Experimental Design and Analysis of Variance

Review: chi square = we want to know whether a data set fits a certain distribution/independence model. We use the chi square distribution, then we check how far away the test statistic is from 0. As data set becomes farther away from what you expect to get, you get larger differences between expected model and actual model (you get a larger test statistic)

Components of ANOVA: Factor – independent variable. We want this variable to be qualitative. Classifications of the factor is called the treatments. (ex. Color of the light vs. response variable ie height of the plant. Light is qualitative, treatments are the kinds of lights such ash red, white, violet, green. In anova, the response variable must be quantitative. If not quantitative, then go back to chi square test)

When we design an experiment, the factors are controlled by you. But sometimes some factors are difficult to control, and if we want to do an experiment on that we will have to just look at observational data. Example of this kind of factor is the weather.

Regardless, usually to test whether a certain factor has an effect on a response variable, we do replication. We look at replicating the experiment on more units. The more the better. If we find differences between the growths (in the mongo seeds) we do not know if this is true for the whole population, so the more elements of sample we have the better.

Gasoline Mileage Case: Factor: Gas Type. Treatments: A B C Response Variable: Mileage (miles/gallon) -> we want to know whether the factor affects this We replicate the experiment 5 times.

Concept of Variation:
We can see a set of data in terms of its variation. Ex we look at 15 identical cars. And then we look at a certain variable, mileage. Would you expect to get the same mileage from all of those cars given that all of us did different things to it? But if none of the things we did had an effect, then we’d have the same mileage. If there’s a certain factor that affects the mileage, then there’d be differences. If we have more than just one variable (the things we did to the car), then we’ll have many kinds of differences.

In Anova, we look at the total variation and then we separate it into different categories. Ex we get total var. and then we look at the variation attributable to gas type. The rest of the variance becomes attributed to “unexplained variation” due to randomness, due to all the other factors that affect mileage. If gas type is the only one that affects mileage, then the whole of total variation can be attributable to gas type. We can say that gas type has no effect if when we look at the variation, only a small part can be attributed to gas type. As this part gets larger, then gas type has more effect. Compare the “between treatment” (effects of gas types on mileage) and “within treatment” (variance among the cars itself) variability. The F test looks at a standardized ratio between the two. F = BT/WT

Assumptions: 1. Constant variance. Homoscedasticity 2. Normal 3. Independence

One Way ANOVA:
Initial assumption: factor has no effect
Alpha: 1, 5, 10%
Calculate test statistic: f test = BT/WT

SSTO (sum of squares total variability)– total variation in the data set
SST (sum of squares treatment) – between treatment variability
SSE (sum of squares error) – within treatment variability (from all other factors, random factors)

Divide SST and SSE by df1, which is in p-1 and df2 which is n-p [p is the number of treatments]
MST = SST/df1
MSE = SSE/df2
MST and MSE are variances
F = MST/MSE