STAT 330 Lecture 36: Review of Course

Experimental Designs

You will need to be able to recognize, in a problem description, the following experimental or sampling designs. For each context i have prepared a list of techniques you are expected to be able to apply:

1. One simple random sample
   1. Tests of with either 1 sided or two sided alternatives or tests of (or ) against a one sided alternative.
      • Using normal tests if by some miracle is known.
      • Using t based tests when is estimated
      • Using normal tests when is estimated and n is large and t based P values are unavailable.

   2. Confidence intervals for the true value of based on the same distributional considerations as for the hypothesis tests.
   3. Confidence intervals for based on the distribution. Don't forget to distinguish between the variance and the standard deviation .
   4. Hypothesis tests for (or with or based on the distribution. You need to be aware that this test requires that the population distribution be normal, even in large samples.
   5. Power calculations. You should know how to compute making a normal approximation for a problem in which is known.
   6. Sample size calculations for one or two tailed hypothesis tests when you are given: specified Type I error rate , specified Type II error rate , specified null hypothesis value , specified discrepancy between the true value and the hypothesized value . (You need to know and you might be told that or you might be told and separately.

2. Replicate measurements under identical circumstances of a single quantity: This is treated the same way as a single sample problem. We use the model

   

   where the are independent and identically distributed from a population with mean 0 and variance . In this case we regard the measurements as a sample from an infinite population whose mean is and

   

   Techniques are all the same as the previous problem. There is one further sort of problem I could ask about. Measuring instruments can be calibrated by making a measurement where the true value is known (such as weighing an object whose weight is accurately known). Testing is the same as testing .

3. Two independent random samples
   1. Test the hypothesis (or with or ) with one or two tailed alternatives.
      • Using a pooled estimate of and a two sample t test if the sample sizes are small. There are degrees of freedom in this case.
      • Using

        

        to estimate the standard error of if the sample sizes are reasonably large and it appears that .

   2. Confidence intervals for with the same considerations for estimating the standard error of .
   3. Tests of the hypothesis based on the F distribution. You need to be aware that this test requires that the population distribution be normal, even in large samples. You need to know how to carry out a two tailed test as well as a one tailed test.
   4. Confidence intervals for again based on the F distribution.
   5. Sample size calculations based on normal theory. You should be aware that when the resulting sample size is small and is not really known you should use the graphs in the Appendix for power of the t test.

4. A sample of pairs

   This is the design we have analyzed using 3 tools:

   1. To test for equality of means use a paired comparisons t-test. You should know how to get confidence intervals for the difference in means and how to do a sample size and power calculation in this setting. (You use 1 sample ideas on the differences .)
   2. To predict Y from X use regression. (See the discussion below for what you need to know about regression.
   3. Correlation analysis. You need to be able to do tests of and get confidence intervals for based on Fisher's z-transform.

5. I samples: the one way layout
   1. ANOVA table: properties, degrees of freedom, sums of squares, mean squares and so on.
   2. Tests of .
   3. Model equation .
   4. Confidence intervals for :
      • One at a time: t intervals.
      • Simultaneous: Tukey.

   5. Diagnostic plots.

6. Two factor designs
   1. Completely randomized. Experimental units are assigned at random to levels of each of two factors.
   2. Randomized (Complete) Blocks. One of the factors is a blocking factor; the level of such a factor to which an experimental unit belongs is generally beyond the control of the experimenter.

7. Two factor designs with replicates
   1. ANOVA table has rows: Factor A, Factor B, Interaction, Error and Total.
   2. Know properties of ANOVA table.
   3. Test for interactions.
   4. If no interactions are detected test for main effects.
   5. Model equation: .
   6. One at a time (t) and simultaneous (Tukey) confidence intervals for .
   7. Diagnostic plots.

8. Two factor designs without replicates
   1. ANOVA table has rows: Factor A, Factor B, Error and Total.
   2. Know properties of ANOVA table.
   3. Test for main effects.
   4. Model equation: .
   5. One at a time (t) and simultaneous (Tukey) confidence intervals for .
   6. Diagnostic plots.

Coverage of course in textbook

1. Chapter 6: all except from foot of page 253 to middle of page 257.
   • Bias
   • Mean Squared Error
   • Standard Error
   • Estimated Standard Error
   • Method of Moments
   • Maximum likelihood estimation

2. Chapter 7: all . Confidence Intervals.
   • For proportions, rates and means in 1 sample.
   • With known and either normal data with small n or large n, using normal multipliers .
   • With unknown and either normal data with small n or large n, using estimated standard errors and t multipliers .
   • For or using the distribution of .

3. Chapter 8: all . Hypothesis tests.
   • Theory: level, power, probability of Type 1, 2 errors, P-values, fixed level tests.
   • Practice: all cases as above for confidence intervals.
   • Power calculations.
   • Sample size determination.

4. Chapter 9: all . Two sample problems.
   • Tests and confidence intervals for differences of two proportions, rates or means.
   • Power and sample size calculations.
   • F tests for equality of variances.
   • Standard errors for differences.
   • Pooled estimate of .
   • Paired comparisons analysis.

5. Chapter 10: omit power, sample size and random effects models
   • One way ANOVA.
   • Tests for .
   • One at a time t type confidence intervals for contrasts.
   • Simultaneous Tukey confidence intervals for differences .
   • Model equations.
6. Chapter 11: Sections 1 and 2 only
   • Two way ANOVA with and without replicates.
   • Tests for interactions and main effects.
   • Model equations.
   • Confidence intervals for .
   • t type confidence intervals of the form .

7. Chapter 12: All except coefficient of determination, Bonferroni intervals. Simple Linear Regression and Correlation.
   • Least squares, least squares estimates.
   • Tests, confidence intervals for slope (intercept).
   • Confidence intervals for .
   • Prediction intervals.
   • ANOVA table.
   • Tests, confidence intervals for .
   • Residual plots.

Final Exam Advice

• Open Book, 3 hours.
• You will need your text for tables.
• You will need a calculator.
• You may bring your notes and other books if you like.
• There will be a number of questions where I provide two or more SAS outputs which purport to analyze a certain experiment. You must pick the correct one use it to answer the question.
• You should use P-values whenever possible.
• There may be a question of the type where the SAS output is edited and you are asked to fill in the rest of the ANOVA table. If so you will have to do fixed level testing.
• The whole course is covered with emphasis on the second half.
• There will be 6 to 8 questions.