# DOE Glossary

**Alias**

Two or more effects are said to be aliased in an experiment if these effects cannot be distinguished from each other. This happens when the columns of the design matrix corresponding to these effects are identical. As a result, the aliased effects are estimated by the same linear combination of observations instead of each effect being estimated by a unique combination.

**ANOVA**

ANOVA is the acronym for Analysis of Variance. It refers to the procedure of splitting the variability of a data set to conduct various significance tests.

**ANOVA Model**

The regression model where all factors are treated as qualitative factors. ANOVA models are used in the analysis of experiments to identify significant factors by investigating each level of the factors individually.

**Balanced Design**

An experiment in which equal number of observations are taken for each treatment.

**Blocking**

Separation of experiment runs based on the levels of a nuisance factor. Blocking is used to deal with known nuisance factors. You should block what you can and randomize what you cannot. *See also* Nuisance Factors, Randomization.

**Center Point**

The experiment run that corresponds to the mid-level of all the factor ranges.

**Coded Values**

The factor values that are such that the upper limit of the investigated range of the factor becomes +1 and the lower limit becomes -1. Using coded values makes the experiments with all factors at two levels orthogonal.

**Confidence Interval**

A closed interval where a certain percentage of the population is likely to lie. For example, a 90% confidence interval with a lower limit of *A* and an upper limit of *B* implies that 90% of the population lies between the values of *A* and *B*.

**Confounding**

Confounding occurs in a design when certain effects cannot be distinguished from the block effect. This happens when full factorial designs are run using incomplete blocks. In such designs the same linear combination of observations estimates the block effect and the confounded effects. *See also* Incomplete Blocks.

**Contrast**

Any linear combination of two or more factor level means such that the coefficients in the combination add up to zero. The difference between the means at any two levels of a factor is an example of a contrast.

**Control Factors**

The factors affecting the response that are easily manipulated and set by the operator. *See also* Noise Factors.

**Cross Array Design**

The experiment design in which every treatment of the inner array is replicated for each run of the outer array. *See also* Inner Array, Outer Array.

**Curvature Test**

The test that investigates if the relation between the response and the factors is linear by using center points. *See also* Center Point.

**Defining Relation**

For two level fractional factorial experiments, the equation that is used to obtain the fraction from the full factorial experiment. The equation shows which of the columns of the design matrix in the fraction are identical to the first column. For example, the defining relation *I*=*ABC* can be used to obtain a half-fraction of the two level full factorial experiment with three factors *A*, *B* and *C*. The effects used in the equation are called the *generators* or *words*.

**Degrees of Freedom**

The number of independent observations made in excess of the unknowns.

**Design Matrix**

The matrix whose columns correspond to the levels of the variables (and their interactions) at which observations are recorded.

**Design Resolution**

The number of factors in the smallest word in a defining relation. Design resolution indicates the degree of aliasing in a fractional factorial design. *See also* Defining Relation, Word.

**Error**

The natural variations that occur in a process, even when all the factors are maintained at the same level. *See also* Residual.

**Error Sum of Squares**

The variation in the data not captured by the model. The error sum of squares is also called the residual sum of squares. *See also* Model Sum of Squares, Total Sum of Squares.

**Extra Sum of Squares**

The increase in the model sum of squares when a term is added to the model.

**Factorial Experiment**

The experiment in which all combinations of the factor levels are run.

**Fractional Factorial Experiment**

The experiment where only a fraction of the combinations of the factor levels are run.

**Factor**

The entity whose effect on the response is investigated in the experiment.

**Fitted Value**

The estimate of an observation obtained using the model that has been fit to all the observations.

**Fixed Effects Model**

The ANOVA model used in the experiments where only a limited number of the factor levels are of interest to the experimenter. *See also* Random Effects Model.

**Full Model**

The model that includes all the main effects and their interactions. In Weibull++ DOE folios, a full model is the model that contains all the effects that are specified by the user. *See also* Reduced Model.

**Generator**

See Word.

**Hierarchical Model**

In Weibull++ DOE folios, a model is said to be hierarchical if, corresponding to every interaction, the main effects of the related factors are included in the model.

**Incomplete Blocks**

Blocks that do not contain all the treatments of a factorial experiment.

**Inner Array**

The experiment design used to investigate the control factors under Taguchi's philosophy to design a robust system. *See also* Robust System, Outer Array, Cross Array.

**Interactions**

Interaction between factors means that the effect produced by a change in a factor on the response depends on the level of the other factor(s).

**Lack-of-Fit Sum of Squares**

The portion of the error sum of squares that represents variation in the data not captured because of using a reduced model. *See also* Reduced Model, Pure Error Sum of Squares.

**Least Squares Means**

The predicted mean response value for a given factor level while the remaining factors in the model are set to the coded value of zero.

**Level**

The setting of a factor used in the experiment.

**Main Effect**

The change in the response due to a change in the level of a factor.

**Mean Square**

The sum of squares divided by the respective degrees of freedom.

**Model Sum of Squares**

The portion of the total variability in the data that is explained by the model. *See also* Error Sum of Squares, Total Sum of Squares.

**Multicollinearity**

A model with strong dependencies between the independent variables is said to have multicollinearity.

**New Observations**

Observations that are not part of the data set used to fit the model.

**Noise Factors**

Those nuisance factors that vary uncontrollably or naturally and can only be controlled for experimental purposes. For example, ambient temperature, atmospheric pressure and humidity are examples of noise factors.

**Nuisance Factors**

Factors that have an effect on the response but are not of primary interest to the investigator.

**Orthogonal Array**

An array in which all the columns are orthogonal to each other. Two columns are said to be orthogonal if the sum of the terms resulting from the product of the columns is zero.

**Orthogonal Design**

An experiment design is orthogonal if the corresponding design matrix is such that the sum of the terms resulting from the product of any two columns is zero. In orthogonal designs the analysis of an effect does not depend on what other effects are included in the model.

**Outer Array**

The experiment design used to investigate noise factors under Taguchi's philosophy to design a robust system. *See also* Robust System, Inner Array, Cross Array.

**Partial Sum of Squares**

The type of extra sum of squares that is calculated assuming that all terms other than the given term are included in the model. The partial sum of squares is also referred to as the adjusted sum of squares. *See also* Extra Sum of Squares, Sequential Sum of Squares.

**Prediction Interval**

The confidence interval on new observations.

**Pure Error Sum of Squares**

The portion of the error sum of squares that represents variation due to replicates. *See also* Lack-of-Fit Sum of Squares.

**Qualitative Factor**

The factor where the levels represent different categories and no numerical ordering is implied. These factors are also called categorical factors.

**Random Effects Model**

The ANOVA model used in the experiments where the factor levels to be investigated are randomly selected from a large or infinite population. *See also* Fixed Effects Model.

**Randomization**

Conducting experiment runs in a random order to cancel out the effect of unknown nuisance factors. *See also* Blocking.

**Randomized Complete Block Design**

An experiment design where each block contains one replicate of the experiment and runs within the block are subjected to randomization.

**Reduced Model**

A model that does not contain all the main effects and interactions. In Weibull++ DOE folios, a reduced model is the model that does not contain all the effects specified by the user. *See also* Full Model.

**Regression Model**

A model that attempts to explain the relationship between two or more variables.

**Repeated Runs**

Experiment runs corresponding to the same treatment that are conducted at the same time.

**Replicated Runs**

Experiment runs corresponding to the same treatment that are conducted in a random order.

**Residual**

An estimate of error which is obtained by calculating the difference between an observation and the corresponding fitted value. *See also* Error, Fitted Value.

**Residual Sum of Squares**

See Error Sum of Squares.

**Response**

The quantity that is investigated in an experiment to see which of the factors affect it.

**Robust System**

A system that is insensitive to the effects of noise factors.

**Rotatable Design**

A design is rotatable if the variance of the predicted response at any point depends only on the distance of the point from the design center point.

**Screening Designs**

Experiments that use only a few runs to filter out important main effects and lower order interactions by assuming that higher order interactions are unimportant.

**Sequential Sum of Squares**

The type of extra sum of squares that is calculated assuming that all terms preceding the given term are included in the model. *See also* Extra Sum of Squares, Partial Sum of Squares.

**Signal to Noise Ratio**

The ratios defined by Taguchi to measure variation in the response caused by the noise factors.

**Standard Order**

The order of the treatments such that factors are introduced one by one with each new factor being combined with the preceding terms.

**Sum of Squares**

The quantity that is used to measure either a part or all of the variation in a data set.

**Total Sum of Squares**

The sum of squares that represent all of the variation in a data set.

**Transformation**

The mathematical function that makes the data follow a given characteristic. In the analysis of experiments transformation is used on the response data to make it follow the normal distribution.

**Treatment**

The levels of a factor in a single factor experiment are also referred to as treatments. In experiments with many factors a combination of the levels of the factors is referred to as a treatment.

**Word**

The effect used in the defining relation. For example, for the defining relation *I*=*ABC*, the word is *ABC*.