Analyzes data from split-split-plot experiments. The whole-plots can be assigned to experimental units using either a completely randomized or randomized complete block design. Function split_split_plot also analyzes split‑split‑plot experiments replicated at several locations.

The type double function is imsls_d_split_split_plot.

Address of a pointer to the memory location of a two dimensional, 20 by 6 array containing the ANOVA table. Each row in this array contains values for one of the effects in the ANOVA table. The first value in each row, anova_tablei,0 = anova_table[i*6], identifies the source for the effect associated with values in that row. The remaining values in a row contain the ANOVA table values using the following convention:

The Source Identifiers in the first column of anova_tablei,j are the only negative values in anova_table[]. Note that the p-value for the F-statistic is returned as 0.0 when the value is so small that all significant digits have been lost. Assignments of identifiers to ANOVA sources use the following coding:

† If n_locations=1 sources involving location are set to missing (NaN).

‡ If IMSLS_CRD is set, entries for blocks within location are set to missing, and its sum of squares and degrees of freedom are pooled into the whole-plot error.

* Split-plot error component calculation varies depending upon n_locations. See Description below for details.

Function imsls_f_split_split_plot is capable of analyzing a wide variety of split-split-plot experiments.

Split-split-plot experimental designs can vary in the assignment of whole-plot factors to experimental units. In some cases, this assignment is completely random. For example, in a drug study the experimental unit might be the subject receiving a treatment. The whole-plot factor, possibly different treatments, could be assigned in one of two ways. Each subject could receive only one treatment or each could receive all treatments over an appropriate period of time. If each subject received only a single randomly selected treatment, then this design constitutes a completely randomized design for the whole-plot factor, and the optional input argument IMSLS_CRD must be set.

On the other hand, if each subject receives every treatment in random order, then the subject is a blocking factor, and this sampling scheme constitutes a randomized complete block design. In this case, it is necessary to assume that there are no carry-over effects from one treatment to another. This sampling scheme is the default setting, i.e. IMSLS_RCBD is the default setting.

This randomization choice occurs often in agricultural field trials. A trial designed to test different fertilizers and different seed lots can be conducted in one of two ways. The whole-plot factor, fertilizer, can be applied to different fields, or each can be applied to sub-divisions of these fields. In either case, a field, or a sub-division of a field, is the whole-plot experimental unit. In the first case, in which only one randomly selected fertilizer is applied to each field, the whole-plot factor is not blocked and this scheme is called as a completely randomized design, and the optional input argument IMSLS_CRD must be set. However, if fertilizers are applied to sub-divisions within a field, then the whole-plot factor is blocked within fields and this assignment is referred to as a randomized complete block design. By default, imsls_f_split_split_plot assumes that levels of the whole-plot factor are randomly assigned within blocks, i.e.IMSLS_RCBD is the default setting for randomizing whole-plots.

The essential distinction between split-plot and split-split-plot experiments is the presence of a third factor that is blocked, or nested, within each level of the whole-plot and split-plot factors. This third factor is referred to as the sub-plot factor.

Contrast the split-split plot experiment to the same experiment run using a strip-split plot design, see Table 26. In a strip-split plot experiment factor B is applied in strip across factor A; whereas, in a split-split plot experiment, factor B is randomly assigned to each level of factor A. In a strip-split plot experiment, the level of factor B is constant across a row; whereas in a split-split plot experiment, the levels of factor B change as you go across a row, reflecting the fact that factor B is randomized within each level of factor A.

In some studies, a split-split-plot experiment is replicated at several locations. Function imsls_f_split_split_plot can analyze these, even when the number of blocks or replicates at each location is different. If only a single replicate or block is used at each location, then location should be treated as a blocking factor, with n_locations set equal to one. If n_locations=1, it is assumed that the experiment was conducted at a single location with more than one block or replicate at that location. In this case, all entries in the anova table associated with location will contain missing values.

However, if n_locations>1, it is assumed the experiment was repeated at multiple locations, with replication or blocking occurring at each location. Although the number of blocks, or replicates, at each location can be different, the number of levels for whole-plot and split-plot factors, n_whole and n_split, must be the same at each location. The locations associated with each of the observations in y are specified in the argument locations[], which is a required input argument when n_locations>1.

By default, locations are assumed to be random effects. Tests involving whole-plots use the interaction between whole-plots and locations as the error term for testing whether there are statistically significant differences among whole-plot factor levels. This assumes that the interaction of whole-plots and locations is not statistically significant. A test of this assumption uses the pooled whole-plot error. If the interaction between location and whole-plots, split-plots or sub-plot is statistically significant, then the nature of that interaction should be explored since it impacts the interpretation of the significance of the treatment factors.

When n_locations >1 are assumed to be random effects, tests involving split-plots do not use the split-plot errors pooled across locations. Instead, the error term for split plots is the interaction between locations and split-plots. The split-plot by whole-plot interaction is tested against the location by split-plot by whole-plot interaction.

Suppose, for example, that a researcher wanted to conduct an agricultural experiment comparing the effectiveness of 4 fertilizers with 3 rates of application and 2 seed lots. One replicate of the experiment is conducted at each of the 3 farms. That is, only a single field at each location is assigned to this experiment.

Each field is divided into 4 whole-plots and the fertilizers are randomly assigned to each of the 4 whole-plots. Each whole-plot is then further sub-divided into 3 split-plots which are each randomly assigned one of the three fertilizer application rates. Finally, each of these sub-divisions assigned a particular fertilizer and application rate is sub-divided into 2 plots and randomly assigned one of the two seed lots.

In this case, each farm is a blocking factor, fertilizers are whole-plots and fertilizer application rate are split plots, and seed lots are sub-plots. The input array rep would contain integers from 1 to the number of farms, with n_whole=4, n_split=3 and n_sub=2.

However, if each farm allocated more than a single field for this study, then each farm would be treated as a different location with n_locations set equal to the number of farms, and fields might be treated as blocking factor. The array rep would contain integers from 1 to the number fields used in a farm, and locations[] would contain integers from 1 to the number of farms.

In summary imsls_f_split_split_plot can analyze 3x2=6 different experimental situations, depending upon the settings of:

The default condition depends upon the value for n_locations. If n_locations>1, locations are assumed to be a random effect. Assignment of experimental units to whole-plots is assumed to use a RCBD design and whole-plots, split-plots and sub-plots are all assumed to be fixed effects.

This example uses data from a split‑split‑plot design consisting of two whole-plots, two-split‑plots and two sub‑plots.