biostatistics-lecture 9 experimental designs ruibin xi peking university school of mathematical...
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Biostatistics-Lecture 9Experimental designs
Ruibin XiPeking University
School of Mathematical Sciences
Two-way ANOVA
• A study investigated the effects of 4 treatments (A, B, C, D) on 3 toxic agents (I, II, III).
• 48 rats were randomly assigned to 12 factor level combinations.
Two-way ANOVA
• Y: the response variable• Factor A with levels i=1 to a• Factor B with levels j = 1 to b• A particular combination of levels is called a
treatment or a cell. There are treatments• is the kth observation for treatment (i,j), k
= 1 to n kjiY ,,
ab
Two-way ANOVA
• One observation per cell (n=1)– Cannot estimate the interaction, have to assume
no interaction
Randomized Complete Block Design
• Useful when the experiments are non-homogenous– Rats are bred from different labs– Patients belong to different age groups
• Randomized Block design can used to reduce the variance
Randomized Complete Block Design
• A “block” consists of a complete replication of the set of treatments
• Block and treatments usually are assumed not having interactions
• Advantages:– Effective grouping can give substantially more precise results– Can accommodate any number of treatments and replications– Statistical analysis is relatively simple– If an entire block needs to be dropped, the analysis is not
complicated thereby
Randomized Complete Block Design
• Disadvantages– The degree of freedom for experiment error are not
as large as with a completely randomized design– More assumptions (no interaction between block and
treatment, constant variance from block to block)– Blocking is an observational factor and not an
experimental factor, cause-and-effect inferences cannot be made for the blocking variable and the response
Randomized Complete Block Design
• The model (similar to additive two-way ANOVA)
• block effect treatment effect
Random effect designs
• Fixed effect models– Levels of each factor are fixed– Interested in differences in response among those
specific levels• Random effect model– Random effect factor: factor levels are meant to be
representative of a general population of possible levels
• If there are both fixed and random effects, call it mixed effect model