Lecture 32

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Example of Experimental Design

A study is currently being conducted at UCONN to quantify the amount of safety improvement which results from specific types of highway reconstruction. In Phase I of the project two types of reconstruction were studied: offset intersection realignment and curve straighening. In addition, the effect of landuse type was studied. To two landuse types considered where rural and urban.

This is typical of designed experiment - the element of such an experiment are as follows

Response (or Dependent) Variable - the variable of interested that is measured in the experiment

What is the response variable for the traffic safety project?

Factors - these are the variables whose effect on the response variable is being investigated. The factors may be quantitative or qualitative variables.

What are the factors for the traffic safety project?

Factor Levels - values of the factors that are used in the experiment

What are the factor levels for the traffic safety project?

Treatments - factor-level combinations for the experiment

How many treatments for the traffic safety project?

What are the treatments for the traffic safety project?

Experimental Units - object on which the response and factors are observed or measured

In an experiment such as the traffic study, the analyst may use a designed experiment or an observational experiment

Designed Experiment - analyst specifies the treatments and control the method of assigning experimental units to each treatment

Observational Experiment - analyst simiply collect a sample of experimental units and observes the treatments and response. There is no control over levels of treatments or which units is assigned to which treatment

Which type of experiment would work best for traffic safety study?

Can be have a complete designed experiment for the traffic safety study?

Lecture 33

A completely randomized design in which independent random samples of experimental units re selected for each treatment

This type of experimental design is either practical or desireable for all types of studies, however, many studies use this design

Today we looking at the method of analysis that is used for this study. One of the objectives for this study is to determine if the different treatments have different effect on the response. In other words, we are comparing the treatment means.

 

We can determine if the treatment means are different by testing the following hypothesis

Ho: pop mean treatment 1 = pop mean treatment 2 = pop mean treatment 3 = … = pop mean treatment p

Ha: at least two of the p treatment means differ

Test for this hypothesis is based on comparing variances (hence, analysis of variance)

Basically we compare

Variability between treatments

versus

Variablity within treatments

Variability between treatments

This is called the Sum of Squares for Treatments (SST)

SST is the sum of the square for all the treatments of the difference between the treatment mean and the overall mean

Variability within treatments

This is called the Sum of Squares for Errors (SSE)

SSE is the sum of the square for all the treatments of the difference between the treatment mean and each response measure in that treatment

 

Mean Squares

The mean squares is the sum of squares divided by the associated degrees of freedom (df)

MS = SS/df

Degrees of Freedom for SST is (p-1)

Therefore, MST = SST/(p-1)

Degrees of Freedom for SSE is (n-p)

Therefore, MSE = SSE/(n-p)

Hypothesis

Ho: pop mean treatment 1 = pop mean treatment 2 = pop mean treatment 3 = … = pop mean treatment p

Ha: at least two of the p treatment means differ

Test Statistics

F = MST/MSE

If F is large the null hypothesis is rejected

How large is large? This is based on the F-distribution

The value of comparison is F(p-1, n-p)

 

If we have only two treatments we can use a t-test for the hypothesis. This is exactly equivalent to the F-test.

Hypothesis

Ho: pop mean treatment 1 = pop mean treatment 2

Ha: at least two of the p treatment means differ

Test Statistics

t=(sample mean 1 - sample mean 2)/sqrt(mse(1/n1 + 1/n2)

The t-test is a two sided test

If t is very large or very small the null hypothesis is rejected

The value for comparison in this case is t(n-p)

These procedures (both t and F for comparing means) are only valid under the following conditions

• The probability distribution of the population of responses for each treatment should be normal

• The probability distribution of the population of responsed for each treatment should have equal variance

• The samples for experimental units selected for each treatment should be random and independent

Lecture 34