Select the method or formula of your choice.

The calculations for the mean square for the factors, interaction, and error follow:

Term | Description |
---|---|

MS | Mean Square |

SS | Sum of Squares |

DF | Degrees of Freedom |

The sum of squared distances. SS Total is the total variation in the data. SS (A) and SS (B) are the amount of variation of the estimated factor level mean around the overall mean. These statistics are also known as the sum of squares for factor A or factor B. SS Error is the amount of variation of the observations from their fitted values. The calculations follow:

- SS (A) = nb Σ
_{i}(y̅_{i..}− y̅ ...)^{2} - SS (B) = na S
_{j}(y̅_{.j.}− y̅ ... )^{2} - SS (AB) = SS Total − SS Error − SS (A) − SS(B)
- SS Error = S
_{i}Σ_{j}Σ_{k}(y_{ijk}− y̅_{ij.})^{2} - SS Total = Σ
_{i}Σ_{j}Σ_{k}(y_{ijk}− y̅...)^{2}

Term | Description |
---|---|

a | number of levels in factor A |

b | number of levels in factor B |

n | total number of trials |

y_{i..} | mean of the i ^{th} factor level of factor A |

y... | overall mean of all observations |

y_{.j.} | mean of the j ^{th} factor level of factor B |

y_{ij.} | mean of observations at the i ^{th} level of factor A and the j ^{th} level of factor B |

For a model with factors A and B, the degrees of freedom associated with each sum of squares follow:

- DF (A) =
*a*− 1 - DF (B) =
*b*− 1 - DF (AB) = (
*a*− 1)(*b*− 1) - With no interaction in the model, DF Error = (
*n*− 1) − (*a*− 1) − (*b*− 1) - With the interaction in the model, DF Error = (
*n*− 1) − (*a*− 1) − (*b*− 1) − (*a*− 1)(*b*− 1) - Total =
*n*− 1

Term | Description |
---|---|

a | number of levels in factor A |

b | number of levels in factor B |

n | total number of observations |

The fitted means are least squares estimates. For a factor level, the least squares mean is the sum of the constant coefficient and the coefficient for the factor level.

For a combination of factor levels in an interaction term, the least squares mean is the same as the fitted value.

The equation that defines the vector of estimated coefficients is as follows:

In the case of balanced data, the fitted means are equivalent to means in the data. For a factor level, the fitted mean is as follows:

For a combination of levels in an interaction term, the fitted mean is the same as the fitted value.

Term | Description |
---|---|

constant coefficient | |

coefficient for the i^{th} level of a factor | |

coefficient for the j^{th} level of the second factor | |

X | design matrix |

X' | transpose of the design matrix |

(X'X)^{−1} | inverse of the X'X matrix |

Y | vector of response values |

y_{ij} | i^{th} value at the j^{th} level of the factor |

n_{i} | sample size for the j^{th} level of the factor |

y_{ijk} | i^{th} value at the j^{th} level of the first factor and the k^{th} level of the second factor |

n_{jk} | sample size for the j^{th} level of the first factor and the k^{th} level of the second factor |

The fitted values are least squares estimates.

The equation that defines the vector of estimated coefficients is as follows:

In the case of balanced data, the fitted values are equivalent to means in the data. If the model has no interaction term, the fitted value is as follows:

If the model has an interaction term, the fitted value is the cell mean, or the mean of observations at the *i*^{th} level of factor A and the *j *^{th} level of factor B.

Term | Description |
---|---|

X | design matrix |

X' | transpose of the design matrix |

(X'X)^{−1} | inverse of the X'X matrix |

Y | vector of response values |

mean of the observations at the i^{th} level of factor A | |

mean of the observations at the j^{th} level of factor B | |

mean of all of the observations | |

mean of the observations at the i^{th} level of factor A and the j^{th} level of factor B |

The F statistic depends on the term in the test. For factor A, the F-statistic is as follows:

For factor B, the F-statistic is as follows:

For the interaction between factor A and factor B, the F-statistic is as follows:

When the lack-of-fit test appears, the F-statistic is as follows:

The numerator and denominator degrees of freedom correspond to the degrees of freedom for the mean square.

Term | Description |
---|---|

MS | Mean Square |

The pooled standard deviation is equivalent to S, which is displayed in the output. The formula follows:

Term | Description |
---|---|

MS | Mean Square |

The degrees of freedom for the F statistic that you use to calculate the p-value depend on the term that is in the test.

When you test a term, the denominator degrees of freedom are always the degrees of freedom for error. The degrees of freedom for error depend on whether the interaction term is in the model or not.

- With no interaction in the model, DF Error = (
*n*− 1) − (*a*− 1) − (*b*− 1) - With the interaction in the model, DF Error = (
*n*− 1) − (*a*− 1) − (*b*− 1) − (*a*− 1)(*b*− 1)

When you test a term, the numerator degrees of freedom depend on the term.

- For F(
*A*), the degrees of freedom for the numerator are*a*− 1 - For F(
*B*), the degrees of freedom for the numerator are*b*− 1 - For F(
*AB*), the degrees of freedom for the numerator are (*a*− 1)(*b*− 1)

For the lack-of-fit test, the degrees of freedom follow:

- Denominator DF =
*n*−*c* - Numerator DF =
*c*−*p*

1 − P(*F* ≤ *f _{j}*)

Term | Description |
---|---|

a | number of levels in factor A |

b | number of levels in factor B |

n | total number of observations |

c | number of unique combinations of factor levels |

p | number of terms in the model |

P(F ≤ f)_{j} | cumulative distribution function for the F distribution |

f_{j} | f statistic for the test |

Term | Description |
---|---|

y_{ijk} | k^{th} response value for the i^{th} level of factor A and the j^{th} level of factor B |

k^{th} fitted value for the i^{th} level of factor A and the j^{th} level of factor B | |

mean of the fitted values for the i^{th} level of factor A and the j^{th} level of factor B |

R^{2} is also known as the coefficient of determination.

Term | Description |
---|---|

y _{i} | i ^{th} observed response value |

mean response | |

i ^{th} fitted response |

Accounts for the number of predictors in your model and is useful for comparing models with different numbers of predictors.

Term | Description |
---|---|

y _{i} | i ^{th} observed response value |

i ^{th} fitted response | |

mean response | |

n | number of observations |

p | number of model parameters |

While the calculations for R^{2}(pred) can produce negative values, Minitab displays zero for these cases.

Term | Description |
---|---|

y _{i} | i ^{th} observed response value |

mean response | |

n | number of observations |

e _{i} | i ^{th} residual |

h _{i} | i ^{th} diagonal element of X(X'X)^{–1}X' |

X | design matrix |

Term | Description |
---|---|

MSE | mean square error |

The vector **x**_{0} defines the factor levels for a fitted mean in the same terms as the design matrix. The vector has 1 for the constant coefficient, the combination of 1, 0, and -1 that defines the factor levels for the term, and 0 for any factor levels that are not in the term. For the highest-level interaction in the model, all of the elements in the vector define factor levels and the standard error of the mean equals the standard error of the fitted value.

Term | Description |
---|---|

s^{2} | mean square error |

X | design matrix |

X' | transpose of the design matrix |

(X'X)^{−1} | inverse of the X'X matrix |

x_{0} | vector that defines the factor levels for the fitted mean |

x'_{0} | transpose of the vector that defines the factor levels for the fitted mean |

Standardized residuals are also called "internally Studentized residuals."

Term | Description |
---|---|

e _{i} | i ^{th} residual |

h _{i} | i ^{th} diagonal element of X(X'X)^{–1}X' |

s ^{2} | mean square error |

X | design matrix |

X' | transpose of the design matrix |