Analyzes a wide variety of split-plot experiments with fixed, mixed or random factors. The whole‑plots can be assigned to experimental units using either a completely randomized or randomized complete block design. Function split_plot also analyzes split-plot experiments replicated at several locations.

The type double function is imsls_d_split_plot.

Address of a pointer to the memory location of a two dimensional, 11 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:

Function imsls_f_split_plot is capable of analyzing a wide variety of split-plot experiments. Whole-plot and split-plot factors can each be designated as either fixed or random, allowing for experiments with fixed, random or mixed treatment effects. By default, imsls_f_split_plot assumes that all treatment factors are fixed effects, i.e. IMSLS_WHOLE_FIXED and IMSLS_SPLIT_FIXED are default settings. Whole-plot or split-plot factors can each be declared as random effects by setting the optional input arguments IMSLS_WHOLE_RANDOM and IMSLS_SPLIT_RANDOM, respectively.

Split-plot experimental designs can also vary in the assignment of the whole-plot factor to its 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.

A similar randomization choice occurs 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 is the whole-plot experimental unit. In the first case in which only a single randomly selected fertilizer is applied to a single 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-plots 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, this function 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 experiments and completely randomized or randomized complete block experiments is the presence of a second factor that is blocked, or nested, within each level of the whole-plot factor. This second factor is referred to as the split-plot factor, see Table 22. If levels of this factor were completely randomized, then two or more treatments with the same split‑plot level could be assigned to the same whole-plot level, see Table 23.

In some studies, a split-plot experiment is replicated at several locations. Function imsls_f_split_plot can also analyze split-plot experiments replicated at multiple locations, even when the number of blocks or replicates at each location are 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, the four entries associated with location in the Anova table 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 location associated with y[i] is specified in location[i], which is a required input argument when n_locations>1.

By default, locations are assumed to be random effects. However, they can be specified as fixed effects by setting the optional argument IMSLS_LOC_FIXED. This setting changes the calculations of the F‑tests for whole-plot and split-plot factors. If locations are assumed to be fixed effects, then the whole-plot and split-plot errors at each location are pooled to form the whole-plot and split-plot errors. This can dramatically increase the degrees of freedom associated with the F‑test for the treatment factors, resulting in smaller p‑values. However, pooling the error terms from different locations requires experimenters to assume that the errors at each location are approximately the same. This should be verified using a test for homogeneity of variance, such as Bartlett’s or Levene’s test.

On the other hand, if locations are assumed to be random effects, then 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. However, 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 whole-plots and locations is statistically significant, then the nature of that interaction should be explored since it impacts the interpretation of the significance of the whole-plot treatment factor.

Similarly, when locations 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 4 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.

The field at each farm 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 divided into 4 split-plots, and the seed lots are randomly assigned to these split‑plots.

In this case, each farm is a blocking factor, fertilizers are whole-plots and seed lots are split-plots. The input array rep would contain integers from 1 to the number of farms.

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 would 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 this function can analyze 3x2x2x2=24 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 both whole‑plots and split‑plots are assumed to be fixed effects.

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