Statistics Lecture 18

zLast Day: ANOVA Example, Paired Comparisons zToday: Re-visit boys shoes...Randomized Block Design

Example (Boys Shoes) zCompany ran an experiment to determine if a new synthetic material is better than the existing one used for making the soles of boys' shoes zExperiment was run to see if the new, cheaper sole wears at the same rate at which the soles wear out

Example (Boys Shoes) z10 boys were selected at random zEach boy was given a pair of shoes zEach pair had 1 shoe with the old sole (Sole A) and 1 shoe with the new sole (sole B) zFor each pair of shoes, the sole type was randomly assigned to the right or left foot

Data

zCan we use a 2-sample t-test or ANOVA here? zWould the 2-sample t-test or ANOVA detect a significant difference?

Paired or Matched Pairs T-test zSituation: yTwo measurements made on same experimental unit yCompute difference (say B-A) in observation on the same experimental unit yAnalyze differences using a 1-sample t-test zBecause we analyze the differences using a 1-sample t-test, what must we assume about the difference?

Data

Analyzing the Data

zJust looked at comparing means for two treatments applied to the same experimental unit (see boys shoes example) zUsed a matched pairs T-test and analyzed the differences to see if there was a significant difference in the treatment means zWhen more than 2 treatments are applied to the same experimental unit, the experiment is called a randomized block experiment

Example zAn experiment was performed to investigate the impact of soil salinity on the growth of salt marsh plants (C. Schwarz, 2001) zPlots of land at 4 agriculture field stations were used to grow plants in this environment zSix different amounts of salt (in ppm) are to be investigated zThe plots of land were divided into 6 smaller plots zEach of the 6 smaller plots were treated with a different amount of salt and the bio-mass at the end of several months recorded

Data

zThe application of the 6 treatments to the smaller plots are done randomly zLike the Boys Shoes Example, each experimental unit has received more than 1 treatment zHere each unit receives 6 treatments

Plot of Bio-mass for Each Treatment

Plot of Bio-mass for each plot

Observations zNotice that the 4 th plot gives smaller results than the other plots zDue to a block (plot effect) zSimilar to the way a boy wears his shoes zAre the observations independent?

Randomized Block Design zSituation: yHave k treatments yHave b blocks yEach of the k treatments appears in each of the b blocks yThe treatments within block are assigned to the within block units in random order

Structure of Data zHave k treatments in b blocks zDenote i th treatment from the j th block as y ij

Model: zModel for comparing k treatments from a randomized block design: y for i =1, 2, …, k and j =1, 2, …, b ywhere is the overall mean, and y is the i th treatment effect y is the j th block effect ye ij has a distribution zWant to test: y

ANOVA Table for the Bio-Mass Example

What are the F-Tests?

Why do this?

zGeneral Rule: Block what you can, randomize what you cannot