# All statistics for Create 2-Level Split-Plot Design

Find definitions and interpretation guidance for every statistic that is provided with create 2-level split-plot (hard-to-change factors).

## Factors

The number shows how many factors are in the design.

### Interpretation

The factors are the variables that you control in the experiment. Factors are also known as independent variables, explanatory variables, and predictor variables. Factors assume only a limited number of possible values, known as factor levels. Factors can have text or numeric levels. For numeric factors, you select specific levels for the experiment, even though many values for the factor are possible.

For example, you are studying factors that could affect plastic strength during the manufacturing process. You include factors for additive and temperature in the experiment. The additive is a categorical variable. Additive can be type A or type B. Temperature is a continuous variable. Because temperature is a factor, only two temperatures settings are in the experiment: 100°C and 200°C. If the design includes a center point, the numeric factor can have three levels (100°C, 150°C, and 200°C).

## Whole plots

The number shows how many whole plots are in the design.

### Interpretation

In a split-plot experiment, levels of the hard-to-change factors are held constant for several experimental runs. The experimental runs with the same hard-to-change settings form a whole plot. The combinations of settings for the easy-to-change factors change inside the whole plot. A design usually has multiple replicates of the same whole plots. These replicated whole plots allow estimation of the statistical significance of the hard-to-change factors.

For example, a quality engineer wants to study the factors that affect the texture of frozen yogurt. The engineer can quickly change the amount of air added to the yogurt mix and the mixing speed by adjusting settings on the machine. However, to change the temperature on a machine, the engineer has to change the temperature setting, then wait for the temperature of the yogurt mix to stabilize. Temperature is a hard-to-change factor that defines the whole plots.

The engineer runs all 4 combinations of settings for air and mixing speed at the high level of temperature, which makes one whole plot. The engineer changes the temperature of the machine, then runs the 4 combinations of the other factors again. These second 4 runs form a second whole plot. To complete the design, the engineer replicates the first two whole plots. The entire design has 4 whole plots.

## Resolution

The design resolution is the length of the shortest word in the defining relation for the design.

### Interpretation

The design resolution describes which effects in a fractional factorial design are aliased with other effects. For more information on aliasing, see the section on Alias structure.

In a split-plot design, the resolution does not account for whole-plot generators. For example, a resolution IV split-plot design can alias a 2-factor interaction with whole plots. Such a 2-factor interaction is not possible to estimate. When the alias table is in the output, Minitab lists all terms aliased with whole plots.

Resolution III, IV, and V designs are most common.
Resolution III
No main effects are aliased with any other main effect, but main effects are aliased with 2-factor interactions.
Resolution IV
No main effects are aliased with any other main effect or 2-factor interactions, but some 2-factor interactions are aliased with other 2-factor interactions and main effects are aliased with 3-factor interactions.
Resolution V
No main effects or 2-factor interactions are aliased with any other main effect or 2-factor interactions, but 2-factor interactions are aliased with 3-factor interactions and main effects are aliased with 4-factor interactions.

A design with higher resolution has less aliasing among lower-order terms. When you create a design, you need to balance the number of runs you can perform with an alias structure that you can accept. Identifying the important effects can be more complicated in a lower resolution design because of the terms that are aliased but lower resolution designs are usually smaller and more affordable.

## Hard-to-change

The number shows how many factors are in the design that are difficult to randomize completely due to time or cost constraints.

### Interpretation

In a split-plot experiment, levels of the hard-to-change factor are held constant for several experimental runs, which are collectively treated as a whole plot. For example, temperature is a common hard-to-change factor because temperature often requires a significant amount of time to stabilize after adjustments are made.

Hard-to-change factors are often confused with blocking variables. However, there are several important differences between blocks and hard-to-change factors:
• In a blocked design, the blocks are nuisance factors that are only included in a design to obtain a more precise estimate of the error term. However, you are interested in estimating the effect of hard-to-change factors, such as how temperature affects the moisture of a cake.
• In a blocked experiment, the interaction between the blocking variable and the factors is not of interest. When you have a hard-to-change factor, you might be interested in interactions between the hard-to-change variable and other factors in the experiment.
• Designs with hard-to-change and easy-to-change factors have two different sizes of experimental units. The hard-to-change factors are applied to a large experimental unit. Within this unit, the observational units are small experimental units used to study the easy-to-change factors. With a block design, the experimental units are all the same size.
• Blocks are usually random factors while hard-to-change factors are usually fixed.
• Blocks are a collection of experimental units. Hard-to-change factors are applied to the experimental units.

## Runs per whole plot

The number shows how many runs are in each whole plot in the design.

For example, pastry chefs at a large-scale bakery are designing a new brownie recipe. They are experimenting with two levels of chocolate and sugar using two different baking temperatures. To save time, instead of baking each tray individually, they decide to bake more than one tray of brownies at the same time. A whole plot represents all the trays of brownies that are baked at the same temperature. The subplots are individual trays of brownies. If there is 1 subplot replicate per whole plot, then the number of runs per whole plot is 4.

Whole Plot Temperature Chocolate Sugar
1 1 1 1
1 1 1 2
1 1 2 1
1 1 2 2

If there are 2 subplot replicates per whole plot, the number of runs per whole plot is 8.

Whole Plot Temperature Chocolate Sugar
1 1 1 1
1 1 1 2
1 1 2 1
1 1 2 2
1 1 1 1
1 1 1 2
1 1 2 1
1 1 2 2

## Fraction

The fraction indicates the proportion of runs from the full factorial design that are in the base design. For example, a full factorial, 2-level design with 4 factors has 16 runs. A ½ fraction of this design has 8 runs.

### Interpretation

The fraction indicates how many different sets of runs exist with a similar alias structure. If an experiment is a ½ fraction, then 2 sets of runs exist with similar alias structures. If an experiment is an 1/8 fraction, then 8 sets of runs exist with similar alias structures.

Before you perform your designed experiment, an important step is to verify that all the runs are feasible to conduct. By default, Minitab uses the principal fraction for a fractional factorial. The principal fraction always includes the run where all the factors are set at the high level. This combination of settings could be unfeasible, unsafe, or too expensive to run. One way to avoid unfeasible settings in a fractional factorial experiment is to change the fraction number of the design. To change the fraction number, go to the Options sub-dialog box.

## Runs

The number shows how many rows of data are in the design.

### Interpretation

A run is an experimental condition or factor level combination at which the response is measured. Each run corresponds to a row in the worksheet and results in one or more response measurements, or observations. For example, you perform a full factorial design with 2 factors, each with 2 levels. Your experiment has 4 runs:

C1 C2 C3 C4 C5 C6 C7
StdOrder RunOrder Blocks CenterPt Factor 1 Factor 2 Response
1 4 1 1 −1 −1 11
2 2 1 1 1 −1 12
3 1 1 1 −1 1 10
4 3 1 1 1 1 9

###### Note

When doing an experiment, the run order should be randomized. The randomized order is shown in the RunOrder column.

The entire set of runs represents the design. Multiple executions of the same factor level settings are considered separate runs and are called replicates.

In the design summary table, Minitab displays the runs for the base design and the total number of runs. For example, you create a fractional factorial design with 3 factors, 2 replicates, and 2 center points. The base design has 4 runs. With the replicates and center points, the final design has 10 total runs.

Design Summary Factors: 3 Base Design: 3, 4 Resolution: III Runs: 10 Replicates: 2 Fraction: 1/2 Blocks: 1 Center pts (total): 2

## Whole-plot replicates

The number shows how many times the base design is run. The base design includes a whole plot for each level combination of the hard-to-change factors and several subplots within each whole plot.

For example, pastry chefs at a large-scale bakery are designing a new brownie recipe. They are experimenting with two levels of chocolate and sugar using two different baking temperatures. To save time, instead of baking each tray individually, they decide to bake more than one tray of brownies at the same time. In this design, temperature is the hard-to-change (HTC) factor.

For the brownie experiment, whole plot 1 has 4 trays baked at the same temperature. Whole plot 2 has 4 trays baked at the other temperature. These 8 runs comprise the base design.

Replicate Whole plot Temperature (HTC) Sugar Chocolate
1 1 1 1 1
1 1 1 1 2
1 1 1 2 1
1 1 1 2 2
1 2 2 1 1
1 2 2 1 2
1 2 2 2 1
1 2 2 2 2

A whole plot replicate contains all of the runs in the whole plots that form the base design.

Replicate Whole plot Temperature (HTC) Sugar Chocolate
2 3 1 1 1
2 3 1 1 2
2 3 1 2 1
2 3 1 2 2
2 4 2 1 1
2 4 2 1 2
2 4 2 2 1
2 4 2 2 2

## Fraction number

The fraction number distinguishes the runs in the design from another set of runs that form the same size fraction. The possible values for fraction number depend on what size fraction of the full design you choose for your base design. For example, if the design is a ¼ fraction, then the possible fraction numbers are 1, 2, 3, and 4. Minitab only displays the fraction number when you change the fraction.

### Interpretation

In Minitab, the principal fraction number equals the denominator of the number displayed as "Fraction". For example, if the design is a 1/8 fraction, then the principal fraction number is 8. The principal fraction is the fraction where all of the signs for the design generators are positive. By default, Minitab uses the principal fraction when creating the design.

If you can't use the principal fraction, it is usually because one or more combinations of factor levels that are in the principal fraction are impractical to run. For example, the principal fraction always includes the run where all of the factors are at their high setting. The other fractions do not. If setting all of the factors to their high levels is expensive or difficult, you can change the fraction number in the Options sub-dialog box.

## Blocks

The number shows how many blocks are in the design.

### Interpretation

Blocks account for the differences that might occur between runs that are performed under different conditions. For example, an engineer designs an experiment to study welding and cannot collect all of the data on the same day. Weld quality is affected by several variables that change from day-to-day that the engineer cannot control, such as relative humidity. To account for these uncontrollable variables, the engineer groups the runs performed each day into separate blocks. The blocks account for the variation from the uncontrollable variables so that these effects are not confused with the effects of the factors the engineer wants to study. For more information on how Minitab assigns runs to blocks, go to What is a block?.

## Subplot replicates

The number shows how many sets of runs for the easy-to-change factors are within each whole plot.

For example, pastry chefs at a large-scale bakery are designing a new brownie recipe. They are experimenting with two levels of chocolate and sugar using two different baking temperatures. To save time, instead of baking each tray individually, they decide to bake more than one tray of brownies at the same time.

The whole plot is all the trays of brownies that are baked at the first temperature. The subplots are each individual tray of brownies.

If each combination of levels for the easy-to-change factor is run once, the whole plot has one subplot replicate.
 Tray 1 (Chocolate 1, Sugar 1) Tray 2 (Chocolate 1, Sugar 2) Tray 3 (Chocolate 2, Sugar 1) Tray 4 (Chocolate 2, Sugar 2)
If each combination of levels for the easy-to-change factor is run twice before the hard-to-change factor changes, then the whole plot has two subplot replicates.
 Tray 1 (Chocolate 1, Sugar 1) Tray 2 (Chocolate 1, Sugar 2) Tray 3 (Chocolate 2, Sugar 1) Tray 4 (Chocolate 2, Sugar 2) Tray 5 (Chocolate 1, Sugar 1) Tray 6 (Chocolate 1, Sugar 2) Tray 7 (Chocolate 2, Sugar 1) Tray 8 (Chocolate 2, Sugar 2)

## Design generators

Design generators are comprised of the factors that are multiplied together to determine the settings for another factor in the design. For example, the design generator D = ABC means that A, B, and C are multiplied together to determine the settings for D.

### Interpretation

The design generators determine which fraction of runs are in the fractional factorial design. For example, to construct a ½ fraction, 4-factor design using the design generator D = ABC, Minitab does the following:
1. Construct the full 3-factor design where –1 and +1 represent the low and high factor levels, respectively.
A B C
–1 –1 –1
+1 –1 –1
–1 +1 –1
+1 +1 –1
–1 –1 +1
+1 –1 +1
–1 +1 +1
+1 +1 +1
2. Generate the runs for factor D by multiplying the settings for factors A, B, and C together. For example, the setting for factor D for the first run is –1 × –1 × –1 = –1 (the low setting).
A B C D = ABC
–1 –1 –1 –1
+1 –1 –1 +1
–1 +1 –1 +1
+1 +1 –1 –1
–1 –1 +1 +1
+1 –1 +1 –1
–1 +1 +1 –1
+1 +1 +1 +1

Because the settings for factor D are equal to the settings for A × B × C, factor D is confounded with the ABC interaction. Because effects that are confounded cannot be estimated separately from each other, design generators should be carefully chosen. By default, Minitab uses the design generators that create the design with the highest resolution for the number of factors in the design. However, if you want to specify a different design generator, use Create 2-Level Factorial Design (Specify Generators).

## Hard-to-change factors

The list of factors shows which factors in the design are difficult to randomize completely due to time or cost constraints.

### Interpretation

Hard-to-change factors stay at the same setting for an entire whole plot. For example, temperature is a common hard-to-change factor because adjusting temperature often requires significant time to stabilize. Use the list to identify the design.

Hard-to-change factors are often confused with blocking variables. However, there are several important differences between blocks and hard-to-change factors:
• In a blocked design, the blocks are nuisance factors that are only included in a design to obtain a more precise estimate of the error term. However, you are interested in estimating the effect of hard-to-change factors, such as how temperature affects the moisture of a cake.
• In a blocked experiment, the interaction between the blocking variable and the factors is not of interest. When you have a hard-to-change factor, you might be interested in interactions between the hard-to-change variable and other factors in the experiment.
• Designs with hard-to-change and easy-to-change factors have two different sizes of experimental units. The hard-to-change factors are applied to a large experimental unit. Within this unit, the observational units are small experimental units used to study the easy-to-change factors. With a block design, the experimental units are all the same size.
• Blocks are usually random factors while hard-to-change factors are usually fixed.
• Blocks are a collection of experimental units. Hard-to-change factors are applied to the experimental units.

## Whole plot generators

The list shows which terms are held constant within each whole plot.

### Interpretation

Whole plot generators are terms that are held constant and determine which runs are grouped together in whole plots.

If an effect other than a hard-to-change factor is a whole plot generator, then that effect is aliased with whole plots. The resolution for the design does not account for any aliasing with whole plots. For example, a 6-factor, half-fraction design with 2 hard-to-change factors, 8 whole plots, and 4 subplots is resolution IV, but the whole plots are aliased with a 2-way interaction. Except for hard-to-change factors, Minitab removes effects that are aliased with whole plots from the model when you use Analyze Factorial Design.

The hard-to-change factors are always listed as whole plot generators, but are not always aliased with whole plots. When a design has multiple whole plots with each level of the hard-to-change factor, the whole plots are not aliased with the hard-to-change factors. For example, this design has 1 hard-to-change factor and 3 easy-to-change factors. The design is in standard order. The low level of the hard-to-change factor is in whole plots 1 and 3. The high level of the hard-to-change factor is in whole plots 2 and 4. In this design, the hard-to-change factor is a whole plot generator because the values are constant for entire whole plots. However, the hard-to-change factor is not aliased with whole plots because the same levels are in different whole plots.

Whole Plots A[HTC] B C D
1 -1 -1 -1 -1
1 -1 1 1 -1
1 -1 1 -1 1
1 -1 -1 1 1
2 1 -1 -1 -1
2 1 1 1 -1
2 1 1 -1 1
2 1 -1 1 1
3 -1 1 -1 -1
3 -1 -1 1 -1
3 -1 -1 -1 1
3 -1 1 1 1
4 1 1 -1 -1
4 1 -1 1 -1
4 1 -1 -1 1
4 1 1 1 1

## Defining relation

The defining relation is the total collection of terms that are held constant to define the fraction in a fractional factorial design. The defining relation is used to calculate the alias structure, which indicates which terms are aliased with each other.

### Interpretation

These results show the defining relation and alias structure for a ¼ fractional factorial design with five factors (A, B, C, D, and E).
* NOTE * Some main effects are confounded with two-way interactions. Design Generators: D = AB, E = AC Defining Relation: I = ABD = ACE = BCDE Alias Structure I + ABD + ACE + BCDE A + BD + CE + ABCDE B + AD + CDE + ABCE C + AE + BDE + ABCD D + AB + BCE + ACDE E + AC + BCD + ABDE BC + DE + ABE + ACD BE + CD + ABC + ADE

Minitab uses the defining relation to calculate each line of the alias table. Any letter multiplied by itself is the identity, I (for example, A × A = I). The identity, I, multiplied by any letter is the same letter (for example, I × A = A). To determine which effects are confounded with a specific term, multiply the term of interest by each term in the defining relation, and then eliminate the squared terms. For example, the following list shows how to use the defining relation to find the terms that BC is confounded with:

(BC)(ABD) = AB2CD = ACD

(BC)(ACE) = ABC2E = ABE

(BC)(BCDE) = B2C2DE = DE

Therefore, BC is aliased with ACD, AE, and DE.

The identity column I is always a column of 1's (in coded units). Therefore, since I = ABD in our example, the product of the columns A, B, D is a column of 1’s. The same is true for ACE and BCDE.

## Alias structure

The alias structure describes the confounding pattern that occurs in a design. Terms that are confounded are also said to be aliased.

Aliasing, also known as confounding, occurs in fractional factorial designs because the design does not include all of the combinations of factor levels. For example, if factor A is confounded with the 3-way interaction BCD, then the estimated effect for A is the sum of the effect of A and the effect of BCD. You cannot determine whether a significant effect is because of A, because of BCD, or because of a combination of both. When you analyze the design in Minitab, you can include confounded terms in the model. Minitab removes the terms that are listed later in the terms list. However, certain terms are always fit first. For example, if you include blocks in the model, Minitab retains the block terms and removes any terms that are aliased with blocks.

You can use the alias structure to verify that important terms are not aliased with each other. If the alias structure is unacceptable, consider taking one of the following actions:
• Create the design again, but enter the factors into Minitab in a different order.
• Use a larger fraction of the design.

To see how to determine the alias structure, go to the section on Defining relation.

### Interpretation

Use the alias structure to verify that important terms are not aliased with each other. For example, researchers at an agricultural station want to learn about controlling the growth of a weed without the use of an herbicide. The researchers design an experiment to study the effect of these 5 factors:
• A: Type of habitat
• B: Introduction of competing plants
• C: Use of molluscicide
• D: Fencing
• E: Use of insecticide
Because the researchers have limited space and creating a particular habitat is time-consuming, the researchers design a split-plot fractional factorial experiment so that the habitat is held constant while the other factors vary. The researchers agree that the interaction between fencing (D) and the introduction of competing plants (E) is likely to be important.

## Design table

The design table shows the factor settings for each experimental run. Because the design table takes up less space than the worksheet, it can be useful for reports with limited space.

Letters represent the factors and follow the order that you used when you created the design. In each row, − indicates that the factor is at the low setting while + indicates that the factor is at the high setting.

### Interpretation

Use the design table to see the factor settings for each run and the order of the runs in the design. In these results, the design table shows that the design includes 32 runs in 1 block. The whole plots and the runs are randomized. In the first run, factors A, B, and C are at their high level while Factor D is at its low level.

### Full Factorial Split-Plot Design

Design Summary Factors: 4 Whole plots: 4 Hard-to-change: 1 Runs per whole plot: 8 Runs: 32 Whole-plot replicates: 2 Blocks: 1 Subplot replicates: 1
Hard-to-change factors: A Whole Plot Generators: A All terms are free from aliasing. Design Table (randomized) Run Blk WP A B C D 1 1 2 + + + - 2 1 2 + - + - 3 1 2 + + - + 4 1 2 + - - - 5 1 2 + + + + 6 1 2 + - + + 7 1 2 + + - - 8 1 2 + - - + 9 1 3 - - + - 10 1 3 - + + + 11 1 3 - + - - 12 1 3 - - - + 13 1 3 - + + - 14 1 3 - - + + 15 1 3 - - - - 16 1 3 - + - + 17 1 1 - - - - 18 1 1 - + + - 19 1 1 - - + + 20 1 1 - + - + 21 1 1 - + + + 22 1 1 - - - + 23 1 1 - - + - 24 1 1 - + - - 25 1 4 + - + - 26 1 4 + - + + 27 1 4 + + + - 28 1 4 + - - + 29 1 4 + - - - 30 1 4 + + - - 31 1 4 + + + + 32 1 4 + + - +
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