Stat Today: Transformation of the response; Latin-squares

Transformations (Section 2.5) Often one will perform a residual analysis to verify modeling assumptions…and at least one assumption fails A defect that can frequently arise in non-constant variance This can occur, for example, when the data follow a non-normal, skewed distribution The F-test in ANOVA is only slightly violated In such cases, a variance stabalizing transformation may be applied

Transformations Several transformations may be attempted: –Y * =

Transformations Analyze the data on the Y * scale, choosing the transformation where: –The simplest model results, –There are no patterns in the residuals –One can interpret the transformation

Example An engineer wishes to study the impact of 4 factors on the rate of advance of a drill. Each of the 4 factors (labeled A-D) were studied at 2 levels

Example Would like to fit an N-way ANOVA to these data (main effects and 2- factor interactions only) Model:

Example

A New Example A scientist wishes to investigate the effect of 5 different ingredients (A- E) on the reaction time of a chemical process The scientist has enough resources to perform 25 trials Each batch of raw material is only large enough to permit 5 runs to be made Each run takes about 1.5 hours, so only 5 runs can be performed in a day

Example How can we run the experiment?

Two Blocking Variables ; 1 Factor Can set up an experiment to remove the effect of 2 blocking variables (e.g., season and time of day) Experiment is an example of a 5x5 Latin Squares Design

Latin Squares Design Situation: –Have 2 blocking factors - one for rows and one for columns –Have 1 experimental factor –Each factor has k levels –Design is arranged so that each level of the experimental factors appears exactly one time in each row and each column –The levels of the two blocking factors are assigned at random to the columns and rows

Model: i, j=1,2,…,k l indicates the index for the Latin letter in the (i,j) th cell The triplet (i,j,l) takes on values

Notes No interaction, since interactions cannot be estimated in an un- replicated experiment Usual assumptions apply

ANOVA Decomposition

ANOVA Table

Hypotheses

Multiple Comparisons

Example

Useful Plots

Example