Transcript Chapter 10
Chapter 10 Design of Experiments and Analysis of Variance Elements of a Designed Experiment •Response variable Also called the dependent variable •Factors (quantitative and qualitative) Also called the independent variables •Factor Levels •Treatments •Experimental Unit 2 Elements of a Designed Experiment Designed vs. Observational Experiment •In a Designed Experiment, the analyst determines the treatments, methods of assigning units to treatments. •In an Observational Experiment, the analyst observes treatments and responses, but does not determine treatments •Many experiments are a mix of designed and observational 3 Elements of a Designed Experiment Single-Factor Experiment Population of Interest Sample Independent Variable Dependent Variable 4 Elements of a Designed Experiment Two-factor Experiment 5 The Completely Randomized Design Achieved when the samples of experimental units for each treatment are random and independent of each other Design is used to compare the treatment means: H0 : 1 2 ...k Ha : At least two of the treatment means differ 6 The Completely Randomized Design • • The hypotheses are tested by comparing the differences between the treatment means to the amount of sampling variability present Test statistic is calculated using measures of variability within treatment groups and measures of variability between treatment groups 7 The Completely Randomized Design Sum of Squares for Treatments (SST) Measure of the total variation between treatment means, with k treatments k 2 Calculated by SST n xi x i 1 i th n number of observations in i treatment group Where i xi mean of measurements in i th treatment group x overall mean of all measurements 8 The Completely Randomized Design Sum of Squares for Error (SSE) Measure of the variability around treatment means attributable to sampling error Calculated by n1 SSE x1 j x1 j 1 x 2 n2 j 1 2j x2 2 nk ... xkj xk j 1 2 After substitution, SSE can be rewritten as SSE n1 1 s n 2 1 s ... nk 1 s 2 1 2 2 2 k 9 The Completely Randomized Design Mean Square for Treatments (MST) Measure of the variability among treatment means SST MST k 1 Mean Square for Error (MSE) Measure of sampling variability within treatments SSE MSE nk 10 The Completely Randomized Design F-Statistic Ratio of MST to MSE MST F , with df (k 1, n k ) MSE Values of F close to 1 suggest that population means do not differ Values further away from 1 suggest variation among means exceeds that within means, supports Ha 11 The Completely Randomized Design Conditions Required for a Valid ANOVA FTest: Completely Randomized Design 1. Independent, randomly selected samples. 2. All sampled populations have distributions that approximate normal distribution 3. The k population variances are equal 12 The Completely Randomized Design A Format for an ANOVA summary table ANOVA Summary Table for a Completely Randomized Design df SS Treatments k 1 SST Error nk SSE Total n 1 SS Total Source MS SST k 1 SSE MSE nk MST F MST MSE 13 The Completely Randomized Design ANOVA summary table: an example from Minitab 14 The Completely Randomized Design Conducting an ANOVA for a Completely Randomized Design 1. 2. 3. 4. 5. Assure randomness of design, and independence, randomness of samples Check normality, equal variance assumptions Create ANOVA summary table Conduct multiple comparisons for pairs of means as necessary/desired If H0 not rejected, consider possible explanations, keeping in mind the possibility of a Type II error 15 Multiple Comparisons of Means •A significant F-test in an ANOVA tells you that the treatment means as a group are statistically different. •Does not tell you which pairs of means differ statistically from each other •With k treatment means, there are c different pairs of means that can be compared, with c calculated as k k 1 c 2 16 Multiple Comparisons of Means •Three widely used techniques for making multiple comparisons of a set of treatment means •In each technique, confidence intervals are constructed around differences between means to facilitate comparison of pairs of means •Selection of technique is based on experimental design and comparisons of interest •Most statistical analysis packages provide the analyst with a choice of the procedures used by the three techniques for calculating confidence intervals for differences between treatment means 17 Multiple Comparisons of Means Guidelines for Selecting a Multiple Comparisons Method in ANOVA Method Tukey Bonferroni Scheffe Treatment Sample Sizes Equal Equal or Unequal Equal or Unequal Types of Comparisons Pairwise Pairwise General Contrasts 18 The Randomized Block Design Two-step procedure for the Randomized Block Design: 1. Form b blocks (matched sets of experimental units) of k units, where k is the number of treatments. 2. Randomly assign one unit from each block to each treatment. (Total responses, n=bk) 19 The Randomized Block Design Partitioning Sum of Squares k SST b xTi x i 1 b SSB k x Bi x i 1 n 2 2 SS (Total ) xi x i 1 2 SSE SS (Total ) SST SSB 20 The Randomized Block Design Calculating Mean Squares Setting Hypotheses SST k 1 SSE MSE n b k 1 MST H 0 : 1 2 ... k H a : At least two treatment means differ Hypothesis Testing MST F MSE Rejection region: F > F, F based on (k-1), (n-b-k+1) degrees of freedom 21 The Randomized Block Design Conditions Required for a Valid ANOVA FTest: Randomized Block Design 1. The b blocks are randomly selected, all k treatments are randomly applied to each block 2. Distributions of all bk combinations are approximately normal 3. The bk distributions have equal variances 22 The Randomized Block Design Conducting an ANOVA for a Randomized Block Design 1. 2. 3. 4. 5. 6. Ensure design consists of blocks, random assignment of treatments to units in block Check normality, equal variance assumptions Create ANOVA summary table Conduct multiple comparisons for pairs of means as necessary/desired If H0 not rejected, consider possible explanations, keeping in mind the possibility of a Type II error If desired, conduct test of H0 that block means are equal 23 Factorial Experiments Complete Factorial Experiment •Every factor-level combination is utilized Schematic Layout of Two-Factor Factorial Experiment … … … … … … Trt.(a-1)b+1 Trt.(a-1)b+2 Trt.(a-1)b+3 … 1 Trt.1 Trt.b+1 Trt.2b+1 b Trt.b Trt.2b Trt.3b Trt.ab … … a … … Factor A at a Levels Level 1 2 3 Factor B at b levels 2 3 Trt.2 Trt.3 Trt.b+2 Trt.b+3 Trt.2b+2 Trt.2b+3 24 Factorial Experiments Partitioning Total Sum of Squares •Usually done with statistical package 25 Factorial Experiments Conducting an ANOVA for a Factorial Design 1. 2. 3. 4. 5. 6. Partition Total Sum of Squares into Treatment and Error components Test H0 that treatment means are equal. If H0 is rejected proceed to step 3 Partition Treatment Sum of Squares into Main Effect and Interaction Sum of Squares Test H0 that factors A and B do not interact. If H0 is rejected, go to step 6. If H0 is not rejected, go to step 5. Test for main effects of Factor A and Factor B Compare the treatment means 26 Factorial Experiments SPSS ANOVA Output for a factorial experiment 27