1. STAT 6304 Final Project
Fall, 2016

2. Statistical Tools vs. Variable Types
Response (output)
Predictor (input)
Numerical
Categorical/Mixed
Simple and Multiple Regression
Analysis of Variance (ANOVA)
Analysis of Covariance (ANCOVA)
Categorical
Categorical data analysis

3. Project Outline
For detail, see
For analysis part, you should
Explore Data. What do the data show? By tables, graphs and descriptive statistics
Build up models: (One-way) ANOVA model or SLR model or both
For categorical data, conduct analysis in Chapter 10
Explain how the explanatory variables (X) affect responses (Y) based on these models/analysis

4. More on One-way ANOVA Method
Use the battery data
Quick review of Lab5-R-one-way
Now we know the mean lifetimes of brands are not all the same. What next?

5. Multiple Comparison Procedures
Once we reject H0: ==...t in favor of H1: NOT all ’s are equal, we don’t yet know the way in which they’re not all equal, but simply that they’re not all the same. If there are 4 columns (levels), are all 4 ’s different? Are 3 the same and one different? If so, which one? etc.
multiple comparisons

6. These “more detailed” inquiries into the process are called MULTIPLE COMPARISON PROCEDURES.
Errors (Type I):
We set up “” as the significance level for a hypothesis test. Suppose we test 3 independent hypotheses, each at = .05; each test has type I error (rej H0 when it’s true) of However,
P(at least one type I error in the 3 tests)
= 1-P( accept all ) = 1 - (.95)3  .14
3, given true
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7. In other words, Probability is
In other words, Probability is .14 that at least one type one error is made. For 5 tests, prob = .23.
Question - Should we choose = .05, and suffer (for 5 tests) a .23 Experimentwise Error rate (“a” or aE)? OR
Should we choose/control the overall error rate, “a”, to be .05, and find the individual test  by 1 - (1-)5 = .05, (which gives us  = .011)?
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8. would be valid only if the tests are independent; often they’re not.
The formula
1 - (1-)5 = .05
would be valid only if the tests are independent; often they’re not.
[ e.g., 1=22= 3, 1= 3
IF accepted & rejected, isn’t it more likely that rejected? ] 1 2 3 1 2 3
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9. Error Rates
When the tests are not independent, it’s usually very difficult to arrive at the correct for an individual test so that a specified value results for the experimentwise error rate (or called family error rate).
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10. There are many multiple comparison procedures. We’ll cover only a few.
Pairwise Comparisons
Method 1: (Fisher Test) Do a series of pairwise t-tests (pooled t-tests), each with specified  value (for individual test).
This is called “Fisher’s LEAST SIGNIFICANT DIFFERENCE” (LSD). Not recommended!!
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11. Pairwise comparisons
Method 2: (Tukey Test) A procedure which controls the experimentwise error rate is “TUKEY’S HONESTLY SIGNIFICANT DIFFERENCE TEST ”. Recommended!!
Note that to avoid being too conservative, the significance level of Tukey test can be set bigger (10%), especially for more than 4 samples.
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13. Summarize the Results: Underline Diagram
Summarize the comparison results. (Lab5-R)
Now, sort the sample means in an ascending order. We will begin at the smallest one.
Compare the smallest and largest and check if they are significant (p-value < .10). No, mark a underline between them and Stop; Yes, continue to compare the smallest and the 2nd largest
Repeat step 2 for the 2nd smallest and so on.
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14. More on Linear Regression
Multiple Linear Regression
One response Y and many predictors X1, …, Xk
Regression Equation is
Y=b0+b1X1+b2X2+…bkXk
Model diagnostics are the same (see Lab 7- R):
Normality of residuals
Equal variance of residuals
Independence of residuals

15. U. S. State Public-School Expenditures (Anscombe dataset in R, Car library)
The Anscombe data frame has 51 rows and 4 columns. The observations are the U. S. states plus Washington, D. C. in 1970.
This data frame contains the following columns:
educationPer-capita education expenditures, dollars. (Y)
incomePer-capita income, dollars. (X1)
youngProportion under 18, per (X2)
urbanProportion urban, per (X3)

16. # import data library(car) data(Anscombe) education<-Anscombe$education income<-Anscombe$income young<-Anscombe$young urban<-Anscombe$urban # draw scatterplots plot(education,income) plot(education,young) plot(education,urban) # Multiple Linear Regression result<-lm(education~income+young+urban) summary(result) rs<-rstandard(result) fits<-fitted.values(result) # check for normality qqnorm(rs) qqline(rs,col=2) shapiro.test(rs) # check for equal variances: residual plots plot(fits,rs,main="residuals vs. fitted values") plot(income,rs) plot(young,rs) plot(urban,rs) # If we add income^2 into the model income2<-income^2 result2<-lm(education~income+income2+young+urban) summary(result2)