Factorial Experiments HSTS312 9 17

Factorial Design 1. Experimental Units (Subjects) Are Assigned Randomly to Treatments Subjects are Assumed Homogeneous 2. Two or More Factors or Independent Variables Each Has 2 or More Treatments (Levels) 3. Analyzed by Two-Way ANOVA HSTS312

2 Factor Model HSTS312

Model HSTS312

model HSTS312

Advantages of Factorial Designs 1. Saves Time & Effort e.g., Could Use Separate Completely Randomized Designs for Each Variable 2. Controls Confounding Effects by Putting Other Variables into Model 3. Can Explore Interaction Between Variables HSTS312

Two-Way ANOVA Tests the Equality of 2 or More Population Means When Several Independent Variables Are Used Same Results as Separate One-Way ANOVA on Each Variable But Interaction Can Be Tested HSTS312

Two-Way ANOVA Assumptions 1. Normality Populations are Normally Distributed 2. Homogeneity of Variance Populations have Equal Variances 3. Independence of Errors Independent Random Samples are Drawn HSTS312

Two-Way ANOVA Data Table Factor Factor B A 1 2 ... b Observation k 1 Y Y ... Y 111 121 1b1 Yijk Y Y ... Y 112 122 1b2 2 Y Y ... Y 211 221 2b1 Y Y ... YX 212 222 2b2 Level i Factor A Level j Factor B : : : : : a Y Y ... Y a11 a21 ab1 Y Y ... Y a12 a22 ab2 HSTS312

Two-Way ANOVA Null Hypotheses 1. No Difference in Means Due to Factor A H0: 1. = 2. =... = a. 2. No Difference in Means Due to Factor B H0: .1 = .2 =... = .b 3. No Interaction of Factors A & B H0: ABij = 0 HSTS312

Two-Way ANOVA Total Variation Partitioning SS(Total) Variation Due to Treatment A Variation Due to Treatment B SSB SSA Variation Due to Interaction Variation Due to Random Sampling SS(AB) SSE HSTS312

Two-Way ANOVA Summary Table Source of Degrees of Sum of Mean F Variation Freedom Squares Square A a - 1 SS(A) MS(A) MS(A) (Row) MSE B b - 1 SS(B) MS(B) MS(B) (Column) MSE AB (a-1)(b-1) SS(AB) MS(AB) MS(AB) (Interaction) MSE Error n - ab SSE MSE Same as Other Designs Total n - 1 SS(Total) HSTS312

ANOVA` HSTS312

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Interaction 1. Occurs When Effects of One Factor Vary According to Levels of Other Factor 2. When Significant, Interpretation of Main Effects (A & B) Is Complicated 3. Can Be Detected In Data Table, Pattern of Cell Means in One Row Differs From Another Row In Graph of Cell Means, Lines Cross HSTS312

Graphs of Interaction Effects of Gender (male or female) & dietary group (sv, lv, nor) on systolic blood pressure Interaction No Interaction Average Average Response male Response male female female sv lv nor sv lv nor HSTS312

Two-Way ANOVA F-Test Example Effect of diet (sv-strict vegetarians, lv- lactovegetarians, nor-normal) and gender (female, male) on systolic blood pressure. Question: Test for interaction and main effects at the .05 level. HSTS312

Linear Contrast Linear Contrast is a linear combination of the means of populations Purpose: to test relationship among different group means with Example: 4 populations on treatments T1, T2, T3 and T4. Contrast T1 T2 T3 T4 relation to test L1 1 0 -1 0 μ1 - μ3 = 0 L2 1 -1/2 -1/2 0 μ1 – μ2/2 – μ3/2 = 0 HSTS312

T-test for Linear Contrast (LSD) Construct a t statistic involving k group means. Degrees of freedom of t - test: df = n-k. Construct To test H0: Compare with critical value t1-α/2,, n-k. Reject H0 if |t| ≥ t1-α/2,, n-k. SAS uses contrast statement and performs an F – test df (1, n-k); Or estimate statement and perform a t-test df (n-k). HSTS312

T-test for Linear Contrast (Scheffe) Construct multiple contrasts involving k group means. Trying to search for significant contrast Construct To test H0: Compare with critical value. Reject H0 if |t| ≥ a HSTS312

Conclusion: should be able to 1. Recognize the applications that uses ANOVA 2. Understand the logic of analysis of variance. 3. Be aware of several different analysis of variance designs and understand when to use each one. 4. Perform a single factor hypothesis test using analysis of variance manually and with the aid of SAS or any statistical software. HSTS312

Conclusion: should be able to 5. Conduct and interpret post-analysis of variance pairwise comparisons procedures. 6. Recognize when randomized block analysis of variance is useful and be able to perform the randomized block analysis. 7. Perform two factor analysis of variance tests with replications using SAS and interpret the output. HSTS312

Fisher’s Least Significant Difference (LSD) Test To compare level 1 and level 2 Compare this to t/2 = Upper-tailed value or - t/2 lower-tailed from Student’s t-distribution for /2 and (n - p) degrees of freedom MSE = Mean square within from ANOVA table n = Number of subjects p = Number of levels HSTS312