Chapter 13 ANOVA - sections 1-5

single-factor experiment

An experiment involving only one factor with k populations or treatments

completely randomized design

An experimental design in which the treatments are randomly assigned to the experimental units

MSE (within-group estimates)

Another estimate of σ^2 is called the mean square due to error - DO NOT calc by hand, use excel

response variable

Another word for the dependent variable of interest

factor

Another word for the independent variable of interest

Test Statistic for the Equality of k Population Means

F = MSTR/MSE The test statistic follows an F distribution with k - 1 degrees of freedom in the numerator and nT - k degrees of freedom in the denominator

Hypothesis

Ho : μ1 + μ2 + μ3 Ha : Not all population means are equal If Ho is rejected, we cannot conclude that all population means are different. Rejecting Ho means that at least two population means have different values.

multiple comparison procedures

Statistical procedures that can be used to conduct statistical comparisons between pairs of population means.

experimental units

The objects of interest in the experiment

partitioning

The process of allocating the total sum of squares and degrees of freedom to the various components.

MSTR (between group estimates)

When sample sizes are equal, one estimate of σ2 is called the mean square due to treatments, MSTR. based on the assumption that the null hypothesis is true - DO NOT calc by hand, use excel

treatment

different levels of a factor

p-value

is area in upper tail of F distribution to right of test statistic "= F.DIST.RT(F-stat, df N, df D)" If the p-value < α = .05, we reject the null hypothesis and conclude that the means for the 3 methods are not all equal.

estimate of σ2

is called the mean square due to treatments and is denoted MSTR

What are the two important principles of all experimental designs?

randomization and replication

If the means for the populations differ...

the stronger the evidence we have for the conclusion that the population means differ supports Ha

If the means for the populations are equal...

we would expect the three sample means to be close together. The closer the three sample means are to one another, the weaker the evidence we have for the conclusion that the population means differ. supports Ho

ANOVA

"Analysis of Variance" - check for the equality of k population means summary: the logic behind ANOVA is based on the development of two independent estimates of the common population variance σ^2. One estimate of σ^2 is based on the variability among the sample means themselves, and the other estimate of σ^2 is based on the variability of the data within each sample. By comparing these two estimates of σ^2, we will be able to determine whether the population means are equal. Ho: μ1 = μ2 = ... = μk H1: At least one mean is different where μj = mean of the jth population We assume that a simple random sample of size n has been selected from each of the k populations or treatments.

Fisher's LSD

- Fisher's LSD allows us to test for a significant difference between means of a pair of populations. - In Chemtech example, we've shown the means are not equal, but which pairs are significantly different from each other? denominator is sqroot(MSE x (1/n1+1/n2)) - use f statistic test # in =F.DIST.2T(#, degrees of freedom)

randomized block design

- In cases where a factor not under consideration accounts for significant variations in the data, the MSE (denominator of F) can become too large. Since the F statistic is the ratio of MSTR to MSE, overestimating the MSE causes the F value to be lower than it otherwise would (and therefore the p-value to be to high). In these cases it's possible to overlook a significant difference. - Randomized Block Design is an experimental design that helps to minimize extraneous sources of variation from the MSE. - A random sample is taken from the population. Each subject is a block and each block is given each treatment, in a random order.

Analysis of Variance and the Completely Randomized Design

- The calculations done in an ANOVA problem are often organized in an ANOVA table - useful when given incomplete information or partially summarized information

Rationale for ANOVA

- When Ho is true, MSTR is an unbiased estimator of σ2. If the k means are not equal (Ho is false), MSTR overestimates σ2. - MSE is based on variation within each treatment, so it does not depend on the truth of Ho. Thus MSE always provides an unbiased estimate of σ2. -Therefore we have two estimates for the same parameter (σ2). One is always unbiased (MSE), while the other (MSTR) is only unbiased if the H0 is true, otherwise it overestimates the parameter. We can compare these two estimates and if they are relatively close to each other, then we can assume the H0 is true. However if they are far apart from each other, it indicates to us the H0 is false. - the sampling distribution for the ratio of 2 independent estimates of σ2 follow an F distribution - if Ho is true, the ratio of MSTR/MSE follows an F distribution with numerator degrees of freedom equal to k-1, and denominator degrees of freedom equal to nT - k - If Ho is not true, the ratio will be larger (because MSTR will be overestimating σ2 while MSE will not) and will yield an extreme value on the F distribution

factorial experiment

- an experimental design that allows simultaneous conclusions about 2 or more factors - Main effect (factor A): Do the preparation programs differ in terms of effect on GMAT scores? - Main effect (factor B): Do the undergraduate colleges differ in terms of effect on GMAT scores? - Interaction effect (A and B): Do students in some colleges do better on one type of preparation program whereas other do better on a different program? We could also ask: Does which program is most effective depend on which college the student attends?

3 Assumptions of ANOVA

1. For each population, the response variable is normally distributed. - Implication: In the Chemitech experiment the number of units produced per week (response variable) must be normally distributed for each assembly method. 2. The variance of the response variable, denoted "σ^2", is the same for all of the populations. - Implication: In the Chemitech experiment, the variance of the number of units produced per week must be the same for each assembly method. 3. The observations must be independent. - Implication: In the Chemitech experiment, the number of units produced per week for each employee must be independent of the number of units produced per week for any other employee.