Final - Chapters

Chapter 9: Design of Experiments and Analysis of Variance 

 9.1 - Elements of a Designed Experiment 

  • The response is the variable of interest in the experiment. This is also refered to as the dependent variable. 

  • True or false? The response variable is quanitative in nature? 

  • Factors are the variables that effect the response/dependent variable. 

    • Quanitative factors are measured numerically 

    • Qualitative factors are measured are not on a naturally occuring numerical scale 

  • Factors are the independent variable 

  • Factor levels are the values of the factor used in the experiment 

  • The treatments of an experiment are the factor-level combinations used. 

    • Ex: the effect of the factor Gender on GMAT scores, then the treatments of the experiment are the two levels of gender - male and female 

  • An experimental unit is the object on which the response and factors are observed or measured 

  • A designed experiment is one where the analyst controls the treatments and the method of assigning the experimental units to each treatment 

  • An observational study is one which the analyst simply observes the treatments and the response 

  • What is the objective of desiging a study? 

    • Maximize the amount of information obtained about the relationship between the treatments and the response 

 9.2 - The Completely Randomized Design: Single Factor 

  • Completely randomized deisgn - a design in which the experimental units are randomly assigned to the k treatments or in which independent random samples of experimental units are selected for each treatment 

  • What is a balanced design? 

    • When an equal number of experimental units are assigned to each treatment 

  • Ho = u1 = u2 = uk 

  • Ha = at least two of the k treatment means differ 

  • How is the variation between the treament means calculated? 

    • Sum of Squares for Treatment (SST) - sum(treatment mean - overall mean)². 

  • How is the sampling variability within the treatments calculated? 

    • Sum of Squares for Error (SSE) - sum(response measurement - corresponding treatment mean)² 

  • What is the Mean Square for Treatments? 

    • MST = SST/(k-1) 

  • What is the Mean Square for Error? 

    • MSE = SST/(n-k) 

  • F= MST/MSE 

  • What must be assumed?

    • random sampling

    • normal distribution

9.3 - Multiple Comparisons of Means

  • How do we know which means are different?

    • we use the pairwise comparisons of treament means

      • c = k(k-1)/2

  • In this case, alpha is called the experimentwise error rate

  • Type I for comparison = comparisonwise eroor rate (CER)

  • Guidelines for Selecting a Multiple Comparsisons Method in ANOVA

    • Tukey - Equal - Pairwise

    • Bonferroni - Equal or Unequal - Pairwise or general contrasts

    • Scheffe - equal or unequal - general contrasts

Tukey method has the smallest width (most confidence)

Bonferroni has the smaller width (less confidence)

Scheffe has the smallest width (least confidence)

9.4 - The Randomized Block Design 

  • Randomized block design uses experimental units that are matched sets (blocks) 

  • What is the procedure? 

    • Matched sets of experimental units called blocks are formed, each block consists of k experimental units (k is the number of treatments). 

    • One experimental unit from each block is randomly assigned to each treatment, resulting in a total of n = bk responses 

  • Use ANOVA test to compare k treatment means of randomized block designs

    • Conditions Required: 

      • The b blocks are randomly selected, and all k treatments are applied 

      • The distributions of observations corresponding to all bk block-treatment combinations are approximately normal 

      • The bk block-treatment distributions have equal variations 

  • If the treatment means are not equal, we then use the multiple comparrisons proceedure: 

    • c = k(k-1)/2 

  • If the treatment means differ, use Tukey or Bonferroni to find pairs of means 

9.5 - Factorial Experiments: Two Factors 

  • A complete factorial experiment is one in which every factor-level combination is employed 

    • the number of treatments in the experiment equals the total number of factor-level combinations 

  • The factorial experiment is often referred to as a two-way classification 

  • Factor interation is used to test whether the factors combine to affect the response 

  • Factor main effect components are used to determine whether the factors separately affect the response 

  • Steps: 

Chapter 10: Categorial Data Analysis

 10.1 - Categorial Data and the Multinomial Experiment

  • What are the properties of the multinomial experiment?

    • The experiment consists of n identical trials

    • There are k possible outcomes to each trial. These outcomes are called classes, categories, or cells

    • The probabilities of k outcome, denoted by p1 + p2…… pk

    • The trials are independent

    • The random variables of interest are the cell counts

10.2 - Testing Category Probabilities: One-Way Table

10.3 - Testing Category Probabilities: Two-Way Contigency Table 

  • One-way table because a single variable is used

  • Chi-Square Test measures the degree of disagreement between the data and the null hypothesis

  • What are the conditions required for a valid chi square test?

    • A multinomial experiment has been conducted. This is generally by taking a random sample.

    • The sample size must be large