# Two Level Factorial Experiments

Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

## 2^{k} Designs

The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as *full factorial experiments*. Full factorial two level experiments are also referred to as **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{k}\,\!}**
designs where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\,\!}**
denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

A full factorial two level design with **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k\,\!}**
factors requires **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{2}^{k}}\,\!}**
runs for a single replicate. For example, a two level experiment with three factors will require **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2\times 2\times 2={{2}^{3}}=8\,\!}**
runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {45}^{o}C\,\!}**
to **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {90}^{o}C\,\!}**
, then the two levels used in the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{k}\,\!}**
design for this factor would be **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {45}^{o}\,\!C\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {90}^{o}\,\!C\,\!}**
.

The two levels of the factor in the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{k}\,\!}**
design are usually represented as **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -1\,\!}**
(for the first level) and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1\,\!}**
(for the second level). Note that this representation is reversed from the coding used in General Full Factorial Designs for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1\,\!}**
for the indicator variable, while the second level was represented using a value of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -1\,\!}**
. For details on the notation used for two level experiments refer to Notation.

### The 2^{2} Design

The simplest of the two level factorial experiments is the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{2}\,\!}**
design where two factors (say factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A\,\!}**
and factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B\,\!}**
) are investigated at two levels. A single replicate of this design will require four runs (**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{2}^{2}}=2\times 2=4\,\!}**
) The effects investigated by this design are the two main effects, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B,\,\!}**
and the interaction effect **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle AB\,\!}**
. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -1\,\!}**
; **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a\,\!}**
represents the treatment combination where factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A\,\!}**
is at the high level or the level of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1\,\!}**
, while the remaining factors (in this case, factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B\,\!}**
) are at the low level or the level of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b\,\!}**
represents the treatment combination where factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B\,\!}**
is at the high level or the level of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ab\,\!}**
represents the treatment combination where factors **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{2}\,\!}**
design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{2}\,\!}**
design is an *orthogonal design*. In fact, all

### The 2^{3} Design

The **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{3}\,\!}**
design is a two level factorial experiment design with three factors (say factors **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C\,\!}**
). This design tests three (**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=3\,\!}**
) main effects, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C\,\!}**
; three (**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (_{2}^{k})=\,\!}**
**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (_{2}^{3})=3\,\!}**
) two factor interaction effects, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle AB\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle BC\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle AC\,\!}**
; and one (**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (_{3}^{k})=\,\!}**
**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (_{3}^{3})=1\,\!}**
) three factor interaction effect, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ABC\,\!}**
. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (1)\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ab\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ac\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle bc\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle abc\,\!}**
. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the *standard order* or *Yates' order*. The **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{3}\,\!}**
design is shown in figure (a) below. The design matrix for the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{3}\,\!}**
design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

The

## Analysis of 2^{k} Designs

The

### Notation

Based on the notation used in General Full Factorial Designs, the ANOVA model for a two level factorial experiment with three factors would be as follows:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\ & & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon \end{align}\,\!}**

where:

- •
**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu \,\!}**represents the overall mean - •
**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\tau }_{1}}\,\!}**represents the independent effect of the first factor (factor**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\tau }_{1}}\,\!}**and**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\tau }_{2}}\,\!}** - •
**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\delta }_{1}}\,\!}**represents the independent effect of the second factor (factor**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\delta }_{1}}\,\!}**and**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\delta }_{2}}\,\!}** - •
**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{(\tau \delta )}_{11}}\,\!}**represents the independent effect of the interaction**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle AB\,\!}**out of the other interaction effects - •
**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\gamma }_{1}}\,\!}**represents the effect of the third factor (factor**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C\,\!}**) out of the two effects**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\gamma }_{1}}\,\!}**and**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\gamma }_{2}}\,\!}** - •
**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{(\tau \gamma )}_{11}}\,\!}**represents the effect of the interaction**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle AC\,\!}**out of the other interaction effects - •
**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{(\delta \gamma )}_{11}}\,\!}**represents the effect of the interaction**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle BC\,\!}**out of the other interaction effects - •
**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{(\tau \delta \gamma )}_{111}}\,\!}**represents the effect of the interaction**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ABC\,\!}**out of the other interaction effects

and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon \,\!}**
is the random error term.

The notation for a linear regression model having three predictor variables with interactions is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\ & & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon \end{align}\,\!}**

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{0}}\,\!}**
can represent the overall mean instead of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu \,\!}**
, and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{1}}\,\!}**
can represent the independent effect, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\tau }_{1}}\,\!}**
, of factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{123}}\,\!}**
can be used to represent the three factor interaction effect, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{(\tau \beta \gamma )}_{111}}\,\!}**
).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in General Full Factorial Designs. Here **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\hat{\tau }}_{1}}\,\!}**
, obtained based on the coding of General Full Factorial Designs, and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\hat{\beta }}_{1}}\,\!}**
, obtained based on the new coding, will be the same but their signs would be opposite).

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & & \text{Factor }A\text{ Coding (two level factor)} \\ & & \end{align}\,\!}**

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\ {} & {} & {} & {} & {} \\ Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\ Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\ \end{matrix}\,\!}**

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:

- • The notation of the regression models is used for the effect coefficients.
- • The coding of the indicator variables is reversed.

### Special Features

Consider the design matrix, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X\,\!}**
, for the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{X}^{\prime }}X\,\!}**
) **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ^{-1}\,\!}**
matrix is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix} 0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\ \end{matrix} \right]\,\!}**

Notice that, due to the orthogonal design of the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X\,\!}**
matrix, the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{X}^{\prime }}X)}^{-1}}\,\!}**
has been simplified to a diagonal matrix which can be written as:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I \end{align}\,\!}**

where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I\,\!}**
represents the identity matrix of the same order as the design matrix, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X\,\!}**
. Since there are eight observations per replicate of the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (X\,\!}**
' **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X{{)}^{-1}}\,\!}**
matrix for **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m\,\!}**
replicates of this design can be written as:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!}**

The **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{X}^{\prime }}X)}^{-1}}\,\!}**
matrix for any

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!}**

Then the variance-covariance matrix for the

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I \end{align}\,\!}**

Note that the variance-covariance matrix for the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{1}}\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{2}}\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{12}},\,\!}**
etc.) for these designs are uncorrelated. This implies that the terms in the

It can also be noted from the equation given above, that in addition to the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle se({{\hat{\beta }}_{j}})\,\!}**
, for all the coefficients is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j \end{align}\,\!}**

This property is used to construct the normal probability plot of effects in

#### Example

To illustrate the analysis of a full factorial **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle psi\,\!}**
and 400 **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle psi\,\!}**
, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

The applicable model using the notation for

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\ & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon \end{align}\,\!}**

where the indicator variable, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{x}_{1,}}\,\!}**
represents factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{x}_{1}}=-1\,\!}**
represents the low level of 200 **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle psi\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{x}_{1}}=1\,\!}**
represents the high level of 400 **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{x}_{2}}\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{x}_{3}}\,\!}**
represent factors **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{0}}\,\!}**
is the overall mean, while **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{1}}\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{2}}\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{3}}\,\!}**
are the effect coefficients for the main effects of factors **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{12}}\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{13}}\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{23}}\,\!}**
are the effect coefficients for the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle AC\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle BC\,\!}**
interactions, while **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\beta }_{123}}\,\!}**
represents the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ABC\,\!}**
interaction.

If the subscripts for the run (**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i\,\!}**
; **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i=\,\!}**
1 to 8) and replicates (**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j\,\!}**
; **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j=\,\!}**
1,2) are included, then the model can be written as:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\ & +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}} \end{align}\,\!}**

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!}****Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!}**

This test investigates the main effect of factor

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!}**

where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M{{S}_{A}}\,\!}**
is the mean square for factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M{{S}_{E}}\,\!}**
is the error mean square. Hypotheses for the other main effects,

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!}****Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!}**

This test investigates the two factor interaction

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!}**

where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M{{S}_{AB}}\,\!}**
is the mean square for the interaction **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M{{S}_{E}}\,\!}**
is the error mean square. Hypotheses for the other two factor interactions,

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!}****Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!}**

This test investigates the three factor interaction

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!}**

where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M{{S}_{ABC}}\,\!}**
is the mean square for the interaction **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M{{S}_{E}}\,\!}**
is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y=X\beta +\epsilon \,\!}**
.

#### Expression of the ANOVA Model as **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y=X\beta +\epsilon \,\!}**

In matrix notation, the ANOVA model can be expressed as:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y=X\beta +\epsilon \,\!}**

where:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y=\left[ \begin{matrix} {{Y}_{11}} \\ {{Y}_{21}} \\ . \\ {{Y}_{81}} \\ {{Y}_{12}} \\ . \\ {{Y}_{82}} \\ \end{matrix} \right]=\left[ \begin{matrix} 90 \\ 90 \\ . \\ 90 \\ 86 \\ . \\ 80 \\ \end{matrix} \right]\text{ }X=\left[ \begin{matrix} 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 \\ . & . & . & . & . & . & . & . \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \end{matrix} \right]\,\!}**

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta =\left[ \begin{matrix} {{\beta }_{0}} \\ {{\beta }_{1}} \\ {{\beta }_{2}} \\ {{\beta }_{12}} \\ {{\beta }_{3}} \\ {{\beta }_{13}} \\ {{\beta }_{23}} \\ {{\beta }_{123}} \\ \end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix} {{\epsilon }_{11}} \\ {{\epsilon }_{21}} \\ . \\ {{\epsilon }_{81}} \\ {{\epsilon }_{12}} \\ . \\ . \\ {{\epsilon }_{82}} \\ \end{matrix} \right]\,\!}**

#### Calculation of the Extra Sum of Squares for the Factors

Knowing the matrices **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta \,\!}**
, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\ = & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \end{align}\,\!}**

where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H\,\!}**
is the hat matrix and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J\,\!}**
is the matrix of ones. The matrix **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{\tilde{\ }A}}\,\!}**
can be calculated using **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!}**
where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{X}_{\tilde{\ }A}}\,\!}**
is the design matrix,

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\ = & 654.4375-549.375 \\ = & 105.0625 \end{align}\,\!}**

Similarly, the extra sum of squares for the interaction effect

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\ = & 654.4375-636.375 \\ = & 18.0625 \end{align}\,\!}**

The extra sum of squares for other effects can be obtained in a similar manner.

#### Calculation of the Test Statistics

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\ = & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\ = & \frac{18.0625/1}{147.5/8} \\ = & 0.9797 \end{align}\,\!}**

where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M{{S}_{AB}}\,\!}**
is the mean square for the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p\,\!}**
value corresponding to the statistic, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{f}_{0}})}_{AB}}=0.9797\,\!}**
, based on the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F\,\!}**
distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\ = & 1-0.6487 \\ = & 0.3513 \end{align}\,\!}**

Assuming that the desired significance is 0.1, since **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p\,\!}**
value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to Multiple Linear Regression Analysis.

#### Calculation of Effect Coefficients

The estimate of effect coefficients can also be obtained:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\ = & \left[ \begin{matrix} 86.4375 \\ 2.5625 \\ -4.9375 \\ 1.0625 \\ -1.0625 \\ 2.4375 \\ -1.3125 \\ -0.1875 \\ \end{matrix} \right] \end{align}\,\!}**

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle se({{\hat{\beta }}_{j}})\,\!}**
. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t\,\!}**
statistic, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{t}_{0}}\,\!}**
, corresponding to the coefficients. The P Value column displays the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p\,\!}**
value corresponding to the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t\,\!}**
statistic. (For details on how these results are calculated, refer to General Full Factorial Designs). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

#### Model Equation

From the analysis results in the above figure within calculation of effect coefficients section, it is seen that effects

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\ = & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}} \end{align}\,\!}**

To make the model hierarchical, the main effect,

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!}**

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

## Replicated and Repeated Runs

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a

## Unreplicated 2^{k} Designs

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the

### Pooling Higher Order Interactions

One of the ways to deal with unreplicated

### Normal Probability Plot of Effects

Another way to use unreplicated **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta =0\,\!}**
) will fall along the straight line representative of the normal distribution, N(**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!}**
). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =2\times \,\!}**
effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of a DOE folio plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\sigma }^{2}}\,\!}**
remains unknown since

#### Example

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D\,\!}**
). The two levels of the factors used in the experiment are as shown in below.

With a **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{4}\,\!}**
design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S{{S}_{E}}\,\!}**
. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

Lenth's method uses these values to estimate the variance. As described in [Lenth, 1989], if all effects are arranged in ascending order, using their absolute values, then **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{s}_{0}}\,\!}**
is defined as 1.5 times the median value:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\ = & 1.5\cdot 2 \\ = & 3 \end{align}\,\!}**

Using **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{s}_{0}}\,\!}**
, the "pseudo standard error" (**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle PSE\,\!}**
) is calculated as 1.5 times the median value of all effects that are less than 2.5 **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{s}_{0}}\,\!}**
:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\ = & 1.5\cdot 1.5 \\ = & 2.25 \end{align}\,\!}**

Using **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle PSE\,\!}**
as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D\,\!}**
and the interaction **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle AD\,\!}**
do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!}**

The **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t\,\!}**
statistic, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{t}_{\alpha /2,d}}\,\!}**
, is calculated at a significance of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha /2\,\!}**
(for the two-sided hypothesis) and degrees of freedom **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=(\,\!}**
number of effects **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle )/3\,\!}**
. Thus:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\ = & 2.015\cdot 2.25 \\ = & 4.534 \end{align}\,\!}**

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D\,\!}**
and the interaction **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle AD\,\!}**
. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

### Center Point Replicates

Another method of dealing with unreplicated

#### Example: Use Center Point to Get Pure Error

Consider a

Since the present **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y_{i}^{c}\,\!}**
, can be used to obtain an estimate of pure error, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S{{S}_{PE}}\,\!}**
. Let **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\bar{y}}^{c}}\,\!}**
represent the average response for the five replicates at the center. Then:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!}**

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\ = & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\ = & 0.052 \end{align}\,\!}**

Then the corresponding mean square is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\ = & \frac{0.052}{5-1} \\ = & 0.013 \end{align}\,\!}**

Alternatively, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M{{S}_{PE}}\,\!}**
can be directly obtained by calculating the variance of the response values at the center points:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} M{{S}_{PE}}= & {{s}^{2}} \\ = & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1} \end{align}\,\!}**

Once **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M{{S}_{PE}}\,\!}**
is known, it can be used as the error mean square, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A,\,\!}**
the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\ = & 0.5625 \end{align}\,\!}**

Then, the test statistic to test the significance of the main effect of factor

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\ = & \frac{0.5625/1}{0.052/4} \\ = & 43.2692 \end{align}\,\!}**

The **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{f}_{0}})}_{A}}=43.2692\,\!}**
, based on the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F\,\!}**
distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\ = & 1-0.9972 \\ = & 0.0028 \end{align}\,\!}**

Assuming that the desired significance is 0.1, since

### Using Center Point Replicates to Test Curvature

Center point replicates can also be used to check for curvature in replicated or unreplicated

#### Example: Use Center Point to Test Curvature

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2^2}**
design augmented by five center point runs"). Let **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{x}_{1}}\,\!}**
be the indicator variable to indicate if the run is a center point:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} {{x}_{1}}=0 & {} & \text{Center point run} \\ {{x}_{1}}=1 & {} & \text{Other run} \\ \end{matrix}\,\!}**

If **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{x}_{2}}\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{x}_{3}}\,\!}**
are the indicator variables representing factors

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!}**

To investigate the presence of curvature, the following hypotheses need to be tested:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\ & {{H}_{1}}: & {{\beta }_{1}}\ne 0 \end{align}\,\!}**

The test statistic to be used for this test is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!}**

where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M{{S}_{curvature}}\,\!}**
is the mean square for Curvature and

**Calculation of the Sum of Squares**

The **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y\,\!}**
vector for this experiment are:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X=\left[ \begin{matrix} 1 & 1 & -1 & -1 & 1 \\ 1 & 1 & 1 & -1 & -1 \\ 1 & 1 & -1 & 1 & -1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ \end{matrix} \right]\text{ }y=\left[ \begin{matrix} 24.6 \\ 25.4 \\ 25.0 \\ 25.7 \\ 25.2 \\ 25.3 \\ 25.4 \\ 25.1 \\ 25.3 \\ \end{matrix} \right]\,\!}**

The sum of squares can now be calculated. For example, the error sum of squares is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\ & = & 0.052 \end{align}\,\!}**

where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I\,\!}**
is the identity matrix and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H\,\!}**
is the hat matrix. It can be seen that this is equal to **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S{{S}_{PE\text{ }}}\,\!}**
(the sum of squares due to pure error) because of the replicates at the center point, as obtained in the example. The number of degrees of freedom associated with **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S{{S}_{E}}\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle dof(S{{S}_{E}})\,\!}**
is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\ & & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\ & = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y \end{align}\,\!}**

where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H\,\!}**
is the hat matrix and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J\,\!}**
is the matrix of ones. The matrix **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{\tilde{\ }Curvature}}\,\!}**
can be calculated using **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!}**
where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{X}_{\tilde{\ }Curv}}\,\!}**
is the design matrix,

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\ & = & 0.7036-0.6875 \\ & = & 0.0161 \end{align}\,\!}**

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\ & = & \frac{0.0161}{1} \\ & = & 0.0161 \end{align}\,\!}**

**Calculation of the Test Statistic**

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\ & = & \frac{0.0161/1}{0.052/4} \\ & = & 1.24 \end{align}\,\!}**

The **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{({{f}_{0}})}_{Curvature}}=1.24\,\!}**
, based on the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F\,\!}**
distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\ & = & 1-0.6713 \\ & = & 0.3287 \end{align}\,\!}**

Assuming that the desired significance is 0.1, since

## Blocking in 2^{k} Designs

Blocking can be used in the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (1)\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ab\,\!}**
were run on the first day and treatments **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b\,\!}**
were run on the second day. Then, the incomplete block design for this experiment is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \text{Block 1} & {} & \text{Block 2} \\ \left[ \begin{matrix} (1) \\ ab \\ \end{matrix} \right] & {} & \left[ \begin{matrix} a \\ b \\ \end{matrix} \right] \\ \end{matrix}\,\!}**

For this design the block effect may be calculated as:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\ & & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\ & = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\ & = & \frac{1}{2}[(1)+ab-a-b] \end{align}\,\!}**

The

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\ & & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\ & = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\ & = & \frac{1}{2}[(1)+ab-a-b] \end{align}\,\!}**

The two equations given above show that, in this design, the

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!}**

where the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\alpha }_{i}}\,\!}**
s are the exponents for the factors in the effect that is to be confounded with the block effect and the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{q}_{i}}\,\!}**
s are values based on the level of the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i\,\!}**
the factor (in a treatment that is to be allocated to a block). For **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\alpha }_{i}}\,\!}**
s are either 0 or 1 and the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{q}_{i}}\,\!}**
s have a value of 0 for the low level of the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i\,\!}**
th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=2\,\!}**
, with **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i=1\,\!}**
representing factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i=2\,\!}**
representing factor

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!}**

The value of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\alpha }_{1}}\,\!}**
is one because the exponent of factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\alpha }_{2}}\,\!}**
is one because the exponent of factor

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\ & = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\ & = & {{q}_{1}}+{{q}_{2}} \end{align}\,\!}**

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (1)\,\!}**
, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L=0\,\!}**
represents block 2 and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L=1\,\!}**
represents block 1. In order to decide which block the treatment **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{q}_{1}}=0\,\!}**
. Similarly, since factor **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{q}_{2}}=0\,\!}**
. Therefore:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & L= & {{q}_{1}}+{{q}_{2}} \\ & = & 0+0=0\text{ (mod 2)} \end{align}\,\!}**

Note that the value of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L\,\!}**
used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L\,\!}**
, treatment

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\ & a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\ & b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\ & ab: & \text{ }L=1+1=2=0\text{ (mod 2)} \end{align}\,\!}**

Therefore, to confound the interaction **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L=0\,\!}**
) should be assigned to block 2 and treatment combinations **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L=1\,\!}**
) should be assigned to block 1.

#### Example: Two Level Factorial Design with Two Blocks

This example illustrates how treatments can be allocated to two blocks for an unreplicated **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{4}\,\!}**
design to investigate the four factors affecting the defects in automobile vinyl panels discussed in Normal Probability Plot of Effects. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ABCD\,\!}**
was not significant and decided to allocate treatments to the two operators so that the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ABCD\,\!}**
interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.

The defining contrast for the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{4}\,\!}**
design where the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ABCD\,\!}**
interaction is confounded with the blocks is:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!}**

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L=0\,\!}**
represents block 2 and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L=1\,\!}**
represents block 1. Then the value of the defining contrast for treatment

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!}**

Therefore, treatment

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!}**

Therefore,

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2^k}**
designs have to be used to identify significant effects.

### Unreplicated 2^{k} Designs in 2^{p} Blocks

A single replicate of the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{p}\,\!}**
blocks where **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p<k\,\!}**
. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2-1=1)\,\!}**
effect is confounded with the blocks. If four blocks are used, then three (**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4-1=3\,\!}**
) effects are confounded with the blocks and so on. For example an unreplicated **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{L}_{1}}\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{L}_{2}}\,\!}**
. Let **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle BD\,\!}**
be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\ & {{L}_{2}}= & {{q}_{2}}+{{q}_{4}} \end{align}\,\!}**

Based on the values of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{L}_{1}}\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{L}_{2}},\,\!}**
the treatments can be assigned to the four blocks as follows:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\ {{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\ {} & {} & {} & {} & {} & {} & {} \\ \left[ \begin{matrix} (1) \\ ac \\ bd \\ abcd \\ \end{matrix} \right] & {} & \left[ \begin{matrix} a \\ c \\ abd \\ bcd \\ \end{matrix} \right] & {} & \left[ \begin{matrix} b \\ abc \\ d \\ acd \\ \end{matrix} \right] & {} & \left[ \begin{matrix} ab \\ bc \\ ad \\ cd \\ \end{matrix} \right] \\ \end{matrix}\,\!}**

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle BD\,\!}**
, the third effect confounded with the block effect is their generalized interaction, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (AC)(BD)=ABCD\,\!}**
.
In general, when an unreplicated **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{p}\,\!}**
blocks, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!}**
). **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {2}^{p}\,\!}**
blocks can then be assigned the treatments using the **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{2}^{p}}-(p+1)\,\!}**
effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the

#### Example: 2 Level Factorial Design with Four Blocks

This example illustrates how a DOE folio obtains the sum of squares when treatments for an unreplicated **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle BD,\,\!}**
with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle BD\,\!}**
and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2^k}**
designs have to be used to identify significant effects.

## Variability Analysis

For replicated two level factorial experiments, the DOE folio provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\ & = & 0.6779+0.2491{{x}_{1}}{{x}_{3}} \end{align}\,\!}**

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{x}_{1}}\,\!}**
should be **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{x}_{3}}\,\!}**
should be **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{x}_{1}}\,\!}**
should be