathit{\text{value}}\u27e9$
Constraint: ${\mathbf{lfac}}\left(i\right)\ge 2$.
On entry, ${\mathbf{maxt}}=\u27e8\mathit{\text{value}}\u27e9$
Constraint: ${\mathbf{maxt}}\ge \u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{mterm}}=\u27e8\mathit{\text{value}}\u27e9$
Constraint: ${\mathbf{mterm}}\ge \u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nblock}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: when ${\mathbf{nblock}}>1$, n must be a multiple of nblock.
On entry, the number of plots per block is incompatible with the number of plot factors.
There are no degrees of freedom for the residual sum of squares. The standard errors and $F$-statistics cannot be computed.
The residual sum of squares is zero. The standard errors and $F$-statistics cannot be computed.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The block and treatment sums of squares are computed from the block and treatment residual totals. The residuals are updated as each effect is computed and the residual sum of squares computed directly from the residuals. This avoids any loss of accuracy in subtracting sums of squares.
8Parallelism and Performance
g04caf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The number of rows in the ANOVA table and the number of treatment means are given by the following formulae.
Let there be $k$ factors with levels ${l}_{i}$ for $i=1,2,\dots ,k$, and let $t$ be the maximum number of terms in an interaction then the number of rows in the ANOVA tables is:
where ${S}_{i}$ is the set of all combinations of the $k$ factors $i$ at a time.
To estimate missing values the Healy and Westmacott procedure or its derivatives may be used, see John and Quenouille (1977). This is an iterative procedure in which estimates of the missing values are adjusted by subtracting the corresponding values of the residuals. The new estimates are then used in the analysis of variance. This process is repeated until convergence. A suitable initial value may be the grand mean. When using this procedure irdf should be set to the number of missing values plus one to obtain the correct degrees of freedom for the residual sum of squares.
For analysis of covariance the residuals are obtained from an analysis of variance of both the response variable and the covariates. The residuals from the response variable are then regressed on the residuals from the covariates using, say, g02cbforg02daf. The coefficients obtained from the regression can be examined for significance and used to produce an adjusted dependent variable using the original response variable and covariate. An approximate adjusted analysis of variance table can then be produced by using the adjusted dependent variable. In this case irdf should be set to one plus the number of fitted covariates.
For designs such as Latin squares one more of the blocking factors has to be removed in a preliminary analysis before the final analysis. This preliminary analysis can be performed using g04bbf or a prior call to g04caf if the data is reordered between calls. The residuals from the preliminary analysis are then input to g04caf. In these cases irdf should be set to the difference between n and the residual degrees of freedom from preliminary analysis. Care should be taken when using this approach as there is no check on the orthogonality of the two analyses.
10Example
The data, given by John and Quenouille (1977), is for the yield of turnips for a factorial experiment with two factors, the amount of phosphate with $6$ levels and the amount of liming with $3$ levels. The design was replicated in $3$ blocks. The data is input and the analysis of variance computed. The analysis of variance table and tables of means with their standard errors are printed.